OPTIMAL CONTROL OF MULTIPLE RESERVOIRS SYSTEM UNDER WATER SCARCITY
By
Iftikhar Ahmad M.Sc (Geology) M.Phil (Hydrology)
A thesis submitted in the fulfillment of requirements for the degree of Doctor of Philosophy
INSTITUTE OF GEOLOGY UNIVERSITY OF THE PUNJAB, LAHORE-PAKISTAN 2009 OPTIMAL CONTROL OF MULTIPLE RESERVOIRS SYSTEM UNDER WATER SCARCITY
By
Iftikhar Ahmad M.Sc (Geology) M.Phil (Hydrology)
Under the Supervision of
Prof. Dr. Nasir Ahmad Ph.D. (U.K), M.Sc. (Pb)
A thesis submitted to the Punjab University in the fulfillment of requirements for the degree of Doctor of Philosophy
INSTITUTE OF GEOLOGY UNIVERSITY OF THE PUNJAB, LAHORE-PAKISTAN 2009
Dedicated to my family and brother
CERTIFICATE
It is hereby certified that this thesis is based on the results of modeling work carried out by Iftikhar Ahmad under our supervision. We have personally gone through all the data/results/materials reported in the manuscript and certify their correctness/ authenticity. We further certify that the materials included in this thesis have not been used in part or full in a manuscript already submitted or in the process of submission in partial/complete fulfillment for the award of any other degree from any other institution. Iftikhar Ahmad has fulfilled all conditions established by the University for the submission of this dissertation and we endorse its evaluation for the award of PhD degree through the official procedures of the University.
SUPERVISOR SUPERVISOR
Prof. Dr. Nasir Ahmad Prof. Dr. Zulfiqar Ahmad Director Institute of Geology Chairman Department of Earth Sciences University of the Punjab Quad-i-Azam University Lahore, Pakistan Islamabad, Pakistan
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ABSTRACT
The use of mathematical programming for short term (10-day) operation of Indus River System under uncertainty was investigated. A two stage mix optimization procedure was proposed for the stochastic optimization of the Indus River System. The first stage of the proposed procedure cycles through three main programs, a transition probability matrix (tmp) computation algorithm, a DDP-SDP (Deterministic-Stochastic Dynamic Programming) model and a simulation program. In DDP-SDP program, four model types and three objective types were investigated for multiresevoir system. These non-linear objectives were calibrated for the large scale complex system to minimize the irrigation shortfalls, to maximize the hydropower generation and to optimize the flood storage benefits. Simulation program was used for the validation of each policy derived through this cycle. The accumulation of these programs is called 10 day reservoir operation model of the multireservoir Indus River System. Various model types in SDP/DDP formulation may produce different results in different reservoir conditions and different hydrologic regimes. The model types are therefore system specific. For the Indus Reservoir System best fit SDP model type was identified, alternate multi objective functions were proposed and analysed. Taking one or two objectives and ignoring other or considering all the objectives to optimize, produced different results in different model types. Especially the results were significantly different in terms of storage contents of the reservoir during simulation. The proposed procedure identifies the best stochastic operational policies for the system under uncertainty. The second stage of proposed procedure uses advantages of the stochastic optimal policies derived in the first stage of the optimization with a Network Flow programming (NFP) model developed for the Indus River System for 10 day operation. The whole system was represented by a capacitated network in which nodes are reservoirs, system inflow locations or canal diversion locations. The nodes are connected with the arcs which represent rivers, canal reaches or syphons in the system. The maximum and minimum flow conditions were defined from the physical data. The NFP model was solved with the help of two main programs, the out of kilter algorithm and on line reservoir operation model with stochastic operating policies. The accumulation of these programs is called 10 day stochastic network flow programming (SNFP) model of the multireservoir Indus River System. The proposed SNFP model provides two main benefits. First, the incorporation of the stochastic operating policies at reservoir nodes controls the uncertainty and improves the system operation performance. The stochastic behaviour of the inputs and non-linear objectives in the linear programming model is incorporated in this way. Second, the complete system is under control and presents acomplete physical picture of the system. The results obtained from the above two stage procedure were verified with help of simulating the system with forecasted inflows and comparing these results with actual historic data record. For this purpose, 10 day forecasting models were investigated, calibrated and verified. The results also proved the methodology effective for the test case. The reservoir operation model is characterized as generalised and flexible model, and can be used for any other reservoir. The SNFP model is system (the Indus River System) specific to and needs minor modifications to be used for other water resource systems.
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The proposed optimization procedure presents the optimum operation of reservoirs for irrigation water supplies, hydropower production and flood protection, optimal allocation of water resources in the canal network of Indus River System and identifies the resource limitations at various locations in the system. While comparing with the historic data records, the model performance was found to be better than the historic data at all locations in the system during simulation. The complete model may be used as a guiding tool for the optimum 10 day operation of the Indus River System. A two stage frame work consisting of a steady state SDP 10 day reservoir operation model followed by a Network Flow model appears to be promising for the optimization of Indus River System. The model has also been used for future planning of water resources in Pakistan. The methodology developed provides a viable way of applying stochastic optimization into deterministic optimization procedure under multireservoir, multiobjective water resource system with 10 day operation under uncertainty.
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ACKNOWLEDGEMENTS
I would like to extend my sincere thanks to my research supervisors, Prof. Dr. Nasir Ahmad (Director, Institute of Geology) and Prof. Dr. Zulfiqar Ahmad (Chairman, Department of Earth Sciences, Quid-e-Azam University, Islambad) for their keen interest, proficient guidance, valuable suggestions, and encouraging attitude during the course of this research work.
Special recognitions go to Dr. S. M. Saeed Shah (Head of hydrology division, Centre of Excellence in Water Resources Engineering, University of Engineering and Technology Lahore) for his insightful suggestions while writing up this thesis. I am extremely grateful to Prof. Dr. Iftikhar Hussain Baloch (Principal, College of Earth and Environmental Sciences, University of The Punjab) for his cooperation and encouragement.
I wish to thank many professional colleagues, specially Dr. Ashraf Malik, ex.Chief Hydrology, NESPAK, Dr. Muhammad Younas Khan, ex General Manager, Tarbela Dam WAPDA, and Dr. Maboob Alam, Director IWASRY WAPDA for their wise comments on the script.
I thank to my University fellows, Mr. Muhammad Akhtar and Mr. Khursheed Alam for their co-operation.
Finally, I would like to express my heartiest gratitude to my wife and children whose cooperation, prayers and well wishes strengthened my confidence to endure hardships faced during this study.
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LIST OF TABLES
Table 4.1 Details of Indus Basin Rivers 124
Table 4.2 Hydraulic characteristics of Indus River and its tributaries 126
Table 4.3 Salient features of Jhelum river and its tributaries 128
Table 4.4 Hydraulic characteristics of tributaries of Ravi joining within Pakistan 132
Table 4.5 Hydraulic characteristics of important tributaries of Sutlej 132
Table 4.6 Water and Power benefits from Tarbela dam 139
Table 4.7 Water and Power benefits from Mangla dam 145
Table 4.8 Water benefits from Chasma reservoir 148
Table 4.9 Loss of reservoir capacities in MAF 149
Table 4.10 Summary of the basic Information of the Barrages located in the Indus Basin 153
Table 4.11 Indus zone and Jhelum Chenab Zone 154
Table 4.12 Average gains and losses of the 46 years of data 156
Table 5.1 Statistics of Annual Flows (Time series Oct-Sep) 161
Data Statistics, Consistency and Outliers in 10 Daily Inflows 1922-2004 Oct- Table 5.2 169 Sep, Jhelum at Mangla Data Statistics, Consistency and Outliers in 10 Daily Inflows 1961-2004 Oct- Table 5.3 170 Sep, Indus at Tarbela
Table 5.4 Serial Correlation Coefficients 173
Table 5.5 Correlation Coefficients between 10 daily flows 174
Table 5.6 Transition Probability Matrix of Period August 1, Indus at Tarbela 176
Table 5.7 Variation of Rescale Range and Hurst Exponent 177
Table 5.8 Results of Gould transitional probability matrix method 179
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Table 5.9 Summary result from Rippl mass curve analysis 182
Table 5.10 Summary results of Sequent Peak Analysis 183
Table 5.11 Selected Regression Models for 10 day forecasting in Indus Rivers 199
Summary results for calibration of stochastic network flow programming model, Table 7.1 254 simulation period 1985-95. Sample result of calibration of SNFP model 10 day time period 10 year Table 7.2 255 simulation for 1985-1995 (values in 1000 x cfs) Summary results for validation of stochastic network flow programming model, Table 7.3 260 simulation period 1994-95 to 2003-04 Sample result of validation of SNFP model 10 day time period 10 year Table 7.4 261 simulation for 1995-96-2003-04 (values in 1000 x cfs)
Table 8.1 Possible cases for conjunctive operation 267
Table 8.2 Summary results of reservoir simulation at Tarbela (calibration case) 271
Table 8.3 Summary results of reservoir simulation at Mangla (calibration case) 274
Table 8.4 Summary results of reservoir simulation at Tarbela (validation case) 277
Table 8.5 Summary results of reservoir simulation at Tarbela (validation case) 280
Summary of mean annual results of 10-daily conjunctive operation Case TMB: Table 8.6 282 2015 Summary of mean annual results of 10-daily conjunctive operation Case TMBA: Table 8.7 283 2020 Summary of mean annual results of 10-daily conjunctive operation Case Table 8.8 283 TMBAK: 2030
Table 9.1 Summary results of Reservoir Operation Model, Mangla Reservoir 285
Annual hydropower generated under various model types. (MKWh) Mangla Table 9.2 289 Reservoir
Table 9.3 Annual irrigation releases under various model types. (MAF) Mangla Reservoir 289
Benefits from water and power under various model types. (Rs. Million) Mangla Table 9.4 290 Reservoir Water wasted through spillage under various model types. (MAF) Mangla Table 9.5 290 Reservoir
Table 9.6 Summary results of Reservoir Operation Model, Tarbela Reservoir 294
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Annual hydropower generated under various model types. (MKWh) Tarbela Table 9.7 298 Reservoir
Table 9.8 Annual irrigation releases under various model types. (MAF) Tarbela Reservoir 298
Benefits from water and power under various model types. (Rs. Million) Tarbela Table 9.9 299 Reservoir Water wasted through spillage under various model types. (MAF) Tarbela Table 9.10 299 Reservoir Comparison of SNFP model performance with historic operation [annual canal Table 9.11 301 withdrawals (MAF)]
Table 9.12 Verification of network flow model 302
Comparison of SNFP model performance with historic operation [Annual canal Table 9.13 303 withdrawals (MAF)] Comparison of Results (mean annual water released from 10-daily conjunctive Table 9.14 304 operation) Comparison of Results (mean annual energy generated from 10-daily Table 9.15 304 conjunctive operation)
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LIST OF FIGURES
Figure 3.1 Typical multistage process 76
Figure 3.2 Structure of the probabilistic dynamic programming 80
Reservoir states [ S(i), i = 1,2,3,. . .,n ]. Volumetric increments between states, Figure 3.3 109 all states equal
Figure 3.4 Flow chart of the optimization program 111
Figure 3.5 Flow chart of the simulation program 112
Figure 3.6 Basic structure of the NFP model 121
Figure 4.1 Indus River System and surface storage 123
Figure 4.2 Schematic Diagram – Indus Basin Irrigation System (WAPDA, 2006) 125
Figure 4.3 The Indus and its tributaries 127
Mean monthly recorded precipitation at Jhelum station of Pakistan Figure 4.4 130 Meteorological Department
Figure 4.5 Tarbela dam auxiliary spillway 134
Figure 4.6 Tarbela reservoir 134
Figure 4.7 Elevation-capacity curves for Tarbela 135
Figure 4.8 Minimum maximum rule curve at Tarbela 136
Figure 4.9 Tarbela Dam from space 140
Figure 4.10 Mangla reservoir at 1040 ft AMSL 142
Figure 4.11 Mangla dam power house and Bong canal 142
Figure 4.12 Elevation-capacity curves for Mangla 143
Figure 4.13 Minimum maximum rule curves at Mangla 144
Figure 4.14 Work in progress on Mangla Raising 147
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Figure 4.15 Elevation-capacity curves for Mangla Raising 147
Figure 4.16 Location of major canals in Indus Basin Irrigation System 155
Figure 4.17 Year wise historic gains and losses from the Indus Irrigation System 156
Figure 4.18 Node Arc Representation and Schematic Diagram 158
Figure 5.1 Annual Recorded Flows in Indus River System 162
Figure 5.2 Annual Recorded Flows in Indus River System at Basha Tarbela and Kalabagh 162
Annual Recorded Flows in Indus River System at Jhelum, Chenab, Ravi and Figure 5.3 163 Sutlej
Figure 5.4 Results of Outliers Testing, Jhelum at Mangla 163
Figure 5.5 Estimated water scarcity in Indus at Tarbela 164
Figure 5.6 Estimated water scarcity in Indus at Kalabagh 165
Figure 5.7 Estimated water scarcity in Kabul at Nowshera 165
Figure 5.8 Estimated water scarcity in Jhelum at Mangla 166
Figure 5.9 Estimated water scarcity in Chenab at Marala 166
Figure 5.10 Estimated water scarcity in Eastern Rivers (Ravi+Sutlej) 167
Figure 5.11 Mean 10-daily recorded flows in Indus River System 168
Computed histogram showing frequency distribution of flows in Indus River Figure 5.12 172 System
Figure 5.13 Computed probabilities of Reservoir States using Gould TPM method 179
Figure 5.14 Sample graph showing dam capacities for different releases for Mangla Dam 181
Estimated reservoir capacities for different releases in Indus River reservoirs Figure 5.15 181 using Ripple mass curve method
Figure 5.16 Sample graph showing results for Mangla dam at water demand 16 MAF 183
Figure 5.17 Precipitation vs evaporation at Tarbela 185
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Figure 5.18 Precipitation-evaporation at Mangla 185
Figure 5.19 Historic operation of Mangla Dam 187
Figure 5.20 Historic operation of Tarbela 187
Irrigation demand from existing and proposed reservoirs as per Water Accord Figure 5.21 189 1991
Figure 5.22 Rule Curve at Tarbela Dam 190
Figure 5.23 Rule Curve at Mangla Dam 191
Figure 5.24 Minimum Rule Curve at Basha Dam 191
Figure 5.25 Minimum Rule Curve at Kalabagh Dam 192
Comparison of Observed and Forecasted 10-daily flows with 3 year weighted Figure 5.26 195 moving average at Mangla Dam for dry, average and wet years
Comparison of observed and forecasting annual flows with 5 year moving Figure 5.27 196 average at Mangla and Tarbela Dams
Comparison of observed and computed annual flows using 10 Day forecast Figure 5.28 200 models, Indus at Tarbela
Comparison of observed and computed annual flows using 10 Day annual Figure 5.29 200 forecast model, Indus at Tarbela
Comparison of observed and computed 10 day flows for 1991-92 to 1992-93, Figure 5.30 201 Indus at Tarbela
Comparison of observed and computed 10 day flows for 1993-94 to 1994-95, Figure 5.31 201 Indus at Tarbela
Comparison of observed and computed 10 day flows for 1991-92 to 1992-93, Figure 5.32 202 Jhelum at Mangla
Comparison of observed and computed 10 day flows for 1993-94 to 1994-95, Figure 5.33 202 Jhelum at Mangla
Comparison of observed and computed 10 day flows for 1991-92 to 1992-93, Figure 5.34 203 Chenab at Marala
Comparison of observed and computed 10 day flows for 1993-94 to 1994-95, Figure 5.35 203 Chenab at Marala
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Comparison of observed and computed 10 day flows for 1991-92 to 1992-93, Figure 5.36 204 Kabul at Nowshera
Comparison of observed and computed 10 day flows for 1993-94 to 1994-95, Figure 5.37 204 Kabul at Nowhera
Figure 6.1 Reservoir mass balance and continuity equation 206
Figure 6.2 Convex loss function in reservoir optimization 219
Optimal policies for June to September from Type One model for Mangla Figure 6.3 220 reservoir
Optimal Policies for June to September from Type One model for Mangla Figure 6.4 221 reservoir with all Objective Functions
Optimal losses for June to September from Type One model for Mangla Figure 6.5 222 Reservoir with all Objective Functions
Optimal Policies for June to September from Type Two model for Mangla Figure 6.6 224 Reservoir
Optimal losses for June to September from Type Two model for Mangla Figure 6.7 225 Reservoir
Sample Optimal Releases/ loss in stage 30 of Type Four Model, Mangla Figure 6.8 226 Reservoir
Comparison of computed reservoir levels and releases in Type One model for Figure 6.9 228 Mangla Reservoir
Comparison of computed reservoir levels and releases in Type Two model for Figure 6.10 229 Mangla Reservoir
Comparison of computed reservoir levels and releases in Type Three model for Figure 6.11 231 Mangla Reservoir (with all objective fucntions)
Comparison of computed reservoir levels and releases in Type Three model for Figure 6.12 232 Mangla Reservoir (minimizing irrigation and power shortfalls)
Comparison of computed reservoir levels and releases in Type Three model for Figure 6.13 233 Mangla Reservoir (minimizing irrigation shortfalls)
Comparison of computed reservoir levels and releases in Type Four model for Figure 6.14 234 Mangla Reservoir (with all objective fucntions)
Comparison of computed reservoir levels and releases in Type Four model for Figure 6.15 235 Mangla Reservoir (minimizing irrigation and power shortfalls)
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Comparison of computed reservoir levels and releases in Type Four model for Figure 6.16 236 Mangla Reservoir (minimizing irrigation shortfalls)
Comparison of computed reservoir levels and releases in Type One model for Figure 6.17 237 Tarbela Reservoir
Comparison of computed reservoir levels and releases in Type Three model for Figure 6.18 238 Tarbela Reservoir (with all objective fucntions)
Comparison of computed reservoir levels and releases in Type Three model for Figure 6.19 239 Tarbela Reservoir (minimizing irrigation and power shortfalls)
Comparison of computed reservoir levels and releases in Type Three model for Figure 6.20 240 Tarbela Reservoir (minimizing irrigation shortfalls)
Comparison of computed reservoir levels and releases in Type Four model for Figure 6.21 241 Tarbela Reservoir (with all objective fucntions)
Comparison of computed reservoir levels and releases in Type Four model for Figure 6.22 243 Tarbela Reservoir (minimizing irrigation and power shortfalls)
Comparison of computed reservoir levels and releases in Type Four model for Figure 6.23 244 Tarbela Reservoir (minimizing irrigation shortfalls)
Stochastic network flow model is repeatedly applied for each 10 day period in Figure 7.1 249 each year
Figure 7.2 Comparison of actual and computed canal allocations during model calibration 252
Comparison of observed and computed discharges in Rohri Canal during model Figure 7.3 253 calibration
Comparison of actual and computed canal allocations during model validation Figure 7.4 259 1995-96 to 2003-04
Figure 8.1 Schematic of Indus multi-reservoir system for conjunctive operation study 266
Figure 8.2 A view of Basha Diamer damsite on Indus River 267
Figure 8.3 Reservoir elevations and power generated during simulation period 1962-2004 269
Figure 8.4 Reservoir levels during operation at Tarbela during simulation period 1962-2004 269
Figure 8.5 Releases from the dam during simulation period 1962-2004 270
Figure 8.6 Average water levels in the reservoir for period 1962-2004 270
Reservoir elevations and power generated at Mangla during simulation period Figure 8.7 272 1922-2004
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Figure 8.8 Reservoir levels at Mangla during simulation period 1922-2004 272
Figure 8.9 Optimal releases from the dam during simulation period 1922-2004 273
Figure 8.10 Average reservoir levels at Mangla during simulation 1922-2004 273
Reservoir elevations and power generated at Tarbela during simulation with Figure 8.11 275 forecasted flows for 42 year period (validation case)
Reservoir levels at Tarbela during simulation with forecasted flows for 42 year Figure 8.12 276 period (validation case)
Optimal releases at Tarbela during simulation with forecasted flows for 42 year Figure 8.13 276 period (validation case)
Average reservoir water levels at Tarbela during simulation with forecasted Figure 8.14 277 flows for 42 year period (validation case)
Reservoir elevations and power generated at Mangla during simulation with Figure 8.15 277 forecasted flows for 82 year period (validation case)
Reservoir levels at Mangla during simulation with forecasted flows for 82 year Figure 8.16 279 period (validation case)
Optimal Releases from Mangla dam during simulation with forecasted flows for Figure 8.17 279 82 year period (validation case)
Average water levels in Mangla reservoir during simulation with forecasted Figure 8.18 280 flows for 82 year period (validation case)
Figure 9.1 Comparison of historic and model hydropower generated, Mangla reservoir 291
Figure 9.2 Comparison of historic and model releases from the reservoir (Mangla reservoir) 291
Figure 9.3 Comparison of historic and model water and power benefits (Mangla reservoir) 292
Comparison of historic and model water wasted through spillage (Mangla Figure 9.4 292 reservoir)
Figure 9.5 Releases from Mangla reservoir, historic vs SDP Type Four (New Data Set) 293
Figure 9.6 Water wasted through spillage, historic vs SDP Type Four (New Data Set) 293
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ABBREVIATIONS AND NOTATIONS
Ac Acres AF Acre feet AMSL Above Mean Sea Level
At(S ,S) Average surface area of the reservoir over the 10day period t in ac. BCM Billion Cubic Meter cfs / cusecs Cubic feet per sec Cms / cumecs Cubic meter per sec CVP United States Central Valley Project DBC Diamer Basha Consultants DDP Deterministic Dynamic Programming
DEMANDt Downstream irrigation demand in 10 day period t dt Expected release at a discrete level k in MAF El. Elevation et Evaporation in 10 day period t in ft. (known) F(S ) Return function fn Total optimal shortfall or lossses in stage n for one of proposed case, m ft Feet
Ht Average productive storage head in ft IBIS Indus Basin Irrigation System IRSA Indus River System Authority KAF Thousand Acre feet
KWHt Kilowatt hours of energy m Index for specifing model, varies 1 to 7 m Meter MAF Million Acre feet MJV Mangla Joint Venture MW Mega Watt MWh Mega Watt Hours NFP Network Flow Programming
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O.F Objective Function PCAP Plant capacity in KW
Pijt Conditional probability of inflows in 10day period t given inflow in t+1
Pit Probability of Inflows in 10 day period t PLANTEF Plant efficiency at a given head PMF Probable Maximum Flood
PWRDEMt Downstream energy demand in 10 day period t
PWRFLOt Flow through turbines required to generate energy equal to demand at average productive storage head and hydropower efficiency in MAF
qt Inflow to reservoir in 10day period t in MAF (known or forcasted) R.L. Reduced level
R2 Coefficient of determination
RFt Accretion to reservoir by rainfall in 10 day period t in ft. (known)
RULMAXt Maximum desirable reservoir content in 10 day period t to mitigate flood control in MAF
RULMINt Minimum desirable reservoir content in 10 day period t to mitigate sedimentation flushing in MAF SDP Stochastic Dynamic Programming
Smax Maximum storage capacity in MAF (known)
Smin Minimum allowable storage or dead storage capacity for sedimentation in MAF (known) Sq. km Square Kilometer Sq. mile Square Miles
St Storage content in the reservoir at the beginning of the 10day period t in MAF tht Hours in period t USBR United States Bureau of reclamation WAPDA Water and Power Development Authority x Hydropower efficiency
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CONVERSION FACTORS
Length
1 Meter = 3.2808 Feet = 39.3696 inches
Area 1 Sq Km = 100 Hectares = 0.3861 Sq Miles 1 Acre = 43560 Sq Feet = 4046.95 Sq meters
Volume 1 Cumecs = 35.31 Cusecs
Factor for converting 1000*Cusecs to MAF • For 11 Days Period 1000 Cusecs = 0.0218 MAF • For 10 Days Period 1000 Cusecs = 0.0198 MAF • For 08 Days Period 1000 Cusecs = 0.0158 MAF
Factor for converting 1000*Cusecs to BCM • For 11 Days Period 1000 Cusecs = 0.0269 BCM • For 10 Days Period 1000 Cusecs = 0.0244 BCM • For 08 Days Period 1000 Cusecs = 0.0196 BCM
Factor for converting Cumecs to MCM • For 11 Days Period Cumecs = 0.9504 MCM • For 10 Days Period Cumecs = 0.8640 MCM • For 08 Days Period Cumecs = 0.6912 MCM
1 MAF = 1233.50 MCM 1 BCM = 0.8107 MAF
CONTENTS
Abstract...... i Acknowledgements...... iii List of Tables ...... iv List of Figures ...... vi Abbreviations and Notations...... xiii Conversion Factors ...... xv
CHAPTER-1 INTRODUCTION...... 1 1.1 Background...... 1 1.2 Objectives of the Study...... 1 1.3 Research and Developments...... 2 1.4 Thesis Layout ...... 2
CHAPTER-2 LITERATURE REVIEW...... 4 2.1 Network Flow Programming ...... 4 2.2 Deterministic Dynamic Programming...... 9 2.3 Stochastic Dynamic Programming ...... 18 2.4 Linear Programming...... 37 2.5 Multiobjective Optimization...... 42 2.6 Other Techniques...... 48 2.7 River Basin System Optimization ...... 52 2.8 River Water Disputes...... 60 2.9 Comparison of Methods ...... 63 2.10 Previous Studies on the Indus Basin...... 65
CHAPTER-3 METHODOLOGY...... 68 3.1 Proposed Procedure ...... 68 3.2 Formulating a Mathematical Model ...... 71 3.3 Dynamic Programming...... 73 3.3.1 Solution procedure...... 74 3.3.2 Characteristics...... 74 3.3.3 Problem formulation...... 75 3.3.4 Stationary policy...... 76 3.4 Deterministic Dynamic Programming (DDP) ...... 77
3.4.1 Discrete approach ...... 77 3.4.2 Computational procedure...... 78 3.4.3 Multistage problem...... 78 3.5 Stochastic Dynamic Programming (SDP) ...... 79 3.5.1 Probabilistic dynamic programming...... 79 3.6 Stochastic Dynamic Programming (Formulation of the Model) ...... 80 3.6.1 Formulation of the SDP recursive equation...... 80 3.6.2 Discount factor...... 82 3.6.3 Discounted stochastic DP model ...... 84 3.6.4 Marcov process...... 84 3.6.5 Marcov chains, inflow process and uncertainty...... 85 3.6.6 Methods of computing Marcov chains ...... 86 3.6.7 Streamflow and stochastic process ...... 91 3.6.8 Determination of the observed inflow transition probabilities ...... 91 3.6.9 Markov process and stochastic dynamic programming (solution 93 procedure) ...... 3.6.10 Solution by value iteration method...... 94 3.6.11 Solution by policy iteration method...... 97 3.6.12 White’s solution procedure...... 99 3.6.13 Method of successive approximation...... 100 3.7 Recent Research Trend in Stochastic Optimization ...... 101 3.8 Mathematical Statement of SDP / DDP Models...... 102 3.8.1 State transformation equation ...... 102 3.8.2 Model constraints...... 103 3.8.3 Multi objective return functions...... 106 3.8.4 Discretization of state variables...... 108 3.8.5 Algorithm for optimizing model...... 110 3.9 System Network Optimization ...... 113 3.10 The Out-of-Kilter Algorithm ...... 114 3.10.1 Basic theory ...... 115 3.10.2 Conditions for optimality...... 116 3.10.3 Minimum cost circulation optimization using Out of Kilter 118 algorithm...... 3.10.4 Algorithm steps...... 118 3.11 Introducing Uncertainty Analysis in Network Flow Optimization...... 119
3.12 Complete River Systems Operation Optimization Model ...... 119 3.13 Contribution to the Research ...... 120
CHAPTER-4 DESCRIPTION OF THE STUDY AREA ...... 122 4.1 Rivers in the System...... 122 4.2.1 The Indus River ...... 126 4.2.2 The Jhelum River...... 128 4.2.3 The Chenab River ...... 130 4.2.4 The Ravi River...... 131 4.2.5 The Beas River...... 132 4.2.6 The Sutlej River...... 132 4.2 Reservoirs/Dams of the Indus River...... 133 4.3.1 Tarbela Dam ...... 133 4.3.2 Mangla Dam ...... 141 4.3.3 Chasma Reservoir...... 148 4.3.4 Loss of Reservoir Capacities ...... 149 4.3 Hydrological and Other Data...... 149 4.4 Barrages in the System ...... 150 4.5.1 Chashma Barrage...... 150 4.5.2 Rasul Barrage...... 150 4.5.3 Qadirabad Barrage ...... 150 4.5.4 Marala Barrage ...... 151 4.5.5 Sidhnai Barrage...... 151 4.5.6 Mailsi Syphon...... 151 4.5.7 Trimmu Barrage...... 151 4.5.8 Balloki Headworks ...... 152 4.6 Canals in the System...... 152 4.7 River Gains and Losses ...... 152 4.8 Complete river basin multi reservoir system ...... 156
CHAPTER-5 STOCHASTIC ANALYSIS OF UNCERTAIN 159 HYDROLOGIC PROCESSES ...... 5.1 General...... 159 5.2 Hydrological Data...... 160 5.3 Statistical Analysis of Annual Flows...... 160
5.4 Water Scarcity and Identification of Drought Periods...... 164 5.5 Statistical Analysis of 10 day Flows...... 167 5.6 Unconditional Probabilities ...... 171 5.7 Serial Correlation Coefficients ...... 173 5.8 Transition Probabilities...... 175 5.9 Hurst Phenomenon ...... 176 5.10 Gould Transition Probability Matrix Method...... 177 5.11 Rippl Mass Curve Analysis ...... 180 5.12 Sequent Peak Analysis...... 182 5.13 Evaporation Losses and Rainfall Accretion to Reservoirs ...... 184 5.14 Characteristics of Hydro Electric Plants...... 185 5.15 Historic Operation of Reservoirs...... 186 5.16 Release Requirements and Operation Objective ...... 186 5.17 Irrigation Demands...... 188 5.18 Power Demand ...... 188 5.18.1 Maximum and Minimum Design Rule Curves...... 189 5.19 Stochastic Control of Reservoir Inflows...... 192 5.19.1 WAPDA Forecasting Procedure...... 193 5.19.2 Forecasting with Moving Average ...... 193 5.19.3 Forecasting using Autoregressive Models...... 194 5.19.4 Forecasting using Multiple Regression ...... 197 5.19.5 Forecasting with expected values ...... 199
CHAPTER-6 RESERVOIR OPERATION OPTIMIZATION ...... 205 6.1 Background...... 205 6.2 Problem Formulation for Reservoir Operation Optimization...... 205 6.2.1 Formulation for Deterministic Optimization...... 207 6.2.2 Formulation for Stochastic Optimization ...... 208 6.3 Problem Formulation for Multiple Objective Reservoirs...... 212 6.4 Model Calibration of Reservoir Operation Optimization...... 216 6.4.1 Calibration of Dynamic Programming models...... 216 6.4.2 Calibration of Multiple Objectives ...... 218 6.4.3 Calibration Results ...... 219 6.5 Model Verification of Reservoir Operation Optimization...... 227 6.5.1 Mangla Reservoir ...... 227
6.5.2 Tarbela Reservoir ...... 230 6.6 Improved Strategies...... 241 6.6.1 Reservoir Operation Model ...... 241 6.6.2 Comments...... 241
CHAPTER-7 STOCHASTIC NETWORK FLOW PROGRAMMING 245 7.1 Background...... 245 7.2 Suggested Approach...... 245 7.3 Problem Formulation for System Network Operation...... 246 7.3.1 The Objective Function ...... 246 7.3.2 Constraints...... 247 7.3.3 Node-Arc Representation ...... 247 7.3.4 Strategy of Out of Kilter Algorithm ...... 247 7.3.5 Structure of the Program...... 247 7.4 Application of the Methodology to Indus River System ...... 248 7.4.1 The Operating Policy Problem ...... 248 7.4.2 Calibration of the SNFP model ...... 249 7.4.3 Verification of the SNFP model ...... 258 7.5 Improved Strategy ...... 258 7.6 Comments...... 259
CHAPTER-8 CONJUNCTIVE OPERATION OF MULTIPLE 264 RESERVOIRS SIMULATION...... 8.1 Background...... 264 8.2 Indus multi-reservoir system for conjunctive operation study ...... 265 8.3 Model Calibration for Conjunctive Operation of Multi Reservoirs...... 268 8.4 Model Validation for Conjunctive Operation of Multi Reservoirs...... 275 8.5 Model Prediction for Conjunctive Operation of Multi-reservoir 281 Simulation with Future Reservoirs ......
CHAPTER-9 RESULTS AND DISCUSSION...... 284 9.1 General...... 284 9.2 Reservoir Operation Model ...... 284 9.2.1 Results and Discussion ...... 284 9.3 Network Flow Model...... 297 9.3.1 Results and Discussions...... 297
9.3.2 Case Study Results ...... 301 9.4 Conjunctive Operation of Mutilple Reservoirs for future Scenarios...... 303
CHAPTER-10 CONCLUSIONS AND SUGGESTIONS FOR FUTURE 305 WORK...... 10.1. Conclusions ...... 305 10.1.1 Theoretical Development ...... 305 10.1.2 Practical Development...... 307 10.2. Recommendations ...... 307
REFERENCES ...... 309
Chapter One 1 Introduction
CHAPTER 1 INTRODUCTION
1.1 Background
Presently, the Indus River System is the main source of water to be used for irrigation and energy generation in Pakistan. However, no substantial efforts have been made to improve the system in order to meet the growing agriculture, domestic, industrial and power needs of the country. (The rapidly growing demand of water and power in the country therefore accentuates the need for careful management of the Indus River System). The traditional and conventional operational methods of the system have proved inefficient for a sustainable supply of water, resulting into a grave shortage of water for both irrigation and power supplies. Hence, the country is presently facing worse crisis of power and food shortage. The September 1992 flood in the Jhelum and Indus Rivers is another striking example of the failure of the convectional techniques used for the operation of the system. This situation demands a comprehensive study to develop a robust and up-to-date methodology for the optimal operation of the Indus River System. Hence, present study envisages the improved procedures for the operational optimization of the multireservoir system, and will provide a precise and accurate database to the policy makers for a judicious apportionment of waters of the Indus River System in accordance with the March 1991 agreement between the four provinces is to withdraw 114.35 MAF (+3.00 MAF) water from the system instead of 107.74 MAF. For the implementation of this accord, a comprehensive study is required. This research also includes such issues.
1.2 Objectives of the Study
• To develop and calibrate a working mathematical model coupled with various optimization techniques for optimal control of surface water resources of the Indus River System for short-term (10-day) operation under uncertainty.
• To find a trade off between the objectives. The primary objective is to minimize irrigation shortfalls. Whereas optimal power generation and flood control are the secondary objectives.
Chapter One 2 Introduction
• To analyze future planning and development for irrigation water supplies and power generation in accordance with 1991 Water Accord between the provinces of Pakistan.
1.3 Research and Developments
Review of literature indicates that reservoir operation optimization models are solved on monthly or annual time step. This research is extended to develop the formulation for a 10- daily time step which is common practice in most of the real time operation of reservoir systems.
The main components of the research and developments in the present study are given below:
• Improved stochastic optimization procedures for their practical application to multireservoir system. • Develop a working computerized mathematical model for optimal control of Indus River System. • Derive operating rules for the system. • Optimum operation of reservoirs for water supply. • Optimization of energy generation from the dams. • Optimum distribution of available water resource between canal network. • Identification of resource of limitations.
An additional aspect of the research was to assess efficiency of the proposed procedure in which dynamic programming, network programming and stochastic optimization are linked together for the analysis of multireservoir systems and to assess the merits and demerits of the methodology.
1.4 Thesis Layout
The thesis is divided into ten chapters. Chapter 1 highlights the rational of the study also pin points the objectives and salient features of the study area. Chapter 2 presents the review of previous work of various researchers for the optimization of water resource system operation. It covers the important work published in various journals during the last 35 years. Chapter 3 presents methodology that could be used for the optimum operation of large scale water resource systems. Chapter 4 describes the study area and data used for the optimal control of the Indus River System operation. Chapter 5 describes the stochastic analysis of uncertain hydrological processes. Chapter 6 presents reservoir operation optimization, model
Chapter One 3 Introduction calibration and validation. Chapter 7 is the application of stochastic network flow programming for Indus River System. In Chapter 8 conjunctive operation of mutlireservoirs simulation is performed. In chapter 9, results are logically discussed. Conclusions and suggestions for future work are given in chapter 10.
Chapter Two 4 Literature Review
CHAPTER 2
LITERATURE REVIEW
One of the most important advances made in water resources/ hydrological sciences is the evolvement and application of optimization techniques for planning, design and management of complex water resource systems. Many successful applications of optimization are made in reservoir studies. But there may still exist a gap between theory and application, particularly in the area of short term reservoir operation. The problem of ineffective operation of reservoirs using outdated technology and highly subjective management practices has been indicated by the researchers (Loucks, 1997; Chen, 2003; John, 2004).This chapter reviews the literature pertinent to reservoir operation and water resource system analysis. The review is onward described in different sections.
2.1 Network Flow Programming
Network flow programming (NFP) and conventional simulation models are two widely used alternative approaches for analysing reservoir/river systems. The NFP technique does not suffer from the dimensionality problem common to Dynamic Programming (DP) (Bellman and Dreyfuss 1962). It is also preferred over conventional linear programming because it is computationally much faster (Sabet and Creel, 1991). In this regards, the work of following authors is of great importance.
Yerramreddy and Wurbs (1994) presented a comparative evaluation of network flow programming and conventional simulation models. They concluded that simulation models typically provide greater flexibility in representing the complexities of real-world systems, while network flow programming helps to search systematically and efficiently through numerous possible combinations of decision variable values with a more prescriptive modelling orientation. They applied "The Water Rights Analysis Program" (TAMUWRAP) developed by Texas A&M University to study the Brazo River basin.
Sigvaldason (1976) used a mathematical model based on Out of Kilter algorithm (a simulation model) and used for assessing alternative operation strategies for the Trent River System in Ontario, Canada. The Trent basin, characterised by numerous reservoirs (48 were
Chapter Two 5 Literature Review
represented in the model) is used for flood control, water supply, hydropower generation and flow enhancement in canal system during the summer period. In mathematical form the out- of-kilter algorithm is stated simply as
min Z = Σ cij qij (2.1) ij
subject to: Σ qij - Σ qji = 0 for all j i j
Lij ≤ qij ≤ Uij for all ij
Where z is the objective function, qij is the flow in the arc from node i to node j, cij is the cost
of each unit of flow qij and Lij and Uij are the lower and upper bounds respectively on qij. The author claimed that the model was efficient, and permitted flexibility in readily using the model for a wide range of reservoir configurations and operating policies.
Barnes and Chung (1986) developed a detailed river basin simulation model to simulate the combined operation of two major water project systems in California, namely the Central Valley Project (CVP), operated by the US Bureau of reclamation, and the State Water Project (SWP), of the California Department of Water Resources. These projects comprise a system of dams, reservoirs, canals, tunnels, pumping plants and power plants designed to serve multi-objectives of flood control, recreation, power generation and water conservation. Although each project operates its upstream reservoirs separately, the release from the upstream reservoirs is intermixed in the Sacramento River and the Sacramento-San Joaquin Delta where southern exports are made by each project. The HEC-3 "Reservoir System Analysis for Conservation" model developed by the US Corps of Engineers has been adopted as a basic tool for reservoir releases and channel routings.
Faux et al. (1986) investigated an interactive river basin network flow model called MODSIM2 and applied to a large irrigation/hydropower system in Philippines. The upper Pampanga River Integrated Irrigation system (UPRIIS) is in Central Luzon. MODSIM2 is based on Out-of-Kilter algorithm. Mathematically it may be expressed as
Chapter Two 6 Literature Review
N N Min Σ Σ Cij Qij (2.2) i=1 j=1
N N Subject to Σ Qij - Σ Qji = 0 for j = 1,..., N i=1 j=1 Lij ≤ Qij ≤ Uij for all i=1,..., N
0 ≤ Lij j = 1,..., N
Where Qij = integer valued variable flow in the link connecting node i to j, Cij = unit cost of the flow in link (ij), Uij = given upper bound on flow in link (ij), and N is total number of network nodes. Nodes can represent storage reservoir or non-storage junctions such as river confluence or diversion structure and links represent river reaches, canals or penstocks. The
hydropower generation rate (POWt) is calculated as
POWt = Qt Eh k [ Ht - HT ] (2.3)
Where POWt = average power during period t, (MW), Qt = flow through the link representing turbine flow during period t as computed by the network algorithm in million cubic meter (mcm), Ht = average reservoir elevation during period t in meters (m), HT =
tailrace elevation in meters (m), Eh = hydropower conservation efficiency and k = unit conversion constant. Energy generation during the period was calculated as
δ = 0.001 POWt * Hrst (2.4)
Where δ = energy generation in period t in gigawatt-hour (GWH) and Hrst = number of hours in period t. Reservoir operating rules, network setup and model calibration were discussed in detail. The study demonstrates the versatility and usefulness of the Out-of-Kilter algorithm.
Chung et al. (1989) optimized the combined operation of California State Water Project (SWP) and the Federal Central Valley Project (CVP) in USA. They extended the water resources planning model (DWRSIM) of California Department of Water Resources by applying network flow algorithm. A least cost network flow programming procedure, the Out-of-Kilter algorithm (OKA) has been incorporated into DWRSIM to perform the operation of the CVP-SWP facilities south of the Sacramento-San Joaquin Delta. The main reservoirs of the system are Clair Engle lake, Shasta lake, Whoskey town reservoir, lake
Chapter Two 7 Literature Review
Oroville, Flosom lake, New Melons reservoir, San Luis reservoir, Milterton lake, Silverwood lake, lake Perris etc. The model (DWRSIM) uses a mass balance accounting procedure as a basic tool for channel routing.
Sabet and Creel (1991) investigated NFP to simulate the operation of the Oroville reservoir system of the California State Water Project (SWP). They simulated the hydraulic operation, hydropower generation and contractual obligations of the California Department of Water Resources (CDWR) to deliver energy to Southern California Edison (SCE). The computer code SUPERK (Barr et al. 1974) was incorporated in the California On-Line Optimizing System for Scheduling and updating Schedules (COLOSSUS). For the purpose, three integrated NFP models have been developed based on the OKA approach.
The simulation was carried out on weekly and daily levels with up-to hourly detail. A real word application of NFP to water resources was demonstrated. The results suggested that although NFP/OKA is a linear model, it was easily adapted to modelling a complex reservoir nonlinear system in conjunction with the linearization techniques and stepwise simulation.
Sabet and Creel (1991) presented model aggregation approach based on the network flow programming (NFP) for the operation of California State Water Project (SWP). NFP was used to model a portion of the California Aqueduct containing 30 aqueduct pools, two reservoirs, four pumping plants and one pump generating plant in the system. The Delta- O'Neil simulation model (DELMOD) was used to modify the schedules at certain locations to ensure the storage in the reservoirs was in allowable limits. For the daily model with hourly detail, the network representation of the system has 4,212 arcs and 1,395 nodes. To improve the efficiency of the computational time, a condensed network of 534 arcs and 123 nodes was used to find the initial solution. The final solution was then found by use of the expanded network. The weekly operation of SWP facilities was determined by use of the California on-line optimizing system for scheduling and updating schedules (COLOSSUS) (Sabet and Coe 1991). The NFP technique does not suffer from the "curse of dimensionality" common to dynamic programming. The out-of-kilter algorithm (OKA) is one method used to solve NFP problems (Ford and Fulkerson 1962). One of the computer code was the Texas Water Development Board incorporated in SYMLYDII. Shafer et al, (1981), Labadie and Pineda (1986) Graham and Labadie (1986) have used SYMLYDII to model a variety of
Chapter Two 8 Literature Review
water resources system. Another OKA code called SUPERK has been used by Farley et al. (1988), Coe and Rankin (1988) Chung et al. (1989). Sabet and Coe (1986) have been used to model water resources system. The weekly operation of DELMOD with a simple example of Dos Amigos was described. The results indicated the model aggregation approach is computationally efficient in term of computer run time.
Sun et al. (1995) investigated generalized network algorithm for water-supply-system optimization. They said that, to date, most algorithms are designed to solve transhipment problem in a pure network setting with total demand being equal to total supply. The non- network type constraints and variables are precluded from the network models. They presented an algorithm, EMNET, for solving the regional water-supply-system optimization that corresponds to a generalized network problem with additional non-network type constraints and non-network type variables. Metropolitan water district of Southern California is a case study. EMNET is 11-117 times faster than standard LP codes such as MINOS (Murtagh and Sounders, 1987).
Khaliquzzaman and Chander (1997) presented a network flow programming model for multireservoir sizing. It is a multiperiod model where all single period networks, representing reservoirs, rivers, canals and demand points are interconnected in adjacent periods by reservoir carryover arcs. Flows in carry over arcs represent reservoir storage and the maximum flow in an arc for a reservoir indicates storage capacity requirement for that reservoir. The carry over arcs are split into multiple arcs representing multiple zones in the storage capacity. Reservoir capacities are obtained by optimizing the flow in carryover arcs. The concept of Performance Matrix (PM) has been introduced to reflect the interreservoir and interzonal competitiveness based on the objective function. Applicability of the model is demonstrated by estimating the capacity of 7 reservoirs in a water transfer scheme in India using the criterion of minimization of forest submergence.
Niedda and Sechi (1997) investigated a mixed optimization technique for a large scale water resources system. It is based on Network Linear Programming and the Subgradient method. Since inside domain, the global objective function is a convex piecewise linear function, a subgradient method is used to obtain the direction of the improvement of design variables at each iteration using the solutions of the network subproblem. The mixed technique permits
Chapter Two 9 Literature Review
an efficient evaluation of the design variables in order to reach a good approximation of the global objective function optimum. The proposed method is applied to a hypothetical system which is made up of two reservoirs, a hydroelectric plant, a diversion dam and two irrigation areas.
2.2 Deterministic Dynamic Programming
Single reservoir problem Young (1967) developed an algorithm based on dynamic programming which solves the problem recursively forward in time from the start. He used a yearly time unit and therefore his methodology finds optimal operating policies for annual usage of a single reservoir. The results are generalized to specify a near-optimal policy as a function of several state variables and a flow forecast. The forward looking deterministic algorithm is given by
n-1 Z* = min { min [ l ( -Sn+1 + Sn + Xn ) + min Σ l (-Si+1 + Si + Xi ) ] } (2.5) Sn+1 Sn S2, S3,...,Sn-1 i=1
th Where Xi ,Xn Inflow in the ith or n year. th Si ,Sn Storage at the start of i or nth year n Total number of items in a sequence l A loss function which is usually equal to zero at a pre-selected target, T. Z* Minimum cumulative loss for a known inflow sequence
The computational technique uses both deterministic dynamic programming and hydrologic simulation. It was suggested that this type of approach be called Monte Carlo Dynamic
Programming or MCDP. The MCDP methodology starts by selecting values for X1 and S1 and generates Xi and Xi+1, i = 1, 2,...,1000 using Thomas-Fiering recursive relation.
After generating Xi and rounding each value of Xi to the nearest integer, integral optimal
storage Si : i = 1, 2, ..., 1001 using algorithm (2.5) are found and then optimal drafts (Di) are computed using Xi and Si by
Di = - Si+1 + Si + Xi (2.6) _ The arranged values (Di ,Si ,Xi ,Xi+1 ): i = 1,2,...900 are finally regressed using least squares regression to estimate the parameters of
Chapter Two 10 Literature Review
_ Di = Reg (Si , Xi , Xi+1) (2.7) _ Where Xi+1 = Forecast of Xi+1
The IBM 7094 computing time for his methodology (as coded by Young) was 0.33 Sm minutes (where Sm is reservoir volume). He presented fifty-two regression functions. For instance an estimated policy for quadratic loss Sm = 10 and no forecasting is
Di = 6.848 + 0.0650 Si + 0.290 Xi (2.8)
(Multiple Correlation Coeff., R2 = 0.64)
He agreed with the finding that for smooth convex loss functions Di = µx (whenever possible) is a near optimal policy. He also concluded that optimal policies are independent of the target for smooth convex loss functions.
Hall et al. (1968) presented technique of analysis by which the dynamic operation policies for planning complex multipurpose reservoir system can be optimized. Using the standard recursive procedures of dynamic programming, the recursive relationship is given by
Vj(S) = max [ Pwi Rj + Pej Epj + Pnj Enj + Vj+1 (S + fi - qi - Ri ] (2.9)
Where j Number of the time period, j=N being the final period.
Vi(S) Total income from sale of water and power from period to the end of the planning period, starting with storage S.
Pej Peak energy price
Pnj Non-peak energy price
Pwj Water price
R,Ep,En Release, peak energy and non-peak energy respectively.
fj Inflow and accretion in period j.
qj Evaporation in period j.
Equation (2.9) is subject to constraints on release, constraints on energy production and constraints on the maximum and minimum allowable storage levels, the latter reflecting flood control reservations and minimum power pool requirements as well as physical limitations. He used monthly time period giving an optimal schedule of release for each
Chapter Two 11 Literature Review
month of a year. He claimed that the dynamic programming computational procedures developed, were extremely fast.
Mobasheri and Harboe (1970) presented a two stage optimization model for the design of a multipurpose reservoir. The technique takes into account the fact that economic returns from a project are a function of both design and operational rules of the project. A dynamic programming (DP) model computes the optimum operation policy of a feasible design. An iterative grid sampling algorithm is then used to compare designs for which optimum operation are already determined and to select the best design. The application is made to a hypothetical single multipurpose reservoir water resource project. A flow chart for the proposed procedure was presented. The limitations of the study were (i) Only the relevant and most important properties of reality were represented, (ii) Only a few development purposes were included and the model was for a single reservoir and (iii) Only deterministic streamflow data and the concept of critical period were used.
A useful theorem in the dynamic programming solution of sequencing and scheduling problems occurring in capital expenditure planning was presented by Morin and Esogbue (1974). The theorem may be followed for reduction of dimensionality, one of the major problems associated in dynamic programming. The theorem is given below
“If in the one dimensional sequencing problem there exists m, 1 ≤ m ≤ N, distinct subsets
of projects φ1, φ2,...,φm, all of whose respective members have either (i) the same capacity Q(i) for j e {1,2,...,m} or (ii) the same capital cost, C(i), then in any optimal sequence these projects will appear in nondecreasing order of the ratio of their respective costs to their respective capacities”.
Chow et al. (1975) analysed the complete time and memory requirements for Dynamic Programming (DP) and Discrete Differential Dynamic Programming (DDDP). They suggested that the computer time required for a water resource optimization problem by DP or DDDP might be considered as the sum of the compiling time of programming language translation, the initiating time for the program, and the execution time TE. They found a relation for execution time
s p TE = Tα MN π Qi π Pj (2.10) t=1 j=1
Chapter Two 12 Literature Review
Where TM Average time for one unit operation.
M No. of iterations involved in optimizing a system
N No. of stages in the problem
Qi, Pj No. of feasible values that state variable i (i = 1, 2, ...,S) and variable j (j =1,2,...,D) respectively can take in each iteration or in the optimization procedure The computer memory required may be considered as the sum of the machine memory, code memory and data memory. They verified the formulas by solving problems for operation of single and multiple-purpose reservoir networks, etc. For example using IBM 360/75, for the operation of a 2 reservoir system for irrigation and hydro-electric power using Multistate incremental DP, with 2 states, 2 decisions, 64 iterations, 144 stages and 3 Nos. of Qi and Pi the execution time TE was 80.26 sec. and Ta was 108 µs.
Klemes (1977) investigated minimum number of discrete states to represent the range of reservoir storage required to get optimal results. It was demonstrated that a too coarse discrete storage representation could not only impede accuracy but may completely distort reality in a most unexpected way. It was shown that the number of storage states was subject to some absolute constraints and that it must increase linearly with the reservoir storage capacity in order that comparability of results is assured. A linear rule for the estimation of the number of storage states has been suggested and given below
no - 2 S n = ║ ( ------S + 2 ) = ║ --- + 2 (2.11) So ∆o
Where n Number of storage states
no Number of storage states in specific case S Reservoir storage capacity
So Reservoir capacity in a specific case
∆o Width of storage zone in a specific case
The symbol ║ is used to indicate the rounding to the nearest integer. So, no and ∆o are
obtained corresponding to a distribution {P}o (log normal and normal input distributions for
Chapter Two 13 Literature Review test case), used in arriving at a result with which the result for a reservoir of capacity S is to be compared.
The value of n obtained from (2.11) must satisfy the constraints given by:
n ≥ 3
S n ≥ ║ ( ------+ 2 ) 2 (D - xmin)
S n ≥ ║ ( ------+ 2 ) 2 (xmax - D)
Where D Reservoir draft (desired outflow) xmin, xmax Reservoir inflow (min, max respectively)
Charts are derived giving the numbers of storage states necessary to obtain stationary probabilities of reservoir emptiness and/or fullness with an error ε ≤ 0.1%.
Karamouz and Houck (1982) derived general reservoir system operating rules by deterministic dynamic programming, regression and simulation. These were tested for 48 cases. The following equations were used to define the loss function
A [exp (Rt /RUP) - exp (1)] , if Rt ≥ RUP
LOSS (Rt) = 0 , if RLOW ≤ Rt ≤ RUP (2.12)
B [exp (- Rt /RLOW) - exp (-1)], if Rt ≤ RLOW
Where A, B Constants that depend upon the price of water and how extensive the property damage is known
Rt Release during time t, m RUP Upper limit of safe range (known) m3 RLOW Lower limit of the safe range (known) m3
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And the objective function becomes
T min Z = Σ LOSS (Rt) t=1
Several physical restrictions were included as constraints in the DP. Annual and monthly rules were generated. Bhasker and Whitlatch Jr. (1980) developed a backward looking dynamic programming algorithm to obtain optimal releases from a single multiple-purpose reservoir. One sided and two sided loss functions were solved and monthly policies were derived for Hoover Reservoir located on Big Walnut Creek in Central Ohio.
Multiple Reservoir Problem
Schweig and Cole (1968) evaluated optimal control of linked reservoirs meeting a common demand for a two storage case. They included a numerical example of the dynamic programming calculation for a system of a finite surface reservoir and a full aquifer having limited pumpage. The computer algorithm converges to optimal control rules for the most part within 5 years of iteration. Given the optimal control rules for an assumed reservoir system, it becomes possible to form transition matrices of contents by an adaptation of Gould's method. The steady state solutions of the matrices show probabilities of each reservoir's contents in the long term. The total direct operation cost, TCR is given by
TCR = (RA x URA) + (RB x URB) + (RAB x URAB) (2.13)
Where RA, RB Release from reservoir A and B respectively in million gallons per month RAB Transfers from reservoir A to B in million gallons per month.
URA, URB Unit cost, in $, of releasing one million gallons per month from reservoir A and B respectively URAB Ditto for transfers from reservoir A to B.
They demonstrated that dynamic programming is far simpler in practice than its algebra might seem to suggest. The hurdles barring wider use of such procedures are more technical
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than conceptual. One of the biggest problems is that of computer running time to obtain optimal policies with large number of state variables. They concluded that research into speeding the convergence of solutions is urgently required.
Heidari et al. (1971) presented an iterative method based on Discrete Differential Dynamic Programming approach (DDDP). It was claimed that the algorithm could ease the two major difficulties: memory requirements and computer time requirements. Their equation for invertible systems is given below
F* [S(n), n] = max { R [ S(n-1), Φ [ S(n-1), S(n), y(n-1) ] , n-1 ] + F* [ S (n-1), n-1 ] } S(n-1) ε D(n-1) (2.14) Where D(n-1) The state subdomain located in the neighbourhood of the trial trajectory at stage n-1 F Optimal sum of the returns for N time periods n Beginning of a time period called a stage R Return from the system in one time increment S(n) m-dimensional state (storage) vector at stage n y(n-1) Inflow at stage n-1 Φ Decision as a function of state only
The method starts with a trial trajectory satisfying a specific set of initial and final conditions and applies Bellman's recursive equation (2.14) in the neighbourhood of this trajectory. At the end of each iteration step a locally improved trajectory is obtained and used as the trial trajectory in the next step. They proved that the method is effective in the case of invertible systems. The proposed approach is applied to a four-unit two-purpose water resources system. The example was restricted to determine inflows.
Trott and Yeh (1973) developed a method to determine the optimal design of system of reservoirs with series and parallel connections. A modified gradient technique is used to determine the set of reservoir sizes which maximizes the net benefits, subject to the imposed constraints. The technique was based on incremental Dynamic Programming algorithm using the reservoir equation
ft [S1(t+1)] = max { min [ {R'(t)+r1(t)}/αj , ft[S1(t) ] } (2.15)
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Where f M-dimensional vector of functions
j monthly period index = 1,2,3,...12
R'(t) Total release from (M-1) constrained reservoirs for month t
ri(t) Release during month t for reservoir i
S(t+1) Storage level at end of month t
Si(t) Sequence of states for reservoir t αj Fraction of yearly demand to be delivered in month j
They applied their successive approximation algorithm with state incremental dynamic programming on a network of six reservoirs named Dos Rias, Pine Mountain, Indian Valley, English ridge, Clear lake and Kennedy Flats. The returns from the system were considered to be derived from the firm water contract at a demand point. The advantage of the method of successive approximation was that the solution of a six-dimensional dynamic programming problem was obtained by solving a series of one-hundred dynamic programming problems. In their example problem, number of iterations for the convergence of successive approximation algorithm (monthly time step) to get optimal policies varies between 1 and 67.
Becker and Yeh (1974) developed a method based on dynamic programming through which optimal timing, sequencing and sizing of multiple reservoirs surface water supply facilities could be found. The recursion formula for a forward DP routine is
k ki -φ(q(i)-z(i)) k fi(qi) = min [ c (y ) (1+r) + fi-1(qi - x i) ] (2.16)
Subject to
qi-1 < qi < D(T) k x i = qi - qi-1
f0( ) = 0 yk < Yk Where i Stage
fi( ) Minimum cost function at stage i
qi Total annual firm water available at i k x i Annual firm water increment resulting from construction at k site at i
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ck Cost function for project at the k site k y i Actual physical capacity of project at k site i stage D(t) Annual firm water demand relationship φ( ) Inverse of demand function, years T Time horizon, years r Annual discount rate (assumed 5% for case study) Yk Maximum possible reservoir capacity at kth site
The methodology was applied to a proposed system of reservoirs associated with the Eel River Ultimate Project in Northern California. The reservoirs, named Dos Rios (1), Pine Mt. (2), Indian Valley (3), English Ridge (4), Clear Lake (5), Kennedy Flats (6) were included in the system.
They concluded that although it is firm water that is demanded or sold, it is reservoir capacity at a particular site that is costed and the two are not simply related, nor is the relationship independent of previously constructed reservoirs and the stream-reservoir configuration. Their technique assumed streamflows to be subnormal correspond to a critical period analysis method and no advantage was taken if any supposed knowledge about future flows. They recommended that the work should be extended to multipurpose multireservoir system.
A new technique, called multilevel incremental dynamic programming (MIDP), is presented by Nopmongcol and Askew (1976). The algorithm can greatly increase the efficiency of solution and permit high-dimensional deterministic problems to be handled with far greater ease comparatively. Hypothetical four reservoir system was investigated.
Kuo et al. (1990) developed a modelling package for the real time operation of Feitsui and Shihmen Reservoirs in the Tanshui River Basin, Taiwan, the Republic of China. The package consisted of a 10-daily streamflow forecast model, a rule curve based simulation model, and an optimization model. With the help of forecasted streamflow for a year, the simulation model is first used to find whether there is severe shortage of water. The DP model is then used to find an improved operating policy. Normal and abnormal periods were considered in the DP. At the end of each 10-day period the forecast is updated using a Kalman filter technique and the observed streamflow during the period. The simulation model and
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optimization model was rerun for the remaining period of the year. The cycle was repeated until the last period was reached. The execution time for 1 year (36 10-day periods) operation was about 45 min. on the PC. It was suggested that the 10-day operating rules could be used as boundary conditions (beginning and ending storage and 10-day releases) for a daily or even an hourly operation.
A modification of differential dynamic programming (DDP) which made that technique applicable to certain constrained sequential decision problems such as multireservoir control problem was presented by Murray and Yakowitz (1979). The authors contended that the technique is superior to available alternative. They supported their belief by experimentation and analysis. They concluded that constrained DDP did not suffer the 'curse of dimensionality' and required no discretization. DDP efficiently solved their 4-reservoir as well as 10-reservoir problems.
2.3 Stochastic Dynamic Programming
Stochastic dynamic programming (SDP) is an optimization technique based on Bellman's principle of optimality (Bellman and Dreyfus 1962, Dreyfus and Law 1977, Nemhauser 1966). Deterministic optimization methods for reservoir system optimization have several computational advantages over stochastic optimization. However, ignoring the stochasticity of the system to be considered not only simplifies the model but also introduces bias, as described by Loucks et al. (1981), Huang et al. (1991). Loucks wrote that deterministic models based on average or mean values of inputs, such as streamflows, are usually optimistic. System benefits are overestimated, and costs and losses are underestimated, if they are based only on the expected values of each input variable. In contrast to the models yielding only "optimal releases" within a fixed deterministic framework, stochastic dynamic programming is of practical interest (Yakowitz 1982, Yeh 1985). Application of SDP to water resource systems has been investigated by many authors. Their work is described in brief in the following paragraphs.
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Single reservoir problem
Torabi and Mobesheri (1973) presented a stochastic Dynamic Programming model useful in determining the optimal operating policy of a single multipurpose reservoir. Stochastic nature of the inflows was taken into account by considering the correlation between the streamflow of the pair of consecutive time intervals.
The model equation moving backward, starting from state K is given by
L Q r r Fk(t-1) = Max Σ Σ {[U k,l + Fl(t)] P q (St =l/St-1 = K) } (2.17) l=1 q=1
Where Fl(t) Expected return, if an optimal stationary policy is followed starting from state l with t intervals
Rt-1 Release of water during month (t-1) l Index for storage level q Index for storage level r U k,l Amount of energy production during the transition from k to l under the policy r r P q Probability of being in state k in month t
St Index for storage in month t (at the beginning of month t) r Index for release levels subject to various constraints
The model was applied to the Folsom Reservoir and Power plant on the American River in Southern California. Reservoir is meant for hydroelectricity water quality and flood control. Firm water supply was treated as parameter. Flood and water quality control downstream from the reservoir were kept in constraints. Discretization in the model written in FORTRAN IV, was taken for 10 storage levels and 12 months. A transition probability matrix approach was used.
The model calculates the optimum monthly release policy for a given level of water supply to the expected annual level of on-peak energy production. By changing the level of water supply, the trade off between energy and water supply was determined. They recommended
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that the model should be extended in order to find optimum operation policy for multi- reservoir systems.
Pegram (1974) investigated the effect of input model, size of reservoir and various input parameters on the probability of zero draft from a discrete reservoir fed by a serially correlated input. In his work a probability theory of reservoirs with serially correlated inputs was presented. Four simple transition probability matrix (tpm) models were presented. These tpm models may find use in reservoir studies since they are specified by very few parameters.
Out of four tpm models one model (T1 ) is given below to conserve space
(1-a) (1-a)(1-b) (1-b)2
T1 = 2a(1-a) a + b - 2ab 2b(1-b) (2.18) a2 ab b2
Where a µ/2(1-r) b r + a µ Mean r First serial correlation coefficient σ2 Variance c Reservoir capacity
The computation was done by using a difference equation method (Pegram 1972) to examine the factors affecting the probability distribution of the draft (ño).
The results showed that if µ, σ and c were fixed (µ = 0.8, σ = 0.48, c = 9 and r was variable (r = 0 ~ 0.8, there was very small amount of variation of ño (0.0001 to 0.0054) between the
models T1, T2, T3 and T4. If only reservoir capacity c is fixed (c = 9) and µ, σ and r are varied, the variation of ño is typical for each model. Similarly the variation of ño with µ, r and c for a fixed variance was identical for each model. He concluded that a change of model
(T1, T2 ...T3) had relatively little effect upon the values of ño. His studies reveal that as long as µ, σ and r are preserved the form of the stochastic model of the input process may be relatively unimportant from the point of view of the draft.
Chapter Two 21 Literature Review
Su and Deininger (1974) studied the regulation of Lake Superior under uncertainty of future water supplies. The levels of Lake Superior have been regulated since 1922. These regulation plans have been modified several times since their first design. The objective of the study was to find operating policies that minimized the expected undiscounted yearly losses over an infinite time horizon. The inflows to the lake were treated as stochastic random variables. The system was modelled as a periodic Markovian decision problem. A new algorithm based on the White's method of successive approximations for solving single chained and completely ergodic Markovian decision problems was developed and was proven to be fairly efficient in terms of computer time and storage. The newly developed policies were tested against the current operating policies by using the historical inflow record as data. The results show that if some of the developed operating policies were adopted, the average yearly losses could be reduced by at least 15%. At the same time the monthly lake level variances could be reduced by 25%.
To derive long-term operating policies for water resource systems, an algorithm is developed by Mawer and Thorn (1974) that uses value iteration dynamic programming and simulation in conjunction with Penalty costs. Two types of problems were solved on monthly time step as given below
i) Determine transition costs lead to a very simple and efficient type of value iteration recurrence relationship
k k Vn(i) = min [ W n(i) + en (i - r ) i = 1,2,3,...N (2.19) k
k k Where en(i-r ) = Σ fn (i-r , j) Vn-1(j) k k fn(i-r ,j) Probability of making a transition from intermediate state (i-r ) to final state j during month n. rk Release to be made if decision k is selected. k W n Immediate cost incurred in month n, i being starting point if decision k is used.
Vn(i) Total expected cost incurred in the next n months starting from state i if the reservoir is operated under the optimal (i.e. cheapest policy).
Chapter Two 22 Literature Review
j Final state for month n, equal to the initial state for month n-1. i Initial state for month n. k Index denoting the decision used in month n.
ii) Another problem was solved for probabilistic transition costs. The value iteration recurrence relationship was given by
k qmax k * k Vn(i) = min { Σ P n(q) [ r n(j/i, q) + Vn-1(j) ] } (2.20) k k = qmin
k Where P n(q) Probability of a net addition to storage of q units during month n if decision k is used. k r n(j/i),q) Expected cost of making a transition from state i to state j during month n using the pumping policy k to give a net addition to storage q.
Each program took approximately 100S to compute and execute. The author claimed that advantage of the optimization algorithm is the facility to include in the simulation items of interest that would otherwise be omitted. The method described had been based on a single reservoir. It was recommended that it may be extended for multireservoir systems and more than one objective function, it is necessary to extend the method to flood control, the maintenance of amenity levels etc.
Maidment and Chow (1981) investigated whether state space models, stochastic dynamic programming and Markov chain analysis can be linked together for the analysis reservoir systems and to assess what the advantages and limitations are of such methodology. The mathematical description of the inflows to the reservoir system and response of the system to inflow and release are formulated as a state space model. This model is incorporated within an explicit stochastic dynamic programming procedure as the state transition equation. By using the model, the transition probabilities of the state variables are found for each decision and tested in the dynamic programming. Those for the optimal decisions are stored for later use in a Markov chain analysis to determine the statistical characteristics of the state and
Chapter Two 23 Literature Review
decision variables if the reservoir is operated for a long period of time with the optimized policy. The procedure is applied to proposed Watasheamu Dam in Nevada.
Esmaeil-Beik and Yu (1984) used the SDP to develop optimal policies for operating the multipurpose pool of the Elk City Lake in the Arkansas River basin in Kansas with serially correlated inflows. A weekly time step was adopted. Using Savarenskiy's scheme, the number of storage states are 13 and number of inflow states are 15. The SDP recursive equation was
(t) (t) FMT-t(i) = min { Σ pij [ Rij + FMT-t+1(j) ] } (2.21) j j In which the period T = 52 weeks and T counts the weeks in a year except for the initial stage
when t=0 was used. At t=0 FMT+1(j) for all j must be specified to start the backward (t) computation. pij = probability of transition to stage j if the system is now in state i and a (t) feasible decision is chosen. Rij = loss function. The operation of the multipurpose pool of the Elk City Lake could be treated as a periodic Markov decision process with finite states and discrete time. The SDP model was used to derive long-term operating policy to minimize the expected average annual loss. Using the developed optimal policy the lake was operated for the period 1966-67. The results showed a marked reduction in the expected annual losses as compared with the historical operation.
Wang and Adams (1986) proposed a two-stage optimization framework which consists of a real time model followed by a steady state model. The intention of the work was to develop an efficient computational procedure based on SDP for real time optimization of single reservoir. In recognition of hydrologic uncertainty and seasonality, reservoir inflows were described as periodic Markov processes. The steady state model that described the convergent nature of the prospective future operations was regarded as a periodic Markov decision process and was optimized with the generalized policy iteration method (Howard 1960, White 1963). The results of steady state model were in fact used as an interim step for deriving the optimal immediate decisions for the current period in the real-time model. The procedure was applied to the Dan-River-Issue Reservoir on a tributary of the Yangtze River in Canada. 30-year of historical inflow records was used in the study. The reservoir live storage was discretized into n states (in this case n=22) with identical increments Ω according to Savarenskiy's scheme (Klemes 1977). If the minimum and maximum live
Chapter Two 24 Literature Review
storage volumes of the reservoir are donated as 0 and C respectively, the storage volume Si corresponding to the discrete state indices i are specified as
Ω = C/(n-2) (2.22)
S1 = 0
Sn = C
Si = (i-1.5).Ω i = 2,3,...,(n-1) The storage state transition probability was derived from the inflow probability distribution. The number of inflow states varied from 6 to 36 in various months. An alternative procedure for deriving the inflow transition matrices if the historic inflow record is limited, was presented. The SDP (value iteration) recursive equation for the real time model is
d d VRi(t) = max [ bi (t) + Σ Pij (t). VRj(t+1) ] (2.23) d ε D d Where Pij (t) = storage state transition probability from state i to state j during period t under d decision d, bi (t) = expected immediate reward per transition for state i under decision d in
month t, VRi(t) = real-time objective function value from state i in month t, D = decision space equal to {d}. The results indicated that the proposed procedure is suited to real-time reservoir operation optimization.
Hashimoto et al. (1982) studied three criteria for evaluating the possible performance of water resources system. These criteria evaluate how likely a system is to fail (reliability), how quickly it recovers from failure (resiliency) and how severe the consequences of failure may be (vulnerability). These measures may be used in assessing alterative design and operation policies of a water resource project. The reliability of a system may be described by the frequency or probability α that a system is in a satisfactory state:
α = Prob [ Xt ε S ] (2.24)
Where S is the set of all possible outputs and Xt is random variable in time t. Resiliency describe how quickly a system is likely to recover or bounce from a failure once the failure has occurred. If system recovery is slow, this may have serious implications for system design. Resiliency may be described as
Chapter Two 25 Literature Review
p: τ = ----- (2.25) 1 - α
1 n : Where P = lim ---- Σ Wt n → ∞ n t=1
= Prob { Xt ε S, Xt+1 ε F }
Wt = transition from a satisfactory to an unsatisfactory state, F = set of all unsatisfactory (failure) outputs. The vulnerability may be defined as
v = Σ sj ej (2.26) jεF
Where sj = a numerical indicator of severity of that state, ej = Prob. [ xj, corresponds to sj, is the most severe outcome in a sojourn in F ]. The reservoir operation example presented by Loucks et al. (1981) in page 138-152 is used to illustrate the proposed criteria. The summer season operating policies were derived by SDP (Loucks et al. 1981) with the objective of minimizing the loss
Min E [ lß(R) ] (2.27)
Where T = target release of 4.5 x 107 m3 R = summer season release lß(R) = 0 when R ≥ T ß lß(R) = [(T-R)/T] when R < T
The exponent ß (0-7 in this case) defines the shape of the loss function. The results indicated that quantitative risk criteria may be useful in determine efficient release policies.
Another criteria, 'robustness' is proposed as a measure of the likelihood that the actual cost of a proposed project will not exceed some fraction of the minimum possible cost of a system designed for the actual conditions that occur in the future. The value of robustness Rß at the level ß is simply the probability that the system's opportunity cost will not exceed ß times the minimum total cost. It is the probability that the design parameter q will have a value within the domain Ωß
Chapter Two 26 Literature Review
⌠ Rß = │ f(q) dq (2.28) ⌡Ωß
The robustness criteria was illustrated by its application to the planning of water supply system in south-western Sweden. The criteria provides a basis for comparing alternatives.
Huang et al. (1991) presented and compared four types of SDP models for on-line reservoir operation, relying on observed or forecasted inflows. The models are different because of the assumption regarding the inflow in the next time period. The objective is to maximize expected annual hydropower generation. A 10-daily period time step is adopted in the study. 21 discrete states on storage were considered. The models were applied to Feitsua reservoir, in northern Taiwan. Optimal operation was derived using four variations of SDP with a specific objective function. The comparison of the four SDP models was made using simulation. The mathematical models used in the study are presented below:
n n-1 f t(St,Qt) = max [ Bt + Σ PQ(t+1)/Q(t) . f t+1 (St+1,Qt+1) ] (2.29) S(t+1) Q(t+1)
n n-1 f t(St,Qt) = max [ Bt + Σ PQ(t+1) . f t+1 (St+1,Qt+1) ] (2.30) S(t+1) Q(t+1)
n n-1 f t(St,Qt-1) = max { Σ PQ(t+1)/Q(t) [ Bt + f t+1 (St+1,Qt+1)] } (2.31) R(t) Q(t)
n n-1 f t(St) = max { Σ PQ(t) [ Bt + f t+1 (St+1) ] } (2.32) R(t) Q(t)
Where t Within-year period N Total No. of running periods considered prior to actual stage
St Initial storage volume in the beginning of period t
Qt Inflow during that period
Rt Water release including spillage during the same period
P(Q(t+1)/Q(t) Inflow transition probability specifying the conditional probability
Results indicate that SDP model (2.29) is preferred, if perfect inflow forecasting model are available. Otherwise (2.31) is more appropriate instead. For the Feitsui Reservoir system
Chapter Two 27 Literature Review
because of the inevitable errors existing in forecasting models, the type-three model is preferred and selected as the most appropriate SDP model for long-term on-line operation.
Karamouz and Vasiliadis (1992) investigated reservoir systems using SDP and Bayesian decision theory (BDT). The proposed model, called Bayesian stochastic dynamic programming (BSDP) includes inflow, storage and forecast as state variables, describes streamflows with a discrete lag 1 Markov process and uses BDT to incorporate new information by updating the prior probabilities to posterior probabilities. The procedure is used to determine optimal reservoir operating rules of Loch Raven Reservoir on Gunpowder River in Maryland. The performance of BSDP is compared with an alternative SDP model and a classical SDP model. The results indicated that BSDP model performed better than alternative and classical SDP models. ARIMA(1,01,) and ARIMA(1,1,1) model was used to forecast inflows into the Loch Raven reservoir. Methods of discrete representation of storage were discussed. For a reservoir of upto 170% of the mean annual flow, 20 storage values could be adequate which is within the range specified by Klemes (1977). The simplified version of the BSDP model is given by
* ft(S,I,H) = min [Bt(St,It,St+1) + α Σ { φ() . Σ [π().f t+1()]}] (2.33)
Where S = storage, I = inflow, H = streamflow forecast, α = discount factor, φ = posterior flow transition probability, π = posterior forecast transition probability function.
Vasiliadis and Karamouz (1994) investigated demand-driven stochastic programming model (DDSP). The objective of the study was to evaluated the usefulness and the hydrologic reliability of the generated operating policies by DDSP model. The model allowed to use the actual variable monthly demand in generating the monthly operating policies. BDT is applied to forecast and capture the uncertainties of the steamflow process. BDT was useful in continuously updating the probabilities for each month. In the procedure, the associated penalty is a function of the release and the expected storage. Time series analysis for forecasting and generated form of cost function was discussed. The application was made to Gunpowder River, Loch Ravern Reservoir in Maryland. The results indicated that a model with a fixed demand in optimization could not perform adequately in simulation of real-time operation when the actual demand is variable.
Chapter Two 28 Literature Review
Raman and Chandramouli (1996) used a dynamic programming (DP) model, a stochastic dynamic programming (SDP) model and a standard operating policy to derive a general operating policy for reservoirs using Neural Network. The policies are derived to improve the operation and efficient management of available water for the Aliyar Dam in Tamil Nadu, India. The objective function was to minimize the squared deficit of the release from the irrigation demand. From the DP algorithm, general operating policies are derived using a neural network procedure (DPN model) and using a multiple linear regression procedure (DPR model). The DP functional equation is solved for 20 years of historic data. The performance of the DPR, DPN, SDP and SOP models are compared for three years of historic data. The neural network procedure based on the dynamic programming algorithm provided better results than the other models. Adamowski (2008) used artificial neural networks in multiple reservoir problem for peak daily water demand forecast modeling. Chaves and Kojiri (2007) and Ghiassi et al. (2008) also applied neural network technique in water resource studies.
Askew (1974) derived optimum operating policies for three reservoir system by using stochastic dynamic programming with the objective of maximizing the expected net dollar benefits over a 50-year design period. The recursive equation for ith year was
n fi(Si) = max Σ P(qi) [ B(r)-C(x) + [ 1/(1+d) ] . fi-1 (Si-1) ] (2.34) 0 ≤ x ≤ Si+qn i=1
Where fi(Si) Maximum expected net benefit over the remaining i years of the life of the system S Initial storage C(x) Cost incurred for a target annual release of x B(r) Benefit from an actual annual release of r
P(qi) Probability of obtaining a net annual inflow of qi
qn Maximum possible inflow d Annual discount rate
r x (for Si + qi - x ≥ 0)
Si-1 Smax (Smax = max storage capacity).
Chapter Two 29 Literature Review
The problem was solved on an annual time step. Inflows, storages and releases were measured in 'units of volume' and only integer quantities were considered. An additional incentive was provided in the model by imposing an extra penalty W that is incurred if the system fails, that is, if r is less than x. The function for the net benefit in the recursive
equation was unaltered if Si + qi ≥ x but was amended in equation (2.34) as follows
[ B(r)-C(x) + { 1/(1+d) } . fi-1 (Si-1) - W ] (2.35)
if Si + qi < x. The penalty, W, has the effect of reducing the expected net benefit associated with releases that give rise to possible shortfalls and therefore, it tends to cause a more conservative release to be chosen as an optimum. His technique provides a combine use of dynamic programming and simulation. Probability of failure was estimated with the aid of system simulation. He concluded that if the goal in operating a water resource system is to maximize its expected net benefits over its design life, then stochastic dynamic programming provides very efficient means of deriving the optimum operating policy.
Askew (1974) presented a model, an extension of his previous model as discussed above. He suggested that an optimum operating policy must be derived subject to relevant chance constraints in order to limit the probability that a water resource system will fail. If a stochastic dynamic program is developed that has the ability to restrict the probability of given variables taking on values outside a fixed range, then it can be said to be capable of handling chance constraints. He suggested that such a stochastic dynamic program could be called a 'chance constrained dynamic program (CCDP). 'The recursive equation used by the CCDP to derive optimum operating policies is given by
n fi(Si) = max Σ P(qi ) [ B(ri) - C(xi) - W1 + { 1/(1+d) } . fi-1 (Si-1 ) ] (2.36) 0 ≤ xi ≤ Si + qn i=1
Where n f1(S1) = max Σ P(qi ) [ B(ri) - C(xi) - W1 ] 0 ≤ xi ≤ S1 + qn i=1
W1 = Hypothetical penalty for system failure. Rest of the symbols are same as defined in equation 2.34.
Chapter Two 30 Literature Review
Also
i) ri = xi, Si-1 = Smax, W1 = 0 [ if Si + qi - xi > Smax
ii) ri = xi, Si-1 = Si+qi-xi, W1=0 [if 0 ≤ Si + qi - xi ≤ Smax
ii) ri = Si + qi, Si-1 = 0, W1=W [if 0 ≤ Si + qi - xi ≤ 0
Where W is constant in function for W1. Optimal value for which is evaluated by iterative search.
Turgeon (1980) presented and compared two possible manipulation methods for solving weekly operating policy of multiple reservoir hydroelectric power systems using SDP. The first method, called the one-at-a-time method, consists in breaking up the original multivariable problem into a series of one-state variable subproblems that are solved by DP. The second method, called the aggregation/decomposition method consists in breaking up the original n-state variable stochastic optimization subproblems of two-state variables that are also solved by DP. The final result is a suboptimal global feedback operating policy for the system of n-reservoirs. The procedure was applied to a hypothetical system of a network of six reservoir-hydroplants. The results indicated that the aggregation decomposition method gave better operating policy than the known successive-approximation-DP method and this with the same processing time and computer memory.
In 1981, Turgeon presented a method for determining the weekly operating policy of a power system of n-reservoirs in series. The stochasticity of the river flows were considered in the method. The stochastic nonlinear optimization problem of n-state variables was transformed to n-1 problems of two state variables which were solved by DP. The release policy obtained with this method for reservoir i is a function of the water content of that reservoir and of the total amount of potential energy stored in the downstream reservoir. The application was made to a power system of four reservoirs. Turgeon (2007) applied optimal reservoir trajectory approach for stochastic optimization of multireservoir operation.
Kelman et al. (1990) developed sampling stochastic dynamic programming (SSDP), a technique that captures the complex temporal and spatial structure of the streamflow process by using a large number of sample streamflow sequences. The best inflow forecast can be included as a hydrologic state variable to improve the reservoir operating policy. To test the
Chapter Two 31 Literature Review
performance of the proposed methodology, a hydroelectric system with 9 reservoirs, 10 power houses located on the North Fork of the Feather River in California has been used.
In SSDP approach, one selects M possible streamflow scenarios for the system to describe the joint distribution of reservoir inflows and local inflows. For a monthly time step (T=12), each scenario is a year of observed monthly streamflow data representing one 12-month realization of the corresponding stochastic process. These streamflow scenarios are used to simulate the reservoir's operation and river basin energy production for all possible combinations of storage and hydrologic state in each month. In SDP, recursive equation is
max E [Bt(Rt,i,k,St+1) + α E { ft+1(St+1,Xt+1) }] (2.37) Rt* Qt/Xt Xt+1/Qt,Xt
On the other hand, in SSDP the DP recursive equation is
M L max Σ Pt(i/l) [Bt(Rt,i,k,St+1)+α Σ Pxt(v/l,i).ft+1(St+1,v,i)] (2.38) Rt* i=1 v=1
Where
EQ(t)/X(t) Conditional expectation of the inflow vector Qt given Xt
St(k) Reservoir storage at stage t, discretized into K values (k=1,...,K) with
St(1)=Smin and St(K)=Smax
Xt(l) Streamflow forecast stage t, discretized into L values (l=1,...L) with
Xt(1) corresponding to a dry forecast and Xt(L) a wet forecast
Qt(i) Vector of inflows throughout the basin at stage t for the ith scenario (i=1,...,M)
Rt*(k,l) Target release in state (k,l) at stage t R Actual release at any stage/state
Bt Return at stage t due to the release R, given the initial and final storage
ft(k,l,i) Benefit of reservoir operation from t through T when the state is (k,l) and the ith scenario occurs
Pt(i/l) Probability of the ith scenario at stage t given streamflow forecast Xt(l)
PXt(v/l,i) Transition probability from forecast Xt(l) and Qt(i) to forecast Xt+1(v) α Monthly discount factor
Chapter Two 32 Literature Review
SSDP (equation (2.38)) uses M streamflow scenarios to describe the distribution of flows over time and space. SDP (equation (2.37)), on the other hand, generally employs a discrete
approximation of a continuous distribution of Qt given Xt, presumably based on the observed historical record. Both models employ a Markov chain for hydrologic state variable. In SSDP, it is conditioned on the scenario i. The probability assigned to scenario i can be calculated by Bayes theorem based on the actual inflow that occurs with sequence i between month t and July as given below. In northern California several sources of information, including snow pack are used to forecast the snowmelt season's (January-July) runoff. For this situation it seems natural to let the hydrologic state variable Yt be the forecast of the remainder of the seasonal runoff
July Yt(i) = Σ QT(i) (2.39) T=t
p[ Xt/Yt(i) ] p(i) Pt(i/Xt) = ------(2.40) M Σ p[ Xt/Yt(j) ] p(j) J=1
The SSDP approach has been useful in hydropower planning of Feather River in California to generate efficient operating policies faster than the traditional trial-and-error reservoir operation.
Braga Jr. et al. (1991) developed an SDP model for the optimization of hydropower production of a multiple storage reservoir system with correlated inflows. The model contains two parts, (i) an off line program which is based on DDP and (ii) an on line program which is based on SDP. The off line program calculates the value of stored water (one time only) in all of the reservoirs as a function of the several reservoirs storages and the month of the year. The calculated value represents potential energy generation and is based on historical flow data. The on line program performs real time operation. Each month a multi- dimension search is made for the optimal set of reservoir releases that maximize system benefits. The search uses the transition probability matrices and the tables of stored-water benefit for the particular month determined by the off line program. The optimization requires knowledge of the previous month's inflows and the starting storages for the current
Chapter Two 33 Literature Review
month; but these, of course, are observable and are readily determined. The following month, the on-line search procedure is repeated to find the optimal releases for that month. Application was made to a subsystem of the Brazilian hydroelectric system. For a single reservoir, their recursive equation of DP is
Itmax - 1 ft(St,It+1) = max { Σ P(It/It+1)[Bt(Rt,St)+ --- ft+1 (St-1,It)]} (2.41) Rt Itmin 1+r
Subject to
St-1 = St + It - Rt - Et + Rut
and the various reservoir and system constraints.
Where ft = expected total return from optimal operation with t time periods to go to the end of the planning horizon, Rt = release during time period t; Rut = deterministic release from upstream reservoir (if any) during time t, It = stochastic inflow during time t, St = storage at - the beginning of time t, Bt = net benefit for time t, S t = average storage during time t, r =
period-to-period discount rate and P(It/It+1) = conditional probability of It given It+1. Because ft is a function of ft-1, the recursive equation has to be solved backward in time.
In on line operation, release policy for any month m is determined by conducting an on-line multidimensional search at the beginning of each month with the previous month inflow and the ending storage being known.
Itmax i i i i i i fm = max { Σ Σ P (I m/I m-1) [ Bm(R m,S m) + V m(Sm,m) ] } (2.42) Rt i Itmin
Where Rm = a vector of releases, Sm = the ending storage vector, Im-1 = the previous month's
inflow, Im = the local river inflow, i = specific reservoir being considered, Vm = future value i of the storage vector as determined by the off-line program. Note that V m is a function of 1 2 S m, S m, ... as well as m for each value of i. Best releases from the upstream reservoir are determined while the storages of the other reservoirs are maintained. The next reservoir release is the considered, using above equation, and the previously determined releases of the other reservoirs are maintained. The one-at-a-time optimization is cycled until there is no
Chapter Two 34 Literature Review
further change in release policy. The combination of off-line and on-line procedures drastically reduces the computational requirements inherent in multidimensional SDP, because results from the one-time-only calculations can be stored beforehand and the real time, on-line SDP only operates one month at a time. SDP convergence was reached in three to four iterations. Model performances were superior to that of the historical operational records. The methodology provides a viable way of applying SDP to multiple reservoir system.
Karamouz et al. (1992) investigated an implicit stochastic optimization scheme to consider multiple reservoir systems. The scheme comprised a three-step cyclic procedure that attempt to improve the initial operating rules for the system. The three-step cycle begins with an optimization of reservoir operations for a given set of streamflows. The optimal operations from the solution are then analyzed in a regression procedure to obtain a set of operating rules. These rules are evaluated in simulation model using a different set of the data. Based on the simulation results, bounds are placed on operations and cycle returns to the optimization model. The cycle continues until one of the stopping rules is satisfied. The methodology was applied to a two reservoir system. The Lock Reservoir is located on the Gunpowder River, while the Liberty Reservoir is located on Patapsco River. These two reservoirs are two of the three principal sources of water supply to the city Baltimore. A discrete dynamic program has been used to determine reservoir operating rules for the multiple reservoir system. The objective function can be expressed as
T x Minimize Z = Σ loss [ Σ Rst ] (2.43) t=1 s=1
Where T = the time horizon, x = the total number of sites and loss ( Σ Rst ) is a hypothetical loss function which is logical and has been used in the literature. The results indicates major improvement in the monthly operating rules over standard operating rules when the operating rules are refined using the proposed algorithm.
Valdes et al. (1992) presented a group of optimization models for the real time operation of a hydropower system of reservoirs. The procedure combines stochastic DP and linear programming. The dimensionality problem usually found in DP was solved by a space-time aggregation/desegregation method. Scheme for discretization of inflows (state variable) was
Chapter Two 35 Literature Review
given. When the stochastic nature of the inflows has to be taken into account, SDP seems to be the preferred technique to solve the optimization problem. The reservoirs in a hydropower system were aggregated in power units rather than in water units. An optimal operating policy for the equivalent aggregated reservoir was found first. The objective function was to minimize the total cost or energy production for a hydrothermal system. The method of successive approximations (Su and Deininger 1974) was used to solve SDP. The aggregated policy is then used in the real time operation of the system to determine the daily releases and power production from each reservoir of the system. Lower Caroni system in Venezula which is composed of four reservoirs in series and a total installed capacity of 17,000 MW was used as a test case. The results are found to be effective in computational sense.
Georgakakos and Yao (1993) stated that in case where data records are insufficient (e.g. extreme events), stochastic methods were inadequate. They presented a control approach where input variables are only expected to belong in certain sets. The solution is based on DP and derived for the case where all sets are convex polyhedra. The objective is to determine control action that the system will remain within desirable bounds. In the set control approach the inputs are unknown but bounded. The method was applied to a three reservoir system in the southern United States. The value of streamflow forcasting in reservoir operation is given by Georgakakos (1989).
Huang and Wu (1993) investigated a procedure to check whether the SDP models for reservoir operation will converge or not. The steady state SDP model has been guaranteed after the SDP convergence has been reached. The study indicates that the stability of a SDP model as it relates to the existence and uniqueness of solutions of transition probabilities for the inflows depends on the rank of the coefficient matrix A. If the rank of A is the same as the number of inflow states, there is precisely one solution for steady state distributions of the inflows and SDP convergence will be reached. Otherwise the SDP model is divergent. In practice, SDP divergence often occurs due to the fact that the available historical records are not of sufficient duration to define flows in all intervals of the flow matrix. This condition may be avoided either by careful discretization of reservoir inflows or by synthetic inflow generation. The method can be illustrated by the following example taken from Huang and Wu (1993). Given transition inflow matrix of two periods (say Rabi and Kharif)
Chapter Two 36 Literature Review
0.7 0.3 0.6 0.4 P1 = P2 = 0.2 0.8 0.0 1.0
The steady state inflow distributions for these two periods are given by
1 2 2 1 PQ1 = PQ1 P P and PQ2 = PQ2 P P (2.44)
where 0.42 0.58 0.5 0.5 P1P2 = P2P1 = 0.12 0.88 0.2 0.8
The condition for convergence is
1 a11-1 a12 ... a1m 1 a21 a22-1 ... a2m A = . . . (2.45) 1 am1 am2 ... amm-1
The form of matrix A becomes
1, 0.42-1 0.58 AP1P2 = 1, 0.21 0.88-1
1, 0.5-1 0.5 AP2P1 = 1, 0.2 0.8-1
The rank of matrix A in both the above matrix is 2 because a two rowed squared submatrix with nonzero determinant exists. Therefore SDP convergence will be attained, resulting in
PQ1 = [0.171, 0.829 ]
PQ2 = [0.286, 0.714 ]
Chapter Two 37 Literature Review
Cardwell and Ellis (1993) presented SDP models for water quality management in Schuylkill River in Pennsylvania. River reaches correspond to stages, water quality parameters are the state variables and control action in the DP represent treatment levels. The SDP objective function was
v(s/n) = min [ rk + Σ P(s*/s,k) . v(s*/n-1) ] (2.46) kεK s* Where P(s*/s,k) is the probability that the system will transition to state s* at stage n-1 if control k is applied to state s in stage n. In addition, constraint relaxation, simulation with uncertainty and regret-based models were presented. The principle importance in the study was to incorporate both model and parameter uncertainty.
Increased withdrawals from the drainage basin of the Dead Sea over the past 50 years resulted elevation differences between the Mediterranean Sea and the Dead Sea. It is proposed that withdrawals from the Dead Sea would be replaced by a controlled inflow of water from the Mediterranean Sea which would generate hydro-electric power. A stochastic dynamic programming model (Weiner and Ben-Zvi 1982) is applied to optimize the annual Mediterranean water inflows, taking into account the high variability of the remaining natural inflows to the Dead Sea and to maximize the discounted expected value of the plant benefit.
2.4 Linear Programming
For the solution of the reservoir optimization problem, LP was used by Mannos (1955), Loucks (1968), Roefs and Bodin (1970). Chance-constrained LP has been used by Revelle et al. (1969), Eisel (1972), Houck (1979), and Houck et al. (1980).
Mohammadi and Marino (1984) presented a model for the real time operation of Folsom reservoir of the California Central Valley Project. The model was a combination of linear programming (used for month by month operation) and dynamic programming (used for annual optimization). Choice of objective function for reservoir operation was discussed. It gave flexibility to select the objective that would best satisfy the needs of the area. The following three objective functions were studied.
i) Maximization of water and energy over the year.
Chapter Two 38 Literature Review
ii) Maximization of water and energy with flood control consideration.
iii) Maximization of water and energy for months with relatively high water and energy demands.
Curry et al. (1973)'s work is an extension of the work of Re Velle for reservoir modelling (1969). Re Velle's linear decision rule (LDR) as applied to a reservoir is given by
x = s - b (2.47)
Where x Release during a period of reservoir operation
s Storage at the end of the previous period
b A decision parameter chosen to optimize some criterion functions
LDR was applied in two contexts
i) The stochastic context where inflow (input) are treated as random variable.
ii) The deterministic context where inflows are specified in advance.
The objective of Curry's model was to minimize the operating cost of the system over a specified planning horizon. They used SIMPLEX method to solve the problem. The decision variables, to be determined were
i) The amount of water released from reservoir-1, 2 and 3.
ii) The amount of water pumped into reservoir-1 from reservoir-2 and 3. The authors claimed that the primary advantage of their model would be in the real time operation of a linked system of multipurpose reservoirs. The model would provide operational guidelines which could either minimize or maximize the selected objective function if both inflow and water demands could be anticipated through forecasting procedures. The authors concluded that the change-constraint formulation places no restrictions on the inflow distribution types and objective function forms. The inflows can be independent, correlated for different lags for each reservoir or even correlated for all reservoirs.
Chapter Two 39 Literature Review
Nayak and Arora's (1974) work is an extension of Revelle's linear decision rule (LDR), 1969,
for release management of reservoirs. According to this rule, a release of xi made during the period i is a function of initial storage Si-1 and a decision parameter b for this period: i.e. xi =
Si-1 - bi. They analyzed that if the minimum required pool volume was assumed to be equal to
Am.C where Am was a fraction between zero and one and C was the optimal capacity of the
reservoir, then the quantity C - Am.C had been defined as control volume. They proved that
control volume C is independent of Am for given flow data. Their derivation (starting from Revelle's model, min C) leads to the following theorem.
For a given inflow data, free board capacity and the minimum and maximum flow requirements, there exists a constant K such that
K = (1 - Am ). C (2.48)
Where C is the optimal capacity of reservoir and 0 ≤ Am ≤ 1. Loucks and Dorfman (1975) compared and evaluated several Linear Decision Rules (LDR) used in chance-constrained models for estimating efficient reservoir capacities and operating policies. They used the following objective function for estimating the trade-off between the release target YR and the maximum storage capacity k required for a given release target including a weight w
min k - wYR (2.50)
They compared the linear decision rules (LDR)
Rt = St - bt
St = It-1 + bt-1 (2.51)
Rt = St + It - bt
St = bb-1
Their results showed that the choice of the LDR substantially influenced to active storage capacity requirement for any given release target, etc. A simulation model was written and used to evaluate the solutions of various chance-constrained models (2.51). It also confirmed the conservative nature of chance-constrained models. One of the principal reasons for the
Chapter Two 40 Literature Review
conservative nature of these chance-constrained models utilizing LDR is that they assume that each flow in each period will be critical.
Reservoir management models were investigated for optimal policies by Sobel (1975). A deterministic model for investigating smallest storage capacity was presented. Inflows and demand were treated as being known in advance. The objective function whose solution was obtained through linear programming is
St = min [[c, St-1 + rt - xt ] (2.52)
Subject to:
0 ≤ St ≤ c 0 ≤ xt ≤ St-1 + rt
Where xt Drawdown during period t rt Inflow during period t St Storage at end of period t c Reservoir capacity
He suggested that it was a Chebyshev optimization problem, namely, a search for the minimum possible value of a constrained maximum. Equation (2.52) was derived for xt as
xt = min [ft , St-1 + rt - Lt ] (2.53)
Where ft Maximum draft Lt Sum of mt mt Minimum storage
A stochastic minmax capacity model was discussed. Another formulation based on DP was derived and given below:
ft(u) = min [ Gt(u,v) + Eft+1 { p(v,Rt) } ] (2.54) u ≤ vc Where ft(u) Expected cost of an optimal policy during t if c-u is in storage at start of t. Gt(u,v) Expected net cost in t of a vector n-u of drawdown if storage levels at the beginning of the period are c-u. E Expectation (of a random variable). p(v,Rt) Storage at end of period if inflow is r and storage was c-n just before inflow occurred. He also discussed some other models with simple numerical examples. These models are valuable for reservoir management.
Chapter Two 41 Literature Review
A sequential explicitly stochastic linear programming model (SESLP) which consists of a nonlinear program and an algorithm for obtaining an approximate solution is presented by Houck and Cohon (1978). The model can be used either to determine for a multipurpose multiple-reservoir system, both a design and a management policy or to determine only a management policy. A discrete lag-one Markov process is explicitly included in the model as the streamflow description.
Benefit transformation curves were derived from a multiple linear programming model (Thampapillai and Sinden 1979). These curves were used to assess the relationship between objectives. The model was illustrated through application to a policy problem in northern New South Wales, Australia.
Houck (1982) presented five different types of objective functions. These were used to solve a real-time daily reservoir operation by mathematical programming. The five objective functions were
N+T-1 2 2 (A) min PENA = Σ [ (St+1-55000) /2025000 + (Rt-500) /12250) ] t=N
N+T-1 b e (B) min PENB = Σ [ (1/C) │St+1 - a │ + (1/f) │ Rt - d │ ] t=N
(C) min PENC = max [ SA(St+1), SB(St+1), RA(Rt), RB(Rt) ]
for t = N,...,N + T - 1
2 2 (D) min PEND = max [ (St+1-55000) /2025000 + (Rt-500) /12250) ] (2.55)
for t = N,...,N + T - 1
N+T-1 (E) min PENE = Σ [ SA(St+1)+ SB(St+1)+ RA(Rt)+ RB(Rt) ] t=N Where St+1 Reservoir storage volume m Rt Reservoir draft based on forecasted inflows for previous days m a thru f Parameters to be adjusted. Best parameter set found is a = 53600, b = 2084, c= 1400000, d = 500, e = 2.8, f = 12250. It produced a loss equal to 11.924. SA, SB Cumulative Distribution Functions value may be 0 to 1.
Chapter Two 42 Literature Review
It was found that model C was easy to use and produced very good results.
Wang et al. (2004) performed optimization of short-term hydropower generation and showed that with the development of a direct search procedure, a reformulated problem with only linear constraints of outflow release and storage content can be solved.
2.5 Multiobjective Optimization
All major water resources systems have the capability of providing a number of water-related benefits. These benefits may include water supply for irrigation, domestic, and industrial use, recreation, hydroelectric power generation, water quality improvement, flood control, fish and wildlife maintenance and navigation. A basic problem is that the various objectives may be conflicting and are not commensurable or affect different groups of people or interests. Sometimes it is important assign priority to that objective which has the greatest monetary benefit. These difficulties are well defined in Starr and Zeleny (1977)
"Decision making is a dynamic process, complex, redundant with feedback and sideways, full of search, detours, information gathering and information ignoring, fuelled by fluctuating uncertainty, fuzziness and conflict; it is an organic unity of both pre-decision and post- decision stages of the overlapping regions of partial decisions."
Multiobjective optimization in reservoir operation studies was carried out by Cohon and Marks (1975), Goicoechea et al. (1979), Haith and Loucks (1976), Yazicigil et al. (1983) and others. Duckstein and Opricovic (1980) suggested that multiobjective may be performed at two levels: first, an engineering level (cost effectiveness approach) and second, a managerial level (compromise solution). The proposed method was applied to the design of a water resource system in the Central Tisza River Basin in Hungary. As proposed, several approaches are possible for the selection of a final alternative. These methods may include, in particular, voting, dominance analysis and group decision making.
The other methods in this group are ELECTRE (David and Duckstein (1976)) and multi attribute utility theory (Keeney and Wood (1977). Vemuri (1974) presented a technique through mathematical derivations to solve Multiple- objective optimization problems naturally arising in resource management projects. The
Chapter Two 43 Literature Review
algorithm optimizes two objectives at the same time. It is desired to minimize water loss due
to evaporation (J1) from the reservoir, to minimize the capital cost (J3) of the project and to maximize the total volume capacity of the reservoir. Let h be the height of the dam, A be the surface area of reservoir and V the volume capacity of the reservoir. The author wrote the following functions heuristically without any rigorous theoretical base for their derivation
x(i) 2 1/α h = [ e (x1) ] a > 0, const.
2 A = K1 π (x2) k1 > 0, const.
2 x(1) 1/α V = hA = k1 π (x2) [ e (x1)2 ] (2.56)
Where x1,x2 = decision variables
The water loss due to evaporation:
2 J1 = K3 A = K3 K1 π (x2) K3 > 0, const. (2.57)
The capital cost is:
J2 = K2 h2 A K2 > 0, const.
2 x(1) 2 2/α J2 = K1 K2 π (x2) [ e (x1) ] (2.58)
The inverse of the volume capacity is
-2/α -x(1)/α J3 = 1/V = [1/k1π] [ (x1) (x2) e ] (2.59)
The model equation for non-inferior index elements was derived by the author as given below
n * wk J i(α1,α2,α3) = C [ wi/αi ] π (αk) (2.60) k=1 That is * -3/4 1/4 1/2 J 1 (α1,α2,α3) = C [ 1/4 ] { (α1) (α2) (α3) }
* -1/4 -3/4 1/2 J 2 (α1,α2,α3) = C [ 1/4 ] { (α1) (α2) (α3) }
* 1/4 1/4 -1/2 J 3 (α1,α2,α3) = C [ 1/4 ] { (α1) (α2) (α3) } (2.61)
The constant C (computed 2.38 for example problem) is evaluated by solving and using one of equation 2.61 for one particular choice of the vector α = (α1, α2, α3).
Chapter Two 44 Literature Review
* * The author showed optimum evaporation J 1, optimum costs J 3 and optimum volume (in
inverse) 1/J against various values of α1, α2, α3 in a tabular form. The technique attempted to emphasize the importance of multiple-objective optimization and presented a computation procedure for calculating a set of the so-called non-inferior vectors to an unconstrained optimization problem. The author recommended extending this method to problems in which decision and state variable constraints are present.
A new method for solving non-commensurable multi-objective functions namely, the surrogate worth trade off method, was developed by Haimes and Hall (1974). The Vemuri (1974) multi-objective problem given in (2.56 - 2.61) had been chosen as an example and was successfully solved via the surrogate trade off method. The multi-objectives general vector optimization problem in water resources system analysis may be given as _ fi(x) = min [ fi(x), f2(x),...,fn(x) ] = min fi(x) (2.62) x x
Subject to
gk(x) ≤ 0 k = 1,2,...,m
Where x is an N-dimensional vector of decision variables; f (x), i = 1,2,...,n. are n objective functions and g, k = 1, 2, ..., m are m constraints.
Reformulating the system (2.62) as follows
min fi(x) (2.63) x
Subject to:
fj(x) ≤ ε j = 2,3,...,n
gk(x) ≤ 0 k = 1,2,...,m
Where _ _ ε = fj(x) + εj j = 2,3,...,n
_ ε > 0 j = 2,3,...,n
Chapter Two 45 Literature Review
_ _
Where fi(x) is defined in (2.62) and εj , j ╪ i, j = 1, 2, ..., n are maximum tolerable levels and can be varied parametrically to evaluate the impact on the single objective function fi(x) in (2.50). They formed the Lagrangian L as
m n L = fi(x) + Σ µk gk(x) + Σ τ1j [ fj(x) - εj ] (2.64) k=1 j=2
Where µk, k = 1,2,...,m and τ1j, j = 2,3,...,n are Langrange multipliers. Trade off function τ1j [ A(εj) ] was derived as
τ1j [ A (εj) ] = - δf1(x) / δfj(x) (2.65)
With the help of satisfying Kuhn-Tucker conditions. A surrogate worth function Wij, i ╪ j, j =
1,2,...,n, can be defined as a function of τ1j for estimating the desirability of the trade off τ1j.
Wij could range from -10 to +10 and where its value is 0, it signifies that the solution belongs to the noninferior solutions which belong to indifference band. The Kuhn-Tucker conditions
for a minimum in (2.65) were solved for various values of ε2 and ε3 via Newton-Raphson method.
A new algorithm for optimal long-term control of a multi-purpose reservoir with direct and indirect users was described by Opricovic and Djordjevic (1976). (Indirect users are those who reuse water after some direct user, e.g. the hydro.). In such cases the usual dynamic programming cannot be used. They developed a three-level algorithm for solving such problems: At the first level, optimize the distribution of available water among time intervals; at the second level, allocate water to direct users in one time interval; at the third level, allocate water already used by direct users to indirect users further downstream. Forward dynamic programming was employed at all three levels, but the recurrence relations were developed in accordance with the decomposition of the control problem. The model was developed to solve certain practical design problems of multipurpose reservoirs in the Vardar Basin (Yugoslavia). The recurrence relation for the second level was given by
Chapter Two 46 Literature Review
dk(ym) = max { Dk,m (Uk,m) + Rk (Uk,m) + dk-1 (Ym - Uk,m )]} (2.66) Uk,m
Where dk(Ym) is optimal profit, Dk,m is the benefit obtained from an allocation of water to the th k user in the mth month, Uk,m the allocation of water to the kth user in the mth month. Rk a function expresses the indirect benefit from the direct allocation Uk,m and Ym is total quantity of water released in the mth month. The optimization results are optimal water storage, the allocation of water to all users and consideration of energy production over a 20-year period. Statistical analysis of the results yields a graph of the probability of optimal water levels in a reservoir during a year.
A new method, namely multiobjective dynamic programming (MODP) was developed by Tauxe et al. (1979). The method can be used to generate the entire non-inferior solution set of the multi-objective problems. It can also be used to generate the trade-off ratios between objectives. The Reid - Vemuri multi-objective problem was first chosen as an example and solved using MODP.
In the next stage of the study, operation of Shasta Reservoir in California was examined by considering three objectives (i) to maximize the cumulative dump energy, (ii) to minimize the cumulative evaporation or loss of the reservoir, and (iii) to maximize the firm energy. The decision variable was the monthly volume of reservoir release. The problem was formulated as follows
fj(Sj, Vj) = max { Ej(qj, Sj, Vj, FEj) + fj-1(Sj-1, Vj-1) } (2.67)
s Sj-1 = T j (qj, Sj, Vj )
v Vj-1 = T j (qj, Sj, Vj )
qj ≥ qmin,j and Smin ≤ Sj ≤ Smax
Where fj ( ) Long range returns (dump energy accumulated through stage j), MWh.
Ej ( ) Short-range returns (stage j dump energy), MWh.
FEj Firm energy required in time period j, MWh. 3 Sj Volume of storage at stage j, 10 ac.ft. 3 Vj Cumulative evaporation through stage j, 10 ac.ft.
qj State variable transformation functions.
Chapter Two 47 Literature Review
Multiobjective analysis of multireservoir operations was carried out with the help of a modified linear programming and dynamic programming (Yeh and Becker 1982). California Central Valley Project (CVP) was the test case. The five purposes (benefits), treated as objectives here in the multiobjective optimization, include (i) hydropower production (ii) fish protection (iii) water quality maintenance (iv) water supply and (v) recreation. Two sets of monthly historical streamflows, one set corresponding to a drought year and the other set to an excess water year are used to develop the non-inferior sets. The recursive equation which is characteristic of the DP is given by
k fi+1(Ec(i+1)) = max { Σ Wk [ ∆ S i+1(Ec(i),Ei+1) ] + fi(Ec(i) ) } i = 1,2,...,N (2.68)
Where i Time period N Number of periods
Ec(i) Cumulative value of benefit through period i.
Ei Value of benefit in period t.
fi Maximum storage (maximum weighted sum of its components) at end of period i. k th ∆S i+1 Change in storage of k reservoir in period i+1 th Wk Weight assigned to k reservoir.
Bras et al. (1983) presented steady state stochastic dynamic programming model with adaptive closed loop control technique was presented to introduce real time streamflow forecasts in reservoir operation. As a case study, streamflow forecasting and adaptive control were used in the High Aswan Dam in Egypt, to derive optimal policies.
Barros et al. (2008) investigated an optimization model for the management and operation of a large-scale, multireservoir water supply distribution system with preemptive priorities. The model considered multiobjectives and hedging rules. During periods of drought, when water supply is insufficient to meet the planned demand, appropriate rationing factors were applied to reduce water supply. Water distribution system is formulated as a network and solved by the GAMS modeling system for mathematical programming and optimization. Method was applied to the São Paulo Metropolitan Area Water Supply Distribution System in Brazil. You (2008) also applied hedging rules for reservoir optimization.
Chapter Two 48 Literature Review
Baltar and Fontane (2008) presented an implementation of multiobjective particle swarm optimization (MOPSO) method for multi-objective problems. The MOPSO solver was used on three applications: (i) test function for comparison with results of other MOPSO and other evolutionary algorithms reported in the literature; (ii) multipurpose reservoir operation problem with up to four objectives; and (iii) problem of selective withdrawal from a thermally stratified reservoir with three objectives.
2.6 Other Techniques
DP and LP have been applied due to simple problem formulation, availability of efficient codes, reaching global optimum and easy consideration of bounds on both the control and state variables. It was recognized that strong simplifications imposed in the LP and dimensionality problem in the DP do not permit application of these methods to many practical situations. Nonlinear programming (NLP) was applied by Lee and Waziruddin (1970) and Chu and Yeh (1978).
A simulation model was developed and used to derive operating policies for the Indus River basin in Pakistan (Malik, 1976). The river system is characterised by three main reservoirs, namely Mangla, Tarbela and Chashma. Mangla and Tarbela are meant for irrigation supplies and hydropower. Whereas Chashma is a buffer reservoir, located downstream of Tarbela to regulate the water supplies of Tarbela reservoir according to downstream irrigation requirements. Alternative operating policies were presented on the basis of irrigation or otherwise power priorities. The model passed through several testings and modifications later. The present form of the model is known as ROCKAT (Reservoir Operation of Chashma, Kalabagh and Tarbela).
Hipel et al. (1979) carried out a survey for the hydrologic generating model selection. They suggested that stochastic models of river flows could be used to generate synthetic traces for use in reservoir design. To select a suitable model, a two tier decision-making procedure was recommended. The first step consists of accepting only those models which pass diagnostic checks. at the second stage, further model discrimination can be done by comparing the economic results of the response surfaces for the various data sources. Available stochastic models and Box-Jenkins models were discussed. South Saskatchewan hydroelectric-reservoir facilities were used to define the procedure for reservoir model selection.
Chapter Two 49 Literature Review
A reliability programming technique (Simonovic and Marino, 1982) which includes the concept of 'reliability or risk' in an optimization is applied to multiple multipurpose reservoir systems. The reliability programming model is nonlinear and can be split into two models; search model and special linear programming model. The technique is illustrated using a portion of the Red River system in Oklahama and Texas, a system of three multipurpose reservoirs. The three reservoirs individually satisfy purposes (flood protection, hydro-electric power generation, water supply and water quality enhancement) and two of the reservoirs work together to satisfy additional water requirements (flood control and water quality enhancement downstream). The reliability programming formulation includes an objective function based on economic efficiency (maximization of the differences between net benefits and the yearly risk losses).
T m j j j max Z = Σ [ WQSBt + Σ ( POWER t + WSUP t + IRRI t t=1 j=1 m jt j j j + WQB + PUMP t ) ] - Σ [ RL (α ) + RL (τ ) - RL (ß) - RL (δ) (2.69) j=1 Subject to usual constraints.
Where WQSBt Total water quality benefits for period t for reservoirs working together dollars/acre.ft.
j POWER t Total power production benefits for reservoir j in period t dollar/kwh.
j PUMP t Total benefits from pumping P from reservoir j to reservoir k in period t dollar/acre ft.
j WSUP t Total water supply benefits for reservoir j in period t, dollar/acre ft.
j IRRI t Total irrigation benefits for reservoir j in period t, dollar/acre ft.
j WQB t Total water quality benefits for reservoir j in period t, dollar/acre ft.
RL Risk losses
αj, ß, Deterministic constraints on reliability τj, δ levels, varies between 0 and 1
Chapter Two 50 Literature Review
Can and Houck (1984) used goal programming for the real time daily operation of Green River Basin (GRB) system, Kentucky comprising four multipurpose reservoirs. The results are compared with the optimization model (LP) specifically developed for the system. In some cases the goal programming results were better.
Papageorgiou (1985) developed an algorithm based on the discrete maximum principle for the optimal control of multireservoir system. Variable metric techniques were used for direct solution of the resulting two point boundary value problem. The efficiency of the proposed procedure was demonstrated by a 10-reservoir hypothetical network system. The objective function was of the following type:
1 k-1 N 2 Min J = --- Σ [ D(k) - Σ vi qi(k) ] (2.70) 2 k=0 i=1 x(k+1) = f [ x(k), u(k), k ] k = 0,...,K-1 (2.71)
Where D(k) represents the energy demand and vi qi(k) is the energy generated by reservoir i at time k. x and u are state and control vectors respectively. The criterion (2.70) minimizes the energy deficit and (2.71) distributes deficits. The algorithm avoids operating with high dimensional matrices, doesn't call for discretization of state variables and requires moderate computer time and storage for its execution.
Yang, et al. (1995) presented comparison of real time reservoir operation techniques using two hydrologic and two optimization models. The first-order autoregressive (AR) model, the GR3(Conceptual rainfall runoff model), the streched thread method (ST) and dynamic programming (DP) method were used to design 3 reservoir operation technique by combining GR3 with ST AR with ST AR with DP (SDP)
The last possibility (GR3+DP) is not computable. From the efficiency viewpoint, the techniques for a daily reservoir regulation are compared using 3 year recorded series and then a general 100 year data series. The comparisons show surprisingly a favourable efficiency for
Chapter Two 51 Literature Review
technique based on the ST method in the Bar-Sur-Seine reservoir upstream from Paris. The study confirms the value of simple optimization methods such as ST and the applicability of scenarios methods in real time reservoir operation. They concluded that the basis of a good reservoir operation system is to view forecast and decision making as a whole unit.
Russel and Campbell (1996) investigated reservoir operating rules with Fuzzy Programming. Fuzzy logic seems to offer a way to improve an existing operating practice which is relatively easy to explain and understand. The application was made to a single purpose hydroelectric project where both the inflows and the selling price for energy can vary. Operation of the system is simulated using both fuzzy and logic programming and fixed rules. The results are compared with those obtained by deterministic dynamic programming with hindsight. The use of fuzzy logic with flow forecast is also investigated. It is concluded that the fuzzy logic approach is promising but it suffers from the curse of dimensionality. It can be useful supplement to other conventional optimization techniques but probably not a replacement. Dubrovin et al. (2002) applied a fuzzy rule-based control model for multipurpose real-time reservoir operation. A comparison between Total Fuzzy Similarity and a more traditional method (the Sugeno method) was done. They concluded that this method can perform generally well and is easy for the operator to understand due to its structure based on human thinking. Jairaj and Vedula (2001) used fuzzy mathematical programming model for the optimization of a three reservoir system in the Upper Cauvery River basin, South India. The model clearly demonstrates that, use of fuzzy linear programming in multireservoir system optimization presents a potential alternative to get the steady state solution with a lot less effort than classical stochastic dynamic programming.
Genetic Algorithm (GA) with a simulation model was developed by Chen (2003) and applied to optimize 10-day operating rule curves of a major reservoir system in Taiwan. The results showed that the proposed technique can be used to optimize the rule curves, not being limited by the type of the objective function and simulation model used. Chang et al., (2005) performed a comparison between binary-coded and real-coded GA in optimizing the reservoir operating rule curves. The results revealed that the new operating rule curves are better than the current operation rule curves, and the real-coded GA is better and more efficient than the binary-coded GA. Akter and Simonovic (2004) combined fuzzy sets and GA for dealing with the uncertainties in short-term reservoir operation. In their work,
Chapter Two 52 Literature Review
uncertainties involved in the expression of reservoir penalty functions and determining the target release value were considered. Kadu et al. (2008) applied GA for optimal design of water networks using a Modified Genetic Algorithm with reduction in search space. Shamir, and Salomons (2008) used GA for the optimal real-time operation of urban water distribution systems in Haifa, Israel. Yang et al. (2007) applied multiobjective GA for planning of surface water resources with Constrained Differential Dynamic Programming in southern Taiwan. Momtahen and Dariane (2007) used direct search approaches using genetic algorithms for optimization of water reservoir operating policies.
Barbaro et al.(2008) investigated the minimum discharge from a dam for the evaluation of the real impact of the reservoir on the catchments downstream. The assessment of this parameter was based on two conflicting objectives: the maximum use of water and environmental protection. The objective of this study was to optimize, apply, and discuss different methods for evaluation of the minimum discharge found in the technical literature to the reservoir on the Menta stream in the province of Reggio Calabria, Italy. Among the methods tested, the one adopted by the Autorità di Bacino del Fiume Serchio (Serchio River Basin Authority, Italy), provided a good indication of the minimum discharge to be adopted.
2.7 River Basin System Optimization
Different techniques have been employed for the optimization of various River Basin Systems operation in the world. The optimization procedures in selected river basins have been discussed as follows.
California Central Valley Project (CVP)
The CVP system in USA consists of nine reservoirs vz Shasta, Clair Engle, Lewiston, Whiskeytown, Keswick, Folsom, Natoma, San Luis and O'Neill Forebay. The total capacity of all the reservoirs is about 12,759 mcm. The system contains three canals Delta-Mendota (Q = 123 m3/s ), Folsom South (Q = 98 m3/s) and San Luis (Q = 364 m3/s). There are four pumping plants and nine power plants in the system. The total generating capacity of the power plants is about 1,692 MW.
The real time optimization procedure of the CVP optimizes, in turn, a monthly model over a period of one year, a daily model over a period of up to one month and an hourly model for
Chapter Two 53 Literature Review
24 hours. Output from one model are used as input into the next echelon model, iterating and updating whenever new information on streamflow predictions becomes available (Yeh 1979). An LP-DP formulation was used in the monthly optimization
n k k k k min Σ ( Ci Ri + C'i R'i ) (2.72) k=1 k Where Ri = effective release for the on-peak energy generation during the ith month for k k k the kth reservoir, R'i = all other releases during the ith month for the kth reservoir. Ci , C'i = cost coefficients that are functions of the energy rate function and average storage during any given month i, and n = total number of reservoirs. The monthly model consists of 22 decision variables and 48 constraits for each month.
The CVP daily model was developed in 1976. Inputs to the model include ending storage levels and daily streamflow prediction. Outputs include daily releases for each power plant for a period of up to one month. LP was used for day-to-day optimization. The model consists of 22 decision variables and 70 constraints (Yeh 1979).
The CVP hourly model allocates the total daily releases at each power plant so that the total daily system power output is maximized with reference to the power demand curve supplied 24 hours in advance for each 24 hour period by the PG&E (Pecific Gas and Energy) department. For optimization LP and IDPSA (Incremental Dynamic Programming with successive approximation has been employed. The three models described above were extensively tested and verified at the Bureau of Reclamation, Sacramento, California.
During 1976 and 1977 California suffered a severe drought, all reservoir levels dropped considerably below normal operating levels. The optimization models are not suited to such situation since some of the model constraints are difficult to adjust for a feasible solution. During the drought, the Bureau developed a simulation model to cope with this abnormal condition. However it is believed that the simulation model can be used in conjunction with the optimization model to produce better results. A simulation model can be used to find out inferior or infeasible solution for optimization model.
Sabet and Creel (1991) used the (NFP) for the optimization of California State Water project (SWP) facilities south, north and west of O'Neill Forebay. (See section 2.2)
Chapter Two 54 Literature Review
Arkansas River Reservoirs System
The Arkansas River begins on the eastern face of the Rocky Mountains near Leadvilla, Colorado in USA. The watershed area is about 160,000 square miles. Mean annual precipitation ranges from 15 inches in the western portion of the basin to 52 inches at the mouth. The basin consists of 20 reservoirs with over 8.5 MAF of flood control storage, 2.5 MAF of hydropower storage and storage for water supply, for water quality, and navigation. The computer model used for simulation provided a daily sequential regulation of a multipurpose reservoir system and resulting hydrologic and economic impacts. Model response was based on the structural and physical characteristics of the reservoirs (Coomes 1979, Hula 1979).
Copley (1979) carried out the Arkansas River System regulation study and discussed the flood protection, navigation, hydropower, water supply aspects of the system.
Tennessee Valley Authority Reservoir System
The drainage basin of Tennessee Valley is 40,900 square miles. In terms of flow, the Tennessee is the fifth largest river in the United States. The mean annual precipitation of Tennessee Valley is about 52 inches while the mean annual runoff is about 22 inches. The Tennessee Valley Authority (TVA) water control system consists of 35 hydro power projects including 21 multi-purpose projects and 14 single-purpose power projects. The main objective of these dams and their reservoirs are flood control, recreation, water supply and quality control, and irrigation. Since 1971 the Division of Water Resources has been modeling the TVA reservoir system to assess the impacts of reservoir operation on flood control, navigation, power generation, water quality and recreation. These models optimize the TVA operation (Shelton 1979). Boston (1979) described the computer applications used in operational planning and real-time economic control of hydro power of TVA system.
Brown and Shelton (1986) investigated the Tennessee Valley Authority (TVA) computer models which are for managing and operating the large TVA reservoirs and power facilities. Thirty six major dams make up the TVA water control system which regulates the Tennessee River and its tributaries. Because of the large-scale scope and operational complexity of the
Chapter Two 55 Literature Review
TVA system, computerized decision support capability has become an important factor in efficient and cost-effective operation of the system. The models for daily operation, reservoir water quality, flood information system and models for planning and assessment of environmental and power generation projects have been discussed. These models have enhanced cost-effective and efficient operation of the TVA system to the extent that important tasks can be performed today that was simply infeasible a few years ago.
Columbia Basin Reservoir System
The Columbia River, located in the Pacific Northwest, is an international river that flows from Canada into United States. It is fourth largest river in North America. It includes most of the areas of Idaho, Washington, Oregon and Montana and small areas of Wyoming, Nevada and Utah. The watershed area is about 259,000 square miles. The Columbia system contains more than 46 MAF of active storage but less than 43 MAF is used directly for power production. Out of 200 reservoirs, most of the capacity (more than 40 MAF) is in 15 largest reservoirs. The remaining storage in the smaller reservoirs is less controllable on a system basis. About 100 of all the dams/reservoirs in the basin are involved in power production but most of the power is produced by 50 reservoirs. One-half of all US hydropower is generated in this region (Green 1979).
The system is operated by Army Corps of Engineers and Bureau of reclamation for multipurpose. The Columbia reservoir system is not only operated for hydropower but also for irrigation, navigation, flood control, fish and wildlife, recreation, municipal and industry water supply, and water quality. The Benneville Power Administration is the largest operator of transmission lines in the region.
What is optimum regulation to one special interest may not be optimum to another. It is often difficult to reduce the conflicts in multi-purpose regulation. In Columbia River regulation there are two agreements, one is Columbia River Treaty between Canada and USA, and Pacific Northwest coordination agreement among the 16 parties that control most of the hydropower. The details of the agreements are given by Green 1979. There are several rule curves to manage Columbia reservoir system. These curves include critical rule curve, refill curves, energy content curve, flood control rule curve and operating rule curve etc. From the annual operating studies a family of rule curves is developed for optimum power production
Chapter Two 56 Literature Review from individual reservoir and from the combined system. During actual day-to-day operation the reservoir operators determine which rule curve is most appropriate under the conditions at that time. Actual operations deviate from the optimum hydroelectric plan for many reasons and efforts are being made continually to bring the system back into balance or to keep it as close to optimum conditions as possible. Computer models are used frequently in several offices to make short-range and longer forecasts and simulations of the operating system. Project and hydrometeorlogical data are collected automatically, then operating simulations are made. Human decisions and judgment have been found to be the most efficient, effective and satisfying means of regulating the Columbia reservoir system when real time information is available (Green 1979).
Jones (1979) investigated the hydro system seasonal regulation program of Columbia system. He described how the model works during the critical period and outside the critical period. Schultz (1979) stated experience with optimizer techniques for regulation of the Columbia River System. Although no operations research techniques are involved in this traditional approach to reservoir management, the operating rules are derived from experience, and analyst's skill. It should itself be viewed as an optimizing technique.
Lower Colorado River System The Colorado River System is divided into upper and lower basins. The upper basin comprised of those parts of the United States of Arizona, Colorado, New Maxico, Utah and Wyoming within and from which waters naturally drain into the Colorado River System above Lee Ferry and also all those parts of the States located without the drainage area of the Colorado River System which are served by waters diverted from the system above Lee Ferry. The Lower Colorado Basin comprised those parts of Arizona, California, Nevada, New Maxico and Utah within and from which waters naturally drain into the Colorado River System below Lee Ferry and those parts of the States located without the drainage area of the Colorado River System which are served by waters diverted from the system below Lee Ferry. The average annual flow of the river above Lake Powell is about 15 MAF over the period 1906-84 (Steven 1986). Over 90% of flow volume originates in the upper basin. Most of this inflow is due to snowmelt. 70% of annual runoff at Lake Powell occurs between the period April and July.
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Glen Canyon Dam (Lake Powell, volume 25 MAF) is the farthest downstream dam in the Upper Colorado River Basin. Downstream of Glen Canyon, Hoover dam which is in Lower Colorado River Basin is located. The Lower Colorado System consists of four major dams vz Hoover Dam (Lake Mead, volume 27.377 MAF), Davis Dam (Lake Mohave, volume 1.810 MAF), Parker Dam (Lake Havasu, volume 0.619 MAF), and Imperial Dam. The US Bureau of Reclamation is responsible for the operation of Hoover, Davis and Parker Dams. Additionally the Corps of Engineers is responsible for developing the flood control operation plan for Hoover Dam.
The mathematical model (Freeny 1979) used to simulate the Colorado River System in accordance with all of the laws of the River has been developed. One criterion is to maintain equal storage in Lakes Powell and Mead and to provide a release of at least 8.23 MAF annually from Glen Canyon Dam. The model helps to study reduction of flood potential, water conservation impacts and increased power generation. The probabilistic analysis is used to meet the objective functions of various interests.
Steven (1986) investigated models which are used for the operation of Lower Colorado River operations. A computer model called the "24-month study" was used for planning monthly and seasonal operation of the reservoir and powerplant system. Another model was used to plan and schedule releases for downstream water delivery, energy generation and flood control requirements on a daily and weekly basis. Finally, water schedulers in Boulder City, Nevada, used the Supervisory Control and Data Acquisition (SCADA) system of the Western Area Power Administration to automatically control hourly releases for the next 24-hour day and for same-day changes at Davis and Parker powerplants.
Central Arizona Project
Gooch and Graves (1986) used a computer based programmable master supervisory control (PMSC) system for the operation and optimization of the Central Arizona Project (CAP), one of the largest conveyance system in USA to deliver water of Colorado River for municipal, industrial and agricultural uses in central and southern Arizona. A dual-computer master station was connected to remote terminal stations units (RTUs) located at each pumping plant, checkgate and turnout. The models were based on a constant volume philosophy of operation to keep the system responsive to changes in demands. The PMSC system consisted
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of Aqueduct Control Software (ACS), Initial Conditions Model (ICM), Simulation Model (SNUSM), Baseload Model (BLM), Gate Setting Model (GSM), Reach 1 Simulation Model (RSM) and Water and Power Optimization Model (WPOM). WPOM was based on LP with a seven-day operation to generate schedules with minimizing the peak-hour pumping while meeting downstream delivery requirements and staying within operating constraints.
Duke Power Hydro System Reservoir Operation
The Duke system dates back about 90 years to the time when Mr. J. B. Duke and his associates began developing a novel like idea of building an electric utility supplied by a series of hydoelectric plants located along an entire river basin with a transmission system connecting the plants to provide electric service to all customers throughout the area. Catawba River in Carolina has become the most electrified river in the world (Sledge 1979). Later the Keowee River was also included in the system. In 1975, there were 13 reservoirs, 4.462 MAF of total storage, 1555 MW of hydroelectric capacity and about 627,041 MWH of stored energy. In addition to power supplies, numerous cities, towns and industries draw their water supplies from Duke Reservoirs. The primary objective of the system is the economic generation of electric energy. The Duke Power Company Reservoir Systems operation has been described by Ridenhour (1979).
Faux et al. (1986) has been used network flow programming (NFP) for the optimization of large irrigation/hydropwer system in Philippines.
Missouri River System
The river is regulated by six major reservoirs in series, Fort Peck, Garrison, Oahe, Big Bend, Fort Randall and Gavins Point. These 6 reservoirs witha total storage capacity of about 91 Km3 (74 MAF) represent about 2.5 times the mean annual flow of the river at Gavins Point. Major purposes of the system include hydropower, flood control, recreation, water supply and navigation. The trade off between different objectives have prompted a review of the operation of the system including the development of reservoir simulation models for the system that incorporate value functions for individual project purposes. The US Army Corps of Engineer's Model (HEC-PRM) is a network-flow-based optimization model intended for application to reservoir system analysis problem. Convex penalty functions for this model may be incorporated as piecewise linear function. Lund and Ferreira (1997) derived the
Chapter Two 59 Literature Review
optimum operating rule for Missouri River Reservoir System. Deterministic optimization and implicit stochastic optimization technique is used to infer optimal opertating policy and tested using a simplified simulation model. Applicability and limitations of applying deterministic optimization to development of strategic operating rules for large scale water resources system are demonstrated. For the system, simple data display and simulation modeling are found to be superior to classical regression techniques for inferring and refining promising operating rules from deterministic optimization results.
Sabarmati System in India
Jain et al. (1999) performed reservoir operation studies of Sabarmati river system which is located in a drought prone region, Rajasthan. The operation of the river system, consisting of four reservoirs and three diversion structures was studied. The system provides water for municipal, industrial, irrigation and water supply. It is also used for flood control. Rule curves were derived for the reservoirs. Using the simulation analysis, the rule curves were fine-tuned to achieve the targets to the maximum possible extent. Software was developed to assist the dam operator in determining the safe release from the Dharoi reservoir during floods.
Operation of South Indian Irrigation Systems
South Indian Irrigation Systems include Krishnagiri reservoir. The reservoir has a capacity of 68.26 million cubic meter (mcm). The runoff contributed to the reservoir is from a catchment of 5397 km2. Two canal off take from the reservoir. There are about 13 tanks fed by these canals. Total command area of the project is about 3600 ha. Ravikumar and Venugopal (1998) developed a model for the optimal operation of the system. It includes three phases; a simulation model to simulate the command area of the reservoir, a stochastic dynamic programming (SDP) model to obtain an optimal release policy, a simulation using the optimal release policy from SDP model. The SDP model considers both demand and inflow as stochastic and both are assumed to follow first order Marcov chain model. The third simulation model is used to study the degree of failure associated with adoption of the optimal operating policy for different reservoir storages at the start of the crop season.
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Hoa Binh reservoir in the Red River basin, Vietnam Ngo (2006) developed a framework in which a simulation model is coupled with a numerical search method for optimising decision variables specifically defined for operation of Hoa Binh reservoir in the Red River basin, Vietnam. The MIKE 11 modeling system for simulating the flows and the AUTOCAL software is selected for optimisation. The framework is tested on the selected reservoir, considering hydropower production and downstream flood control. The results show that optimised rule curves significantly improve the reservoir performance in terms of hydropower production without reducing the downstream safety against flooding.
Bhadra reservoir system in Karnataka, India Reddy and Kumar (2006) applied Multi-objective Genetic Algorithm (MOGA) to Bhadra reservoir system for developing suitable operating policies. The Bhadra dam is located at latitude 130 42’ N and longitude 750 38’ 20” E in Chikmagalur district of Karnataka state, India. It is a multipurpose project providing for irrigation and hydropower generation, in addition to mandatory releases to the downstream to maintain water quality. The objectives of the model were minimization of irrigation deficits and maximization of hydropower generation. These two are mutually conflicting objectives, since the one that tries for minimization of the irrigation deficits, requires more water to be released to satisfy irrigation demands and the other tries to maximize hydropower production, requiring higher level of storage in the reservoir to produce more power. This study successfully demonstrated the efficacy and usefulness of Multi-objective Evolutionary Algorithm (Deb 2001) for evolving multi-objective reservoir operation policies.
2.8 River Water Disputes
The disputes may occur among individuals, within society, in a country or among countries. River water are no exception. But in the case of water, a precious natural resource, when disputes are not resolved quickly in an amicable manner, there is long-term incalculable damage not only to the provinces involved but to the nation as a whole (Shah 1994). The water disputes are common especially in developing countries where the resource is limited.
In Pakistan, the history of disputes between provinces on sharing of river waters is long and bitter. After Indus Water Treaty between Pakistan and India in 1960, the source of supply of
Chapter Two 61 Literature Review
many canal systems had been changed and burden was imposed to the three rivers vz Indus, Jhelum and Chenab, left to Pakistan. In 1991 a WATER ACCORD was signed between the four provinces of Pakistan to resolve the dispute in a spirit of accommodation, cooperation and understanding and to allocate additional water out of flood river supplies, over and above the existing canal uses. The aim was to provide additional irrigation supplies to each province for rapid development. Many disputes and issues are raised perhaps due to lack of information. For example, the following disputes/issues were raised (Ali, 1995) for any new large storage on Indus (like Kalabagh dam project).
i. No enough surplus water is available for new reservoirs.
ii. Indus delta, the mangrove forests and river fish need massive water of flood to servive. iii. Stoppage of floods would destroy the agriculture in riverain areas. iv. Additional upstream canals must ensure protection of rights riparian to guard against their becoming deserts. v. The persons displaced by the reservoir lake would be rendered homeless and jobless. vi. Nowshera and Peshawar valley upstream would be subjected to larger flood heights. vii. Mardan and Swabi areas would be get water logged. In many cases the problems have risen due to lack of information or hear say. Resolution of inter-province water disputes is a highly complex and intricate problem. Due consideration was given to these issues while formulating the problem in the present study.
Inter-state river water disputes are common in India where about 80% of water resources are derived from inter-state rivers. For example, the entire waters of Narmada basin (28 MAF) have been flowing wastefully into the sea for many years without any utilization on account of an earlier dispute between Madhya Pradesh and Gujarat, both having large drought-prone areas (Shah, 1994). Under the Indian Constitution, every State Government has the power to legislate in respect of water and this can be exercised for the whole or any part of the state. During British rule, in 1866, the irrigation works were under the control of Central Government. In 1921, irrigation became a provincial but reserved subject. In 1935, irrigation became a provincial subject wholly within the legislative competence of the province. In 1941 Indus commission was set up to investigate the dispute between Punjab and Sind.
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Shah (1994) discussed water disputes in India. The major disputes were on the distribution/allocation of water of Narmada Basin, Krishna Basin, Godavari Basin, Ravi-Bias Rivers and Cauvery Basin. He concluded that, a water dispute of inter-state, usually settled in a period of 5 to 10 years through courts in India. Lack of proper understanding sometimes lead to impracticable decisions as happened in the case of Cauvery Basin dispute in which it was directed a specified quantum of water to be ensured by Karnataka in every month of the year. While the flows ordered by the tribunal were historically not available for about 30% of the period in different months. This award was later modified. He suggested that instead of resolving the disputes through judicial tribunals, these needed to be resolved through mutual agreement with a spirit of give and take and mutual accommodation.
Rao and Prasad (1994) investigated the water resources disputes of the Indo-Nepal region. Large projects have been discussed between the two counties for several decades. However none of these projects has reached even the design stage. This region is one of the poorest in the world. They studied the hydrologic system of Indo-Nepal rivers, projects undertaken by India and Nepal and reasons for delay in project execution. They concluded with a set of recommendations.
Iyer (1994) studied the Indian federalism and water resources. He stated that in India there is no real river basin authority and there has been no basin-wide planning. The conflict- resolution mechanism needs some improvements. He discussed and criticised in detail the Inter-state River water disputes. He also concluded with a set of recommendations.
Six water authorities share the Sefidrud watershed in the northwestern region of Iran and resolving the conflict among them is one of the major challenges of the water division of the government. Zarghami et al. (2008) investigated a group decision support system developed for identifying the criteria and their weights, needed for ranking water resource projects in this watershed. The model was based on extending the ordered weighted averaging (OWA) as an aggregation operator. Using extended OWA, 13 water resources projects in the Sefidrud watershed was ranked.
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2.9 Comparison of Methods
Some investigators compared the various system analysis techniques to see their relative performance. Meier and Beightler (1967) presented a dynamic programming technique for efficiently allocating water to a variety of storage reservoirs in a branching river system. In their enthusiasm for dynamic programming the authors tried to conclude that other techniques such as linear and nonlinear programming are not generally useful for water resources system analysis. Louks (1968) in reply to Meier and Beightler compared dynamic programming, non-linear programming and linear programming with alternate models. He concluded, "Probably our own limitations rather than those of any particular programming method restrict us from examining and including all we would wish to in our analysis". In 1970, Louks and Falkson (1970) again compared some dynamic, linear and policy iteration methods for reservoir operation. They found that the information derived from each of the three model types yielded identical policies, but the computational efficiencies of each model differed and discussed as follows:
1. Dynamic programming models yield transient and steady state policies directly and once the computer program for solving the model has been written and debugged, its solution takes the least amount of computer time.
2. Policy iteration methods again require the writing and debugging of computer programs and the models take somewhat longer to solve.
3. Linear programming does not determine transient policies. No computer programming or debugging is usually necessary since linear programming codes exist, but linear programming takes a greater amount of computational time as compared to dynamic programming.
A survey was carried out by Yakowitz (1982) to review various dynamic programming models for reservoir operation/water resource problems and to examine computational techniques which have been used to obtain solutions to these problems. Discrete dynamic programming, differential dynamic programming, state incremental dynamic program and policy iteration method were among the techniques reviewed.
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Different investigators used various system analysis techniques for reservoir optimization problem. For example linear programming (LP) was used by Mamos (1955), Loucks (1966), Roefs and Bodin (1970), Revelle et al. (1969), Eisel (1972), Houck (1979), Houck et al. (1980), Simonovic and Marino (1982). Dynamic Programming (DP) was selected by Hall et al. (1961, 1968), Young (1967), Meredith (1975), Collins (1977) and Bhaskar el at. (1980). While Incremental DP or discrete differential DP (DDP) was used by Bornholtz et al. (1960), Hall et al. (1969), Heidar et al. (1971). Differential DP with successive approximations has been used by Yakowitz (1982), Trott and Yeh 1973, Yeh et al. (1979). Stochastic DP was used by Butcher et al. (1968), Buras (1983), Su and Deninger (1974), Weiner and Ben-Zvi (1982), Stedinger et al. (1984), Mclaughlin et al. (1990). Chance constrained DP has been used by Askew (1974). Simulation techniques were used by various researchers including Sigvaldason (1976), Malik (1976) and O'Mara 1984.
Turgeon (1982) showed, with two examples that incremental dynamic programming may converge to a non-optimal solution if the same state increment is used for every stage. He also showed how to adjust the increment sizes in each stage to obtain the optimal results. A study was carried out by Karamouz and Houck (1987) to compare stochastic and deterministic dynamic programming for reservoir operating rule generation. Their findings are given below
1. Reservoir operating rules generated by deterministic dynamic programming model are more effective in the operation of medium to very large reservoirs (capacities exceeding 50 percent of the mean annual flow).
2. Whereas rules generated by stochastic dynamic programming model are more effective for the operation of small reservoirs (capacity 20 percent of the mean annual flow).
3. While comparing dynamic programming models, they also compared linear programming (LP) versus dynamic programming (DP). They suggested that because DP might not require the approximation of the objective function as needed by the LP (piecewise linearization of the loss function) and because a relatively small number of discrete storage values in the optimization might produce the best real time operations, it might be possible for DP to be computationally
Chapter Two 65 Literature Review
feasible even for multiple reservoir systems and to produce better operating rules.
Simulation techniques (Sigvaldassan 1976 and others) are very useful when someone is interested to get near optimal results. Experience shows that it is somewhat difficult to get exactly optimal results with this technique as compared to LP or DP. However, in some cases exactly optimal results are sometimes not required and simulation techniques providing several near optimal results give a wide range of choices for decision making. Allocation of shortages were carried out in several simulation to obtain the most desirable result. However, changing system - operating rules may be time consuming and costly.
It is apparent from the review that system analysis techniques are system specific to some extent. The choice of better technique which depends on the kind of objective functions and several system constraints may lead to an efficient algorithm solving the problem more accurately with less computer time and memory. Keeping these limitations in view, new multiobjective functions with several physical constraints are proposed in this study and a new algorithm is searched to get optimal solution for these multiobjective problems. Based on the critical review and comparison of system analysis techniques, a dynamic programming approach coupled with a regression model and simulation model for the problem imposed was found to be best suited to derive optimal operating policies for multiobjective reservoir simulation as proved herein. Optimal operating policy for multiobjective reservoir simulation can be done in a single run with the help of the DP algorithm described herein and it saves a lot of computational efforts.
2.10 Previous Studies on the Indus Basin
A number of studies were carried out by different Organizations and researchers on Indus Basin (Chaudhry et al. 1974, Malik 1976, O'Mara 1984, etc.). A location map of Indus Basin and its tributaries is shown in Figure 4.1.
A numerical model of the Indus Basin irrigation system has been developed by Harza Engineering Company International for WAPDA (1966). The model simulates the operation of the reservoirs, canal and tubewell fields of the Indus Basin for the period 1969-85. It is referred to as "Comprehensive Model of the Irrigation System" or the COMSYM Model.
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World Bank developed a model for Indus Basin. The model is known as IBM (Indus Basin Model, 1989). The model is basically an agro-climatological model.
Chaudhry et al. (1974) described a working mathematical model for optimizing the conjunctive use of surface and groundwater resources of the Indus basin including Mangla reservoir sub-system. The technique to solve the mathematical formulation was based on dynamic programming and efficient direct search method. Direct solution of the outer problem by standard dynamic programming would require several hours of computer time. To overcome this difficulty, a systematic search algorithm was developed. It deleted obviously non-optimal solutions reducing computer time to approximately 1.0 percent of the direct search time. The author suggested that for detailed planning and design, it was necessary to carry out further research on the conjunctive use of surface and groundwater resources of the whole system (Indus Basin).
Malik (1976) described a simulation model to find out an acceptable operation policy under fixed system for Indus Basin Water Resources System. The material based results give insight into the system over time and space. Considering the economical, social and political factors with irrigation and power priorities an operation policy for the Indus Basin was suggested.
O'Mara et al. (1984) described a simulation model to examine alternative policies for achieving more efficient conjunctive use in the Indus Basin. They suggested that large gains in agricultural production and employment are possible if given more efficient policies.
Zahid (1986) carried out a probabilistic analysis of Mangla Reservoir using Gould probability matrix storage yield approach and suggested irrigation releases on the basis of probability of failure (reservoir emptiness) but had completely ignored the power sector.
As a part of normal activities Water Resources Management Directorate (WRMD) WAPDA is responsible for preparing an operation criteria for Mangla, Chashma and Tarbela reservoirs. These traditional criteria in which minimum and maximum rule curves are suggested have been prepared for the two seasons Rabi and Kharif individually every year. These traditional criteria are usually derived with the help of economical, social and political factors and forecasted inflows.
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Ahmad (1990) investigated multiobjective dynamic programming and regression analysis for the derivation of operating rules for Mangla reservoir. Ahmad (2006) developed computer codes for reservoir operation studies.
Hydro Electric Projects Organization (HEPO) and Planning and Investigation Organization (P & I) of WAPDA conducted feasibilities for various future dams on Indus River. It includes feasibility studies of Kalabagh (1987, 1988), Mangla Raising (2003), Akhori (2005) and Diamer Basha dam (2004, 2007). These reports include the source of basic data and physical parameters of the various projects on Indus River.
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CHAPTER 3
METHODOLOGY
This work describes the application of the various optimization techniques for the optimization of a large scale multipurpose multireservoir system. The primary objective is to develop a procedure for optimizing the operation of the large scale system with uncertain inputs. The various techniques developed by different researchers have been discussed in chapter 2. However this chapter is restricted to the algorithms and approaches that could be used for the efficient operation of the system. To make it an easily understandable, each algorithm is discussed in detail. Section 3.1 presents the steps of a proposed procedure applicable to a system of large reservoirs like Indus Basin under uncertainty. Section 3.7 will address the underlying theory of the algorithm of network flow programming (NFP) and demonstrates its use and modeling versatility in river system optimization. Section 3.3 demonstrates the methods and concepts of the procedure with a combined use of network flow programming and stochastic process in the area of multireservoir system. The modeling effort will also provide the best mix of programs to achieve a balanced operation of the system and will be a marvelous tool for decision makers in planning short and long term operation under uncertainty.
3.1 Proposed Procedure
It was important to find an efficient and computationally feasible procedure for the optimization of the large scale multireservoir system. Several schemes have been identified and an attempt has been made to select the best fit procedure among the various identified schemes. Optimization methods applied to water resource system analysis have been described in the previous chapter. The proposed procedure is a mixed optimization procedure in a two stage framework. It is based on the dynamic programming and Network Flow Programming (NFP) for the stochastic optimization of the large scale water resource system. The mixed optimization procedure is selected due to the following reasons.
1. A water resource system usually includes reservoirs, hydropower plants, and diversion points such as barrages, link canals taking water from one river and
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draining it into another river, water supply canals for irrigation, industries or domestic use. An example of such type of a system is the Indus River System. At reservoirs and hydropower plants locations the objective functions are usually nonlinear while at diversion and canal network locations these may be linear.
2. Complete system is a large multidimensional system and may not be efficiently optimized with only stochastic dynamic programming (SDP) due to dimensionality problem associated with dynamic programming.
3. NFP is much faster optimization technique for the systems that can be represented in networks. Literature indicated that large water resource systems had been optimized with this technique. But NFP is based on linear objective functions (or piece wise linear functions). Therefore optimizing that part of the system having nonlinear objective functions (such as reservoir operation), forcing them with linear functions will result a non-optimal results.
4. NFP is a special kind of the linear programming technique, which solves the problem with only the integer values of the variables involved with deterministic inputs such as river flows. But the river flows are random process, their occurrence is uncertain. Therefore considering them as a deterministic process leads to some optimistic results. An alternate way to handle such situation is to consider a stochastic problem where inputs are located in the system. It is shown that it can be handle by mix optimization procedure using simultaneously SDP and NFP.
5. Optimization of a water resource system is a system specific problem. Several tests have been made about the selection of the appropriate solution procedure for the selected system. It is found that a mix optimization procedure develop in this study is effective for the test case.
The proposed procedure is defined in very simple steps for an easy understanding. These steps are given below:
1. Determine the probabilities and transition probabilities of the inflows at each input location in the system. Also determine the statistics (e.g. mean, standard deviation lag 1 correlation etc.) of the inputs.
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2. Propose several stochastic and deterministic optimization model types.
3. Propose the objectives of the reservoir operation in mathematical form.
4. Identify the physical constraints of the reservoir system.
5. Calibrate type one model proposed in step 2 with different objectives in step 3 using the historic data and the probabilities computed in step 1, derive optimal operating rules and identify their relative performance in the optimum reservoir operation problem.
6. Repeat step 5 with the next model type. Continue to repeat step 5 until all the model types in step 2 are calibrated.
7. Verify the calibrated models in step 6 using simulation with historic and forecasted data. The verification results identify the bestfit model for the system.
8. Select the best fit model from step 7 and its operating rules already derived in step 5 for the reservoir simulation in the system.
9. Select the next reservoir of the system and repeat step 1 to 8. Stop when operating rules for all the reservoirs in the system have been derived through this cycle.
10. Represent the water resource system as a capacitated network in which nodes (control points) are representing reservoirs, local inflow locations, diversion locations, system input locations or any other locations where flows or flow limits need to be specified. The nodes are connected by the arcs which represent river and canal reaches, tunnels, or any water transportation facility in the system.
11. Define the maximum and minimum capacities of the river and canal reaches or any water transportation facility in the system from the physical data.
12. Define the water demand of the irrigation areas in the system from the historic data.
13. With the help of step 10 to 12, construct the Network Flow programming model of the system. Optimize the water resource system with NFP algorithm except the nodes where reservoirs are located. These nodes are operated using the derived operating rules in step 8.
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14. Incorporate SDP policies from step 9 with NFP for optimum operation of the system. Optimize and calibrate the short term (10 day) operation with the proposed working mathematical model.
15. Calibrate the NFP model with long term historic data and summarize the results in terms of probability of shortfalls and other statistics.
16. Verify the long term scheduling obtained from step 14 with the help of simulation of the system. Compare these results with the actual historic operation and evaluate their effectiveness.
17. Verify the short term operation optimization with simulation.
18. Illustrate the applicability and limitations of applying such methodology.
3.2 Formulating a Mathematical Model
Models are idealized representations of reality. In case of reservoir storage water resources system, if there are n related quantifiable decisions to be made, they are represented by a decision variable (say x1 , x2 , . . . , xn ) whose respective values are to be determined. The problem is then expressed as a mathematical function (known as Objective Function) of these variables like
P = 2x1 + 4x2 + . . . + 3 xn
The function P is a benefit to maximize. Any restrictions to these variables are constraints as
x1 + x2 ≤ 3
x3 + 2x5 = 5
The constants in the right hand sides of the above equations are called model parameters. If the mathematical functions appearing in both the Objective Functions and the constraints are all linear functions, the model is a linear programming model. As the model is an idealization of the problem, it is required that there is a high correlation between the prediction by the model and what would actually happen in the real world. This can be illustrated by an example such as
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A quantity of water R can be released from a storage reservoir to 3 canals. Determine the
allocation of water (xj) to each canal that maximizes the net benefit (NB). The benefit bj per unit of water is known for each canal.
Maximize NB(x) = b1 x1 + b2 x2 + b3 x3
3 = ∑ (bj xj) j=1
Subject to x1 + x2 + x3 = R
3 or ∑ xj = R j=1
and xj ≥ 0
Several techniques have been derived to solve optimization problem. The selection of the technique depends upon the type of the objective function and its constraints. For example, the above model is a simple linear programming model and can be solved by linear programming method. Commonly used techniques are Lagrange multipliers, linear programming (LP), network flow programming (NFP), integer programming, non-linear programming and dynamic programming (DP)
In water resources / reservoir operation optimization problems, the objective function is usually non-linear. Therefore the solution may be obtained with the help of non-linear methods (e.g. dynamic programming).
Nonlinear Programming One Variable Unconstrained Optimization f(x) d f(x) / dx If the function f(x) is concave as shown in the Figure (in right) and to be maximized. The solution is global optimal if
d f(x) ------= 0 x* dx
At this condition x = x*
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The one dimensional search procedure
d f(x) Step 1 Evaluate ------= 0 at x0 = x dx
d f(x) Step 2 If ------≥ 0 Reset x1 = x dx
d f(x) Step 3 If ------≤ 0 Reset x2 = x dx
Step 4 Select a new x = ( x1 + x2 ) / 2
Stopping Rule if ( x2 - x1 ) ≤ 2ε so the new x must within error ε of x* otherwise return to the step 2. Some other methods used in calculus are given by Kreyszig, E. (2007).
3.3 Dynamic Programming
The technique used to derive optimal operating rules for multiobjective reservoir simulation contains an optimization algorithm based on mathematical programming and a simulation model. The algorithm for the optimization of multiobjective reservoir system is designed to handle situations where two primary and one secondary objectives are involved. The primary objectives are
• To maximize the water supply for irrigation to meet the downstream demand.
• To maximize the power and energy generation for the system.
Whereas the secondary objective is flood control for the downstream areas.
The optimization problem is solved by developing a mathematical model based on dynamic programming. The multiobjective reservoir operating rules are then derived with optimal releases obtained from the dynamic programming algorithm. The release policies are then verified and compared through simulation.
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3.3.1 Solution procedure
An algorithm is developed by which the dynamic operation policies for planning complex reservoir system producing hydroelectric power and providing water may be optimized for the maximum return for assured water supply for irrigation and hydropower generation. The algorithm is based upon dynamic programming (DP), which is a powerful optimization approach to solve a wide variety of problems in many fields and is well suited to micro computers. The DP model is designed to analyze multiobjective problems as an extension of the single-objective optimization method. Several additional abilities are added to the model to handle complex water resources system. The program has been designed to provide maximum flexibility to solve sequential decision problems only.
3.3.2 Characteristics 1. The problem may be divided into various stages (time intervals). At each stage, a policy decision is required. 2. There are a number of states in each stage. 3. The effect of the policy decision at each stage is to transform the current state into a state associated with the next stage (possibly according to a probability distribution). 4. The solution procedure is developed to determine an optimal policy for the overall problem (a prescription of optimal policy decision at each stage for each of the possible state). 5. Given the current state, an optimal policy for the remaining stages is independent of the policy adopted in previous stage (This is called the principle of optimality for dynamic programming). 6. The solution procedure begins by finding the optimal policy for the last stage. 7. A recursive relationship that identifies the optimal policy for stage n given the optimal policy for stage (n+1) is available
* * fn (s) = min Cs xn + fn+1 (xn) (3.1)
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8. When we use the above recursive relationship, the solution procedure moves backward stage by stage, each time finding the optimal policy for that stage, until it determines the optimal policy starting at the initial stage.
3.3.3 Problem formulation
Different types of objective functions depending upon the type of the problem can be formulated. For example, a reservoir is built to supply water for irrigation and hydroelectric
power generation. Therefore it is desired (i) to release certain amount of water dt from the reservoir in each time t and (ii) to maintain the head of water (or storage) in reservoir, Lt at
maximum level (or maximum storage) St to obtain maximum power generation. Optimal releases that minimize the sum of squared deviations are also determined from (a) release
value rt and demand dt and (b) storage value St and target storage Lt
While formulating the model, the objective function becomes
2 2 Z = min [ ( dt - rt ) + ( Lt - St ) ] (3.2)
If we have 10-day time period for reservoir operation we have 36 time steps in a year and dt and Lt is known for the 36 time steps. The time steps in this problem are stages in terms of dynamic programming formulation.
Constraints are evaluated through state transformation equation
St+1 = St + qt - r t (3.3) where qt are known inflows in this case
The storage in reservoir should be less or equal to the reservoir capacity ka as indicated
St+1 , St ≤ ka (3.4)
The multistage process in reservoir operation problem is shown in Figure 3.1. A detail description of the physical constrains are given in section 3.8.2.
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Decision Variable x1 xn+1 xk
Input Output State State Stage n Stage 1 Stage k S2 Sk Sn Sn+1 S1 Sk+1
f1 (S1, x1) fk (Sk, xk) fn (Sn, xn) Return in Stage k
Figure 3.1 Typical multistage process
3.3.4 Stationary policy
n+T n The maximum annual net benefit will be equal to ft (St)-ft (St ) for any value of St and t. It n+T n can be found that the stationary policy has been identified when the values ft (St) - ft (St ) n+T n are independent of St and t. When stationary policy is reached, the term ft (St) - ft (St ) n+T n becomes constant for the next iterations. Therefore, ft (St) - ft (St ) is the minimum annual sum of squared deviation that can be obtained by following the derived sequential operating policy.
It can be stated that if the returns at any stage are dependent on the decisions made at other stages in a way not captured by the state variables, then DP is not an appropriate solution technique, except perhaps a rough approximation.
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3.4 Deterministic Dynamic Programming (DDP)
3.4.1 Discrete approach
Because the deterministic models are optimistic, a deterministic as well as stochastic approach is adopted in the dynamic programming model. The Discrete Deterministic dynamic programming model is first defined. It may be termed as DDP and Stochastic Dynamic programming as SDP. In general, the problem formulated may be as follows. Find a sequence of decisions, (x, x,..., x) such that an objective function Z is optimized with states (S, S,..., S).
Optimize Z = g { St , St+1 ,...,St+n :x x ,...,x ) (3.5)
Subject to S = T (St, x ) k = 1,...,n
St ε Sm
x ε x (St, k)
Involving Bellman's principal of optimality (1957), the return function equation in terms of f(S) can be written as
fn (St) = opt { g(St ,x ) o f n-1T(St+1 ,xt ) } (3.6) for k = 2,...,n)
With fn (St ) = opt { g (St , x ) }
Subject to similar constraints as in (3.5) and 'o' is a binary isotonic operator (Esogbue 1989)
The formulation given in (3.6) implies that we can observe the system and make decision only at a discrete set of stages therefore it may be called as discrete system. If no random variable (independently distributed with probabilities) is attached with function equation (3.6), the problem is solved in a deterministic way or otherwise. The algorithm developed herein is flexible therefore SDP or DDP model can be run at the same time by switching the choice of SDP or DDP on/off. Optimization with a stochastic model in comparison with deterministic one will increase the computer time by a factor N (Weiner and Ben-Zvi 1982).
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3.4.2 Computational procedure
Among several computational procedures to solve (3.6), the one adopted in the present study is described as follows. The methodology, adopted herein, may be known as conventional DP for DDP model. A predictor-corrector algorithm is solved in two steps. For example, it is
required to compute f(S) in equation (3.6) at any stage, given the optimized values of f (.), St, x at a finite number of grids. The number of discretization on state variables St is subject to some physical constraints. The optimal fn (St) is determined by backward solution of the functional equation and by straight forward comparison over discretized points of x and stored on each of the predetermined grid points.
This is the predictor move in the next step, the state transition equation is solved and any other value of f ( ) not pre-computed at previously identified grid points are obtained by an appropriate interpolation to improve fn(P). This step is the corrector move (Esogbue, 1989).The solution of this procedure via digital computer is handled with the code consisting of three nested DO loops (for more details see Section 3.9 which simplifies the problem).
The other computational procedures e.g. state increment DP (SIDP), Discrete Differential DP(DDDP), Decomposition methods, Lagrange multipliers, etc. as described elsewhere (Esogbue, 1989) are not chosen to solve the Dynamic Programming Problem due to following reasons
1) SIDP may yield non-optimal results" It was shown by Turgeon (1982). 2) DDDP guarantees a global solution only when a problem's structure is such that any local solution is also a global solution. Although DDDP is very efficient to overcome the limitations due to the curse of dimensionality, it is in general difficult to implement because it requires certain differentiability properties which may not be guaranteed in certain situations.
3) Decomposition methods require successive approximation and are not so simple to code as compared to the simple technique adopted herein.
3.4.3 Multistage problem
A typical multistage problem may be defined where the solution is required from a stage to
stage (e.g. 10 days or months) transition with state variables St and independent decisions xt
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exercised at each stage. A return fn (St) obtained at stage k must be communicated from stage to stage. A control on state space may be made through constraints associated with decisions selected in each stage. The algorithm not only computes the returns but also a set of decision policies d (previously defined as xt ) feasible to state transformation equation. The algorithm solves multistage problems in a backward solution mode. The multistage problem described herein is illustrated in Figure 3.1.
3.5 Stochastic Dynamic Programming (SDP)
This work reviews past attempts for stochastic optimization of large scale complex systems operation, then the new methodology is presented in detail. The idea of Marcov process and forecasting for inflow uncertainty is introduced and some methods to capture uncertainty in the optimization are explained. Ways of finding optimal operating rules are discussed and the particular form of SDP used is shown in detail. The modified algorithm has been devised to alleviate the problem of dimensionality associated with DP. Finally the results of the proposed Stochastic Optimization procedure are critically examined and discussed.
3.5.1 Probabilistic dynamic programming
If the state at the next stage is not completely determined by the state and policy decision at the current stage, it may be solved through probabilistic dynamic programming.
In deterministic DP, the well known DP recursive equation (3.1) is rewritten as
* fn (Sn , xn ) = Max Bs , xn + fn+1 (Sn+1 )
S No. of possible states at stage n+1
Pi Probability that the system goes to state i given the state Sn and decision xn at stage n ( i = 1, 2, 3, . . . , S)
Bi The contribution of the stage n to the Objective Function if the system goes to state i
* The relationship between fn (Sn , xn ) and the fn+1 (Sn+1 ) is complicated due to probabilistic structure. The probabilistic form of the above relation is
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S * fn (Sn , xn ) = Max ∑ Pi [ Bi + fn+1 (i ) ] (3.7) i=1
* Where fn+1 (i ) = Max fn+1 (i , xn+1) xn+1
The maximization is taken over the feasible values of xn+1. The structure of the calculations in probabilistic dynamic programming is given in Figure 3.2 (Hillier and Lieberman, 2005).
Contribution Optimal Probability from Benefit P1 stage n * C1 1 fn+1 (1)
* Decision P2 C2 fn+1 (2) State Sn x 2 n : : fn (Sn , xn ) Ps : : C s S
* fn+1 (S ) Stage n Stage n+1
Figure 3.2 Structure of the probabilistic dynamic programming 3.6 Stochastic Dynamic Programming (Formulation of the Model)
3.6.1 Formulation of the SDP recursive equation
Often the next state that will be occupied is not a deterministic function of the current state and decision (like reservoir levels). It may depend on uncertain events like rainfall, t streamflow or political decisions. To model such a situation, let Pij (k) equal the probability that the state in period t+1 is Sj given that the state in period t is St and decision k is made:
t t+1 t Pij (k) = P (S = Sj / S = Si and decision k ) (3.8)
t Where Pij (k) is transition probability of a time-dependent decision-dependent Marcov t chain. Methods to calculate Pij (k) transition probability of Inflow or storage state has been described in detail in section 3.7.6.
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The DP recursive equation (3.1) for deterministic case has been defined as
* ft (Si ) = Max Bt (Si, Sj , k ) + ft+1 (Sj ) k
Where ft (Si ) Maximum net benefits abtainable from time period t onward starting in state
Si in period t. In the deterministic case, the subsequent state (or downstream state) Sj is a deterministic function of k and of the initial state Si. For example in reservoir operation problem:
t+1 t t t Sj = Si + Qi - ki
Where Qi is inflow in period t
The DP recursive equation for stochastic case can be defined as
m t * ft (Si ) = Max ∑ Pij (k) [ Bt (Si, Sj , k ) + ft+1 (Sj ) ] (3.9) k j=1
Another form of DP recursive equation for stochastic case may be written as
n t n-1 ft (k,i ) = Max B k,i,l,t + ∑ Pij ft+1 (l,j ) (3.10) l j For all k,i,l feasible.
For the first stage with only one period is remaining,
1 ft (k,i ) = Max ( B k,i,l,t ) for all k,i,l feasible (3.11) l Where Bkilt value of system performance with an initial reservoir storage volume of Skt and inflow in period t equals Qi,t and a release of Rk,i,l,t and a final storage value of Sl,t+1.
t Pij Probability of inflow Qj, t+1 in period t+1 when the inflows in period t equals Qi,t
l Index to represent final storage volume Sl, t+1
t Within year period
n Stages or total number of periods remaining before reservoir operation terminates.
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The recursive equation given above is solved for each period in successive years, after certain iteration, the policy l(k,i,t) defined in each particular period t will repeat in each successive year. At this condition the policy has reach a steady state condition and
n+T n ft (k,i) - ft (k,i ) = constant (3.12)
For all k,i and all period of t within a year t The steady state condition will reach only if the constant, Bkilt, and the probability, Pij , do not change from one year to the next. At this condition, the steady state reservoir operating policy remains the same. Another form of SDP recursive equation is when the present value, PV, of the discounted net benefits is to be maximized (or cost is minimized). The discount factor and the related terms are defined as follows.
3.6.2 Discount factor
Discount factor is related to the interest rate and can be explained as one lends an amount PV at the beginning of period 1, then at the end of period 1 one should receive the principle PV plus the interest r x PV where r is the interest rate or discount factor. Therefore the value of one’s assets, V, at the end of period 1 as defined by Loucks et al. (1981) is
V = PV + r PV = ( 1 + r ) PV (3.13)
If at the end of period 1, one immediately reinvests these assets at the end of period 1 one could lend (1+r)PV and hence would have at the end of period 2.
V2 = PV + r PV + r (PV + r PV)
= (1 + r + r + r2 ) PV = (1 + 2r + r2 ) PV
= (1 + r )2 PV = (1 + r ) (1 + r ) PV
Similarly always reinvesting, one would have assets at the end of period t
t Vt = (1 + r ) PV (3.14) Here PV is called the present value of net benefit NBt = Vt at period t
t NBt = (1 + r ) PV (3.15) or -t PVt = (1 + r ) NBt (3.16)
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The total present value of the next benefits generated by plan P
Tp -t PVp = ∑ (1 + r ) NBt (3.17) t=1
The above equation can be written as
T -t PV = ∑ (1 + r ) NBt (3.18) t=1
Where NB is average annual net benefits
(1 + r )T - 1 PV = ------NB (3.19) r (1 - r )T
Average annual benefit NB is thus
r (1 + r )T NB = ------PV (3.20) (1 + r )T - 1
Where in above equation the term in brackets is CRFT
r (1 + r )T ------= CRFT (3.21) (1 + r )T - 1
CRFT is the capital recovery factor and the TAC (Total annual cost )
TAC = (CRFT ) C0 + OMR (3.22)
Where C0 is fixed initial construction cost and OMR is a constant annual operation maintenance and repair cost. The discount factor is the interest rate
1 α = ------(3.23) 1 + r
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3.6.3 Discounted stochastic DP model
One type of the SDP recursive model when α is included, can be expressed as:
n t n-1 ft (k,i ) = Max B k,i,l,t + α ∑ Pij ft+1 (l,j ) (3.24) l j
For all k, I,: l feasible
Where Pij is a probability transition matrix derived from the historical inflow record. See other types of SDP in chapter 5. When above equation is solved recursively, a situation will eventually reached where
n+T n ft-T (k,i) = ft (k,i ) (3.12)
For all k,i: t At this situation steady state solution has been found. The solution procedure for SDP models are described in detail in section 3.7.8.
3.6.4 Marcov process
Marcov process may be defined as follows:
Consider a system which at any particular time is in one of a finite number of states (I=1, 2, . . . , n ) and assume that at the discrete times t = 0, 1, . . . , T the system changes from one of these admissible states to another. In place of supposing that this change is deterministic, we assume that it is stochastic, ruled by a transition matrix.
P = Pij (3.25)
Where Pij is probability that the system is in state j at time t+1 given that it was in state i at time t. If P (transition matrix) is independent of time then
xt (i) is Probability that the system is in state i at time t (i = 1, 2, 3, . . . , n) and t = 0, 1, . . .
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The rules of probability theory then yielded the equation
n xt+1(j) = ∑ Pij xt(i) j = 1, 2, . . . , n (3.26) i=1
x0 (i) = ci
A special kind of Marcov process is one whose state S(t) can take only discrete values. Such a process is called Markov chain.
The theory of Markov process is devoted to the study of the asymptotic behavior of the function xi(i) as t → ∞. If all of the transition probability Pij are positive, these functions will converge to quantities x(i) as t → ∞ which satisfy the steady state equation
n x(j) = ∑ Pij xt(i) j = 1, 2, . . . , n (3.27) i=1 Therefore the limiting values are independent of the initial state of the system, the values of
x0(i). In many stochastic water resources models, a common assumption is made, that the
stochastic process xt is a Marcov process.
3.6.5 Marcov chains, inflow process and uncertainty
It is recognized that hydrological uncertainty and seasonality is usually present in the inflow process. The analysis of the inflow process represented by historical inflow records provides the basis for the reservoir operation optimization. To control the uncertainty following two procedures may be adopted.
1. The control method is based on the supposition that if a forecast of future inputs to the system can be produced then the system can be controlled efficiently using this forecast. Obviously, if the forecast is perfectly accurate then the system can be operated optimally with respect to certain criteria. Forecasts are never perfect and hence the system is operated by the control rule found for one time period only. The system is then updated and a new forecast is made.
2. Inflow in time period t can be represented by the discrete inflow distribution vector.
The serial correlation coefficient rk between inflows in time period t and t-1 guides
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the selection of models for describing the inflow process. If the value of serial
correlation rk for all k is not significant, the independent model could be selected. Otherwise a first order Marcov model may be accepted. Wang and Adams (1986) stated that Markov dependence is tenable for the dry seasons and independence is tenable for wet seasons where a mixed model could be used.
3.6.6 Methods of computing Marcov chains
The computation procedure is illustrated by the following example:
Example: Let we determine the elements of the one-step annual transition matrix (Marcov chains) of reservoir states. Reservoir capacity Ka = 4 units. Annual demand = 4 units including losses. The inflows to the reservoir are random process and in one year it may be 1 unit and in another year it may be 5 units. Therefore the annual flow may be considered one of the following units:
1, 2, 3, 4, 5
The annual inflows Qt , t = 1, 2, 3, . . . to the reservoir are serially independent and follows
normal distribution. The mean and standard deviation of Qt are known and from the tables of normal distribution following probabilities are calculated.
P(xt = 1) = 0.10 P(xt = 2) = 0.25 P(xt = 3) = 0.30 P(xt = 4) = 0.25 P(xt = 5) = 0.10 ------∑ P = 1.00
Solution procedure for Marcov Chains of storage states
Step 1. Let we divide the reservoir storage into 5 states, the one-step transition probabilities q(i,j) are represented by the Q matrix given below. Q is a square matrix of order c+1 in which the element of the ith row and jth column is q(i,j)
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Previous State j (St ) 0 1 2 . . . m 0 q(0,0) q(0,1) q(0,2) . . . q(0, ka ) 1 q(1,0) q(1,1) q(1,2) . . . q(1, ka ) Q = Current 2 q(2,0) q(2,1) q(2,2) . . . q(2, ka ) State I : ...... (St+1) m q(ka,0) q(ka,1) q(ka,2) . . . q(ka,ka)
Where square matrix Q is order ka + 1 and ka is capacity of the reservoir
Step(2) Determine the elements of transition matrix Q. Use mass balance equation to determine probabilities, q(i,j). Calculate first column.
St+1 = St + xt - dt
Where St+1 and St Storage in the reservoir at time t and t+1 xt Inflow at time t dt Demand / release at time t.
For example q(0,0), St = 0 and St+1 = 0
Using mass balance equation St+1 = St + xt - dt
St+1 = 0 + xt - 4
⇒ St+1 = 0 if xt ≤ 4 therefore we will find P(xt) ≤ 4
P(xt = 0) = -- P(xt = 1) = 0.10 P(xt = 2) = 0.25 P(xt = 3) = 0.30 P(xt = 4) = 0.25 ------∑ P(xt ≤ 4) = 0.90 = q(0,0)
To calculate q(1,0), St+1 = St + xt - dt = 0 + xt - 4 from above if St+1 = 1 xt = 5 and P(xt = 5) = 0.10 = q(1,0)
To calculate q(2,0), St+1 = St + xt - dt = 2 + xt - 4 ⇒ xt = 6 Therefore q(2,0) = P(xt = 6) = 0.00
To calculate q(3,0), St+1 = St + xt - dt = 3 + xt - 4 ⇒ xt = 7 Therefore q(3,0) = P(xt = 7) = 0.00 and q(4,0) = P(xt ≥ 8) = 0.00
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Therefore the first column of the transition matrix comes to be
q(0,0) = P(xt ≤ 4) = 0.10 q(1,0) = P(xt = 5) = 0.10 q(2,0) = P(xt = 6) = 0.00 q(3,0) = P(xt = 7) = 0.00 q(4,0) = P(xt ≥ 8) = 0.00
Step (3) Determine second column of the transition matrix. The element q(0,1) means 1 unit of water is already available in the reservoir and we want to go to state 0.
Therefore St+1 = St + xt - dt St+1 = 1 + xt - 4 St+1 = xt - 3
Therefore St+1 = 0.0 if xt = 3, 2, 1 or 0 Therefore q(0,1) = P(xt ≤ 3 ) = 0.10 + 0.25 + 0.30 = 0.65
To calculate q(1,1), St+1 = 1 + xt - 4 = xt - 3 Therefore for St+1 = 1 xt = 4 q(1,1) = P(xt = 5) = 0.25
Similarly to calculate q(2,1), St+1 = 1 + xt - 4 = xt - 3 For St+1 = 2 xt = 5 q(2,1) = 2 , P(xt = 5) = 0.1 q(3,1) = 2 , P(xt = 6) = 0.0
To calculate q(4,1), St+1 = 1 + xt - 4 = xt - 3 To keep the reservor full, i.e. St+1 = 4 xt should be 7 or more, the rest of the amount will be spill. Therefore q(4,1) = P(xt ≥ 7) = 0.00
Step(4) Determine third column of the matrix
In a similar way the following transition probabilities are calculated. q(0,2) = P(xt ≤ 2 ) = 0.35 q(1,2) = P(xt = 3) = 0.30 q(2,2) = P(xt = 4) = 0.25 q(3,2) = P(xt = 5) = 0.10 q(4,2) = P(xt ≥ 6) = 0.00
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Step(5) Determine fourth column of the matrix In a similar way the following transition probabilities are calculated. q(0,3) = P(xt ≤ 1 ) = 0.10 q(1,3) = P(xt = 2) = 0.25 q(2,3) = P(xt = 3) = 0.30 q(3,3) = P(xt = 4) = 0.25 q(4,3) = P(xt ≥ 5) = 0.10
Step(6)Determine fifth column of the matrix In a similar way the following transition probabilities are calculated. q(0,4) = P(xt ≤ 0 ) = 0.00 q(1,4) = P(xt = 1) = 0.10 q(2,4) = P(xt = 2) = 0.25 q(3,4) = P(xt = 3) = 0.30 q(4,4) = P(xt ≥ 4) = 0.25 + 0.10 = 0.35
Step (7) The one step transition, matrix, when the flows are independent, is given below:
Previous State j (St ) 0 1 2 3 4 0 0.90 0.65 0.35 0.10 0.00 1 0.10 0.25 0.30 0.25 0.10 Q = Current 2 0.00 0.10 0.25 0.30 0.25 State I 3 0.00 0.00 0.10 0.25 0.30 (St+1) 4 0.00 0.00 0.00 0.10 0.35 ------ka ∑ q(i,j) 1.00 1.00 1.00 1.00 1.00 I=1
N-Step transition matrix Q(1) is the one step transition probability matrix of storage state. If the power (1) of Q(1) increases, the elements of matrix Q(n) approaches under ergodic conditions (steady state conditions) .This condition represent long run or invariant distribution.
T If Pt = [ Pt(0), Pt(1), . . . , Pt (ka ) ] is unconditional probability of reservoir states at time t and T denotes transpose of column vector, the probability Pt+1 at time t+1 is given by
Pt+1 = Q Pt Where Pt = P(St = j) (3.28)
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Similarly Pt+2 = Q Pt+1
or = Q ( Q Pt )
2 = Q Pt
(2) = Q Pt
Where Q(2) is square of the Q matrix.
In general, if the process start from t=0 and reaches upto n (year) period
(n) Pn = Q P0 (3.29)
Where Qn = Q . Q. . . . Q n-step transition matrix of conditional probabilities. The following identity is known as Champman - Kolmogorov identity
Q(l+m) = Q(l) . Q(m)
If the states do not communicate, steady state conditions cannot be reached. For example
1 0 Q = 0 1
1 0 Q(n) = (non ergodic conditions) 0 1
In the following example, steady state conditions hold for all n
0.5 0.5 Q = 0.5 0.5
0.5 0.5 Q(n) = 0.5 0.5 However such type of conditions rarely happen in reservoir storage problem.
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3.6.7 Streamflow and stochastic process
The annual volume of streamflow in a river may be represented by discrete random variable.
Let Qt = Discrete random variable of streamflow in year t. qi = Possible value of Q Pi = Unconditional probability of streamflow
m ∑ Pi = 1 (3.30) i=1
It is frequently seem that the value of Qt+1 is not independent of Qt , Such dependence can be modeled by Marcov chains with the help of a transition probability matrix (tpm).
Pij = P (Qt+1 = qj / Qt = qi ) (3.31)
m ∑ Pij = 1 for all I j=1
Where m is the number of the inflow states in the tpm. 3.6.8 Determination of the observed inflow transition probabilities
Procedure 1 The inflow transition probabilities (of a period t conditioned on the preceding period (t-1) inflow) may be calculated using the following procedure.
1. Divide the historic inflows into current class interval (or some discrete values) 2. Take the inflow sequence and its preceding period inflow sequence to compute transition probability. Count number of times the flow remains in state (1) in period 1
(=N1). Count number of times the flow of period (2) remains in state (1) when the
period 2 flow is also (1) (=N2). The Pij will be equal to N2 / N1. In a similar way probabilities of the rest of the states can be calculated.
Another way of computing transition probabilities is to compute the joint probability of A and B in the two periods and divide it by the independent probability of period 1.
P(A,B) Transition Probability = Pij = ------(3.32) P(A)
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Transition probabilities can be easily calculated with the help of computer using the algorithm given by Harbaugh et al. 1970.
DO 10 K = 2, N I = INFLOW(PERIOD, 1) J = INFLOW(PERIOD, 2) 10 P(I, J) = P(I, J) + 1 (for all values of inflows (1,2,...,n)
DO 15 I = 1, NSTATE (NSTATE = No. of the states) SUM = 0.0 DO 20 J = 1, NSTATE 20 SUM = SUM + P(I, J) DO 25 J = 1, NSTATE 25 P(I, J) = P(I, J) / SUM (P(I, J) = Transitional Probability)
Procedure 2 If the historic data is limited, it is better to derive tpm with the help of fitting the probability
distribution. Normal and lognormal distributions to calculate Pij for SDP has been used by various researchers (Wang and Adams 1986).
1. The historical inflow records are fitted with some theoretical probability distribution functions where the distribution parameters are estimated from the inflow records.
2. The goodness of fit is judged. The commonly used distributions are normal, lognormal and gamma (Pearson type III) distributions
3. Determine P(qt) and P(qt , qt-1) with the help of the bestfit model and tpm is estimated as
P(qt , qt-1) Pij = ------(3.33) P(qt)
Where P(qt , qt-1) Multivariate distribution of 2 consecutive periods. Another procedure for calculating tpm, if the inflow record is limited, is given by Wang and Adams 1986. It is described below.
u l 4. Based on the discretizing scheme, the upper and lower bound (qj , qj ) of the inflow volume Q(m) are calculated as
l qj = qj - ∆ / 2 for all j (3.34)
u qj = qj + ∆ / 2 for all j
Chapter Three 93 Methodology
Where ∆ is the discrete interval of the inflow state.
5. Fiering and Jackson (1971) equation is used to determine zu and zl (assuming normal distribution is fitted)
q(m) = µ(m) + ρ(m) δ(m) [ [ q(m-1) - µ(m-1) ] / σ [m-1] ] + 2 1/2 z σ(m) [ 1- ρ (m) ] (3.35)
Where µ(m) is mean, σ(m) is standard deviation and ρ(m) is the serial correlation coefficient between the inflows q(m) and q(m-1). By putting the values of qu and ql one by one, value of zu and zl is determined.
u i 6. Pij = φ (z ) - φ (z )
Where φ ( ) Standard normal cumulative distribution function and can be obtained by normal distribution tables. At terminal conditions (like for the month January) when m = 1 (first month of the year) then m-1= M (i.e. December). The case for lognormal and Gamma distribution can be found in Wang and Adams (1986).
3.6.9 Markov process and stochastic dynamic programming (solution procedure)
If a system makes a transition from i to j, it will earn the amount rij (benefit) plus the amount it expects to earn if it starts in state j with one move fewer remaining. The above definition may also be used to write the recurrence relation.
N Vi(n) = ∑ Pij [ rij + Vj (n-1) ] (3.36) j=1
i = 1,2,...,N , n=1,2,3,....
N N or = ∑ Pij rij + ∑ Pij Vj (n-1) j=1 j=1
i = 1,2,...,N , n=1,2,3,....
N Let qi = ∑ Pij rij i = 1, 2, 3, . . ., N (3.37) j=1
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N or Vi(n) = qi + ∑ Pij Vj (n-1) (3.38) j=1
i = 1,2,...,N , n=1,2,3,....
or in vector form V(n) = q + PV (n-1) (3.39)
Where V(n) column vector with n components Vi(n) called the total-value vector.
q = P.R 3.6.10 Solution by value iteration method.
Define the following variables di(n) = No. of alternatives in the ith state that will be used at stage n (Decision in state i at nth stage)
Vi(n) = Total expected return in n stages starting from state I if an optimal policy is followed.
Therefore using DP recursive relation we get
N k k Vi(n+1) = Max ∑ Pij [ rij + Vj (n) ] (3.40) k j=1 For above equation we can write N k k Vi(n+1) = Max [ qi + ∑ Pij Vj (n) ] (3.41) k j=1 The above equation will tell us that which alternative to use in each state at each stage and will also provide us with expected future earnings at each stage of the process. To apply this relation, we must specify Vj(0) the boundary condition for the process. Let we solve the
proceeding example and suppose that V1(0) and V2(0) = 0. To illustrate the procedure a simple example is presented.
It can be shown from the following example that the iteration process will converge on best alternative for each state as n becomes very large. Therefore the policy converges. The
method is called the value iteration method because the value Vi(n) are determined iteratively.
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Example Given data ______Reservoir Inflow tpm Benefits Expected immediate state Release ------Benefits k k k k k k k (i) (k) Pi1 Pi2 ri1 ri2 qi = Pij rij ______1 1 0.5 0.5 9 3 6 2 0.8 0.2 4 4 4
2 1 0.4 0.6 3 -7 -3 2 0.7 0.3 1 -19 -5 ______Solution: ______
n = 0 1 2 3 4 . . . ______
V1(n) 0 6 8.2 10.22 12.222
V2(n) 0 -3 -1.7 0.23 2.223
d1(n) - 1 2 2 2
d2(n) - 1 2 2 2 ______
Discounted case in value iteration method In the previous section, policies have been derived on the bases of (long run) expected average cost (or benefit) per unit time or the (long run) actual average cost (or benefit) per unit time. Discount factor associated with the interest rate has been defined in section 3.7.2. It may be expressed as: (See equation 3.23)
1 α = ------(α < 1) 1 + r Where, r is current interest rate. A discount factor specifies that the present value of 1 unit of the cost, m periods in the future is αm. Therefore an alternative measure is to find the policies with the help of long run total discounted cos. Let we rewrite the DP recursive equation for stochastic optimization.
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N Vi(n) = ∑ Pij [ rij + Vj (n-1) ] j=1
i = 1,2,...,N , n=1,2,3,....
If Vi(n) is defined as the present value of the total expected reward for a system in state I with n transitions remaining before termination, we get
N Vi(n) = ∑ Pij [ rij + α Vj (n-1) ] j=1
N N or = ∑ Pij rij + ∑ Pij α Vj (n-1) j=1 j=1 N We know qi = ∑ Pij rij i = 1, 2, 3, . . ., N j=1
N or Vi(n) = qi + α ∑ Pij Vj (n-1) (3.42) j=1 The solution procedure for the discounted case is same as for that used in the no discounting case. Let α = 0.90, we solve the same example with the following equation
N Vi(n+1) = qi + 0.90 ∑ Pij Vj (n) j=1 Following results are achieved: ______
n = 0 1 2 3 4 . . . n-1 n ______
V1(n) 0 6 7.78 9.136 10.463 . . . 22.2 22.2
V2(n) 0 -3 -2.03 -0.647 0.581 . . . 12.2 12.2
d1(n) - 1 2 2 2 . . . 2 2
d2(n) - 1 2 2 2 . . . 2 2 ______
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3.6.11 Solution by policy iteration method
We know that the matrix qi = Pij rij (3.37) If there is only one recurrent chain in the system so that, it is completely ergodic (steady state), then all rows of Pij are the same and equal to the limiting state probability distribution for the process ∏.
N Let g = ∑ ∏i qi (3.43) i=1 Let we consider a simple example with data:
Pij (steady state) = 0.444 0.556 = ∏i 0.444 0.556
and qi = 6 -3
N g = ∑ ∏i qi = 0.444 0.556 6 i=1 0.444 0.556 -3
= 1 1
Considering Vi the asymptotic intercepts of Vi(n). There when n → large
Vi(n) = n gi + Vi (I=1,2, . . ., N) (3.44)
If the system is ergodic then gi become constant and therefore gi = g
Therefore Vi(n) = n g + Vi (I=1,2, . . ., N)
Knowing the value of n, g and Vi , Vi(n) can be found. We know (See equation 3.38) N Vi(n) = qi + ∑ Pij Vj (n-1) (3.45) j=1 i = 1,2,...,N , n=1,2,3,....
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Combining (3.44) and (3.45) we get N or ng + Vi = qi + ∑ Pij Vj (n-1) j=1 i = 1,2,...,N , n=1,2,3,....
We can write the first equation as: Vi(n-1) = (n-1)g +Vi
Putting value of Vj(n-1) in the above equation we get
N n g + Vi = qi + ∑ Pij [ (n-1)g + Vj ] j=1
N N n g + Vi = qi + (n-1)g ∑ Pij + ∑ Pij Vj j=1 j=1
N Since ∑ Pij = 1 The above equation becomes i=1 N n g + Vi = qi + n g - g + ∑ Pij Vj j=1
N or g + Vi = qi + ∑ Pij Vj j=1 Therefore there are two steps to find the optimal value of a problem.
Step(1) Value determination operation
Using Pij and gi for a given policy to solve
N g + Vi = qi + ∑ Pij Vj (i=1,2,3 . . ., N) (3.46) j=1
for all relative values Vi and g by setting Vn = 0
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Step(2) Policy Improvement Routine For each state i find the alternative k that maximizes
N k k qi + ∑ Pij Vj (3.47) j=1 Use the values of the previous policy to update the solution until an error limit (stopping rule is satisfied.
An alternative procedure is presented by White (1969) by modifing the Howard’s algorithm. This method is more efficient but only applicable when there is non-zero probability of the k state in the probability matrix Pij .
Discounted case in policy iteration method The method with discounting is similar as defined for the case without discounting in the above section except g in the above equation is now not present. Stopping rule in the iteration process is to stop when decision in n and n+1 is same.
3.6.12 White’s solution procedure
Step (1) Calculate n k k Vi = qi + ∑ Pij Vj (i=1,2,3 . . ., N) (3.48) i=1
* Find Vi
* k Vi = Vi If(Vi-1 ≥ Vi ) (3.49) * k-1 Otherwise Vi = Vi If(Vi-1 < Vi ) (3.50)
Cost = g(n) = V*(NSTATE)
Step (2) Update the solution by new values
* Vj = Vi - g(n) (3.51)
Error = g(n) - g(n-1) (3.52)
If ( Error < 0.0001 ) Stop
Otherwise go to step (1) by using new values of Vj as calculated in step (2). The computer program is prepared to evaluate the effectiveness of the method.
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3.6.13 Method of successive approximation
It is a finite-period Markovian process. If the total discounted cost of a system starting in state I and evolving for n time period is considered for optimal policy determination, the solution may be obtained via deterministic dynamic programming except that Marcov system evolves according to some probabilistic laws of motion rather than evolving in a deterministic fashion. Using the principle of optimization, the DP recursive relationship is
M n+1 k k n Vi = min [ Ci + α ∑ Pij Vj ] (3.53) k j=0
i = 0,1, . . . , M
n Where Vi the expected total discounted cost of a system starting in state i and evolving for n time periods when an optimal policy is followed.
k Ci the cost (or benefit) in state i and stage k Using the above equation, the calculations move backward period by period - each time 0 0 0 finding the optimal policy for that period. It is usually assumed that for n = 0, V0 , V1 ,V2 , 0 . . . , VM = 0.
1 k Therefore Vi = min [ Ci ] (3.54) and M 2 k k 1 Vi = min [ Ci + α ∑ Pij Vj ] k j=0
i = 0,1, . . . , M 1. It should be noted that α can be set equal to one (no discounting ) for finite period problems in which case the cost criterion becomes the expected total cost. 2. The method is same as explained for value iteration procedure. n 3. When discounted costs are used, Vi converges to Vi as n approaches infinity, where
Vi = expected (long run) total discounted cost of a system starting in state I and continuing indefinitely when an optimal policy is followed . M k k Vi = min [ Ci + α ∑ Pij Vj ] k j=0
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4. There is no procedure for deciding when to terminate the method of successive approximation. Some stopping rule like n+1 n Error = Vi - Vi ≤ 0.0001 may be followed. 5. This method (sometimes may lead to near optimal policy) has one advantage over the policy iteration and LP techniques that it never requires the solution of a system of simultaneous equations. Each iteration can be performed simply and quickly.
3.7 Recent Research Trend in Stochastic Optimization
There are two main directions. The first is to use implicit stochastic optimization for the operation optimization of the multiple reservoir system. The second direction is the improvement of models describing reservoir inflows by means of various forecasting technique. (Hillier and Lieberman, 2005)
Selecting the best and efficient solution procedure to solve SDP for the large scale system
1. Howard (1960) explains that “ If we are dealing with a discrete system and if we wish to maximize the total expected benefits over only a few stages of the process, then a value iteration approach is preferable. 2. If we expect the process to have an indefinite duration, the policy iteration method is preferable.
3. Hellier and Lieberman (2005) explain that in policy iteration method, even if the solution converges rapidly, completing step (2) requires considerable calculation for systems with a large number of states. Loucks and Falkson (1970) showed that the policy iteration method takes somewhat longer time to solve.
4. Using Linear Programming (LP) (say with 50 states and 25 decisions lends to 1250 variables and 51 constraints (excluding the non-negativity constraint) which represent a large linear program. LP takes a greater amount of computational time as compared to DP (Loucks and Falkson (1970).
5. When the cost (or benefit) criterion is the expected discounted cost, the method of successive approximation provide a valuable tool for approximating the optimal solution. Much simple and short time calculations are required for this algorithm than for the Policy iteration and LP method.
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6. Withstanding the utility of recent research trends, method of successive approximation still hold potential advantages for problem of operations optimization of both single and multiple reservoir systems, provided that further improvement are made in the computational efficiency of the generalized procedure.
7. The intention of this work is to establish a framework of stochastic models for describing the problem of optimizing of multiple reservoir system operation and to develop an efficient computational procedure for this optimization.
8. A two stage optimization frame work, which consists of a steady state stochastic dynamic programming model (SDP) and a Network Flow Programming (NFP) model, is presented.
9. SDP is used for the optimization of the multiple reservoirs as the objective function is nonlinear. NFP is then applied to the complete system using the results obtained from the SDP to avoid the curse of dimensionality.
10. The merit of this two stage optimization method is supported by the significant reduction of the computing requirements over conventional methods of optimizations. It is also found that this method is better than that in which inflow forecasting is introduced into the optimization procedure directly.
3.8 Mathematical Statement of SDP / DDP Models
The objective of the analysis is to determine, for a given initial state of the system and sequences of inflows, the set of decisions regarding release of water from the reservoir for whatever purpose that will minimize the deficits between the actual release and demand. Therefore, it will maximize the total return from operations subject to physical, social and legal constraints imposed for a sequence of N time intervals.
3.8.1 State transformation equation
A stage transformation equation is required to shift the multistage process from one stage to another such as
St+1 = g (St , xt ) (3.55)
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Equation (3.55) is non-inverted and used in the DP model. However, an inverted form of (3.55) can be optimized directly over S so that d becomes a dependent variable. In some cases inverted form may be useful.
dt = g (St , St+1 ) (3.56)
For a reservoir operation problem, Figure 3.1, the state transformation equation may be expressed as
St+1 = St + qt - dt - A (St ,St+1) [ et + RFt ] (3.57) where St Storage in reservoir at the beginning of period t. qt unregulated inflow during the period t. dt Amount of water released during the period i. et Evaporation rate (e.g. ft/10 day). A Average surface area of the reservoir over the month (e.g. acres).
A (St ,St+1 ) = [ A (S t ) + A (S t+1 ) ] / 2
A(S t ) Reservoir surface area at storage level St can be obtained from elevation-area-capacity data for the reservoir site.
RF t Rainfall over the reservoir during the month (e.g. ft./10 day). 3.8.2 Model constraints
In addition to state transformation equation (3.9), the constraints associated with reservoir operations are listed below:
i) Constraints on storage
Smin < St < Smax (3.58)
Minimum Initial Maximum Allowable < Storage < Capacity Storage Volume
Smin = DEADCAP (3.59)
Minimum Dead Storage Allowable = Capacity for Storage Sedimentation
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St - Sr + Dt - Et = 0 (3.60)
Initial Target Deficit Excess Storage - Storage + Storage - Storage = 0 Volume Volume Release
ii) Constraints on release of water
dmin < dt < dmax (3.61)
Minimum Release Maximum Release < in < Release for period k period k period k
k dmax = St + qt - Smin - Losst (3.62)
Maximum Storage Inflow Minimum Loss Release = in + Accretion - Storage - due to for period k period k period k Evaporation
dmin = max [ St + qt - Smax - Loss , d ] (3.63)
Minimum Initial Inflow Maximum Loss Mandatory Release = max { Storage + + - Storage - due to , Release for in period k Accretion for k period evaporation period k. iii) Constraints on Energy Generation
KWHt - ((C) qt ) . H(K ,St ,St+1 ) . et = 0 (3.64)
Kilowatt Unit Flow Average Hydro Hours of - Conser- . Through . Productive . Power = 0 Energy vation Turbines Storage Efficiency Constant Head
KWHt - (P) . (t ) . (P ) < 0 (3.65)
Kilowatt Plant Hours in Plant Hours of - Capacity . Period . Factor < 0 Energy KW K
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KWHt - FIRMt + Dt - Et = 0 (3.66)
Kilowatt Firm Deficit Excess Hours of - Energy + Energy - Energy = 0 Energy Target Production Production
E = KWHt IF KWHt < E (3.67) E = 0
Kilowatt Peak Peak Kilowatt IF Hour of < Energy Energy = Hour Energy Capacity of Energy
Non Peak = 0 Energy
EP = min [ Pt (St ) , Pt (St +qt -losst-dt ) . hk ] (3.68)
Peak Power Power Number of Energy = min Capacity . Capacity . Peak Hours Capacity Function Function in period k I II IF KWHt > E Et = EP ;
Peak Peak Energy = Energy ; Capacity
EN = E - Et (3.69)
Non- Total Peak Peak = Energy - Energy Energy Capacity
If E < KWH < E
E = E ; E = KWH - E (3.70)
E = min P (S ),P (S +q -d -loss) h (3.71)
Where h = total number of hours in month k.
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Equations (3.58) and (3.61) state that the allowable storage volume and releases must be greater than or equal to the minimum storage capacity and minimum release and vice versa respectively. Equation (3.62) indicates that maximum possible release may be equal to storage + inflow by subtracting losses and minimum storage capacity. Equation (3.63) states that minimum release in each stage should be greater than either the flexible release for that stage or the mandatory release. Equation (3.64) states that energy generated in (kwh) is equal to the product of flow through turbines, storage head, hydropower efficiency and a unit conversion factor. Equation (3.65) indicates that energy generated should be less than or equal to plant capacity, hours in period and a plant factor usually given by the manufacturer. (3.66) is a mass balance equation in terms of energy. (3.67) shows that if energy function is greater than the capacity limit on peak energy, then all the energy that can be produced will be on peak energy. (3.69) indicates that if energy function is greater than the capacity limit on total energy then the maximum possible peak energy is produced and the excess energy is the peak or dump energy. (3.70) explains that if the energy function is greater than the capacity limit on peak energy but less than or equal to the capacity limit on total energy, then there will be two energy values, peak and non-peak energy (Hall 1969).
3.8.3 Multi-objective return functions
Our objective of water resources development for the test case is to minimize the losses due to irrigation deficits to meet downstream water requirements and to minimize the losses due to energy deficits produced to meet the energy requirements for the country as desired from the project. Another objective is the flood protection. The reservoir levels should have to be drawn down to reduce the intensity of floods in certain months. There are some standard minimum and maximum rule curves published by the planners during project design. These curves may not be violated during the operation period. Therefore, the next objective is to minimize the losses between optimal policy and envelope of minimum and maximum design curves.
Both two sided and one sided loss functions are formulated. To analyze the multi- objective complex water resources problem and for its development, a very flexible methodology is adopted. (1) the technique may take priority for irrigation to derive optimal operating
Chapter Three 107 Methodology
policies (2) whereas in an other run, optimal policies based on power priority can be obtained (3) the algorithm is designed to get optimal operating policies also for both irrigation and power multiple objective with a preference level on power deficits. At the same time, in above three cases, the policy can be further improved by minimizing deviations between the releases and the envelope of maximum and minimum design levels (curves) of the storage reservoir. The objective functions can be expressed as:
Case (1) Irrigation priorities n 2 F1t (St ) = Min [ DEMANDt - dt ] (3.72)
Case (2) Power priorities
n 2 F2t (St ) = Min [ PWRFLOt - dt ] (3.73)
PWRFLOt = PWRDEMt / 1.024 * Headt * PLANTEFt ) n n If F2 t < 0 F2 t = 0.0
n Otherwise [ F2 t (St ) ]
Case (3) Storage loss functions
n F3 t (St ) = Min [ RULMAXt - St] (3.74)
n n If F3 t (St ) > 0 , F3 t (St ) = 0
n F4 t (St ) = Min [ RULMINt - St] (3.75)
n n If F4 t (St ) < 0 , F t (St ) = 0
Case (4) Irrigation + design levels
n n n n F5 t = F1 t + F3 t + F4 t (3.76)
Case (5) Irrigation + power
n n n F6 t (S ) = F1 t + η.F2 t (3.77)
where η is preference level constant or trade off coefficient for each period, varies between 0 and 1.
Case (6) Irrigation + power + design levels
n n n n n F7 t = F1 t + η.F2 t + F3 t + F4 t (3.78)
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The optimal return in the preceding stage is then added to the models in each of the above case:
* n-1 ft (. ) = Fmt (. ) + f t+1 (. ) (3.79)
Where m varies between 1 and 7. The trade-off between agricultural vs. hydropower can be obtained with the help of the methodology described herein. The Dynamic Programming model optimizes in a recursive fashion according to Bellman's principle of optimality (Bellman et al 1962). The discrete state variables in the model are monthly storage and monthly inflow. The model then takes successive steps back in time from an initial point, searching for an optimal release decision that will minimize the losses as described above for the initial point to the end of the horizon. For example, the computation begins at the end of the time period that is at month n and proceeds backwards through time (k = n-1, n-2, n-3,...,1). In this case, the stages are monthly intervals. 10-daily intervals for time horizon are possible with the same model if very high speed computer is available.
3.8.4 Discretization of state variables
A uniform discretization interval DEL and DEL2 can be selected for the state variables S and decision variables d respectively. Interval DEL and DEL2 must be the same for all stages. It is inappropriate to be in uniform discretization interval for certain problems, a non-uniform interval can be specified accordingly with the help of flexible input operations available in the model described herein.
Selection of DEL is extremely important since it affects execution time, computer storage requirements and solution accuracy (Labadie 1989, Klemes 1977). Klemes(1977) found that a coarse discrete representation of storage could cause a collapse of the solution so the results obtained might be in complete contradiction to reality as far as the probability distribution of storage was concerned.
Due to the reasons discussed above, special consideration is given to the selection of DEL. The method proposed by Klemes (1977) is adopted herein to select the number of storage states. It illustrates a linear relationship between storage ratio S and number of storage states 'NSTATE' and a lognormal distribution input with the indicated value of the
Chapter Three 109 Methodology coefficient of variation C . The selection of the threshold values S and n is made from Figure 3.3 to be used in the proposed linear rule.
For C = 0.75 and So = 0.7 ; no = 9
no - 2 NSTATE = ------S + 2 (3.80) So For example for Mangla Reservoir (one reservoir in test case) with specified active storage capacity of S = 4.81 MAF, the number of storage states (NSTATE) required for the proposed DP procedure has been computed as follows:
9 - 2 NSTATE = ------4.81 + 2 = 50 0.7
The number of states estimated herein ensures to obtain the stationary risk of failure and stationary probability of spillage with an error e < 0.001. A
Reservoir full Inflow qt
Maximum State Sn Sn-1
Storage St
S3 S2 S1 Conduit Release dt Dead Storage Minimum Pool Level Reservoir bed
Figure 3.3 Reservoir states [ S(i), i = 1,2,3,. . .,n ]. Volumetric increments between states, all states equal
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3.8.5 Algorithm for optimizing model
Bellman's theory of dynamic programming is used herein. Among several types of dynamic programming algorithm (e.g. (i) recursively computations forward in time (ii) recursively computations backward from a given end to the start (iii) computations forward as in (i) with stochastic problem as probability distribution for the inflows or (iv) computations backward as in (ii) with stochastic problem), a type (ii) formulation is developed here by manipulation of the return functions [equations (3.72) through (3.79)]. The computations necessary to solve the deterministic problem posed herein are coded with FORTRAN-90 (Ver.5.1) and flow chart of the entire computation is shown in Figure 3.4. Flow chart of the simulation program is shown in Figure 3.5.
Chapter Three 111 Methodology
Option 1 Select option Option 2 2. Derive DDP or SDP rules. 3. Verify rules Call Read data for SDP or DDP Simulate (B) Select model type Reservoir Select Objective Function Simulation model
Call DP Read physical data Driver Keep of the reservoir Optimal optimal from an external Policy policy in file
Print / save Read memory for the DP optimal policy
Save the policy in a separate t = 1 data file to be retrieved by the NFP model in t = t + 1
Simulate the reservoir with historic and forecasted data
yes t < N
No
Print / save results
Stop
Figure 3.4 Flow chart of the optimization program
Chapter Three 112 Methodology
Program Simulate
Read Data Inflows, demand, evaporation, Area capacity curves, tail race ele., etc
Generate Call TF Inflows Thomas Fiering model
t = 1
Compute storage S(t+1)=S(t)+q(i)-R(i)
Compute average storage S =[S(t+1)+S(t)]/2
t = t + 1 Call XINTERP Interpolate area and head
Compute evaporation losses and solve continuity equation
Spill = S(t)-Smax
Compute hydropower and energy if turbine present in dam
Print results
yes t < N Plot simulation No graphs Stop
Figure 3.5 Flow chart of the simulation program
Chapter Three 113 Methodology
Refer to Figure 3.4, the main steps required for computations are given in the following section. These steps are identical to those described by Smith et al. (1983) and not included here to conserve space.
3.9 System Network Optimization
Network models and analysis are widely used in operations research for diverse applications, such as the analysis and design of large-scale irrigation system, transportation systems and communication networks etc.
Network and network analyses are playing an increasingly important role in the description and improvement of operational systems primarily because of the ease with which the system can be modeled in network form. This growth in the use of networks can be attributed to:
1. The ability to model complex systems by compounding simple systems.
2. The mechanistic procedure for obtaining system-of-merits from networks.
3. The need for a communication mechanism to discuss the operational system in terms of its significant features.
4. The means for specifying the data requirements for analysis of the system
5. The starting point for analysis and scheduling of the operational system.
Item 5 was the original reason for network construction and use. The real strength of the network approach lies in the fact that it can be successfully applied to almost any problem when the modeler has enough knowledge and insight to construct the proper network representation. The advantages of using network models are:
1. Network models accurately represent many real-world systems. 2. Network models seem to be more readily accepted by non-analysts than perhaps any other type of models used in the system analysis. Additionally, since network models are often related to the physical problems, they can be easily explained to the people with little quantitative background. 3. Network algorithms facilitate extremely efficient solution to some large-scale models. 4. Network algorithms can often solve problems with significantly more variables and constraints than can be solved by other optimization techniques.
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A multireservoir system with link canals, barrages, powerhouses and other facilities can be considered in a framework of capacitated, network flow problem. Because there are certain variables like inflows which are random process, the optimization problem could be considered stochastic. The original algorithm was applicable to the deterministic network flows (Ford and Fulkerson 1962, Chung et al. 1989). A procedure is developed to incorporate the stochasticity in the optimization problem.
3.10 The Out-of-Kilter Algorithm
The out-of-kilter algorithm (OKA) is an iterative procedure to find the circulation in a given capacitated network which minimizes the total cost of all flows passing through the arcs of the network. The basic terminology/definition relevant to the algorithm is given below (Phillips and Garcia-Diaz 1981)
1. A network diagram is represented by nodes and arcs. A node is usually a physical origin or termination point such as reservoir, barrage, diversion from a river etc. A
source node (such as a reservoir) generates flow fij and a sink node consumes flow. A set of nodes in a network can be designated as N. 2. Arcs are the lines that connect the nodes in a network. These are sometimes called branches. A directed arc is one in which flow is only permitted in a predesigned direction. The set of arcs in the network is represented by S. 3. A network is a connected set of arcs and nodes. It represents a physical process in which units move from source(s) to sink(s). 4. If an arc can only tolerate certain magnitude of flow, such as maximum capacity of a
canal (upper bound Uij) or an allowable minimum capacity of a canal (lower bound
Lij), the arc is said to be capacitated. 5. If an arc from node i to node j is traversed from node i, it is called forward arc. Otherwise it is called reverse arc if it is traversed from node j. 6. A capacitated network is a network with capacitated arcs. 7. A circulation is an assignment of flow to arcs such that flow is conserved at each node. The total flow entering the node is equal to the total flow leaving the node. The algorithm deals with circulation, therefore it is often necessary to modify the original networks to provide circulation.
Chapter Three 115 Methodology
8. In a capacitated network there are always lower and upper bounds on each arc.
3.10.1 Basic theory
The network flow problem can be represented by a special linear programming (LP) problem that may be called minimum-cost-circulation. Since the cost of flow of 1 unit across arc (i,j)
is cij the LP problem may be written as: (Phillips and Garcia-Diaz 1981)
Minimize ∑ cij fij (3.81) (i,j) ∈ S Subject to 1. Capacity constraints
fij ≤ Uij (i,j) ∈ S (3.82)
fij ≥ Lij (i,j) ∈ S (3.83)
2. Conservation of flow constraint to ensure that any flow entered a node must also leave.
∑ fij - ∑ fji = 0 all i ∈ N, i =/ j (3.84) j ∈ N j ∈ N
3. Nonnegativity
fij ≥ 0 for all arcs (i,j) (3.85) The out-of-kilter algorithm is derived using duality theory of LP. To do this, the above primal problem is written in a more convenient way:
Maximize ∑ - cij fij (3.86) (i,j) ∈ S Subject to 1. Capacity constraints
fij ≤ Uij (i,j) S (3.87)
-fij ≥ -Lij (i,j) S (3.88)
2. Conservation of flow constraint to ensure that any flow entered a node must also leave. ∑ fij - ∑ fji = 0 all i N, i =/ j (3.89) j ∈N j ∈ N
Chapter Three 116 Methodology
3. Nonnegativity (See equation 3.85)
fij ≥ 0 for all arcs (i,j) (3.90) A LP problem result is that a for every primal problem there exist a corresponding dual problem that may be expressed as:
Minimize ∑ Uij αij - Lji δji (3.91) (i,j) ∈S Subject to
π i - π j + α ij - δ ij ≥ -cij (3.92)
π i unrestricted for all i ∈ N
αij 0 for all (i,j) ∈ S
δij 0 for all (i,j) ∈ S Where, variables are associated with conservation of flow constraints of the primal problem. The variables of the dual problem are associated with the upper bound constraints of the primal. The variables are associated with the lower bound constraints. For each primal variable fij there is a dual constraint.
3.10.2 Conditions for optimality
The primal and dual problems solutions in LP are optimal for their respective problems if and only if:
1. Both solutions are feasible.
2. For every positive (nonzero, nonnegative) dual variable, the corresponding primal constraint is tight.
3. For every dual constraint that is not tight, the corresponding primal variable is equal to zero. The conditions 2 and 3 are the complementary slackness conditions and when applied to the minimum-cost circulation problem along with the feasibility constraints, they yield the following necessary and sufficient conditions for optimality (Phillips and Garcia-Diaz 1981).
Primal feasibility as given by Phillips and Garcia-Diaz 1981 follows:
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P1: ∑ fij - ∑ fji = 0 for all i ∈ N (Conservation of flow) (3.93) j ∈N j ∈N
P2: Lij ≤ fij ≤ Uij for all i,j (capacity constraints) (3.94)
Dual feasibility
D1: π i - π j + α ij - δ ij ≥ -cij for all (i,j) ∈ S (3.95)
D2: α ij ≥ 0 for all (i,j) ∈ S
D3: δij ≥ 0 for all (i,j) ∈ S Complementary Slackness C1: if π i - πj + α ij - δ ij -cij then fij = 0 (3.96)
C2: if αij ≥ 0 then fij = Uij
C3: if δ ij ≥ 0 then fij = Lij An equivalent formulation of the conditions for optimality is given by the following relationships:
I. If πi - πj > cij then α ij > 0 and fij = Uij (3.97)
II. If πi - πj < cij then δ ij > 0 and fij = Lij (3.98)
III. If πi - πj = cij then Uij ≥ fij ≥ Lij (3.99) provided that we choose
IV. α ij = max [ 0; πj - π i - cij ] (3.100) V. δij = max [ 0; - πj + π i + cij ] (3.101) and VI. ∑ fij - ∑ fji = 0 for all i ∈ N (3.102) j ∈N j ∈N Assuming that conditions IV and V are satisfied and using the definition: _ cij = cij + πi - πj (3.103) conditions I, II, III and VI can be expressed in a more convenient way:
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_ k1: if cij < 0 then fij = Uij (3.104) _ k2: if cij > 0 then fij = Lij (3.105) _ k3: if cij = 0 then Lij ≤ fij ≤ Uij (3.106) k4: conservation of flow is satisfied If the optimality conditions k1, k2, k3 are satisfied by two nodes i and j and their connecting arc, that arc is said to be in kilter. Otherwise it is called out of kilter. An optimal results is arrived when all arcs are in kilter and the conservation flow (equation k4) is satisfied. If no such set of flows exists, the problem has no feasible solution.
3.10.3 Minimum cost circulation optimization using Out of Kilter algorithm
The procedyre computes the arc flows fij in each arc and values of i's that satisfy the optimality level given by relationships k1 through k4. To start the computations any initial values of fij and i are assigned. There are nine mutually exclusive states of an arc which might be possible as desccibed by Phillips and Garcia-Diaz, 1981. These states are not included here to conserve space.
3.10.4 Algorithm steps
The algorithm may be summarized in five steps as given by Phillips and Garcia-Diaz, 1981.
1. Find an out of kilter arc, (i,j). At this stage any arc may be chosen. If none, stop. 2. To bring the arc in kilter, find, whether the flow should be increased or decreased. If it should be increased, go to step 3. If it should be decreased, go to step 4. 3. Using the labeling procedure, find a path in the network from node j to node i along which the flow can be passed without causing any arcs on the path to become further out of kilter. If a path is found, adjust the flow in the path and increase the flow in (i,j). If (i,j) is now in kilter, go to step 1. If it is still out of kilter, repeat step 3. If no path can be found, go to step 5. 4. Determine a path from node i to j along which the flow can be passed without causing any arcs to become further out of kilter. If a path is found, adjust the flow in the path
Chapter Three 119 Methodology
and decrease the flow in (i,j). If (i,j) is now in kilter, go to step 1. If (i,j) is still out of kilter, repeat step 4. If no path is found, go to step 5. 5. Change the π values (using procedure define in section 3.7.3) and repeat step 2 for arc (i,j) keeping the same labels on all arcs already labeled. If the node numbers become, stop. There is no feasible solution.
For large networks, the manual computations can become quite tedious however the steps are well defined and have been computerized (Phillips and Garcia-Diaz, 1981). An extended computer program for the algorithm was prepared. The important quality of the procedure is that the algorithm remains the same for various varieties of problems; it is the network configuration which has to be changed for different problems. The OKA may be used to solve a wide range of network-flow problems. To begin the procedure, the algorithm does not require an initial feasible solution. Only the conservation of flow equations are to be satisfied. an additional benefit is that the problem can be easily visualized, a property not present in the linear programming formulations in more than two dimensions. Further, the computations are more efficient as compared to that of LP and DP.
3.11 Introducing Uncertainty Analysis in Network Flow Optimization
NFP is much faster technique if the system can be represented in networks. But the solution procedure is based only on integer values of the variables with deterministic inputs and linear objectives. But the river flows are random process and their occurrence depends on chance. An alternate way to handle such problem is to consider a stochastic problem where inputs are located in the system. A mix optimization procedure is proposed to introduce uncertainty in NFP problem. The use of SDP and NFP is demonstrated in chapter 6. The inputs in the system are stochastically derived using SDP and these are coupled with the NFP model. It is shown in Figure. 3.6.
3.12 Complete River Systems Operation Optimization Model
Based on the steps 1 to 18, described in section 3.2 a working 10 day mathematical model for the stochastic optimization of the multireservoir multipurpose water resource system is developed. The model is calibrated with different objectives and different model types. The best fit is then identified. The working model is verified through simulation. The
Chapter Three 120 Methodology
performance of the model with historic operation of the system is compared. It is checked whether the model is superior to others or not. For details please see chapter 6. The use of available computer packages, such as HEC-5 (US Army Corps of Engineers, 2000), HEC- ReSim (US Army Corps of Engineers, 2007), MODSIM (Labadie at al., 2007) and CSUDP (Labadie, 2004) was limited due to the system specific nature of Indus River and the types of techniques investigated in the present research.
3.13 Contribution to the Research
1. A new approach based on mixed optimization methods is introduced for large scale multireservoir, multiobjective system.
2. A 10 day time step is selected for all the proposed models. Previous studies in the literature were mostly on monthly time step.
3. Various types of SDP models are developed and their performance is evaluated.
4. Multiple objectives are formulated and procedure to find a trade off between the objectives is presented.
5. Simple approach to calculate tpm is defined for easy understanding.
6. Uncertainty analysis is indirectly introduced in NFP through SDP.
7. Models for 10 day inflow forecast were investigated.
8. Long term scheduling using stochastic procedure for a large water resource system is presented.
9. Short term operation of the system with the working mathematical model and continuously updating the inflow forecast 10 day before is demonstrated.
10. Applicability and limitations of the proposed procedure is determined.
Improvement in Practical Procedures 1. Optimal operating rules for the system are derived. 2. Optimum operation of reservoirs for irrigation, hydropower generation and flood protection is presented 3. Optimum allocation of water resource in the canal network is identified. 4. Identification of resource limitation.
Chapter Three 121 Methodology
Option 1 Select option Option 2 1. Reservoir operation Call model. Simulate 2. Network flow model (B) Read data for NFP Reservoir Read arcs, nodes, capacities Simulation and cost/benefits model
Read physical data Call OKA of the reservoir Out of from an external kilter
Read SDP Read memory for operating policy SDP optimal policy from a file made for each reservoir bthfit dl di t i t
t = 1 t = 1
t = t + 1 t = t + 1 Simulate the Compute for the reservoir with canal network historic and operation forecasted data optimization of the yes yes t < N t < N
No No Print / save results, plot Print / save Keep results, plot optimal policy in Stop
Figure 3.6 Basic structure of the NFP model
Chapter Four 122 Description of the Study Area
CHAPTER 4 DESCRIPTION OF THE STUDY AREA
The working mathematical model described in chapter 3 has been developed for the optimum operation of the Indus River multireservoir system of Pakistan. The Indus River System comprises of 3 storage reservoirs (Tarbela, Mangla and Chasma), 16 barrages, 12 inter-river link canals, 2 major syphons and 43 canals as shown in Figure 4.1. It is the largest integrated irrigation network in the world, serving about 34.5 million acres of contiguous cultivated land. The total length of main canal alone is about 58,500 km. The flows of the Indus River System constitute the dominant surface water resources of Pakistan having total area of 803,900 km2 (310,000 mi2). Indus River basin is comprised of about 94,600 km2 (364,700 mi2) of catchment area.
The system plays an important role in the agricultural sector of the country, which employs about 55% of the country's labour force, accounts for 26% of the gross domestic product (GDP) and contributes about 26% of the export earnings (WAPDA 2004). The population of Pakistan is growing rapidly. Therefore there is an urgent need to develop a viable and efficient mechanism for the optimal utilization of water from the Indus River System for a sustainable supply of water to irrigation. The methodology described in chapter 3 is applied to Indus River System and tested for its effectiveness for the optimal utilization of water from the system for a sustainable supply of water to irrigation.
4.1 Rivers in the System
Indus Basin drains Himalayan water into the Arabian sea. It consists of Indus River and its five left bank rivers including Jhelum, Chenab, Ravi, Sutlaj and Bias in addition to Kabul river lying at the right bank. Some pertinent details of the Indus Basin rivers are given in Table 4.1.
Chapter Four 123 Description of the Study Area
Figure 4.1 Indus River System and surface storage
Chapter Four 124 Description of the Study Area
Table 4.1 Details of the Indus Basin rivers (WRMD 1981, WAPDA 2007)
Rivers Catchment Mean annual Minor Rivers Area (sq mi) Runoff (MAF
Western Rivers
Indus at Kalabagh 110,500 91.82 Siran, Kunar,
Kabul at Nowshera 2600 21.26 Swat, Pangkora,
Jhelum at Mangla 12,900 22.80 Kohat, Kurram,
Chenab at Marala 11,400 25.85 Gomal, Zhob,
Panjnad, Nari,
Eastern Rivers
Ravi (in Pakistan) 3,100 5.00 Bolan, Streams
Sutlaj (in India) 18,550(47100) 3.97 of Kactchi Plains
Bias (in India) 6,500 -
The Indus Basin Irrigation system is schematically shown in Figure 4.2. A number of small tributaries also join these rivers. The catchment area of Indus River is unique and includes 7 worlds’ highest ranking peaks such as K-2 (28,253 feet), Nanga Parbat (26,600 feet) and Rakaposhi (25,552 feet) in addition to 7 glaciers including Siachin, Hispar, Biafo, Batura, Barpu and Hopper.
Chapter Four 125 Description of the Study Area
Figure 4.2 Schematic Diagram – Indus Basin Irrigation System (WAPDA, 2006)
Chapter Four 126 Description of the Study Area
A brief description of the major Indus Rivers is given below.
4.2.1 The Indus River
Indus River originates from the north side of the Himalayas at Kaillas Parbat in Tibet having altitude of 18000 feet. Traversing about 500 miles in NW direction, it is joined by Shyok river near Skardu (elevation 9000 feet). After about 100 miles in the same direction, it reaches Nanga Parbat and joined by the Gilgit river at an elevation of 5000 feet. Flowing about 200 miles further in SW direction, the river enters into the plains of the Punjab province at the Kalabagh (800 feet). The Kabul river, a major western flank tributary, joins with Indus near Attock. The Kunar which is also called Chitral river joins Indus below Warsak. About five miles below Attock, another stream Haro river drains into the Indus River. About seven miles upstream of Jinnah Barrage, another stream called Soan river joins with Indus. The tributatries of Indus rirves are detailed in Figure 4.3. Its hydraulic characteristics are presented in Table 4.2
Table 4.2 Hydraulic Characteristics of Indus River and its Tributaries (Ahmad 1993)
Observation Catchment Slope in Average Annual Length River Station Area Mountain Discharge Sediment (miles (Sq miles) (ft/mile) (MAF) (acre-ft) Indus Darband 800 103,800 35 59.5 85,441
Indus Kalabagh 925 151,200 - 89 143,744
Kabul Warsak 200 2,600 30 12.5 24,741
Kabul Munda(Swat) - 1,600 - 2.36 2,424
Siran Thapla 60 1,100 35 0.141 644
Haro HasanAbdal 30 2,400 66 0.82 1,044
Mukhad Soan Road 75 4,800 14 0.8 4,934
Kurram Kurram 117 2,663 25 0.47 12,879
Gomal Kot Murtaza - 13,900 - 0.435 550
Chapter Four 127 Description of the Study Area
Figure 4.3 The Indus and its tributaries (WAPDA 2007)
Chapter Four 128 Description of the Study Area
4.2.2 The Jhelum River One of the important Eastern river draining into the Indus River System is Jhelum river which originates from Pir Panjal and flows parallel to the Indus at an elevation of 5500 feet (See Figure 4.3). About 2300 sq mlies of the alluvial land of Kashmir Valley is draining into the Jhelum river. The river flows through Dal and Wular lakes. On emergence from Wular lake near Baramula, it runs through an eighty miles long gorge at an average slope of 33 ft per mile. Near Muzafarbad, at Domel, it joins with Nelum river which is comprised of about 2800 sq miles of hilly area lying on the eastern side of Nanga Parbat.
Another tributary called Kunhar river joins with Jhelum river about five miles below Domel. Two other small rivers (Kanshi and Punch) join with Jhelum between Domel and Mangla, and Punch enters into it about seven miles above Mangla at Tangrot. Below Mangla, several flood water streams join with the Jhelum river. Salient features of Jhelum river along with its tributaries are given in Table 4.3.
Table 4.3 Salient features of Jhelum river and its tributaries (Ahmad 1993) Observation Length Catchment Average Annual River Station Area Discharge Sediment (miles) (Sq miles) (MAF) (acre-ft) Jhelum Domel 180 5,250 11.4 10,172 Jhelum Mangla 350 13,180 23 44,071 Kishan Ganga Muzaffarbad 150 2,600 6.1 5,224 Kunhar GarhiHabibullah 100 1,080 2 2,861 Kanshi Gujar Khan 30 - 0.36 293 Punch Palak 80 1,520 2 5,678 Kahan Rohtas 40 470 0.037 425
Topography and Landform The catchment lies on the southern slopes of the Himalaya mountains. About 82 percent area is higher than 400 ft. above mean sea level (AMSL) whereas about 28 percent area is higher than 10,000 ft. AMSL. The catchment is bounded by Muree Hill range (8000 ft. AMSL) on the western side, whereas in the north it is bounded by the Great Himalaya mountains and contains the Vale of Kashmir. North of the Vale of Kashmir the mountains lead upward towards the snows and glaciers of Nanga Parbat (stands at 26,660 ft. AMSL). At the damsite the river passes through the foothills of Siwalik range and enters the Punjab plains.
Chapter Four 129 Description of the Study Area
Basin Characteristics The length of the main river from the most remote point to the outlet has been estimated to be about 260 miles. Basin shape is numerically calculated with the help of Horton's method and estimated as 0.190. This value indicates an irregular basin with comparatively moderate peaks. Using different methods commonly used in drainage basin studies, various dimensionless catchment parameters, useful in predicting inflow in a river have been estimated for the basin. As a result elongation ratio, compactness coefficient and circularity ratio is worked out to be 0.700, 1.413 and 0.501 respectively.
Climate The climate of the basin may be divided into four seasons, the winter monsoon (December - February), the hot weather period (March - May), the summer monsoon (June - September) and the transition period (October - November). In winter monsoon the precipitation over the major part of the basin occurs in the form of snow. It accumulates until temperatures rise in April, May and June. The snow melt contribution to the river flows at Mangla is normally maximum in June. The months of heaviest rainfall are August and September (Figure 4.4). Mean annual precipitation at Jhelum has been estimated to be 31.20 inches (1950-2008) (Data Source: Surface Water Hydrology Project; WAPDA). Heavy floods due to higher rainfall are witnessed and maximum was recorded 1,100,000 cusecs on August 1929. However, Mangla dam on Jhelum river was designed on a Probable Maximum Flood (PMF) of 2,600,000 cusecs with a return period 240-years.
Chapter Four 130 Description of the Study Area
Precipitation at Jhelum (1950-2008)
10 9 8
7 nch) I 6 on (
i 5 t a t 4 pi
eci 3
Pr 2 1 0 Jan Feb Mar Apr May June July Aug Sep Oct Nov Dec
Figure 4.4 Mean monthly recorded precipitation at Jhelum station of Pakistan Meteorological Department
4.2.3 The Chenab River Chenab River is one of the major rivers which contributes appreciable water into the Indus River System. It originates from Kulu and Kangra districts of Himachal Pardesh, a province in India. In the upper reaches, Chandra and Bhaga are the two main streams of Chenab, rise on opposite side of Baralcha pass at an elevation of about 16000 feet AMSL. These streams join at Tandi located in Jammu and Kashmir. Here the elevation is 9090 feet AMSL. The river after traversing about 400 miles of mountain regions opens out into the plains near Akhnur. The river enters in Pakistan near Diawara village located in Sialkot district. Chenab flows through alluvial plains of the Punjab province covering a distance of about 3398 miles. It joins with Jhelum river at Trimmu. And finally the Jhelum and Chenab, after meeting Ravi and Sutlej rivers, drain into Indus at Mithankot about 40 miles below Punjnad.
There are 12 major tributaries of the Chenab namely Chandra, Bhaga, Bhut, Maru and Jammu in India and Tawi, Manawar Tawi, Doara, Halse, Bhimber, Palkhu and Aik and Bhudi Nallah join in Pakistan. Length of Chenab is about 772 miles and its catchment is about 26,079 sq. miles. About 10,875 sq. miles lie in Jammu ans Kashmir state, 1735 sq.
Chapter Four 131 Description of the Study Area
miles in India and 13,469 sq. miles in Pakistan. Chenab river is life line of the Punjab province. Dependable supplies can be withdrawn while the river remains at a high stage from June to September. Chenab starts rising in the later part of May and the flow becomes over 50,000 cfs in June. The high flows continue till the middle of September, the peak discharge months being July and August.
On Chenab river, no dam is constructed by Pakistan due to topographic conditions. India has constructed a dam at Salal for hydro electric in Jammu territory about 40 miles upstream of Marala barrage. In Pakistan, following barrages are located on the Chenab river.
• Marala Barrage - Feeds upper Chenab canal and Marala Ravi link • Khanki Barrage - Feeds Lower Chenab canal. • Qadirabad Barrage - It is a level crossing for Rasul Qadirabad and Qadirabad Balloki link. • Trimmu Barrage - Feeds Haveli link, Rangpur canal and Trimmu Sidhnai link. • Punjnad Barrage - Feeds Panjnad and Abbasian canals.
4.2.4 The Ravi River Ravi is one of the 5 Eastern tributaries of the Indus River System. Its catchment is about 3100 sq.miles. According to Water Treaty 1960 between India and Pakistan, India has full right to divert all its flows for the development. Therefore arrangements have been made by India to utilise the water of Ravi. The river originates from the basin of Bangahal and drains the southern slopes of the Dhanladhar. Below Bangahal, the river flows through the valley of Chamba. The river leaves the Himalayas at Baseeli. In the mountains area of 130 miles long, the total drop is 15000 feet about which is 115 feet per mile. Its average slope is 45 feet per mile. The Ravi enters Pathankot at Chaundh and forms a boundary between India and the state of Jammu and Kashmir for 23 miles. The important tributaries of Ravi river are given in Table 4.4.
Chapter Four 132 Description of the Study Area
Table 4.4 Hydraulic Characteristics of Tributaries of Ravi joining within Pakistan (Ahmad, 1993)
Catchment Average Slope Maximum Length Tributary Area per 1000 ft Discharge
(miles) (sq miles) (miles) (cfs) Ujh 80 675 31.6 249,000 Bein 48 346 5.9 128,000 Basantar 45 224 6.4 83,000 Degh 160 456 7.4 100,000 Hudiara 62 53 0.25 10,000 4.2.5 The Beas River It is one of the Eastern tributary of the Indus. The length of the river is about 247 miles. It is the shortest river of the system and its flows are under the control of India as per Indus Water Treaty 1960. Pandoh and Pong dams have been built over it by India. The catchment area is about 6500 sq miles.
4.2.6 The Sutlej River It originates from Western Tibet in the Kailas mountain range in India and flows through the Panjal and Siwalik mountains ranges. Then it enters the plains of Indian Punjab. The length of the river is about 964 miles and its catchment is about 47100 sq miles. According to Indus Water Treaty 1960, India has full right to use the flows of Sutlej river. India has built dams and barrages after Independence. Barrages existing in Pakistan were built before Independence. Important tributaries, dams and barrages of Sutlej river are given in Table 4.5.
Table 4.5 Hydraulic Characteristics of Important Tributaries of Sutlej (Ahmad 1993)
Catchment Average Highest Length Tributary Area Slope Altitude (miles) (sq miles) (ft/mile) (feet) Spati 115 3915 89 20,000 Gambhar 40 342 114 6,000 Soan 50 495 46.8 3,340 Sirsa 32 280 83.5 3,660 White Bein 88 1485 11.5 10,700 Black Bein 90 945 13.5 1,900 Beas 290 6200 42.7 13,050 Rohi 24 715 - -
Chapter Four 133 Description of the Study Area
4.2 Reservoirs/Dams of the Indus River
The major reservoirs/dams of the Indus River System are Tarbela dam, Mangla dam and Chasma reservoir.
4.3.1 Tarbela Dam Tarbela dam is world's biggest earth and rock fill dam and was completed in 1974-75 and is located on the Indus River. The dam is 485 feet high and 9000 feet long. A 100 sq. mile lake is capable of conserving gross quantity of 11.7 MAF of water. Installed power generation capacity is 3500 MW. It has two spillways (see Figure 4.5), four tunnels for power generation on the right bank and one for irrigation on the left. The gross capacity has now reduced to 9.745 MAF from its original capacity 11.7 MAF (WAPDA and NEAC, 2004). Tarbela reservoir is shown in Figure 4.6. Elevation – capacity curves of the reservoir are one of the important input parameters in reservoir simulation and it is shown in Figure 4.7.
The main objectives of the Tarbela dam are i) To augment and regulate the supply of Indus River water to irrigate the land of Indus Basin System. ii) Hydropower generation. iii) Incidental Flood Regulation. Based on the irrigation demands, reservoir operation studies were conducted by WAPDA to develop operating rule curves for the dam operation (Tippetts-Abbett McCarthey-Stratton consulting engineers, 1984).
Chapter Four 134 Description of the Study Area
Figure 4.5 Tarbela dam auxiliary spillway (Photo taken on 09-May-2008)
Figure 4.6 Tarbela reservoir (Photo taken on 09-May-2008)
Chapter Four 135 Description of the Study Area
1550
2040
2030 1500 2020
2015 on (ft) ti a v
e 2012 l 2002 E
1450
1400 0123456789
Capacity (MAF)
Figure 4.7 Elevation-capacity curves for Tarbela (Wapda, 2004)
WAPDA Operation Rule • Tarbela reservoir should be lowered to reservoir elevation (El.) 1300 feet by 20 May of each year.
• The reservoir should be held at El. 1300 feet until 20 June unless inflows exceed low level outlet capacity and after that allowed to fill El.1505 feet.
• Above El.1505 feet, the reservoir should be filled at a rate of 1 foot per day in so far as permitted by inflows and irrigation demands. Minimum maximum rule curve is shown in Figure 4.8.
• Drawdown of the reservoir should be in accordance with the irrigation demands balanced against the amount of water available from inflows plus storage.
Chapter Four 136 Description of the Study Area
1600.0
1550.0
1500.0
1450.0 ION (Ft) T
1400.0 ELEVA
1350.0
1300.0
1250.0 Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug MONTHS
Figure 4.8 Minimum maximum rule curve at Tarbela (WAPDA, 2004)
Reservoir Levels • The minimum reservoir level is 1300 feet. This level will assure the required minimum net head of 179 feet on the turbines with a margin of safety open and the tailwater level is at El. 1115 feet or lower.
• The maximum operating reservoir level is El. 1550 feet (normal full pool level). The maximum water level for spillway design flood is El. 1552.2 feet which allow adequate free board.
• After satisfying irrigation requirement, the reservoir may be kept as high as possible to maximize power production.
• The rate of filling should not exceed 10 feet per day. The allowable rate of rise should be determined according to operation experience.
• The normal releases for irrigation should be made through the turbines whenever possible so that power can be generated. Each turbine can produce 175,000 KW (or 239,000 Horse-power) when the wicket gates are 95% open and the net head on the
Chapter Four 137 Description of the Study Area
turbine is 378 feet. Under these conditions discharge from each turbine is 6,450 cfs making a total of 25,000 cfs for the four.
• The irrigation tunnel will be used when the irrigation demand is higher than the turbine discharge.
• The irrigation tunnels should not be used with water level above El1505 when the spillway provides sufficient release. Minimum discharge is 50000 cfs for the service spillway and 70,000 cfs for the auxiliary spillway.
• The sill level of irrigation tunnel is El.1160, 65 lower than the power intakes. Thus until the delta encompasses the intakes, most of the heavier suspended sediment would go through this tunnel and not through the power waterways.
• Rapid variations in the downstream flow should be avoided.
• Every year the reservoir should be drawn down to El.1300, (minimum pool level) to effect sediment flushing. Previous Benefits from the Reservoir The project has been instrumental in achieving self-sufficiency in food through timely water releases for irrigation. Billions of units of electrical energy generated at Tarbela dam saved the country's foreign exchange required otherwise for thermal power generation. The total project cost was Rs.18.5 billion. During the past 18 years of its operation, the dam contributed over 68.332 billion in terms of direct benefits from water releases and power generation. The total cost has been repaid three times and over.
Benefits from Water From 1975 to 1993 about 154.65 MAF of water has been released from the dam for the development. The benefits obtained from these releases was Rs.31,561 million. About 6.31 MAF of water was released from the storage during 1992-93 which is worth Rs.1893 million calculated at a rate of Rs.300 per acre feet.
Benefits from Power From 1975 to 1993 about 122,570 MKWH of energy was produced from the dam and the benefits obtained from this Rs. 36,766 million. About 13,955 MKWH of energy was produced during 1992-93 which is amount to Rs. 4,186 million calculated at a rate of Rs.0.30
Chapter Four 138 Description of the Study Area per KWH. Water and power benefits from the 18 years of dam operation are listed in Table 4.6
Flood Mitigation Benefits Additional benefits were achieved from the project with incidental flood control. Most of the floods occur during the summer monsoon season. The flood discharge is composed of snowmelt flood (base flow) plus storm flood. It has been estimated 1,773,000 cfs (a constant snowmelt flood, 6000 cfs + PMF, 1,173,000 cfs). Assuming discharge through the turbines and one irrigation tunnel, the probable maximum flood, when routed through the reservoir, showed surcharge of 2 feet above full reservoir level of 1550 feet AMSL. The maximum discharge over the spillway is 1,495,000 cfs. The maximum and minimum design curves ensure to take care of incidental floods.
On the basis of flood predictions, the reservoir can be lowered to a pre-determined elevation considerably below the normal pool level. Drawdown to El.1505 for example, would provide storage of about 2.4 MAF of flood water, equivalent to a flow of 400,000 cfs for a period of 3 days. Since immediate refilling is assured, this lowering of reservoir water level would not result in loss of water to irrigation and power.
Chapter Four 139 Description of the Study Area
Table 4.6 Water and Power benefits from Tarbela dam (WAPDA, 1993, 2001, 2004) Water Power Year Total Storage Benefits Generation Benefits Benefits (July-June) Release (MAF) (Rs.million) (MKWH) (Rs.million) (Rs.million) 1975-76 3.33 666 -- -- 666 1976-77 9.07 1814 138 42 1856 1977-78 10 2000 3367 1010 3010 1978-79 8.71 1742 3726 1118 2860 1979-80 9.91 1982 4123 1237 3219 1980-81 10.63 2126 4129 1239 3365 1981-82 11.33 2266 4200 1260 3526 1982-83 9.12 1824 5228 1569 3393 1983-84 9.18 1836 7451 2235 4071 1984-85 9.24 1848 7254 2176 4024 1985-86 9.76 1952 7994 2398 4350 1986-87 9.98 1996 8121 2436 4432 1987-88 7.52 1504 9403 2821 4325 1988-89 11.12 2224 10378 3114 5338 1989-90 7.32 1464 9982 2995 4459 1990-91 6.19 1238 11356 3407 4645 1991-92 5.93 1186 11765 3530 4716 1992-93 6.31 1893 13955 4187 6080 1993-94 9.41 2823 12956 3887 6710 1994-95 5.39 1617 14765 4430 6047 1995-96 8.17 2451 14822 4447 6898 1996-97 9.15 8235 14230 4269 12504 1997-98 8.06 7254 15084 4525 11779 1998-99 9.04 8136 16377 4913 13049 1999-00 8.708 7837 14747 4424 12261 2000-01 8.689 7820 12811 3843 11663 2001-02 8.3 7470 14390 4317 11787 2002-03 9.1 8190 15110 4533 12723 2003-05 8.7 7830 13400 4020 11850 Total 247.4 101224 281261 84379 185603
Chapter Four 140 Description of the Study Area
Figure 4.9 Tarbela Dam from space (Wikipedia, The free encyclopedia 2009)
Chapter Four 141 Description of the Study Area
4.3.2 Mangla Dam Mangla dam on river Jhelum which is a 12th largest earthfill dam in the world has been completed in 1967. Jhelum river at Mangla has a catchment area of about 12,870 sq. miles. Dam height is 380 feet. The original gross storage capacity of the reservoir was 5.35 MAF in 1967. Live storage capacity was 4.81 MAF which was about 90 percent of gross capacity whereas dead storage capacity was 0.54 MAF. Capacity of main spillway is 1,100,000 cusecs while of emergency spillway is 2,300,000 cusecs. The lake area of reservoir at maximum pool level (1202 feet. above sea level) is estimated to be 100 sq. miles. Reservoir of Mangla dam is shown in Figure 4.10. The main objectives of the dam are (i) water storage for supplementing irrigation supplies (ii) hydropower Generation (WAPDA, 1989). Before 1991 hydropower capacity of Mangla dam was 800 MW with 8 units. In 1991, hydropower capacity of the dam was increased to 1000 MW with 10 units. Figure 4.11 shows power house and Bong canal at Mangla dam.
The primary objectives from the reservoir are assured water releases for agriculture and hydropower generation therefore, no space is particularly reserved for flood control. However storage between reservoir levels 1202 feet and 1228 feet (1.5 MAF) is reserved to achieve incidental flood benefits. Recreation and fish production are additional benefits from the reservoir.
Reservoir capacity is depleted due to sediment inflows which were averaged 73 MST (million short ton) per year from 1967 to 2002 (WAPDA and MJV, 2003). Elevation capacity curves showing depletion in storage due to sediments are shown in Figure 4.12.
Chapter Four 142 Description of the Study Area
Reservoir
Mangla Fort
Embankment
Figure 4.10 Mangla reservoir at 1040 ft AMSL (Photo taken on 22-Nov-2005)
Figure 4.11 Mangla dam power house and Bong canal (Photo taken on 08-May-2008)
Chapter Four 143 Description of the Study Area
Elevation Capacity Curves Mangla Dam Before Raising
1250
) 1200 L S M 1150 1967 A t e
e 1983
(f 1100 n 1988 o i t a 1993 v
e 1050 l
E 1997 r i
o 1000 2000 rv e s e
R 950
900 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Reservoir Gross Capacity (MAF)
Figure 4.12 Elevation-capacity curves for Mangla (WAPDA and MJV, 2003)
WAPDA Operation Rule • Mangla reservoir should be lowered to reserevoir elevation (El.) 1050 feet by 10 May of each year.
• The reservoir should be held at El.1050 to El 1040 feet until 31 March unless inflows exceed low level outlet capacity and after that allowed to fill El.1202 feet.
• Mangla reservoir should be filled upto its maximum conservation level 1202 feet before 1 September if permitted by inflows and irrigation demands. Minimum maximum rule curve is shown in Figure 4.13.
• Drawdown of the reservoir should be in accordance with the irrigation demands balanced against the amount of water available from inflows plus storage.
Chapter Four 144 Description of the Study Area
1230
1210
1190
Maximum rule curve 1170 L
S 1150 M
A Minmum rule curve
eet f 1130 ( n o i at ev
l 1110 E
1090
1070
1050
1030 Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug
Figure 4.13 Minimum maximum rule curves at Mangla (Wapda, 2004)
Previous Benefits from the Reservoir
Benefits from Water The Mangla reservoir has been impounded in 1967. According to an estimate on water releases for agriculture, industrial or domestic use from the reservoir, total benefits have been computed about Rs.24,179 million from 1967 to 1993 (WAPDA 1993). Therefore, average annual benefits from water, estimated from 26 year operation (1967-1993) comes to be about Rs.930 million whereas the benefits during (1991-92) and (1992-93) have been estimated to be Rs.936 million and Rs.969 million respectively. These estimates, as reported, were carried out on the basis of a unit return of Rs.200.00 per acre foot volume of water.
Benefits from Power WAPDA reported (1993) that total power generated from Mangla Dam between period 1967 and 1993 was 103,076 MKWH which gave a return of Rs.30,922.89 million to the country. An average annual power generation from the project is 3964 MKWH. During 1992-93 power generated was 5780.09 MKWH, while during 1991-92 it was 5944.04 MKWH. Mangla power station during 1988-89 has been able to touch maximum generation of 950 MW which is about 18.75 percent more than its installed capacity of 800 MW during that
Chapter Four 145 Description of the Study Area period. Average annual returns from Mangla dam has been estimated about Rs.1189 million. The returns/benefits during year (1992-93) due to power generation have been reported to be Rs.1734.03 million whereas during year 1991-92 the benefits were Rs.1783.2. The highest benefits during the entire operation period in the dam history was in 1987-88 when the power generation was reported to be 6039.65 and its benefit was Rs. 1811.9 Million. All these returns/benefits estimated here in monetary units were computed at the rate of Rs.0.30 per unit (WAPDA, 1993). Water and power benefits from the 26 years of dam operation are listed in Table 4.7.
Table 4.7 Water and Power benefits from Mangla dam (WAPDA, 1993, 2001, 2004) Water Power Year Total Storage Benefits Generation Benefits Benefits (July-June) Release (MAF) (Rs.million) (MKWH) (Rs.million) (Rs.million) 1967-80 58.32 17046.3 37150.5 13371.0 30417.3 1980-81 4.15 1458.8 3877.6 1535.0 2993.8 1981-82 5.30 1881.5 4090.3 1625.0 3506.5 1982-83 4.82 2210.8 4917.0 2523.7 4734.5 1983-84 5.35 2587.8 4162.5 2252.7 4840.5 1984-85 5.39 2961.6 3883.6 2388.3 5349.9 1985-86 4.56 2821.8 4637.6 3211.0 6032.8 1986-87 4.84 3083.4 5937.2 4232.0 7315.4 1987-88 4.88 3220.5 6039.7 4459.0 7679.5 1988-89 4.97 3821.8 5307.3 4556.0 8377.8 1989-90 5.03 3952.5 5621.3 4992.0 8944.5 1990-91 3.76 3343.0 5738.2 5708.0 9051.0 1991-92 4.68 4232.9 5944.0 6015.0 10247.9 1992-93 3.23 3490.7 5780.1 6989.0 10479.7 1993-94 5.37 5939.8 5022.5 6215.0 12154.8 1994-95 5.10 6282.9 6809.7 9386.0 15668.9 1995-96 3.94 5684.9 6977.3 11254.0 16938.9 1996-97 4.98 7888.7 5665.3 10041.0 17929.7 1997-98 4.36 7805.9 6103.7 12225.8 20031.7 1998-99 5.10 6462.7 4778.5 9920.0 16382.7 1999-00 4.21 8774.6 3184.7 7425.0 16199.6 2000-01 4.13 9523.3 2799.9 7223.0 16746.3 Total 156.47 24179.0 103076.3 30922.9 55101.9
Chapter Four 146 Description of the Study Area
Flood Mitigation Benefits Incidental flood control is an additional benefit which was achieved from the project. Most of the floods occur during the summer monsoon season. Their duration is short but their rate of rise and fall can be extremely rapid. The maximum and minimum design curves (Figure 4.13) ensures to take care of incidental floods. Available storage (1.5 MAF) between reservoir level 1202 feet and 1228 feet is reserved to achieve incidental flood benefits. The project was designed on a Probable Maximum Flood (PMF) of 2,600,000 cusecs. Total benefits from water and power activities from Mangla dam comes to Rs.55,101.89 million since 1967 whereas total benefits in financial years 1991-92 and 1992-93 from water and power has been estimated to be Rs.2719.21and Rs.2703.03 million respectively.
Although these benefits are quite high, recovering the total cost of the project several times over, but it is however, a limited source. The country has been facing the major problem of rapidly increasing population and food requirements. These problems seriously affected the existing policies and it is essentially needed to design a policy which may overcome these issues by expanding irrigated agriculture and increasing power generation.
Mangla Raising Project At the time of construction of Mangla Dam, Government of Pakistan, requested the World Bank that a provision should be made in the design and construction of the Mangla Dam to facilitate its raising at a later stage by another 30-40 ft. The Government of Pakistan agreed that the incremental cost of the provision for raising would not be charged to the Indus Basin Development Fund. The World Bank accepted this proposal and hence, all the impounding structures of the Mangla Dam Project were designed and constructed in 1967 for raising it by another 30 ft. In year 2003, work on Mangla raising was started. It was proposed to raise the Mangla dam by 30 feet. (WAPDA and Mangla Joint Venture, 2003). This will raise the present maximum reservoir conservation level of 1202 ft to 1242 ft. The work on Mangla raising is in progress as shown in Figure 4.14. About 70% construction work has been completed on Mangla raising till May 2008. The project is expected to be completed in year 2009. This would increase the average annual water availability by 2.9 MAF. Power generation from the existing power plant would also increase by about 11%. Elevation capacity curves after Mangla raising showing depletion in storage due to sediments for the period 2007 to 2082 are shown in Figure 4.15 (WAPDA and MJV 2003).
Chapter Four 147 Description of the Study Area
Figure 4.14 Work in progress on Mangla Raising (Photo taken on 22-Nov-2005)
Elevation Capacity Curves Mangla Dam After Raising
1300
1250 2007-08 L S 2011-12 M 1200 A
t 1016-17 e e f 1150 2021-22 n ( o
i 2026-27 t
a 1100 v
e 2031-32 l E 1050 r 2036-37 oi v
r 2041-42
e 1000 s
e 2046-47 R 950 2081-82 900 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Reservoir Gross Capacity (MAF)
Figure 4.15 Elevation-capacity curves for Mangla Raising (WAPDA and MJV, 2003)
Chapter Four 148 Description of the Study Area
4.3.3 Chasma Reservoir
Located on the Indus River downstream of Tarbela dam, this reservoir acts as a buffer reservoir to re-regulate the releases from Tarbela. It was constructed in 1971 as barrage cum reservoir providing diversion facilities for Chasma Jhelum link Canal on its left side and Chasma right bank canal on the right side. The reservoir acts as a re-regulatory storage for the releases from Tarbela which enable the reservoir to store 2.59 MAF of water and releases 2.52 MAF during 1992-93. According to 1986-87 hydrographic survey by WAPDA, the gross storage capacity of Chasma reservoir has been reduced from 0.87 MAF (originally in 1971) to 0.497 MAF.
Upto 1993, about 100.44 MAF of irrigation water was received in the reservoir. About 94.97 MAF was released downstream of Chasma barrage and 3.794 MAF in Chasma Jhelum Link Canal (CJ Link) and 1.607 MAF in Chasma right bank canal (CRBC) The benefits obtained from the reservoir are listed in Table 4.8
Table 4.8 Water benefits from Chasma Reservoir ______Year Storage (July-June) Release Benefits (MAF) (Rs. million) ______Upto 1980-81 8.41 1682 1981-82 0.74 148 1982-83 0.70 140 1983-84 0.49 98 1984-85 0.49 98 1985-86 0.49 98 1986-87 0.49 98 1987-88 0.49 98 1988-89 0.45 90 1989-90 0.28 56 1990-91 0.46 92 1991-92 2.70 540 1992-93 2.52 756 ______Total 18.71 3994 ______
Chapter Four 149 Description of the Study Area
4.3.4 Loss of Reservoir Capacities
One of the important factors for future water scarcity in Pakistan is due to loss of existing reservoir capacities by sediment inflows. It is a natural process and every reservoir has its useful life. The solution is to make new dams to overcome water crisis. Hydrographic surveys were carried out time to time by WAPDA to determine the loss of reservoir capacities. Following table shows the depletion in gross capacities in Tarbela, Mangla and Chasma.
Table 4.9 Loss of reservoir capacities in MAF (WAPDA, 2004)
Reservoir Gross Storage Capacity Gross Storage Loss Original Year 2004 Year 2004 2012 2025 Tarbela 11.62 8.36 3.26 4.17 5.51 (1974) 72% -28% -36% -47% Mangla 5.88 4.64 1.24 1.72 1.97 (1967) 78% -22% -29% -34% Chashma 0.87 0.48 0.39 0.48 0.5 (1971) 55% -45% -55% -57% Total 18.37 13.48 4.89 6.37 7.98 (73%) -27% -35% -43%
4.3 Hydrological and Other Data
Data used for this study was collected from Water and Power Development Authority (WAPDA), Indus River System Authority (IRSA), Irrigation Department, Punjab and field visits at damsites. The data collected from WAPDA includes inflows of Indus and its tributaries, outflows from the reservoirs, rainfall data and pan evaporation data from the climatological stations. The data also includes the basic information about the physical, legal, social and economical features of the reservoirs and the hydropower generation from the dams. The data collected from Irrigation Department, Punjab are the downstream water requirements from the reservoirs and the historic canal withdrawals.
Chapter Four 150 Description of the Study Area
4.4 Barrages in the System There are 16 barrages located in the water resource system. These barrages receive water from the upstream reservoirs or from the run of the river and diverts into the canals as per requirement.
4.5.1 Chashma Barrage Location: Near village Chashma about 35 miles downstream of Jinnah Barrage on the Indus. Purpose: 1. To divert water released from Tarbela dam into Jhelum river through Chashma Jhelum link canal (CRBC, maximum capacity = 21700 cfs). 2. To feed Paharpur canal (maximum capacity = 500 cfs) taking of from the right side. Salient Feature: River valley is 6.5 feet wide, Barrage is 3536 feet long with 3120 feet of clear water way to pass a maximum discharge of 950,000 cfs. The pond with water level of R.L. 649 extends 14 miles upstream. This is for storage of water in Chashma reservoir. The normal pond level is R.L. 642.
4.5.2 Rasul Barrage Location: In left side of Khadir, about 45 miles downstream of Mangla dam on the Jhelum river. Purpose: 1. To divert water released from Mangla dam into Chenab river through RQ link canal (RQ Link, maximum capacity = 19000 cfs). 2. To feed Lower Jhelum canal feeder (maximum capacity = 5300 cfs) taking of from the left side. 3. This supplements the discharge of water coming from upper Jhelum Canal through the power canal of Rasul Hydro-electric power station. 4. Provision has been made in the right abutment for a proposed right bank Jalalpur Canal. Salient Feature: Barrage is 3209 feet long with 2800 feet of clear water way to pass a maximum discharge of 850,000 cfs.
4.5.3 Qadirabad Barrage Location: About 20 miles below Khanki Head Works on Chenab river.
Chapter Four 151 Description of the Study Area
Purpose: 1. To receive water from Rasul Qadirabad link canal (RQ Link, maximum capacity = 19000 cfs) and to divert Qadirabad Balloki canal (capacity 18,600 cfs). Salient Feature: Barrage is 3373 feet long with 3000 feet of clear water way to pass a maximum discharge of 900,000 cfs 45 bays with 5 bays as undersluices.
4.5.4 Marala Barrage Location: Confluence of Chenab and Tawi on the Chenab river constructed in 1910-12. Purpose: 1. To supply water to Upper Chenab Canal (UCC, maximum and to divert Qadirabad Balloki canal (capacity 18,600 cfs). 2. A feeder canal supplying water to Balloki headworks. 3. To feed Marala Ravir Link Canal (MR Link, capacity 22,000 cfs). Salient Feature: Barrage is 4472 feet long to pass a maximum discharge of 1,100,000 cfs 66 bays with 10 bays as undersluices.
4.5.5 Sidhnai Barrage Location : Located on Ravi river, it was constructed in 1886. Purpose : 1. To supply water Sidhnai feeder canal (4,005 cfs). 2. To feed Sidhnai Mailsi link canal (10,100 cfs). Salient Feature: Barrage is 712 feet long with 600 feet of clear water way to pass a maximum discharge of 150,000 cfs 14 bays with 5 bays as undersluices.
4.5.6 Mailsi Syphon Sidhnai Mailsi link was to supply Mailsi-Bahawal Link Canal. It was decided to transfer the waters of Sidhnai Mailsi Link through a syphon built under River Sutlej about 25 miles downstream of Islam barrage. The syphon was constructed in 1964. The length of the syphon is 2231 feet.
4.5.7 Trimmu Barrage Location: Confluence of Jhelum and Chenab constructed in 1939. Purpose: 1. To supply water to Trimmu Sidhnai Link (TS Link, maximum capacity 11,000 cfs) and to feed Haveli Canal in the left bank. 2. To feed Rangpur Canal in the right bank (maximum capacity 2700 cfs). Salient Feature: Barrage is 3026 feet long to pass a maximum discharge of 645,000 cfs.
Chapter Four 152 Description of the Study Area
4.5.8 Balloki Headworks Location: Located on Ravi river. It was constructed in 1965 Purpose: 1. To supply water from QB link to BS Link. 2. To feed Lower Bari Doab canal (maximum capacity 7000 cfs). Salient Feature: Barrage is 1647 feet long to pass a maximum discharge of 225,000 cfs.
4.6 Canals in the System There are 43 canals which supply water for agriculture in different command areas in the water resource system. These canals receive water from the reservoirs / barrages and divert either into the small distributaries as per requirement. There are two zones of water source for these canals.
1. Indus Zone 2. Jhelum Chenab Zone
Indus zone consists of Tarbela command canals. These are 28 canals in this zone. Jhelum Chenab zone comprised Mangla command canals. There are 16 canals in this zone. The capacity of these canals and their names are presented in Table 4.11. Location of major canals is shown in Figure 4.16.
4.7 River Gains and Losses There is always some gains or losses between the head and tail of river and canal reaches in the Indus Basin. The losses usually occur during the rising stage in period April to June and flood months of July and August. The gains usually occur from September to March. Both the gains and losses in Indus Basin is a complex phenomena. Various studies are carried out and seasonal historic gains are losses are computed by WAPDA for the period 1940-86. The average of gains and losses for the period are given in Table 4.12. Year wise estimated gains and losses are shown in Figure 4.17. (Wapda Loose Files)
Chapter Four 153 Description of the Study Area
Table 4.10 Summary of the basic Information of the Barrages located in the Indus Basin
Sr. River Name Year Width Designed Crest Offtaking No. of of Between Maximum Level Canals Barrage Completion Abutments Discharge (feet) (cusecs) (feet) Indus 1 Kalabagh 1946 3797 950,000 678 Thal 2 Chashma 1971 3556 1,100,000 622 CJ, CRBC, DG Khan 3 Taunsa 1959 4346 750,000 428 Muzafarghar TP Link 4 Guddu 1962 4445 100,000 236 Pat Feeder, Desert Beghari, Ghotki 5 Sukkur 1932 4725 1,500,000 177 NW, Rice, Dadu Rohri, Nara, Khanpur East, Khanpur West 5 Kotri 1954 3034 875,000 48 Kalri, Pinyari Fuleli, Lined Jhelum 5 Rasul 1901 4400 875,000 708 Lower Jhelum 1967 3209 850,000 703 RQ Link 7 Trimmu 1939 3026 645,000 477 TS Link, Rangpur 8 Punjnad 1932 3400 700,000 325 Panjnad, Abbasia Chenab 9 Marala 1912 4475 718,000 800 MR Link 1968 4475 1,100,000 800 Upper Chenab 10 Khanki 1891 4414 750,000 721 Lower Chenab 11 Qadirabad 1967 3373 900,000 684.5 QB Link LCC Feeder Ravi Ravi Syphon Central Bari Boab Upper Depalpur 12 Balloki 1913 1647 139,500 622.4 BS-I, BS-II 1965 1647 225,000 624.5 Lower Bari Doab Lower Depalpur 13 Sidhnai 1965 712 167,000 454 Sidhnai, Haveli Sutlej 14 Sulemanki 1926 2223 325,000 560 Upper Pakpattan Fordwah, Sadiqia 15 Islam 1927 1621 275,000 441 U.Mailsi, Qaimpur U.Bahawal 16 Mailsi 1965 1601 429,000 415.5 L.Pakpattan, L Mailsi Syphon L.Bahawal
Chapter Four 154 Description of the Study Area
Table 4.11 Indus zone and Jhelum Chenab Zone
Chapter Four 155 Description of the Study Area
Figure 4.16 Location of major canals in Indus Basin Irrigation System (WAPDA, 1988)
Chapter Four 156 Description of the Study Area
Table 4.12 Average Gains and Losses (MAF) of 46 Years of data ______
Reach Kharif Rabi Total ______1. JC Zone Mangla-Rasul 1.578 0.413 1.991 Rasul-Trimmu -0.033 0.500 0.467 Trimmu-Panjnad -1.510 0.332 -1.178 Marala-Khanki 0.398 0.020 0.418 Khanki-Trimmu 0.228 0.775 1.003 Balloki-Sidhnai -0.071 0.189 0.118 Total 0.590 2.229 2.819 2. Indus Zone Attock-Kalabagh -2.401 -0.405 -2.806 Kalabagh-Taunsa -0.183 0.840 0.657 Taunsa-Guddu 2.178 0.186 2.364 Guddu-Sukkur -0.379 0.186 -0.193 Sukkur-Kotri -7.417 0.687 -6.730 Total -8.202 1.494 -6.730 ______
Historic Gains and Losses In Indus Irrigation System 1941-2003
15.0 10.0
) . 5.0 F
A 0.0
-5.0 41 44 47 50 53 56 59 62 65 68 71 74 77 80 83 86 89 92 95 98 01 (-) (M s -10.0 sse 1940- 1943- 1946- 1949- 1952- 1955- 1958- 1961- 1964- 1967- 1970- 1973- 1976- 1979- 1982- 1985- 1988- 1991- 1994- 1997- 2000- o
L -15.0 d -20.0 ) an
+ Post
s ( -25.0 Mangla Post Tarbela in a -30.0 Period Period G -35.0 -40.0
Figure 4.17 Year wise historic gains and losses from the Indus Irrigation System
4.8 Complete river basin multi reservoir system The Indus River System comprises of 3 storage reservoirs (Tarbela, Mangla and Chasma), 16 barrages, 12 inter-river link canals, 2 major syphons and 43 canals. It is the largest integrated
Chapter Four 157 Description of the Study Area
irrigation network in the world. The complete system is represented by nodes and arcs. In each time step there are 67 nodes and 119 arcs. The complete river basin multi-reservoir system of Indus Basin is taken in this study and it is shown schematically in Figure 4.16. The unregulated flows in the selected system are from the seven rivers namely Indus, Jhelum, Chenab, Kabul, Gomal, Ravi, Sutlaj. Three existing reservoirs are located in the system namely
Mangla on Jhelum [Irrigation flows + power generation] Tarbela on Indus [Irrigation flows + power generation] Chashma on Indus [Irrigation flows only] The flows are regulated from these reservoirs and diverted to the canals. In addition there are 16 barrages, located on different locations of the river reaches to divert water to different irrigation canals. These barrages receive water from two zones. One is called Indus zone and other is called Jhelum Chenab zone. In the Indus zone, there are 7 barrages with 23 major irrigation canals of different capacities. In Jhelum Chenab zone there are 7 barrages and 2 syphons diverting water to 23 number of major irrigation canals.
Indus Zone Jinnah Thal Taunsa Dera Ghazi Khan, Muzaffar Ghar, TP Link Guddu Pat Feeder, Desert, Beghari, Ghotki sukkur Nara, Khanpur E & W, Rohri, Dadu, Rice, NW Kotri Lined, Fuleli, Pinyari, Kalri (Ghulam Muhammad) Trimmu Rangpur, TS Link Punjnad Panjnad, Abbasia Jhelum Chenab Zone Rasul Lower Jhelum, RQ Link Marala MR Link, BRBD, UC Link, Khanki LC Qadirabad QB Link, LCC Feeder Ravi Syphon U.Depalpur, Central Bari Doab Balloki Lower Bari Boad, BS I & II, L Dipalpur Sulemanki U.Pakpattan, Fordwah, East Sadiqia Islam U.Mailsi, Qaimpur, U.Bahwal Mailsi Syphon L.Pakpattan, L Mailsi, L.Bahawal
The remaining water passing through Kotri(Ghulam Muhammad) Barrage is drained into the Arabian sea and the Indus River System is completed.
Chapter Four 158 Description of the Study Area
This Page is Kept Blank for
Figure 4.18 Node Arc Representation and Schematic Diagram
See File name NFINDUS-v3
Chapter Five 159 Stochastic Analysis of Uncertain Hydrologic Process
CHAPTER 5
STOCHASTIC ANALYSIS OF UNCERTAIN HYDROLOGIC PROCESSES
5.1 General
Uncertainty is always an element in hydrologic processes. Values of many components that affect the performance of water resource system cannot be known with certainty. Hydrological process such as evaporation, rainfall and streamflow behave uncertain due to stochastic nature. Stochastic nature of streamflows is one of the most important factors affecting the design and operation of water resource system and its failure. To deal with uncertainty, one of the approaches is to replace the uncertain quantities either by their expected value or some critical value. An assumption sometimes included to many water resources models is that the stochastic process is a Markov process. This chapter deals with the analysis and evaluation of stochastic nature of the uncertain events. The following steps are proposed and adopted to identify the stochastic process under uncertainty of all the time series in water resource system
1. Collect the long term hydrologic data of uncertain events for the test case. 2. Choose one river of the system and its inflow time series. 2. Perform outliers tests on the time series and detect outliers if any. 3. Perform consistency test to evaluate the consistency/homogeneity of the data. 4. Determine the unconditional probabilities to be used in chapter 6. 5. Determine serial correlation coefficients. 6. Determine transition probability matrix of the events to be used in SDP models in chapter.6 7. Determine Hurst Phenomenon of the events, identify the stationarity of the process. 8. Perform Gould Probability Matrix method for the probabilistic storage reservoir analysis. 9. Verify step 7 with Rippl Mass Curve analysis. 10. Verify step 7 and 8 with sequent peak analysis. 11 Evaluate different forecasting procedures suitable for the forecasting of uncertain events of the test case.
Chapter Five 160 Stochastic Analysis of Uncertain Hydrologic Processes
12. Select next river in the system and repeat step 2 to 11 to evaluate the process. Stop when all the main rivers are stochastically evaluated.
5.2 Hydrological Data
Indus Basin consists of Indus River and its five left bank rivers of Jhelum, Chenab, Ravi, Sutlej and Bias. It consists of one major river in right bank called Kabul river. Necessary details of Indus Basin rivers are given in chapter 4. Data used for this study was collected from Water and Power Development Authority (WAPDA), Irrigation Department, Punjab and Indus River System Authority (IRSA). The data collected from WAPDA includes inflows of Indus and its tributaries, outflows from the reservoirs, rainfall and pan evaporation data from the climatological stations. The data also includes the basic information about the physical, legal, social and economical features of the reservoirs and the hydropower generation from the project. The data collected from IRSA are the downstream water requirements from the reservoirs. The data set is a long term time series for at least 43 years (1961-2004).
Before to derive some reservoir operating rules in a multireservoir system, it is necessary to review the hydrology of the system which may be the back bone of the whole system. With the help of the hydrologic analysis, full control, as possible, be made on the water resources of the system for the best operation. Stochastic analysis leads to control the uncertain events.
5.3 Statistical Analysis of Annual Flows
In many studies of reservoir management, synthetic records are used because the historic natural record is often too short (10-50 years) to provide the analyst with sufficient data necessary for study. However, in the case of Indus Basin Rivers 40-80 years historical sequences of inflows from 1922 to 2004 are available. For example, the inflow data of the Jhelum River at Mangla is for 82 years (1922-23 to 2003-04) and that of the Indus at Tarbela is for 43 years (1961-62 to 2002-04).
For statistical analysis of the data a computer program (as a part of the main program) has been developed. The program calculates mean, standard deviation, variance, coefficient of variation, coefficient of skewness and serial correlation coefficients. The program also carries out Rippl mass curve and Hurst phenomenon analyses. SPSS (Statistical Package for
Chapter Five 161 Stochastic Analysis of Uncertain Hydrologic Processes
Social Sciences) widely acceptable computer program is used to verify the results. All these estimates, to compute, are a pre-requisite to design an optimal operating policy and for the derivation of operating rules. Summary results are placed in Table 5.1 Table 5.1 Statistics of Annual Flows (Time series Oct-Sep)
Mean annual flow (1922-23 to 2003-04) in Jhelum river above Mangla has been estimated 22.90 MAF with a standard deviation of 4.85 MAF. Whereas the coefficient of variation and coefficient of skew worked out to be 0.212 and 0.022 respectively. In Indus at Tarbela, the mean flow is 63.88 MAF, standard deviation is 7.71 MAF and coefficient of variance is 0.121. The coefficient of skew (0.572) indicates that the annual time series may tend to follow nearly normal distribution. Mean annual flow between (1975-76 and 1979-80) in Jhelum at Mangla has been estimated to 22.97 MAF (O'Mara and Duloy, 1984). On the other hand the mean annual flow of the first 50-year data (1922-23 to 1971-72) at the same location comes to be 22.625 MAF with standard deviation of 4.181 MAF. These estimated test statistics for various sets of data indicate that the time series may be at least first order stationary. Long term annual recorded inflows in Indus River System (1961-2004) are compared in Figure 5.1. Flows in Indus River at Basha, Tarbela and Kalabagh are reaches in series. These are drawn on a bar chart in Figure 5.2 to compare compared availability of water at three points along Indus River. Figure 5.3 shows the comparison of water availability at Jhelum, Chenab, Ravi and Sutlej. Outlier testing detected outlier in Jhelum at Mangla and it is shown in Figure 5.4.
Chapter Five 162 Stochastic Analysis of Uncertain Hydrologic Processes
Long Term Annual Recorded Flows in Indus River System 1961-2004 140
120 Indus at Basha
100 Indsu at Tarbela )
AF Indus at Kalabagh M 80 ( Jhelum at Mangla e m
u Chenab at Marala l 60
Vo Kabul at Nowshera w o
l Ravi at Balloki F 40 Sutlej at sulaimanki Indus at Kotri Below 20
0 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 01 03 2 - 4 - 6 - 8 - 0 - 2 - 4 - 6 - 8 - 0 - 2 - 4 - 6 - 8 - 0 - 2 - 4 - 6 - 8 - 0 - 2 - 196 196 196 196 197 197 197 197 197 198 198 198 198 198 199 199 199 199 199 200 200
Figure 5.1 Annual Recorded Flows in Indus River System
Bar Chart showing Annual Recorded Flows in Indus at Basha, Tarbela and Kalabagh 1961-2004 140
120
100 ) AF M
( 80 Indus at Basha e m
u Indus at Tarbela l 60
Vo Indus at Kalabagh
w o l
F 40
20
0 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 01 03
1962 - 1964 - 1966 - 1968 - 1970 - 1972 - 1974 - 1976 - 1978 - 1980 - 1982 - 1984 - 1986 - 1988 - 1990 - 1992 - 1994 - 1996 - 1998 - 2000 - 2002 -
Figure 5.2 Annual Recorded Flows in Indus River System at Basha Tarbela and Kalabagh
Chapter Five 163 Stochastic Analysis of Uncertain Hydrologic Processes
Bar Chart showing Annual Recorded Flows in Jhelum, Chenab, Ravi and Sutlej 1961-2004 35
30
25 ) AF
M Jhelum at Mangla
( 20 e Chenab at Marala m u l 15 Ravi at Balloki Vo Sutlej at sulaimanki w o l
F 10
5
0 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 01 03
1962 - 1964 - 1966 - 1968 - 1970 - 1972 - 1974 - 1976 - 1978 - 1980 - 1982 - 1984 - 1986 - 1988 - 1990 - 1992 - 1994 - 1996 - 1998 - 2000 - 2002 -
Figure 5.3 Annual Recorded Flows in Indus River System at Jhelum, Chenab, Ravi and Sutlej
Outlier in Annual Recorded Flows, Jhelum at Mangla 1922 to 2004 40 Jhelum at Mangla 35 Outlier 30
) F
A 25 (M e
m 20 u
l Outlie o
V 15 w o l F 10
5
0
3 7 1 5 7 1 5 9 3 7 1 5 9 3 7 1 5 9 3 2 2 3 3 4 5 5 5 6 6 7 7 7 8 8 9 9 9 0 39 43 - - 2 - 6 - 0 - 4 - 6 - 0 - 4 - 8 - 2 - 6 - 0 - 4 - 8 - 2 - 6 - 0 - 4 - 8 - 2 - 38 42 192 192 193 193 194 195 195 195 196 196 197 197 197 198 198 199 199 199 200 19 19
Figure 5.4 Results of Outliers Testing, Jhelum at Mangla
Chapter Five 164 Stochastic Analysis of Uncertain Hydrologic Processes
5.4 Water Scarcity and Identification of Drought Periods
Identification of water scarcity is one of the important elements to determine the causes of water, food and electricity shortages in a country. Analysis was carried out to identify drought at various Inflow reaches in Indus River System. A 10 day time step with long term recorded flows 1922-2004 (for Mangla) and 1961-2004 (Tarbela and other rivers) was chosen for identification of natural drought in Indus water resources system. Results of present analysis are presented in Figure 5.5 through 5.10. During the period 1961-2004 at least three drought periods were identified. The most severe drought in the system is the recent one which prolonged for at least 4 years from year 2000 to 2004. This condition causes water scarcity and food shortages in the country. To over come natural droughts, optimal operation of the water resource system was carried out under water scarcity.
Water Scarcity of Inflows, Indus at Tarbela
25.0 ) F 20.0 (MA
s 15.0 Recent
w Drought Drought
o Drought
fl 10.0 In
n 5.0 a 0.0 Me m -5.0 fro n
o -10.0 i t a i
v -15.0 e D -20.0 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 00 02 04
1 - 3 - 5 - 7 - 9 - 1 - 3 - 5 - 7 - 9 - 1 - 3 - 5 - 7 - 9 - 1 - 3 - 5 - 7 - 9 - 1 - 3 - 196 196 196 196 196 197 197 197 197 197 198 198 198 198 198 199 199 199 199 199 200 200 Water Year (Oct-Sep)
Figure 5.5 Estimated water scarcity in Indus at Tarbela
Chapter Five 165 Stochastic Analysis of Uncertain Hydrologic Processes
Water Scarcity of Inflows, Indus at Kalabagh
30.0
) F A 20.0
(M Recent s Drought Drought Drought
w Drought o 10.0 fl In n a
e 0.0 M
m -10.0 fro n o i t a
i -20.0 v e D -30.0 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 2 4 6 7 7 7 7 7 8 8 8 8 8 9 9 9 9 9 0 0 ------2000 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99- 2001 2003 Water Year (Oct-Sep)
Figure 5.6 Estimated water scarcity in Indus at Kalabagh
Water Scarcity of Inflows, Kabul at Nowshera
15.0
) F
A 10.0 M ( Drought Drought Recent
ow Drought l
f 5.0 n I n a
e 0.0 M om r -5.0 on f i t a -10.0 vi e D -15.0 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 02 2000 67- 69- 71- 73- 75- 77- 79- 81- 83- 85- 87- 89- 91- 93- 95- 97- 99- 2001- Water Year (Oct-Sep)
Figure 5.7 Estimated water scarcity in Kabul at Nowshera
Chapter Five 166 Stochastic Analysis of Uncertain Hydrologic Processes
Water Scarcity of Inflows Jhelum at Mangla
15.0 ) F A
M 10.0 Recent ( s Drought Drought Drought ow l
f 5.0 n an I
e 0.0 M
m o
r -5.0 on f i t
a -10.0 vi e D -15.0 23 26 29 32 35 38 41 44 47 50 53 56 59 62 65 68 71 74 77 80 83 86 89 92 95 98 01 04 22 - 25 - 28 - 31 - 34 - 37 - 40 - 43 - 46 - 49 - 52 - 55 - 58 - 61 - 64 - 67 - 70 - 73 - 76 - 79 - 82 - 85 - 88 - 91 - 94 - 97 - 00 - 03 - 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 20 20 Water Year (Oct-Sep)
Figure 5.8 Estimated water scarcity in Jhelum at Mangla
Water Scarcity of Inflows, Chenab at Marala
8.0
6.0 Recent ) Drought F Drought Drought A 4.0 (M 2.0 n a 0.0 Me
m -2.0 fro
n -4.0 o i t a i
v -6.0 e
D -8.0 -10.0 0 2 4 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 0 0 6 7 7 7 7 7 8 8 8 8 8 9 9 9 9 9 ------0 2 - 01 03 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 20 20 Water Year (Oct-Sep)
Figure 5.9 Estimated water scarcity in Chenab at Marala
Chapter Five 167 Stochastic Analysis of Uncertain Hydrologic Processes
Water Scarcity of Inflows, Western Rivers
40.0 ) F 30.0 A Drought Recent
(M Drought 20.0 Drought s w
o 10.0 fl In
n 0.0 a e
M -10.0 m -20.0 fro n o
i -30.0 t a i v
e -40.0 D -50.0 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 6 7 7 7 7 7 8 8 8 8 8 9 9 9 9 9 0 0 ------0 2 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 - 01 99 20 Water Year (Oct-Sep)
Figure 5.10 Estimated water scarcity in Eastern Rivers (Ravi+Sutlej)
5.5 Statistical Analysis of 10 day Flows
A 10 day time step is chosen for the optimization of the water resources system operation. This time step is the actual time step adopted for the operation of Indus River System by WAPDA. A detailed statistical analysis is carried out to evaluate the statistical characteristics of 10 day flows of Indus Rivers. The analysis includes the determination of mean, standard deviation, coefficient of skew, kurtosis, range, variance and standard errors of these estimates of each 10 day period in a year. Therefore there are 36 values of each variable of each river. A program has been developed for the analysis. To test the accuracy of the computer model, another package called SPSS has been used to reconfirm the results. Outliers and consistency tests were performed on each 10 day period to check the quality of the data. Summary results are placed in Table 5.2 and 5.3. Mean 10-daily recorded flows of rivers in Indus River System were graphically plotted in Figure 5.11 for their comparison of flows with each other. In 36 periods, the mean values vary between 0.319 and 5.745 MAF in Indus at Tarbela and 0.159 to 1.352 MAF in Jhelum at Mangla. The variation of mean is due to seasonality in the years. The same variation is observed in standard deviations. However the coefficient of skew in the two data sets is variable phenomena. In Indus the skew values ranges between -
Chapter Five 168 Stochastic Analysis of Uncertain Hydrologic Processes
0.242 and 2.657. Out of 36 values about 12 skew values are near or less than 0.3. This shows the possibility that the uncertain phenomena may flow normal distribution in these periods. In Jhelum the skew values ranges between 0.013 and 5.297. All the data is positively skewed. However the variation is comparatively high. About six values of skew coefficients are less than 0.3. The applicability of normal distribution seems limited when selecting 10 day time step in case of Jhelum river. A lot of other inferences can be drawn from these results.
The Outlier testing program scanned the data for outliers. The test was carried out on each 10 day period of the data of Indus and Jhelum rivers. Results are placed in Table 5.2 and 5.3. In case of Jhelum, 12 outliers have been detected in each 10 day period of November and December. While 1 in July and August and 2 in September are found. Rest of the data is found to be free of outliers.
Consistency of the data of each 10 day period is evaluated using T-test and results are placed in Table 5.2 and 5.3. In Jhelum river, the data is consistent except some period in March and June.
Mean 10-Daily Recorded Discharges of Indus River System 1961-2004 400.000 Indus at Basha 350.000 Indus at Tarbela Indus at Kalabagh 300.000
)
s Jhelum at Mangla c e s
u 250.000 Chenab at Marala c Kabul at Nowshera 200.000 1000
( Ravi at Balloki
e
g Sutlej at Sulaimanki 150.000 har c s i
D 100.000
50.000
0.000 1 1 1
1 1 3 2 3 2 3 2 3 2 3 2 3 2 C B T R N
P E E
C U
F J A AUG 1 D O
Figure 5.11 Mean 10-daily recorded flows in Indus River System
Chapter Five 169 Stochastic Analysis of Uncertain Hydrologic Processes
Table 5.2 Data Statistics, Consistency and Outliers in 10 Daily Inflows 1922-2004 Oct- Sep, Jhelum at Mangla
Chapter Five 170 Stochastic Analysis of Uncertain Hydrologic Processes
Table 5.3 Data Statistics, Consistency and Outliers in 10 Daily Inflows 1961-2004 Oct- Sep, Indus at Tarbela
Chapter Five 171 Stochastic Analysis of Uncertain Hydrologic Processes
5.6 Unconditional Probabilities
A random variable whose value changes through time according to probabilistic laws is called a stochastic process. The inflow process is a random process. There are many ways to deal with uncertainty depending on its severity and how the uncertainty will effect the operation of the system. One method is to replace the uncertain quantities either by their expected value or some critical (e.g. worst-case) value and then proceed with a deterministic way. Use of the expected value or median value of an uncertain quantity can be acceptable if the uncertainty or variation in a quantity is reasonably small and does not critically effect the performance of the system (Loucks et al. 1981).
Assuming that the inflow process is independent first order stationary process and its statistics remains constant with respect to time, the uncertain inflow process may be represented by the unconditional probability. If the data is limited and extreme values are to be determined, usually a probability distribution is fitted and the extreme values are determined. In case of long time historic data (say 70 years), relative frequency estimated from the actual data has of great value. The frequency of an event may be estimated as:
ni fi = ------(5.1) N Where ni is the number of events in class i. N Total number of events fi Frequency of events in class i. i. Class limits (i =1,2, . . .)
Cumulative frequency has been determined as m F = ∑ fi (5.2) i=1 Where m is the number of class limits. The determination of these probabilities is an important feature in stochastic dynamic programming. A computer subroutine has been developed for the automatic computations of these probabilities of each 10 day period in the SDP model. These tables presents class limits, number of events in each class, their frequencies and cumulative frequencies of each 10 day period. Computed histogram of annual inflows at different inflow locations in Indus River is shown in Figure 5.12. Normal probability distributions curves are also plotted in the figure.
Chapter Five 172 Stochastic Analysis of Uncertain Hydrologic Processes
14 12
12 10
10
8 y
ncy 8
nc ue
q 6 eque e r
Fr 6 F
4 4
2 2 Mean = 50.377 Std. Dev. = 6.21073 Mean = 22.8031 N = 40 Std. Dev. = 4.88807 10 N = 82 0 14 0 30.00 40.00 50.00 60.00 70.00 10.00 15.00 20.00 25.00 30.00 35.00 Basha Mangla 12 8
10
6 ncy
ncy 8 e
Freque Frequ 6 4
4
2 2 Mean = 63.8838 Mean = 25.8505 10 Std. Dev. = 7.70895 10 Std. Dev. = 4.24042 N = 42 N = 42 0 0 50.00 60.00 70.00 80.00 90.00 15.00 20.00 25.00 30.00 35.00 Tarbela Chenab 8 8
6 y 6
nc que equency Fr
Fre 4 4
2 2
Mean = 91.8157 Std. Dev. = 11.61865 Mean = 21.255 N = 42 Std. Dev. = 5.25797 0 N = 36 60.00 70.00 80.00 90.00 100.00 110.00 120.00 0 10.00 15.00 20.00 25.00 30.00 35.00 Kalabagh Kabul Figure 5.12 Computed histogram showing frequency distribution of flows in Indus River System
Chapter Five 173 Stochastic Analysis of Uncertain Hydrologic Processes
5.7 Serial Correlation Coefficients
Several estimates of the serial correlation coefficients have been suggested. The computer
program computes serial correlation coefficient (rk ) upto a maximum lag of 50 using the following simple and satisfactory estimate recommended by Kottegoda (1980).
N-k ∑ [ (xt - xm)(xt+k - x m) ] t rk = ------(5.3) N 2 ∑ (xt - xm ) t
Where k = lag between flow events xt , xt+1 and so on.
Serial correlation coefficients (rk ) indicates how strongly one event is affected by a previous event and reflects the degree of persistence in the data. The computed serial correlation coefficients of annual data are given in Table 5.4.
Table 5.4 Serial Correlation Coefficients ______Serial Correlation Indus at Jhelum at Lag Tarbela Mangla ______1 -0.2130 0.2779 2 -0.0731 0.0883 3 -0.0736 -0.0208 ______
Estimated rk is decreasing as the lag is increasing. it shows the persistence is low as the lag increases. For example, for lags 1 the values are -0.21, 0.28 and for lag 3 it is about -0.02.
The rk values from different sets of the same data vary from one set to another. It reflects that the persistence in the data is a variable phenomena and does not follow a stationary process although the sample was found to be a first order stationary stochastic process due to the unbiased estimates of mean, standard deviation, coefficient of variation and coefficient of skewness.
A characteristic feature of the Indus River System is the variation in the 10 day flows. The correlation of one period with its previous period(s) is an important phenomena if one is
Chapter Five 174 Stochastic Analysis of Uncertain Hydrologic Processes interested to forecast the flow of one period with the help of the recorded previous periods flows. The correlation coefficients between each 10 day period have been estimated and the results are placed in Table 5.5a and 5.5b. The results indicated that the correlation of one period (say 11-20 days of October) with previous period (1-10 days of October) in Indus and Jhulem varies mostly between 0.5 and 0.7. In few periods it is upto 0.88. In these periods the flows may be predicted with the help of regression models.
Once the size of the flood is known in first 10 day of October one can predict the pattern of flow with considerable accuracy for second and third 10 day periods of October. These first order autocorrelation coefficients are important tool to model the river flows for short term forecasting.
Table 5.5a Correlation Coefficients between 10 daily flows, Indus at Tarbela
Chapter Five 175 Stochastic Analysis of Uncertain Hydrologic Processes
Table 5.5b Correlation Coefficient between 10 daily flows Jhelum at Mangla
5.8 Transition Probabilities
If the process has the property that the dependence of future values of the process on past values is summarised by the current value, such property can be modelled by transition probabilities. The process is called Marcov process. A special kind of Markov process is one whose state xt can take on only discrete values. Such a process is called Markov chain. Transition probabilities are a kind of Markov chain. It is the conditional probability that the next state is qj given that the current state is qi . The transition probabilities satisfy
n ∑ Pij = 1 for all i (5.4) j=1 The computational methods have been explained in chapter.3. Transition probability matrices (tpm) are needed in SDP when the stochastic process is considered as Markov chain. The incorporation of tmp in SDP was made. The subroutine (tpm.for) for the computations is similar as developed by Harbaugh et al. 1970. The program is run for 10 day time step and 36 tpm were obtained. A sample tpm for Aug-1 Indus at Tarbela for 5 states is given below.
Chapter Five 176 Stochastic Analysis of Uncertain Hydrologic Processes
Table 5.6 Transition Probability Matrix of Period August 1, Indus at Tarbela ------State 1 2 3 4 5
Midpoint 3.09 3.92 4.76 5.59 6.42 ------3.93 0.2857 0.4286 0.1429 0.1429 0.0000 4.87 0.0000 0.3636 0.3636 0.0909 0.1818 5.81 0.0769 0.2308 0.3846 0.1538 0.1538 6.75 0.0000 0.1000 0.3000 0.4000 0.2000 7.70 0.0000 0.0000 0.0000 1.0000 0.0000 ------5.9 Hurst Phenomenon
Annual streamflow data is used to perform Hurst Analysis (1965). The well-known method involves to find a statistic called the 'range of cumulative departures from the mean' which equals the require storage volume of a reservoir which, for a given inflow sequence, can release every year the mean inflow. The accumulated departure of the flows from the sample mean after t years can be computed as: (Hurst et al. (1965) and Kotegoda (1980).
Z = ( x1 + x2 + . . . + xi - ixm ) ( 5.5 ) Average of the cumulative departures from the mean is:
* rN = max ( Z ) - min ( Z ) ( 5.6 ) 1≤ i ≤ N 1≤ i ≤ N
* + - rN = d N + | dN |
Where xi Sequence of inflows; x1 ...,xi xm Mean flow + d N Max accumulated departure from the sample mean - dN Min accumulated departure from the sample mean
* Hurst found that the average value of rN changes as a function of N:
** * K rN = rN / sN = (N/2) ( 5.7 )
** Where rN Rescaled range. N Number of data values K Hurst exponent (average value 0.73) (ranges between 0.46~0.96). sN Standard deviation The value of k normally decreases as n tends to increase. To analyse the time series, historic data have been used. The results of the historic time series for Indus and Jhelum showed that values of Hurst coefficient 'k' as given below:
Chapter Five 177 Stochastic Analysis of Uncertain Hydrologic Processes
Table 5.7 Variation of Rescale Range and Hurst Exponent
** * The results in Table 5.7 indicated that rescaled range rN = rN /s is a function of N and increases as N increases. Non-stationarity is an explanation of Hurst phenomenon. It is found that segments of historic as well as simulated flows from River Indus at Basha, Tarbela and Kalabagh and River Jhelum at Mangla have widely different k values e.g. minimum and maximum k values are 0.51 and 0.94. The same has been shown by Hurst on Nile river and Potter (1976) on segments of some precipitation records from United States. (Hurst (1956))
5.10 Gould Transition Probability Matrix Method
The method was developed by Gould to estimate the probabilities of reservoir states for a given simulation period and for a given demand (McMahon et al. 1978). These alternative
Chapter Five 178 Stochastic Analysis of Uncertain Hydrologic Processes
studies which is a pre-requisite for evolving some criteria, enables us to derive some accurate and efficient operating rules while using the proposed methodology described in Chapter.3. The computer program is designed to carry out Gould transition probability matrix (TPM) storage yield computations on 10 day time step to find the possible yield from the given uncertain inflows and specified storage capacity. The matrix of one-step transition probabilities is estimated after dividing the number of entries in each cell of the transition year matrix:
n P = ----- (5.8 ) N Where P Probability matrix N Number of years of the record. n Number of points in a box representing initial state i and final state j.
Some excellent details about the computational procedure are given by McMahon et al. (1978) and it is not included here to conserve space. However, the program indicates some details about the computations and its procedure. Several runs have been made on 10 day basis for Tarbela and Mangla reservoir to determine a possible release pattern with an acceptable probability of failure. A sample result of the selected trial is presented in Table 5.8 and Figure 5.13. Present study indicated that yield from Tarbela, Mangla and Mangla Raising may be of the order of 59.90, 20.08 and 23.10 MAF respectively. Figure 5.13 indicated that probabilities of reservoir states being empty are within acceptable limits (e.g. it is 2.3%, 1.2% and 8% for Tarbela, Mangla and Mangla Raising respectively).
Chapter Five 179 Stochastic Analysis of Uncertain Hydrologic Processes
Table 5.8 Results of Gould transitional probability matrix method
Probability of Reservoir Storage
0.5 0.45 Mangla
e t Mangla Raised a 0.4 t S
r Tarbela i 0.35 o v r
e 0.3 s e
R 0.25 f
o 0.2 ility
b 0.15 a b o
r 0.1 P 0.05 0 12345678910 Reservoir State
Figure 5.13 Computed probabilities of Reservoir States using Gould TPM method
These estimates are considered to be of preliminary nature due to the following limitations:
i) Annual serial correlation is assumed to be zero whereas in actual, it was estimated to be 0.28 for 82 years inflow sequence at Mangla and -0.21 for 43 years inflow sequence at Tarbela.
Chapter Five 180 Stochastic Analysis of Uncertain Hydrologic Processes
ii) No constraint, except mass balance equation (4.9) can be included in the model. Further the power sectors and incidental flood protection benefits are also ignored. 5.11 Rippl Mass Curve Analysis
The method was developed by Rippl in 1883. (Kottegoda (1980)). It is used to find minimum storage required to meet the given demand. The method was employed in Indus River System. The computer program was designed to estimate possible draft from the Indus River reservoirs with given inflows. The technique involves finding the maximum positive cumulative difference dt between a sequence of specified reservoir releases Rt and known historical or synthetic inflows Qt during an interval of time beginning in period t and extending upto period T: (Kottegoda (1980))
j * dt = maximum ∑ dm ( 5.9 ) t ≤ j ≤ T m =1 The required storage capacity S for a specified release is the maximum of the maximum * cumulative differences dt .
* S = maximum (dt ) (5.10) t ≤ j ≤ T or j S = maximum [ ∑ ( Rt - Qt ) ] (5.11) 1 ≤ t ≤ j ≤ T t=i Figure 5.14 shows a sample graph showing dam capacities against various releases at Mangla dam using Rippl mass curve analysis. Figure 5.15 shows the reservoir capacities against various releases at Basha, Tarbela, Kalabagh , Mangla and Mangla Raising dams. Summary results from Rippl mass curve analysis are presented in Table 5.9.
Chapter Five 181 Stochastic Analysis of Uncertain Hydrologic Processes
Mangla Raising
Figure 5.14 Sample graph showing dam capacities for different releases for Mangla Dam
Rippl Mass Curve Analysis
70
) 60 F A M
( 50 y t Tarbela aci
p 40
a Basha C
e kalabagh
v 30 i L
Mangla ed
r 20 i u q e
R 10
0 0 20406080100 Annual Water Released (MAF)
Figure 5.15 Estimated reservoir capacities for different releases in Indus River reservoirs using Ripple mass curve method
Chapter Five 182 Stochastic Analysis of Uncertain Hydrologic Processes
Table 5.9 Summary results from Rippl mass curve analysis
Possible Storage Release Live Dam (MAF) Basha 7.340 46 Tarbela 7.226 58.2 Kalabagh 6.100 71 Mangla 4.590 14.7 Mangla Raising 7.26 16
Although the approach used in Rippl analysis is simple and still widely acceptable, but it may not be adaptable to a reservoir where variable releases are important considerations with several physical or legal constraints. However, such type of analysis is good to have some idea about the environments before to start some complex optimization analysis as described in Chapter.3.
5.12 Sequent Peak Analysis
The method assumes that the record repeats to take care of case when the critical sequence of flows occurs at the end of the streamflow record. Mathematically the simulation procedure can be expressed as (Loucks et al. (1981))
⌈ Rt - Qt + St-1 if positive (5.12) St = | ⌊ 0 Otherwise
Where Rt Required release in period t Qt Inflow St Storage capacity required at the beginning of period t.
The maximum of all St is the required storage capacity for the specified release Rt. A subroutine is prepared to compute the required storage at different trial demands. The program is tested with example problem given by Loucks et al., 1981. The model is executed for each existing and proposed dam on Indus River. Sample graph for Mangla dam is shown in Figure 5.16. Summary results are shown in Table 5.10. Because the method mainly focused on low flows, therefore conservative values of releases are achieved for the given capacities of the dams.
Chapter Five 183 Stochastic Analysis of Uncertain Hydrologic Processes
Max. Peak shows required dam capacity at given demand
Sequence of low flows determine the dam capacity
Figure 5.16 Sample graph showing results for Mangla dam at water demand 16 MAF
Table 5.10 Summary results of Sequent Peak Analysis
Chapter Five 184 Stochastic Analysis of Uncertain Hydrologic Processes
5.13 Evaporation Losses and Rainfall Accretion to Reservoirs
Evaporation is one of the important losses from reservoir in case of arid region with scarce water. Pan evaporation data have been collected for the period 1961 to 2000 at Tarbela and Mangla climatological stations (WAPDA 2003). The data is used in the study to subtract the evaporation losses during optimization study. Mean annual evaporation at Tarbela and Mangla reservoir is worked out to be 92.83 and 80.28 inches respectively. Mean monthly evaporation was computed at two stations and shown in Figure 5.16. Figure indicated that maximum evaporation takes place from reservoir in the month of May and June and minimum in the months of January and December. Pan evaporation data is converted to shallow lake evaporation with a pan coefficient 0.70 commonly used in this part of the country.
Rainfall falling over the reservoir produces some accretion in the reservoir contents. Rainfall data at Tarbela and Mangla climatological stations was collected for the period 1961 to 2000 (WAPDA 2003). The data is used in the study to include accretion in water content due to rainfall during reservoir operation. Mean annual rainfall at Tarbela and Mangla has been estimated 35.93 and 33.27 inches respectively. Mean monthly precipitation was computed at two stations and shown in Figure 5.17. Figure indicated that maximum precipitation occurs in the month of July and August and minimum in the months of May and November- December.
In contrast to evaporation losses from reservoir, rainfall falling over the reservoirs will replenish the storage. Therefore both elements are plotted together to asses such effect. It is presented for Tarbela and Mangla reservoirs in Figures 5.17 and 5.18 respectively. These figures indicated that high rates of evaporation in monsoon months are compensated by direct accretion by rainfall over the reservoir area. At every period, rainfall accretion is less than evaporation. It is to be noted that in most of the previous studies on Indus Basin these components (i.e. evaporation and rainfall) were ignored. It may affect the accuracy of results during critical periods.
Chapter Five 185 Stochastic Analysis of Uncertain Hydrologic Processes
Mean Monthly Pan Evaporation Mean Monthly Precipitation 1961-2000 1961-2000
16.0 10.0 9.0 14.0 8.0
12.0 7.0 h) nch) I nc
I 10.0 6.0
n ( Tarbela Dam
Tarbela Dam o i on ( 5.0 i 8.0 t
at t
a Mangla Dam Mangla Dam i r
p 4.0 i
6.0 c e apo 3.0 v Pr E 4.0 2.0 2.0 1.0 0.0 Figure 5.15 Mean monthly pan evaporation Figure0.0 5.16 Mean monthly Precipitation Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Precipitation Vs Evaporation at Tarbela Dam Precipitation Vs Evaporation at Mangla Dam
20.0 20.0 18.0 18.0 16.0 16.0 .
. 14.0 14.0
) h h) 12.0 12.0 c
nc
I Evaporation Evaporation
10.0 (In 10.0 h h ( t t Precipitation Precipitation p p e 8.0 e 8.0 D 6.0 D 6.0 4.0 4.0 2.0 2.0 0.0 0.0 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Figure 5.17 Precipitation vs evaporation at Tarbela Figure 5.18 Precipitation-evaporation at Mangla
5.14 Characteristics of Hydro-Electric Plants
As electric energy cannot be economically stored on large scale therefore, the time pattern of generation must duplicate the time pattern of demand. Base load is energy required for many hours whereas peaking power is the fluctuating portion of the energy requirement. In response to fluctuating energy usage, the generation rates can be varied quickly and inexpensively by simply regulating the flow of water through the plant. The generating capacity must exceed the peak rate of energy use to prevent periodic shortages. If possible, it is better to keep storage reservoirs fairly full because peaking capacity depends on the available head.
In case of a hydroelectric project, the cost of water is minor because the water, after passing through the turbines can be used for other purposes. The firm power of a hydroelectric plant
Chapter Five 186 Stochastic Analysis of Uncertain Hydrologic Processes
can be defined as the maximum annual rate at which energy can be generated without interruption during the critical dry period whereas secondary power cannot be guaranteed but is available more than half the time, while dump power is available less than half the time. The firm energy is a function of reservoir capacity, streamflow hydrology and available head.
Load factor is the ratio of the energy generated produced to the energy which would be produced were the plant run continuously at the peak demand rate, the ratio of average demand to peak demand. Hydroelectric plant efficiency is equal to the product of the mechanical efficiencies of the individual components (penstock, turbine and generator), usually about 85-75 percent. The plant factors during previous years of operation (1972 to 2003) are given in Power System Statistics 1993, 2003, WAPDA. Characteristic curves, e.g. head v/s storage and capability of power plant are used in the optimization model. The basic data is obtained from WAPDA and Techno Consult, 2005.
5.15 Historic Operation of Reservoirs
Mangla and Tarbela reservoirs were impounded in 1967 and 1974 respectively. Since that the reservoirs are under operation to supply water for irrigated agriculture and power generation. The data for historic operation has been obtained from WRMD, WAPDA. Reservoir elevations during historic operation of Tarbela and Mangla dams were plotted and presented in Figure 5.19 and 5.20. These levels indicated that the reservoirs act as carry over reservoirs and does not follow a fix reservoir levels. However from these curves minimum and maximum rule curves may be drawn.
5.16 Release Requirements and Operation Objective
The multipurpose dams of Indus River System holds a key position for the agricultural and power sectors of Pakistan economy. There are basically two types of demands which must be fulfilled from the project. These are (i) irrigation demand and (ii) power demand.
Chapter Five 187 Stochastic Analysis of Uncertain Hydrologic Processes
Reservoir Levels During Historic Operation (1976-2002) Tarbela Dam
1600
) 1550 L S
M 1500 A eet f 1450 n ( o i
evat 1400 l E r 1350 voi r e s e
R 1300
1250 1 1 1 1 3 2 3 2 3 2 3 2 3 2 3 2 1 1 - - - - B C T R UN- FE J AP DE OC AUG-
Figure 5.19 Historic operation of Mangla Dam
Reservoir Levels During Historic Operation (1976-2002) Mangla Dam
1250
) 1200 L S M A t e 1150 (fe n io t a v e l 1100 r E i o rv se e
R 1050
1000 1 1 1 1 1 3 2 3 2 3 2 3 2 3 2 3 2 1 - - - - C G- R- N FEB JU AP DE OCT AU
Figure 5.20 Historic operation of Tarbela
Chapter Five 188 Stochastic Analysis of Uncertain Hydrologic Processes
5.17 Irrigation Demands
Several studies (Harza 1968, IBP (Indus Basin Project) 1966) were carried out to compute the cropping pattern in the area and to find the irrigation requirements in the Indus Basin. A schematic diagram of the whole irrigation system of Indus Basin has been shown in Figure 4.2. The diagram illustrates that water requirement of all irrigation and link canals on the Chenab and Jhelum rivers above the Qadirabad Barrage and the Rasul barrage respectively will be met from the supplies of Jhelum and Chenab rivers supplemented by storage water from the Mangla Reservoir. Surplus supplies from these rivers, whenever available, will be utilised at Trimmu, Panjnad and the lower Indus projects. The Northern canal system encompassing canals on Jhelum, Chenab, Ravi and Sutlej rivers includes five linked canals viz UJC, LJC, UCC, LCC and LBDC and Eastern river canals and Trimmu and Panjnad canal systems. As the Eastern rivers Ravi, Sutlej and Beas are under control of India under the Act of Water Treaty, 1960, therefore, their flows are unregulated depending upon the Indian surplus water. A further constraint was imposed by Indus Basin Project, WAPDA (IBP) on Mangla dam that the total reservoir outflow shall in no case be less than 13,500 cusecs to provide for the maximum power generation as well as meet the needs of UJC and LJC. Based on all these constraints and consumptive use studies 10-daily indents of total reservoir releases are furnished by Irrigation Department (ID) to Tarbela and Mangla Dam organisation. The demand collected from I.D. Punjab is an overall demand required from these dams. It was derived from several consumptive use studies on the irrigable areas which are irrigated by reservoir supplies plus Eastern rivers inflows and avoids many complications. (See Figure 5.21) The canal network demand belongs to Water Accord (1991) demand accepted by the four provinces.
5.18 Power Demand
The power demand based on the analysis of consumers and pertinent data has been forecasted by WAPDA in publication 'Power System Statistics', 14th issue, October 1993. The 10 day power demand used in this study is adopted from the WAPDA power statistics. Previous studies (Harza Engineering, July 1962) indicated that about 4570 million KWH of hydroelectric energy (3480 (firm) + 1090 (Secondary) may be obtained from the Mangla dam with maximum pool elevation of 1202 (4.80 MAF of active storage). However previous
Chapter Five 189 Stochastic Analysis of Uncertain Hydrologic Processes
operations during (1967-95) show that a maximum of 6040 million KWH energy was generated from the Project. This energy is about 27 percent less than the required electric energy from the Project. One way to increase power production is to keep the reservoir as high as possible consistent with satisfying irrigation requirements. And also upto the available capacity of the turbines, water released from the reservoir should be made through the turbines. In this study the capacity of the hydropower generation is for Mangla is considered to be 1000 MW and for Tarbela 3500 MW. The present study maximizes the power production by optimizing the release pattern and reservoir contents for the project during a complete water year after a detailed analysis of trade offs between power production and irrigation supplies.
Irrigation Demand
14.0
12.0
. 10.0 )
F Mangla Dam A 8.0 Tarbela Dam (M d n 6.0 Basha Dam ma
e Kalabagh Dam
D 4.0
2.0
0.0 Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep
Figure 5.21 Irrigation demand from existing and proposed reservoirs as per Water Accord 1991
5.18.1 Maximum and Minimum Design Rule Curves These maximum and minimum rule curves are actually design guides and provide a corridor before to optimize some release pattern for a project. These maximum and minimum design curves have been drawn to maintain certain levels in the reservoir in various months to take care of incidental floods, monitoring of the dam to effect sediment flushing and many other physical aspects. For example sometime it is required that the storage capacity of the
Chapter Five 190 Stochastic Analysis of Uncertain Hydrologic Processes
reservoir should be gradually depleted because of sedimentation. Also, lowering the reservoir a day or two prior to a flood would permit the peak discharge to be lower than peak inflow thus adding some flood control benefit. It is usually desired that the derived optimal release pattern, should not violate, if possible, the envelope of maximum and minimum design curves during the operation period of a project. Therefore it is another objective in the mathematical model to minimize the deviation, if any, between the optimized releases and maximum and minimum design curves during operation of the project. The design rule curves of Tarbela, Mangla, Bash and Kalabagh dams have been derived with the help of historic operation and shown in Figure 5.22 through 5.25.
Rule Curve Tarbela Dam
1600
1550
1500 Maximum rule curve t) 1450
ION (F Minmum rule curve
VAT 1400 E L E 1350
1300
1250 Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug MONTHS
Figure 5.22 Rule Curve at Tarbela Dam
Chapter Five 191 Stochastic Analysis of Uncertain Hydrologic Processes
Rule Curve Mangla Dam
1250
1200
Maximum rule curve
Minmum rule curve
) 1150 m N ( O I AT V E
L 1100 E
1050
1000 Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug
Figure 5.23 Rule Curve at Mangla Dam
M inimum Rule Curve Basha Dam
3900 3850 3800
3750
(ft) 3700 N IO
T 3650 A
V 3600 E L
E 3550
3500 3450 3400 Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug
Figure 5.24 Minimum Rule Curve at Basha Dam
Chapter Five 192 Stochastic Analysis of Uncertain Hydrologic Processes
M inimum Rule Curve Kalabagh Dam
920 910 900
890 ) t f 880 ON ( I 870
VAT 860 E L
E 850 840 830 820 Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug
Figure 5.25 Minimum Rule Curve at Kalabagh Dam
5.19 Stochastic Control of Reservoir Inflows Reservoir inflows are uncertain events. The better operation depends on the accuracy of forecast if the operating policy is dependent on the current inflow. Several forecast models have been developed in previous research (Kuo et al.1990, Kottegoda 1980, Hipel et al. 1979). Some of the forecast models are physical and others are statistical models. In physical model streamflow can be forecasted from simplified hydrologic equation. In statistical models, inflows are estimated by equations and parameters derived from statistical characteristics of streamflow records. These models are usually of Box Jenkins types. (Kuo et al.1990).
The Flood Forecasting Centre, Lahore and WAPDA has developed a physical model for Indus River System with the help of Candian consultants. The model is only applicable for flood forecasting during flood season and it cannot be used for the reservoir operation. Alternatively WAPDA is predicting 6 month (say Kharif) total volume of flow with the help of previous 6-month (say Rabi) recorded flow. The method is based on just comparing the same volume of flow in the historic record and noting what was the next 6 month flow with
Chapter Five 193 Stochastic Analysis of Uncertain Hydrologic Processes
all the similar values. The steps and limitation of this method are described in the following section.
5.19.1 WAPDA Forecasting Procedure
The method adopted by Water Resources Management Directorate (WRMD) of WAPDA for forecasting seasonal river flows is illustrated below: Step 1. Consider the actual inflow of Indus (say at Tarbela) from October 1 to February 20. Step 2. Comparing historic data, (Rabi season in the past), with the flow in (1), note the Rabi inflow equal or close to flow in (1). Step 3. Calculate probability of inflow of the seasons in (2), and determine average percent. Step 4. At this probability percent, determine average seasonal flow. Step 5. Comparing historic data with the flow in (4), list up flows in kharif following flow equal or close to flow in (4). Step 6. Calculate probability percentage of inflows of each Kharif listed in (5) Step 7. Exclude extreme (wet or dry) seasons and find out average probability of inflow of the remaining seasons. Suppose it comes out to be 46%. Round it to safe multiple of 5 on the conservative side which is 50% . Step 8. Forecast for Kharif (at Tarbela) will be that of 50% probability (say 50.13 MAF)
Limitations
• The method is used to predict 6-month total volumes. It is difficult to get 10 day flow from 6 month forecasted volume. • The R2 of the method is not known. • The reliability of the method may be low due to rounding the value of probability and excluding the extreme events to get average probability. • The steps are based on the persistence between Rabi and Kharif. 5.19.2 Forecasting with Moving Average The moving average models are used to smooth out random fluctuations for both seasonal and non-seasonal data. The estimated value for a period is based on the average of n prior periods. Simple average and weighted averages can be included. In the simple average the all past periods are weighted equally. In the weighted averages variation, each past period is
Chapter Five 194 Stochastic Analysis of Uncertain Hydrologic Processes
assigned a fractional weight with the most recent periods receiving the most weight. For example, the calculation of an estimate based upon a 3 10 day period average would be
y = xt-1 * 3/6 + xt-2 * 2/6 + xt-3 * 1/6 (5.13) To evaluate the moving average models for Indus Rivers, 3-years weighted moving average have been done for 10 days flows using (5.13) and shown in Figure 5.26. Alternatively annual flows are also estimated with 5 year moving average model and shown in Figures 5.27 for Mangla and Tarbela dams. It is inferred from the results that these models are an approximation and cannot be used for accurate forecasting in this case. 5.19.3 Forecasting using Autoregressive Models An autoregressive model may be expressed as
qt = a1 qt-1 + a2 qt-2 + ... + b0 et + b1 et-1 + ... (5.14)
Where a & b Coefficients
qt Standardised streamflows = ( Qt - Qm ) / σ
et Random errors assumed to be normally distributed with a mean of zero and standard deviation one. t Time period
The 10 day inflow data is standardised and the resulting data is with zero mean and one standard deviation. Using this data, autoregressive models have been calibrated for different periods and different rivers. For the test case these models seem to be not as good as developed by regression (see next section) due to random errors. Therefore these are not used for forecasting purposes.
Chapter Five 195 Stochastic Analysis of Uncertain Hydrologic Processes
Observed and Computed 10 Day 3 Year Weighted Moving Average Model 2000-2001, Dry Year Mangla Dam
0.9 0.8 0.7
) 0.6 F A 0.5 Forecasted
s (M Observed w 0.4
flo 0.3 In 0.2 0.1 0.0 . . r c b P t L R G v N Jan Fe De MA Ma JU Oc SE JU AP No AU Observed and Computed 10 Day 3 Year Weighted Moving Average Model 1988-89, Average Year Mangla Dam
2.0 1.8 1.6
) 1.4 F
A 1.2 Forecasted 1.0 s (M Observed w 0.8 flo
In 0.6 0.4 0.2 0.0 . c n b t P L G R N ov. Ja Fe De MA Mar JU Oc SE JU AP N AU Observed and Computed 10 Day 3 Year Weighted Moving Average Model 1990-91, Wet Year Mangla Dam
2.5
2.0 ) F 1.5 Forecasted (MA s Observed w
o 1.0 fl In 0.5
0.0 . . r c n t P L b G R v N Ja Fe De MA Ma JU Oc SE JU AP No AU Figure 5.26 Comparison of Observed and Forecasted 10-daily flows with 3 year weighted moving average at Mangla Dam for dry, average and wet years (Model Not ok due to shift)
Chapter Five 196 Stochastic Analysis of Uncertain Hydrologic Processes
Forecasting with 5 Year Moving Average Tarbela Dam 90
80 ) F
A 70 (M w o l
f 60 l In a
u 50 n n A 40 Observed Forecasted 30 63 66 69 72 75 78 81 84 87 90 93 96 99 02 2 - 5 - 8 - 1 - 4 - 7 - 0 - 3 - 6 - 9 - 2 - 5 - 8 - 1 - 196 196 196 197 197 197 198 198 198 198 199 199 199 200
Forecasting with 5 Year Moving Average Mangla Dam 40 35
)
F 30 A
M 25 ( ow l
f 20 n I 15
nnual 10 A Observed 5 Forecasted 0
3 23 28 33 38 48 53 58 63 68 73 78 83 88 93 98 03
4 2 - 7 - 2 - 7 - 7 - 2 - 7 - 2 - 7 - 2 - 7 - 2 - 7 - 2 - 7 - 2 - 2 - 192 192 193 193 194 195 195 196 196 197 197 198 198 199 199 200 194
Figure 5.27 Comparison of observed and forecasting annual flows with 5 year moving average at Mangla and Tarbela Dams
Chapter Five 197 Stochastic Analysis of Uncertain Hydrologic Processes
5.19.4 Forecasting using Multiple Regression Short term forecast may be performed by constructing regression models
qt = f (qt-1, qt-2, , ..., )
Where qt ,qt-1 Inflow in month i, lagged inflows in month i-1, i-2,. respectively.
A linear form of the model may be represented as
Y = a + b1 xt + b2 xt-1 + ... (5.15) The parameters and b’s can be determined by multiple regression a least square statistical technique. The parameters of the models for different periods and different rivers have been determined with the help of regression module in EXCEL Spread sheet.
In contrast to linear models, non linear models were also tested. For this purpose a computer program have been developed due to the following reasons.
• Multiple regression in spread sheets only uses linear regression
• The software TableCurve may be used for non-linear models but it accepts only one independent variable.
The program developed herein can used to derive linear and non- linear equations in a single run saving time and cost of computer. The model may be named as 'MATHFIT'. The following four types of models have been included in 'MATFHFIT':
1. (Linear Model) Y = a + b x + b x + ... (5.16) 2. (Exponential Model) Y = a.e . e . ... (5.17) 3. (Logarithmic Model) Y = a + b log x + b log x + ... (5.18) 4. (Power Model) Y = a.x . x . ... (5.19)
Statistical technique was applied to derive the parameters and coefficient of determination (R2) was taken as a criteria to check the fitness of the relation. The best model was also evaluated from some other statistical tests e.g. sum of squares of deviation and regression, mean squares deviation and F-Test.
The algorithm attempts to find the coefficients of a polynomial regression equation as given below in a model form having 'n' independent variables (< 20).
Y = a + b1 x1 + b2 x2 + ... + bn xn (5.20)
Chapter Five 198 Stochastic Analysis of Uncertain Hydrologic Processes
For nonlinear functions (i.e exponential, logarithmic and power), the model finds the solution in three phases. In Phase-I it transforms the equation to a linear form. In Phase-II it attempts to find the model coefficients in a linear case. In Phase-III the model converts the derived equation into nonlinear form just by reversing the procedure adopted in Phase-I.
Error Measures: Standard Error of Estimated The average difference between the observed and estimated values known as standard error is used to determine how well the regression equation describes the relationship between the dependent and independent variables.
Coefficient of Determination, R2: R2 is a ratio of explained variation to the total variation. Explained variation represents the sum of the squared deviations between the estimated values and the mean value of the sample whereas total variation is the sum of the squared deviations between the actual values and the mean value of the sample. R2 which is also known as goodness of fit. In addition to R2 , sum of squares of the difference between observed and forecasted values and F test are additional tests that adopted for the error measures in the parameter estimation.
Several runs have been performed and bestfit models on the basis of the error measures were evaluated. The finally selected models are listed in Table 5.11. Their performance is shown in Figure 5.28 through 5.37.
Chapter Five 199 Stochastic Analysis of Uncertain Hydrologic Processes
Table 5.11 Selected Regression Models for 10 day forecasting in Indus Rivers
2 River / Period Constant x1 Coeff. x2 Coeff. R Indus at Tarbela Oct 1 to Nov 3 0.227 0.06 0.441 0.747 Dec 1 0.087 0.474 0.195 0.716 Dec 2 0.005 0.602 0.289 0.663 Dec 3 0.032 0.422 0.513 0.424 Jan 1 to Jan 2 0.095 0.172 0.453 0.095 Jan 3 -0.01 0.465 0.681 0.411 Feb 1 0.026 0.125 0.708 0.775 Feb 2 to Feb 3 -0.01 0.887 0.084 0.442 Mar 1 to Mar 3 0.069 0.308 0.779 0.442 Apr 1 to July 3 0.409 0.059 0.961 0.84 Aug 1 to Sep 3 -0.18 -0.01 0.882 0.713 One model for all periods 1. (Linear) 0.159 -0.24 1.149 0.863 2. (Power) 0.999 -0.315 1.264 0.902 Chenab at Marala Oct 1 17.115 0.0277 0.0489 Oct 2 to Nov 3 4.341 0.0536 0.3915 0.611 Dec 1 to May 3 2.089 0.2792 0.7364 0.701 Jun 1 to Aug 1 34.732 0.0349 0.5718 0.419 Aug 2 to Sep 3 5.26 0.1045 0.6298 0.513 Kabul at Nowshera Oct 1 to Feb 3 3.389 -0.04 0.671 0.524 Mar 1 to Apr 3 9.306 0.248 0.878 0.655 May 1 to May 3 13.93 0.163 0.648 0.511 Jun 1 to Jun 3 20.38 -0.06 0.85 0.565 Jul 1 to Sep 3 1.067 -0.08 0.965 0.764
5.19.5 Forecasting with expected values An expected or a critical value can be forecasted for an uncertain event. The expected values are determined from the conditional or unconditional distributions. It is done in section 5.6 and 5.8.
Chapter Five 200 Stochastic Analysis of Uncertain Hydrologic Processes
Forcasted Annual Flows using 10 Day Forecast Indus at Tarbela
90.0
80.0
70.0
60.0 ) F A 50.0 Observed 40.0 Computed flows (M n
I 30.0
20.0
10.0
0.0 1961-62 1967-68 1973-74 1979-80 1985-86 1991-92
Figure 5.28 Comparison of observed and computed annual flows using 10 Day forecast models, Indus at Tarbela
Forcasted Annual Flows using 10 Day Annual Forecast Model Indus at Tarbela
90.0
80.0
70.0
60.0 ) F
A 50.0 Series1
s (M 40.0 Series2 ow l
Inf 30.0
20.0
10.0
0.0 1961-62 1967-68 1973-74 1979-80 1985-86 1991-92
Figure 5.29 Comparison of observed and computed annual flows using 10 Day annual forecast model, Indus at Tarbela
Chapter Five 201 Stochastic Analysis of Uncertain Hydrologic Processes
Observed and Computed 10 day Flows for 1991-92 to 1992-93 10 day forcasting models,Indus at Tarbela
8.0
7.0
6.0
) 5.0 F
A Observed M
( 4.0
w Forcasted o fl
n 3.0 I
2.0
1.0
0.0 Oct Dec Feb Apr Jun Aug Oct Dec Feb Apr Jun Aug 10 Day Period
Figure 5.30 Comparison of observed and computed 10 day flows for 1991-92 to 1992-93, Indus at Tarbela
Observed and Computed 10 day Flows for 1993-94 to 1994-95 10 day forcasting models,Indus at Tarbela
8.0
7.0
6.0
) 5.0 F A Observed
(M 4.0
w Forcasted o fl
In 3.0
2.0
1.0
0.0 Oct Dec Feb Apr Jun Aug Oct Dec Feb Apr Jun Aug 10 Day Period
Figure 5.31 Comparison of observed and computed 10 day flows for 1993-94 to 1994-95, Indus at Tarbela
Chapter Five 202 Stochastic Analysis of Uncertain Hydrologic Processes
Observed and Computed 10 Day Flows for 1991-92 to 1992-93 10 Day Forcasting Model, Jhelum at Mangla
3.0
2.5
2.0 ) F Observed 1.5 (MA
w Forcasted o l f
In 1.0
0.5
0.0 OctDecFebAprJunAugOctDecFebAprJunAug 10 Day Period
Figure 5.32 Comparison of observed and computed 10 day flows for 1991-92 to 1992-93, Jhelum at Mangla
Observed and Computed 10 Day Flows for 1993-94 to 1994-95 10 Day Forcasting Model, Jhelum at Mangla
3.0
2.5
2.0 ) F
A Observed
(M 1.5
w Forcasted o l f
In 1.0
0.5
0.0 Oct Dec Feb Apr Jun Aug Oct Dec Feb Apr Jun Aug 10 Day Period
Figure 5.33 Comparison of observed and computed 10 day flows for 1993-94 to 1994-95, Jhelum at Mangla
Chapter Five 203 Stochastic Analysis of Uncertain Hydrologic Processes
Observed and Computed 10 Day Flows for 1991-92 to 1992-93 10 Day Forcasting Model, Chenab at Marala
180 160 ) 140 s f
c
120 x 0
0 100 Computed 0 1 80 Observed ( s 60 w o l
F 40
20 0 OCT DEC FEB APR JUN AUG OCT DEC FEB APR JUN AUG 10 Day Period
Figure 5.34 Comparison of observed and computed 10 day flows for 1991-92 to 1992-93, Chenab at Marala
Observed and Computed 10 Day Flows for 1993-94 to 1994-95 10 Day Forcasting Model, Chenab at Marala
200
180 160 )
s
f 140 c
x 120
0 Computed 100
1 0 Observed (
s 80
o w 60 l F 40 20
0 OCT DEC FEB APR JUN AUG OCT DEC FEB APR JUN AUG 10 Day Period
Figure 5.35 Comparison of observed and computed 10 day flows for 1993-94 to 1994-95, Chenab at Marala
Chapter Five 204 Stochastic Analysis of Uncertain Hydrologic Processes
Observed and Computed 10 Day Flows for 1991-92 to 1992-93 10 Day Forcasting Model, Kabul at Nowshera
140
120 )
s
f
c 100
x 80 Computed Observed 1 0 60 (
40 o w F l 20
0 OCT DEC FEB APR JUN AUG OCT DEC FEB APR JUN AUG 10 Day Period
Figure 5.36 Comparison of observed and computed 10 day flows for 1991-92 to 1992-93, Kabul at Nowshera
Observed and Computed 10 Day Flows for 1993-94 to 1994-95 10 Day Forcasting Model, Kabul at Nowshera
120
100 )
s
c f 80 x
0 Computed 0 60
1 0 Observed
(
40 w o l F 20
0 OCT DEC FEB APR JUN AUG OCT DEC FEB APR JUN AUG 10 Day Period
Figure 5.37 Comparison of observed and computed 10 day flows for 1993-94 to 1994-95, Kabul at Nowhera
Chapter Six 205 Reservoir Operation Optimization
Chapter 6
RESERVOIR OPERATION OPTIMIZATION
6.1 Background
Relatively little of the research on the water resources systems operation has found its way into actual practice. One reason is that operators are uncomfortable with complex optimization models and reluctant to use procedures that they do not fully understand (Russell & Campbell, 1996). The procedures described in this chapter are well explained and easy to understand. A mixed optimization procedure is developed to overcome the problem of dimensionality due to large scale system and the nonlinear objectives in reservoir operation optimization. The procedure describes the use of dynamic programming (DP) model where the reservoirs with nonlinear objectives are located and the use of network flow programming for the rest of the canal/barrage network system. Four types of DP models (three numbers of SDP type and one number of DDP type) have been investigated for reservoir operation optimization and out of the four models, the best fit model for the test case was selected. In each four models there are three types of multiple objectives. Therefore, the number of models [(1 DDP+3 SDP)*3 Multiple Objectives] becomes 12 for the multiple reservoir case. The best fit models have been evaluated. The results of these models are the input to another model which optimizes the whole large scale multi objective mutlireservoir Indus River System. The steps required to perform the proposed procedure are described in chapter 3.
6.2 Problem Formulation for Reservoir Operation Optimization
For the simple understanding, consider a single reservoir fisrt (Figure 6.1). The reservoir is recieving inflows qt and making releases rt in each period t. In deterministic case the sequence of Inflows are known while in stochastic case either the inflow probability, inflow conditional probability or forecasted flow is known.
The reservoir capacity K ( = Smax) and the dead stoarge Smin is given. The reservoir problem involves finding the sequence of releases (or storage level at the end of each period )
Chapter Six 206 Reservoir Operation Optimization
that maximizes the total net benefits (or the system performance). The system performance may be a function of the storage volume as well as of the release. Therefore the system performance may be expressed as
B[St, St+1, Rt] (6.1)
Where St = St + qt - rt - Lt (mass balance equation)
Lt = Losses due to evaporation or seepage from the reservoir
St ≤ K for each period B = Benefits
Rainfall Evaporation accretion et RFt Inflow qt
Storage St
Release dt
S t+1 = St + qt - dt - A( St, S t+1 ) * [ et - RFt ]
Figure 6.1 Reservoir mass balance and continuity equation
The reservoir operating problem can be viewed as a multistage decision-making process. It is shown in Figure 3.1 (Chapter 3). The stages are the time periods and the states are the storage volumes. The constraints on the release Rt limit it to the water available and force a spill if
Chapter Six 207 Reservoir Operation Optimization
the available water exceeds the reservoir capacity K. The general recursive equation for each period t with n (n > 1) periods remaining, backward in time can be written as follows:
n n-1 ft (St) = maximize [ Bt ( St, St + qt - rt, rt ) + ft+1 (St + qt - rt ) ] (6.2) Rt
Where n Proceeds from 1 and increases at each successive stage. t Cycles backward from the last period T to 1 and then to period T again.
To obtain the release policy rt for each period t (10 day), associated with each discrete value of the initial storage volume, three to four iterations are usually required to get a stationary
policy. At this stage the rt with each St will be the same as the corresponding Rt and St in the previous years. The maximum annual net benefit resulting from this policy will be equal to the
n+1 n = ft ( St) - ft (St) (6.3)
n+1 n It is important to note that even if the values of [ ft (St) - ft (St) ] are independent of the
St and rt, the stationary policies has been identified. The details of the stationary policies and their solution techniques have been described in chapter 3. Four types of DP models (One DDP and three SDP) have been proposed and formulated. Solution procedure for each model type is determined. Each model is calibrated, verified and bestfit model type is identified for the Indus reservoir system.
6.2.1 Formulation for Deterministic Optimization
Determination consideration of reservoir operational problems has many advantages in terms of computational efficiency. The general recursive equation for each period t with n (n > 1) periods remaining, backward in time in deterministic optimization is same as formulated in equation (6.2) and the equation is rewritten as:
Type One
n n-1 ft (St) = minimize [ Losst ( St, St + qt - rt, rt ) + ft+1 (St + qt - rt ) ] (6.4) rt
Chapter Six 208 Reservoir Operation Optimization
State transfer equation is
St+1 = St + qt - rt (6.5)
Where n Proceeds from 1 and increases at each successive stage. t Cycles backward from the last period T to 1 and then to period T again.
Stationary policies are obtained solving the above equation for the 36 ten day periods in each year. Three to four iterations are rqeuired to get steady state policies. In the deterministic case the current 10 day inflow qt is considered to be known in advance. The steady state operating policy is derived as a function of the initial reservoir storage the current natural or forecasted inflow and the optimal release target. Raman and Chandramouli (1996) showed that DDP provides better results than SDP and SOP (Standard Operating Policy) models. The finer volume discretization of storage improves the performance marginally.
6.2.2 Formulation for Stochastic Optimization
As described by Huang et al. (1991) and Loucks et al.(1981), determination consideration of reservoir operational problems has many advantages over stochastic problems in terms of computational efficiency. But ignoring the stochasticity of the system introduces bias in the optimization. Further deterministic models based on average values of the inputs such as streamflows are usually optimistic producing overestimated system benefits or underestimated system losses/costs. In comparison to the deterministic models yielding optimal releases, stochastic dynamic programming is of practical interest (Yakowitz 1982, Yeh 1985). Huang et al. (1991) compared four types of SDP models for Feitsui reservoir. They noted that the results are not universal but dependent on the characteristic of the particular reservoir system. The appropriate SDP model might be different under different hydrological regimes. Three SDP models have been described in this study to choose the appropriate model for the particular case. These models are similar to those used by Kottegoda(1980), Loucks et al.(1981), Stedinger et al. (1984) Wang and Adams (1986), Huang et al. (1991), Abdul Kader et al.(1994), and Raman and Chandramouli (1996).
Chapter Six 209 Reservoir Operation Optimization
Type Two
SDP with Marcov Chains
The probabilities of reservoir inflows may be used directly in order to obtain the expected values of the objective function. A more logical way is to weight the DP recursive equation with transition probability matrix (tpm) of storage states instead of inflow states. tpm represents the possible underlying Markov property of the reservoir states as explained in chapter 3. It replaces the unconditional probabilities of reservoir inflows to the conditional one. Assume that the process is ergodic which means that all states communicate and steady state probabilities exists. The recursive equation of the backward DP for Type Two is adopted from Kottagoda 1980 and given below:
m fn(i) = maximize [ Bn,r + ∑ qn,r(i,j) fn-1(j) ] (6.6) rij j=0
Where fn(i) Expected benefit at the start of the nth stage under an optimal policy, given that the storage at the start of the stage is i units.
fn-1(j) Expected benefit at the start of the (n-1)th stage under an optimal policy when the reservoir is in state j. j = i + q - r in which q is inflow during the nth stage and r is the release including losses.
Bt Benefit in stage t qn,r(i,j) Probability of the storage j in the reservoir at the end of the nth stage, conditional to a storage i at its start in which r denotes the release made during the stage.
It is P [St+1 / St ], the method of computations is described in chapter 3.
(e.g. in stage say Sep the probability of P (SOct / SSep ) will be employed) Type Three
At each stage or time period t, the optimal final storage volume Sl,t+1, depends on two state
variables: the initial storage volume Skt and the current inflow Qit. Let Bkilt is the Benefits
Chapter Six 210 Reservoir Operation Optimization
(or system performance) associated with an initial storage volume Skt and inflow Qit,a
release rkilt and a final storage volume Sl, t+1.The SDP recursive equation is given as follows ( Loucks et al. (1981) and Raman and Chandramouli (1996)). m n t n-1 ft (Skt, Qkt) = minimize [ Bkilt + ∑ P ij ft+1 (Sl,t+1, Qj )] (6.7) Si,t+1 j=1
Where Pij = P [Qt+1 / Qt ]
Probability of inflow Qt+1 occurs in time period t+1 given a known inflow of Qt in period t
and m is the number of inflow states. (e.g in stage say September the probability of P (QOct /
QSep ) will be employed) State transfer equation is
St+1 (beginning ) = St (ending)
In this case the probability of inflow is conditional (i.e. correlation between two consecutive inflows exists) and the current inflows are known perfectly. The steady state operating policy is derived on the basis of initial storage, current forecasted inflow, and the optimal final reservoir storage target. The discrete interval of inflow state may be different as discrete interval used in storage and release states of the model. This is due to the reason that inflow states are not incorporated in the state transformation equation.
Type Four
In Type Four model, the probability of inflow is conditional and that current inflows are still unkown. Inflows of the previous time step are employed as a state variable instead of the current inflow. The steady state operating policy is derived. The policy is a function of the initial storage, previous period inflow and the optimal release target. (Huang et al.1991, Abdelkader et al. 1994)
To model uncertain events such as streamflows, transition probabilities of time dependent
and decision dependent Marcov chain is determined. Let Ptij(r) equal the probability that the
inflow in period t is Qt given that the state in period t-1 is Qt-1..
t P ij = P [Qt | Qt-1 ]
Chapter Six 211 Reservoir Operation Optimization
Probability that inflow Qt occurs in time period t given a known inflow of Qt-1 in period t-1 and m is the number of inflow states. (e.g in stage say Sep the probability of P (QAug / QSep) will be employed)
The SDP recursive equation is given by Loucks et al. (1981) m n t n-1 ft (St,, Qt-1 ) = minimize ∑ P ij [ Losst + ft+1 (St+1 ,Qt ) ] (6.8)
rt j=1 Where m Number of class intervals considered in the tpm State transfer equation is
m St+1 = St + Qt - Rt
It is important to note that the discrete interval of inflow state should be same as discrete interval used in storage and release states of the model. This is due to the reason that inflow states are incorporated in the state transformation equation. Otherwise the results will be misleading.
Type one, type two and type three models needs inflow forcast model for verification. A forcast model for each time period is derived with the help of multiple regression or AR(x,x) model and updated for each 10 day period when the next period inflow is available. The updating procedure is called Kalman filter. Out of many forcast models, a 10 day forcast model(ARX(1,0)) is given by Kuo et al.( 1990) for Tanshui river basin reservoirs:
qt = a qt-1 + bpt + et (6.9)
Where q is inflow p is precipitation and e is error. In our case 10 day precipitation in advance in the basin is not available. Therefore the applicability of the model is limited. Therefore the forecast models developed of each 10 day period is based only on the lagged inflows of period t-1, t-2, . . ., t-n.
qt = a qt-1 + bqt + c (6.10) The models for Jhelum at Mangla, Indus at Tarbela and Chenab at Marala have been developed in chapter 5.
Chapter Six 212 Reservoir Operation Optimization
6.3 Problem Formulation for Multiple Objective Reservoirs
There are three main objectives in the operation of the multiple reservoir Indus River System.
• Firm water supply for irrigation
• Firm energy generation from the reservoirs.
• Incidental flood protection.
Multiple objectives are often conflicting and noncommensurable. Alam 1992 considered a single objective optimization problem of firm water supply and firm energy for Tarbela reservoir. He developed a trade off between water supply and energy by combining optimization and multicriteria decision analysis. Some other objectives, such as recreational volume, monthly period volume change and minimum and maximum release are considered as goals to be satisfied.
In this study the proposed dynamic programming model is flexible to include multiobjective purposes of the reservoirs with alternative non-linear objective functions. The algorithm solves multi-stage problem in a backward solution mode. A 10 day time step is selected for the analysis. Therefore, each stage represents a 10-day period t (t = 1,2 3 in September, t = 4,5,6 in August, ..., t = 34, 35, 36 in October). The stage transformation equation which shifts the multistage process from one stage to another is expressed as :
St+1 = St + qt - dt - A(St, St)[ et + RFt ] (6.11) Where
qt Inflow to reservoir in 10 day period t in MAF (known).
dt Expected release at a discrete level t in MAF.
St Storage content in the reservoir at the beginning of the 10 day period t in MAF.
et Evaporation in 10 day period t in ft. (known).
RFt Accretion to reservoir by rainfall in 10 day period t in ft.(known).
A(St ,St+1) Average surface area of the reservoir over the 10 day period t in acres.
Chapter Six 213 Reservoir Operation Optimization
Alternate objectives of water resources development were considered for the analysis. These were to minimize the shortfalls due to irrigation deficits between the irrigation supplies and downstream irrigation water requirement, to minimize the losses due to energy deficits between the energy generated and energy demand. Another objective was considered to be flood protection. The reservoir levels should have to be drawdown to reduce the intensity of floods during flood season every year. Minimum and maximum reservoir levels to mitigate the flood control and sedimentation were published by the planners for the test case. Therefore, another objective becomes to minimize, the deviation between actual resrvoir levels and envalope of minimum and maximum design levels. A flexible algorithm is adopted to observe the behaviour of various proposed objectives by choosing different combination of these functions. The proposed objective functions may be expressed as : Case (1) Minimizing Irrigation Shortfalls :
Ft1(St) = Min [ D1t - dk ]2 (6.12) Where
dk Expected release at a discrete level k in MAF.
Fmt (St)Return function where m is 1 in (6.12) m Index for specifing model, varies 1 to 7.
D1t Downstream irrigation demand in 10 day period t.
Case (2) Minimizing Power Generation Shortfalls :
2 Ft(St) = Min [D2t - Dk] (6.13)
D3t D2t = ------(1.024 * Ht* Eft)
if Ft2 < 0 Ft2 = 0
Chapter Six 214 Reservoir Operation Optimization
2 2 Otherwise [Ft (St) ] Where
Ht Average productive storage head in ft.
D2t Flow through turbines required to generate energy equal to demand at average productive storage head and hydropower efficiency in MAF.
D3t Downstream energy demand in 10 day period t.
EFt Plant efficiency at a given head.
Case (3) Minimizing Deviation between Storage & Design Rule
3 F t (St) = Min [ RMXt - St ] (6.14) If F (S ) > 0 ; 3 Ft (St) = 0 2 otherwise [ F3t (St) ]
4 Ft (St) = Min [ RMNt - St ] (6.15)
If F4t (St) < 0 ; 4 Ft (St) = 0 4t 2 otherwise [ F (St) ] Where
RMXt Maximum desirable reservoir content in 10 day period t to mitigate flood control in MAF.
RMNt Minimum desirable reservoir content in 10 day period t to mitigate sedimentation flushing in MAF.
Case (4) Combination of Equations (6.12), (6.14) and (6.15) :
5 Ft (St) = F1t + F3t + F4t (6.16)
Chapter Six 215 Reservoir Operation Optimization
Case (5) Combination of Equations (6.12) and (6.13) :
6 Ft (St) = F1t + ηF2t (6.17) where η is preference level constant or trade off coefficient for each 10 day period, varies between 0 and 1.
Case (6) Combination of Equations (6.16) and (6.17) :
7 Ft (St) = F1t + F2t + F3t + F4t (6.18)
The optimal return in the preceding stage is then added to the models in each of the above case: n n-1 ft (St,i) = Fmt (St) + f* t+1 (St ,qt ,dk ) (6.19)
Where m varies between 1 and 7
ft Total optimal shortfall or lossses in stage t for one of proposed case, m.
In addition to state transformation equation (6.11) the above proposed alternative objective functions are subject to physical and legal constraints. Some of these constraints are listedbelow:
Smin Sk < Smax (6.20)
dmin < dk < dmax (6.21)
dmax = St + qt - Smin - A(St, St+1) et + A(St, St+1) Pt (6.22)
KWHk < C . thk . (6.23) Where
Smin Minimum allowable storage or dead storage capacity for sedimentation in MAF (known).
Chapter Six 216 Reservoir Operation Optimization
Smax Maximum storage capacity in MAF (known). Hydropower efficiency.
KWHk Kilowatt hours of energy. C Plant capacity in KW.
thk Hours in period k.
The DP model solves the problem by taking successive steps back in time from an initial point, searching for an optimal release decision associated with one of the model in equations (2), (6), (7) or (8) each time. Selection of Discretization of stage variables is extremely important since it affects computer storage requirements, excecution time, and accuracy of results (Klemes 1977, Lobadie 1989). Klemes (1977) showed that a coarse discrete representation of storage could cause a collapse of solution. Further, with finer discretization of reservoir storage and release, higher objective function value could be obtained, along with more computation time (Bogardi et al. 1988). The scheme proposed by Klemes (1977) is adopted to select the number of discrete intervals for state variables. A
uniform discretization intervals were selected for the state variables Si and decision variable dk. These intervals are same for all stages. The computations required to solve the proposed deterministic problem with Bellman's principal of optimality are coded with FORTRAN + Wintrator on micro-computer. The flow chart of the basic part of the entire computation is shown in Figure 3.4.
6.4 Model Calibration of Reservoir Operation Optimization
Huang et al. (1991) stated, in fact, the most suitable type of DP for reservoir operation may be different case by case depending upon the circumstances. We can identify which type of DP model is best for different conditions by performing model tests with specific criteria. Therefore, model tests of these four types of DP models are required to judge the appropriate model for the case of Indus River multiple reservoir system.
6.4.1 Calibration of Dynamic Programming models
Yeh (1985) explained that only backward-moving stochastic dynamic programming can be applied when deriving the steady operating policy because of the consideration of stochastic inflow. In the meantime, stationary transition probabilities of inflows are assumed. A 10 day
Chapter Six 217 Reservoir Operation Optimization
time step has 36 periods in a year. For 36 periods in the 10 day time step in a year, there are 36 transition probability matrices (tpm). These are the conditional probabilities of the current inflows conditioned on the previous period inflow. To derive 36 number of tpm, an algorithm is developed on the basis of the procedure described in chapter 3. In dynamic programming the time step is backward. Therefore the input data is considered to be backward in time (e.g. Sep, Aug,. . ., Oct). The tpm program is made flexible to accept the inflow data backward in
time and then to compute the tpm for Qt+1 / Qt or Qt / Qt-1 employed in type three and type four models respectively. The computations for tpm were performed with the help of 82 years and 42 years of historical inflow data of Jhelum and Indus Rivers respectively. In the first instance 10 inflow states were considered for tpm. But zero rows were found in different inflow states in a number of tpm’s in different 10 day period. This is due to the fact that in very small states there may be some states where no flow occurs causing zero rows in the matrix. Five inflow states were found to be the suitable number for Jhelum at Mangla. But in case of Indus at Tarbela with five inflow states, there are still zero probabilities rows in tpm of following periods:
10 day period number 1, 14, 21, 23, 25, 28, 34, 35
Therefore in case of Indus at Tarbela, the inflow states have to reduce to 4. This is also due to the fact that the discrete interval was considered to be 0.2 MAF in storage and decision state. The same has to be adopted for inflow state for type IV model.
Type Two model requires unconditional inflow probability. It assumes the inflows are independent of the previous period inflows. For each of the 36 period in a year, there is unconditional inflow probability vector of every period. To derive 36 numbers of unconditional probabilities, an algorithm is developed on the basis of the procedure described in chapter 3. The program is coded in FORTRAN + Winteractor.
After calculating the recursive equation of each model type, a steady state solution is obtained for each DP model type. Usually three to four iterations are required to compute a steady state policy in the successive approximation procedure for each model type. This is due to the fact that number of stages is enough for convergence. Otherwise for example, to solve Loucks el. al. (1981) example in page 329 with the same program, about 21 iterations
Chapter Six 218 Reservoir Operation Optimization
are required to get steady state policy. This may be due to the fact that it is a two stage problem.
6.4.2 Calibration of Multiple Objectives
The algorithm in Figure 3.4 searches to find how to allocate water between years and to compute 10 day pattern of releases in order to optimize the objective functions in equations (6.12), (6.16), (6.17), and (6.18). The alternate functions are solved for mean 10 day inflows or transition probabilities of historic sequence by making a 36-stage problem. The performance under different alternatives is then analysed to identify the trade off among conflicting operation objectives of the reservoir. Water for agriculture is considered a 'primary objective' as it is more critical and sensitive for the development of the region. Therefore the function with minimizing irrigation shortfall is attached with all the proposed alternatives. In two of the functions (equations (6.12), and (6.16)) hydropower generation is considered only a residual which would follow the pattern of discharges for agriculture water use. Whereas in (6.17) and (6.18) a trade off coefficient ( 0 < n < 1 ) is imposed to observe the performance of the functions at various preference levels of n. To solve the problem, the reservoir states and decision vector were discretized with 49 discrete levels with a constant increments of 0.1 MAF in case of Mangla reservoir. In case of Tarbela, the reservoir states and decision vector were discretized with 44 discrete levels with a constant increments of 0.2 MAF.
Stationary policies for different alternatives were analysed to find best alternative for the real time 10 day reservoir operation. The criteria to find best alternative may be based on the amount of 10 day losses between the model operation and ideal operation (Houck 1982), where ideal operation is assumed to be that operation when 100 percent demand is met for all the operation periods. Optimal losses are function of demands and releases from the reservoir as shown in Figure 6.2.
Chapter Six 219 Reservoir Operation Optimization
s
e
s
s
o
L
l
a
m ir i o t rv
p e es
O R gla an W M a om te fr r es Re as qu le ir re em le e sib nt s in Po d iff er en t pe ri od s Figure 6.2 Convex loss function reservoir operation optimization
6.4.3 Calibration Results:
Type one Selecting each reservoir in the system, and selecting the objective function of various cases the steady state optimal policies are derived. Sample policies are shown in Figure. 6.3 and 6.4. These policies suggest optimal releases at various initial storage in the reservoir. Sample optimal losses for calibration case are shown in Figure 6.5.
Chapter Six 220 Reservoir Operation Optimization
Policies from Type One Model Mangla Reservoir, Only Irrig Shortfalls, Steady state policies for August to September
1.2
1.0 )
F Sep 3 A 0.8 M Sep 2 ( e s
a Sep 1
le 0.6 e Aug 3 l R Aug 2
ma 0.4 ti
p Aug 1 O 0.2
0.0 0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.8 Intitial Storage (MAF)
Figure 6.3a Optimal Policies for August and September from Type One model for Mangla Reservoir
Policies from Type One Model Mangla Reservoir, Only Irrig Shortfalls, Steady state policies for June to July
1.4
1.2 ) F 1.0 Jul 3 A M
( Jul 2
e
s 0.8
a Jul 1 le e 0.6 Jun 3 R l Jun 2 ma
ti 0.4 p Jun 1 O 0.2
0.0 0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.8 Intitial Storage (MAF)
Figure 6.3b Optimal Policies for June and July from Type One model for Mangla Reservoir
Chapter Six 221 Reservoir Operation Optimization
Policies from Type One Model Mangla Reservoir, All O.F. Steady state policies for August to September
1.2
1.0
)
F Sep 3 A 0.8 M Sep 2 ( e s
a Sep 1
le 0.6 e Aug 3 R l Aug 2
ma 0.4 ti
p Aug 1 O 0.2
0.0 0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.8 Intitial Storage (MAF)
Figure 6.4a Optimal Policies for August and September from Type One model for Mangla Reservoir with all Objective Functions
Policies from Type One Model Mangla Reservoir, All O.F. Steady state policies for June to July
1.8 1.6
) 1.4 F Jul 3 A 1.2 M Jul 2 ( e
s 1.0
a Jul 1 le e 0.8 Jun 3 l R Jun 2
ma 0.6 ti
p Jun 1
O 0.4 0.2 0.0 0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.8 Intitial Storage (MAF)
Figure 6.4b Optimal Policies for June and July from Type One model for Mangla Reservoir with all Objective Functions
Chapter Six 222 Reservoir Operation Optimization
Optimal Losses from Type One Model Mangla Reservoir, All O.F. Steady state policies for August to September
100 90 80
) 70 Sep 3 AF
M Sep 2
( 60 s
e Sep 1 s
s 50 o Aug 3 L l 40 a Aug 2 m i t 30 Aug 1 Op 20 10 0 0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.8 Intitial Storage (MAF)
Figure 6.5a Optimal losses for August and September from Type One model for Mangla Reservoir with all Objective Functions
Optimal Losses from Type One Model Mangla Reservoir, All O.F. Steady state policies for June to July
80
70
) 60 F Jul 3 A
M 50 Jul 2 Jul 1
sses ( 40 o Jun 3 L l a 30 Jun 2 m i t p 20 Jun 1 O
10
0 0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.8 Intitial Storage (MAF)
Figure 6.5b Optimal losses for June and July from Type One model for Mangla Reservoir with all Objective Functions
Chapter Six 223 Reservoir Operation Optimization
Type Two
The same procedure as for type one model is repeated employing the appropriate model. Selecting each reservoir one by one and selecting the objective function of various cases the steady state optimal policies are derived. Sample policies are shown in Figure 6.6. These policies suggest optimal releases at various initial storage in the reservoir. Sample optimal losses for calibration case are shown in Figure 6.7.
Type Three
The model is calibrated employing the appropriate formulation and tpm. Selecting each reservoir, and selecting the objective function of various cases the steady state optimal policies are derived. These policies are function of two variables, current inflow state and storage state. The inflow in this case may be the current forecasted flow. The computer output for one policy (e.g.Mangla reservoir and O.F. case 1) contains about 200 pages of the output.
Chapter Six 224 Reservoir Operation Optimization
Polices from Type Two Model (SDP) Mangla Reservoir 10 day Operation Aug-Sep
1.4 ) 1.2 F A M 1 ( Sep 2 e
a s 0.8 Sep 1 e l Aug 3 e 0.6 R Aug 2
Aug 1 a l
m 0.4
i t p
O 0.2
0 0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3 3.3 3.6 3.9 4.2 4.5 4.8 Initial Storage (MAF)
Figure 6.6a Optimal Policies for Sep and Aug from Type Two model for Mangla Reservoir
Polices from Type Two Model (SDP) Mangla Reservoir 10 day Operation Jun-July
1.6
) 1.4
F A
M 1.2
(
Jul 3 e 1 Jul 2 a s
e
l Jul 1 0.8 e Jun 3 R 0.6 Jun 2 a l Jun 1 m i
t 0.4 p
O 0.2
0 0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3 3.3 3.6 3.9 4.2 4.5 4.8 Initial Storage (MAF)
Figure 6.6b Optimal Policies for Jun and Jul from Type Two model for Mangla Reservoir
Chapter Six 225 Reservoir Operation Optimization
Optimal Losses from Type Two Mangla Reservoir 10 day Operation Aug-Sep
500
450
) F
400
350 M A ( Sep 2 e 300 Sep 1 a s
e 250 Aug 3 e l
R Aug 2
200
l Aug 1 a 150 m i t p
100 O 50
0 0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3 3.3 3.6 3.9 4.2 4.5 4.8 Initial Storage (MAF)
Figure 6.7a Optimal losses for Sep and Aug from Type Two model for Mangla Reservoir
Optimal Losses from Type Two Mangla Reservoir 10 day Operation Jun July
750
)
F 700
A
M 650 ( Jul 3
e Jul 2 s 600 a Jul 1 e l
e Jun 3 550 R
Jun 2 a l 500 Jun 1 m
i
t p 450 O
400 0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3 3.3 3.6 3.9 4.2 4.5 4.8 Initial Storage (MAF)
Figure 6.7b Optimal losses for Jun and Jul from Type Two model for Mangla Reservoir
Chapter Six 226 Reservoir Operation Optimization
Type Four
Type four model is calibrated employing the appropriate formulation and tpm of P(Qt/ Qt-1) . Selecting each reservoir and selecting the objective function of various cases the steady state optimal policies are derived. These policies are function of two variables, previous period inflow and storage state as shown in Figure. 6.8. The inflow in this case may be the previous period flow. The computer output for one policy (e.g. Mangla reservoir and O.F. case 1) contains about 200 pages of the output and not included here to conserve space.
Optimal Losses
Optimal Release oir serv Re e in P rag rev Sto iou s P er iod In flo w
Figure 6.8 Sample Optimal Releases/ loss in stage 30 of Type Four Model, Mangla Reservoir
Chapter Six 227 Reservoir Operation Optimization
6.5 Model Verification of Reservoir Operation Optimization
The operating polices derived by mathematical programming are only guides. Once developed they should be simulated and evaluated prior to their use in practice (Bhasker and Whitlach Jr., 1980, Karamouz and Houck 1982, Huang et al 1991). Therefore to evaluate the reliability of the derived operating policies by the Four types of the DP models, the Indus reservoirs system was simulated using the inflow data. A simple simulation model which acts on the basis of mass balance equation (6.11) is written and used to examine the derived policies. Historic streamflow records as well as forecasted inflow sequence generated by multiple regression models (already developed in chapter 5) were used in the verification. A 10 day 10 year period is selected for the simulation of all model types.
6.5.1 Mangla Reservoir
Type one
Calibrated models for type one in previous step are verified with the help of forecasted inflows because it requires current inflow which is unknown. Their performance under various objectives is evaluated if these are used in real time operation of the system. Selecting the policy of various cases the steady state optimal policies are verified. The verification results are shown in Figure 6.9.
Type Two
The same procedure as for type one model is repeated selecting the appropriate policy. Again a forecast model is incorporated because the current inflows are unknown. Selecting the calibrated policy for each objective type, the simulation of 10 year period was carried out to verify the type two model. Verification results are shown in Figure. 6.10. From the results, it is inferred that Type Two is comparatively conservative releasing zero flow in March 20- 31.
Chapter Six 228 Reservoir Operation Optimization
Comparison of Type One Rule with Design Rule Mangla Reservoir, Minimizing All O.F. Stationary policies, Simulation with Forecasted Flow
6.0
5.0
) 1985-86 F
A 4.0 1986-87 M 1987-88 e ( ag
r 3.0 1988-89 o t
S 1989-90 l a
m Max Rule i 2.0 t p Min Rule O
1.0
0.0 Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep 10 Day Period
Figure 6.9a Comparison of computed reservoir levels in Type One model for Mangla Reservoir
Optimal Releases under Type One Model, Verification Case Mangla Reservoir, Minimizing All 3 O.F. Stationary policies, Simulation with Forecasted Flow
2.00 1.80 1.60
) 1985-86 F
A 1.40 1986-87 M ( 1987-88 e 1.20 s a 1988-89 le
e 1.00 1989-90 R l
a 0.80 Irrig.Demand tim
p 0.60 O 0.40 0.20 0.00 Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep 10 Day Period
Figure 6.9b Comparison of computed reservoir releases in Type One model for Mangla Reservoir
Chapter Six 229 Reservoir Operation Optimization
Comparison of Type Two Rule with Design Rule Mangla Reservoir, Minimizing Irrigation Shortfall and Design Rules, Simulation with Forecasted Flow
6.0
5.0
) 1985-86 F
A 4.0 1986-87 M 1987-88 e ( ag
r 3.0 1988-89 o t
S 1989-90 l a
m Max Rule i 2.0 t p Min Rule O
1.0
0.0 Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep 10 Day Period
Figure 6.10a Comparison of computed reservoir levels in Type Two model for Mangla Reservoir
Optimal Releases under Type Two Model, Verification Case Mangla Reservoir, Minimizing Irrigation Shortfall and Design Rules, Simulation with Forecasted Flow
1.60
1.40 1985-86
) 1.20 1986-87 F
A 1987-88 1.00 (M
e 1988-89 s a
le 0.80 1989-90 e
l R Irrig.Demand 0.60 ma ti p
O 0.40
0.20
0.00 Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep 10 Day Period
Figure 6.10b Comparison of computed reservoir releases in Type Two model for Mangla Reservoir
Chapter Six 230 Reservoir Operation Optimization
Type Three Calibrated models for type three in previous step are verified with the help of forecasted inflows because it requires current inflow which is unknown. Their performance under various objectives is evaluated if these are used in real time operation of the system. Selecting the policy of various cases the steady state optimal policies are verified. The verification results are shown in Figure 6.11, 6.12 and 6.13.
Type Four
Type four requires previous period inflow which is known. Therefore no forecast is required. The policies generated by Type four are verified using 10 year historical inflows of Jhelum at Mangla. The verification results are shown in Figure 6.14, 6.15 and 6.16. The discussion of the results were made in chapter 9.
6.5.2 Tarbela Reservoir
Type one
For Tarbela reservoir models are calibrated. These are verified with the help of forecasted inflows because it requires current inflow which is unknown. Their performance under various objectives is evaluated if these are used in real time operation of the system. Selecting the policy of various cases the steady state optimal policies are verified. The verification results are shown in Figure 6.17.
Type Two
Experience with the results obtained from Mangla reservoir the model reveals that Type two is comparatively not successful for the test case. Therefore it is not used for Tarbela reservoir.
Type Three
Calibrated models for type three in previous step are verified with the help of forecasted inflows because it requires current inflow which is unknown. Their performance under various objectives is evaluated if these are used in real time operation of the system. Selecting the policy of various cases the steady state optimal policies are verified. The verification results are shown in Figure 6.18, 6.19 and 6.20.
Chapter Six 231 Reservoir Operation Optimization
Comparison of Type Three Rule with Design Rule Mangla Reservoir, Minimizing All O.F. Stationary policies, Simulation with Forecasted Flow
6.0
5.0
) 1985-86 F
A 4.0 1986-87 M 1987-88 e ( g a
r 3.0 1988-89 o t
S 1989-90 al
m Max Rule i 2.0 t p Min Rule O
1.0
0.0 Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep 10 Day Period
Figure 6.11a Comparison of computed reservoir levels in Type Three model for Mangla Reservoir
Optimal Releases under Type Three Model, Verification Case Mangla Reservoir, Minimizing All 3 O.F. Stationary policies, Simulation with Forecasted Flow
2.00 1.80
1.60 1985-86 ) F 1986-87
A 1.40
(M 1987-88
e 1.20 s
a 1988-89 le e 1.00 1989-90 R l 0.80 Irrig.Demand tima p 0.60 O 0.40 0.20 0.00 Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep 10 Day Period
Figure 6.11b Comparison of computed reservoir releases in Type Three model for Mangla Reservoir
Chapter Six 232 Reservoir Operation Optimization
Comparison of Type Three Rule with Design Rule M angla Reservoir, M inimizing Irrigigation and Power Shortfalls. Stationary policies, Simulation with Forecasted Flow
6.0
5.0
) 1985-86 F 4.0 A 1986-87 M 1987-88 age (
r 3.0 1988-89 o t
S 1989-90 l a
m Max Rule i
t 2.0 p Min Rule O
1.0
0.0 Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep 10 Day Period
Figure 6.12a Comparison of computed reservoir levels in Type Three model for Mangla Reservoir
Optimal Releases under Type Three Model, Verification Case Mangla Reservoir, Minimizing Irrigigation and Power Shortfalls. Stationary policies, Simulation with Forecasted Flow
2.00
1.80
1.60 1985-86 ) F 1.40 1986-87 A
M 1987-88 ( 1.20 1988-89 ease l
e 1.00 1989-90 R
al 0.80 Irrig.Demand m i t p 0.60 O 0.40
0.20 0.00 Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep 10 Day Period
Figure 6.12b Comparison of computed reservoir releases in Type Three model for Mangla Reservoir
Chapter Six 233 Reservoir Operation Optimization
Comparison of Type Three Rule with Design Rule Mangla Reservoir, Minimizing Irrigigation Shortfalls. Stationary policies, Simulation with Forecasted Flow
6.0
5.0
) 1985-86 F 4.0 A 1986-87 M
( 1987-88 e g a
r 3.0 1988-89 o t
S 1989-90 l
ma 2.0 Max Rule ti p Min Rule O
1.0
0.0 Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep 10 Day Period
Figure 6.13a Comparison of computed reservoir levels in Type Three model for Mangla Reservoir
Optimal Releases under Type Three Model, Verification Case Mangla Reservoir, Minimizing Irrigigation Shortfalls. Stationary policies, Simulation with Forecasted Flow 1.60
1.40
1985-86 ) 1.20 F
A 1986-87
(M 1.00 1987-88 ase
e 1988-89 l
e 0.80 1989-90 R al Irrig.Demand m 0.60 i t p O 0.40
0.20
0.00 Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep 10 Day Period
Figure 6.13b Comparison of computed reservoir releases in Type Three model for Mangla Reservoir
Chapter Six 234 Reservoir Operation Optimization
Comparison of Type Four Rule with Design Rule Mangla Reservoir, Minimizing All O.F. Stationary policies, Simulation with Forecasted Flow
6.0
5.0
) 1985-86 F
A 4.0 1986-87 M 1987-88 e ( ag
r 3.0 1988-89 o t
S 1989-90 l a
m Max Rule i 2.0 t p Min Rule O
1.0
0.0 Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep 10 Day Period
Figure 6.14a Comparison of computed reservoir levels in Type Four model for Mangla Reservoir
Optimal Releases under Type Four Model, Verification Case Mangla Reservoir, Minimizing All 3 O.F. Stationary policies, Simulation with Forecasted Flow
2.50
2.00 )
F 1985-86 A
M 1986-87 1.50 e ( 1987-88 eas l
e 1988-89
R 1989-90 al 1.00 m i
t Irrig.Demand p O 0.50
0.00 Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep 10 Day Period
Figure 6.14b Comparison of computed reservoir releases in Type Four model for Mangla Reservoir
Chapter Six 235 Reservoir Operation Optimization
Comparison of Type Four Rule with Design Rule M angla Reservoir, M inimizing Irrigation and Power Shortfalls. Stationary policies, Simulation with Forecasted Flow
6.0
5.0
) 1985-86 F 4.0 A 1986-87 M 1987-88 age (
r 3.0 1988-89 o t
S 1989-90 l a
m Max Rule i
t 2.0 p Min Rule O
1.0
0.0 Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep 10 Day Period
Figure 6.15a Comparison of computed reservoir levels in Type Four model for Mangla Reservoir
Optimal Releases under Type Four Model, Verification Case Mangla Reservoir, Minimizing Irrigation and Power Shortfalls. Stationary policies, Simulation with Forecasted Flow
2.50
2.00 )
F 1985-86 A 1986-87 M
( 1.50 e
s 1987-88 a le
e 1988-89 R l 1989-90 a 1.00
tim Irrig.Demand p O 0.50
0.00 Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep 10 Day Period
Figure 6.15b Comparison of computed reservoir releases in Type Four model for Mangla Reservoir
Chapter Six 236 Reservoir Operation Optimization
Comparison of Type Four Rule with Design Rule Mangla Reservoir, Minimizing Irrigation Shortfalls. Stationary policies, Simulation with Forecasted Flow
6.0
5.0
) 1985-86 F 4.0 A 1986-87 M
( 1987-88 e g a
r 3.0 1988-89 o t
S 1989-90 l
ma 2.0 Max Rule ti p Min Rule O
1.0
0.0 Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep 10 Day Period
Figure 6.16a Comparison of computed reservoir levels in Type Four model for Mangla Reservoir
Optimal Releases under Type Four Model, Verification Case Mangla Reservoir, Minimizing Irrigation Shortfalls. Stationary policies, Simulation with Forecasted Flow
2.00 1.80 1.60
) 1985-86 F
A 1.40 1986-87 M
( 1987-88 e 1.20 s a 1988-89 le
e 1.00
R 1989-90 l 0.80 Irrig.Demand tima p 0.60 O 0.40 0.20 0.00 Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep 10 Day Period
Figure 6.16b Comparison of computed reservoir releases in Type Four model for Mangla Reservoir
Chapter Six 237 Reservoir Operation Optimization
Comparison of Type One Rule with Design Rule Tarbela Reservoir, All O.F. Stationary policies, Simulation with Historic Flows
12.0
10.0
)
F 8.0 1985-86 A
M 1987-88 1988-89 age ( 6.0 or
t 1989-90 S l
a Design Rule m i t 4.0 Series6 p O
2.0
0.0 Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep 10 Day Period
Figure 6.17a Comparison of computed reservoir levels in Type One model for Tarbela Reservoir
Optimal Releases under Type One Model, Verification Case Tarbela Reservoir, All O.F. Stationary policies, Simulation with Historic Flows
3.5
3.0
2.5 ) F
A 1985-86 2.0 (M 1987-88 s a
le 1988-89 e
R 1989-90
l 1.5 a Irrig.Demand m i t p
O 1.0
0.5
0.0 Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep 10 Day Period
Figure 6.17b Comparison of computed reservoir releases in Type One model for Tarbela Reservoir
Chapter Six 238 Reservoir Operation Optimization
Comparison of Type Three Rule with Design Rule Tarbela Reservoir, Minimizing All O.F. Stationary policies, Simulation with Forecasted Flow
12
) F
10 A M
1985-86
( 8
e 1986-87 g
a 1987-88 r 6 1989-90 t o S Max Design Rule l
4 Min Design Rule i m a
t 2 p
O 0 Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep 10 Day Period
Figure 6.18a Comparison of computed reservoir levels in Type Three model for Tarbela Reservoir
Optimal Releases under Type Three Model, Verification Case Tarbela Reservoir, Minimizing All 3 O.F. Stationary policies, Simulation with Forecasted Flow
3.5
)
F 3 A
M 2.5
( 1985-86 e s 2 1986-87 a l e
1987-88 e 1.5 1989-90 Irrig.Demand l R 1
t i m a 0.5 p O 0 Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep 10 Day Period
Figure 6.18b Comparison of computed reservoir releases in Type Three model for Tarbela Reservoir
Chapter Six 239 Reservoir Operation Optimization
Comparison of Type Three Rule with Design Rule Tarbela Reservoir, Minimizing Irrig and Power Shortfalls. Stationary policies, Simulation with Forecasted Flow
12
)
F 10 A
M 1985-86 (
8 e
1986-87 g
a 1987-88 r 6 o 1989-90 t S 4 Max Design Rule l
a Min Design Rule
i m 2 t p O 0 Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep 10 Day Period
Figure 6.19a Comparison of computed reservoir levels in Type Three model for Tarbela Reservoir
Optimal Releases under Type Three Model, Verification Case Tarbela Reservoir, Minimizing Irrig and Power Shortfalls. Stationary policies, Simulation with Forecasted Flow
3.5
)
F 3 A
M 2.5 1985-86 e ( 2 1986-87 a s e
l 1987-88 e 1.5 1989-90
R Irrig.Demand 1 a l m i
t 0.5 p O 0 Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep 10 Day Period
Figure 6.19b Comparison of computed reservoir releases in Type Three model for Tarbela Reservoir
Chapter Six 240 Reservoir Operation Optimization
Comparison of Type Three Rule with Design Rule Tarbela Reservoir, Minimizing Irrig. Shortfalls. Stationary policies, Simulation with Forecasted Flow
12
) F
10 A M
1985-86
( 8
e 1986-87 g
a 1987-88 r 6 1989-90 t o S Max Design Rule l
4
a Min Design Rule i m
t 2 p
O 0 Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep 10 Day Period
Figure 6.20a Comparison of computed reservoir levels in Type Three model for Tarbela Reservoir
Optimal Releases under Type Three Model, Verification Case Tarbela Reservoir, Minimizing Irrig. Shortfalls. Stationary policies, Simulation with Forecasted Flow
3.5
) 3 F A
M 2.5 (
1985-86 e
s 2 1986-87 a
l e 1987-88 e 1.5 1989-90
R Irrig.Demand l
a 1 t i m
p 0.5 O
0 Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep 10 Day Period
Figure 6.20b Comparison of computed reservoir releases in Type Three model for Tarbela Reservoir
Chapter Six 241 Reservoir Operation Optimization
Type Four Type four requires previous period inflow which is known. Therefore no forecast is required. The policies generated by Type four are verified using 10 year historical inflows of Indus at Tarbela. The verification results for Tarbela reservoir are shown in Figures 6.21, 6.22. and 6.23. The discussion of the results is made in chapter 9. 6.6 Improved Strategies 6.6.1 Reservoir Operation Model • In previous studies on Indus reservoirs, only simulation type models have been used e.g. ROCKAT, COWRM and COMSIM. • In this study optimization techniques are employed in stead of simulation to improve the policies. • The model IBMR developed by World Bank and transferred to WAPDA, is an agroclimatological model. Less emphasis is made to the hydrology. Further for optimization it uses linear programming and may not be suited for reservoir optimization. • Stochastic Dynamic programming is first time applied to the Indus reservoirs for the improved strategies. • Various model types in DDP and SDP environments are formulated computerized and policies are derived to obtain best fit reservoir operation rules for the system. • Multiple objectives are employed and performance of the reservoirs under these objectives is evaluated. This procedure indicates the best mixture of objectives to be used for the operation policies of the reservoirs
6.6.2 Comments
The study presents a means of analysing alternative DP model types and alternative water development objectives and provides a procedure to identify an alternative which may be better than others. A range of alternate models have been proposed, formulated, computerized and tested in a multiple reservoir, multiobjective environment. The methodology is found to be effective for the test case. It reveals that the SDP model type four may be superior to others.
Chapter Six 242 Reservoir Operation Optimization
Comparison of Type Four Rule with Design Rule Tarbela Reservoir, Minimizing All O.F. Stationary policies, Simulation with Forecasted Flow
12
) F
10 A M
1985-86
( 8
e 1986-87 g
a 1987-88 r 6 1989-90 t o S Max Design Rule l
4
a Min Design Rule i m
t 2 p
O 0 Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep 10 Day Period
Figure 6.21a Comparison of computed reservoir levels in Type Four model for Tarbela Reservoir
Optimal Releases under Type Four Model, Verification Case Tarbela Reservoir, Minimizing All 3 O.F. Stationary policies, Simulation with Forecasted Flow
3.5
)
F 3 A
M 2.5
( 1985-86 e s 2 1986-87 a l e
1987-88 e 1.5 1989-90 Irrig.Demand l R 1
t i m a 0.5 p O 0 Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep 10 Day Period
Figure 6.21b Comparison of computed reservoir releases in Type Three model for Tarbela Reservoir
Chapter Six 243 Reservoir Operation Optimization
Comparison of Type Four Rule with Design Rule Tarbela Reservoir, Minimizing Irrig and Power Shortfalls. Stationary policies, Simulation with Forecasted Flows
12
)
F 10 A
M 1985-86 (
8 e
1986-87 g
a 1987-88 r 6 o 1989-90 t S 4 Max Design Rule l
a Min Design Rule
i m 2 t p O 0 Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep 10 Day Period
Figure 6.22a Comparison of computed reservoir levels in Type Four model for Tarbela Reservoir
Optimal Releases under Type Four Model, Verification Case Tarbela Reservoir, Minimizing Irrig and Power Shortfalls. Stationary policies, Simulation with Forecasted Flows )
3.5 F A 3 M (
2.5
e 1985-86 s
a 2 1986-87 l e . 1987-88 e 1.5 1989-90 R
l 1 Irrig.Demand a
t i m 0.5 p
O 0 Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep 10 Day Period
Figure 6.22b Comparison of computed reservoir releases in Type Three model for Tarbela Reservoir
Chapter Six 244 Reservoir Operation Optimization
Comparison of Type Four Rule with Design Rule Tarbela Reservoir, Minimizing Irrig Shortfalls. Stationary policies, Simulation with Forecasted Flow
12
) F
10 A M
1985-86
( 8
e 1986-87 g
a 1987-88 r 6 1989-90 t o S Max Design Rule l
4
a Min Design Rule i m
t 2 p
O 0 Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep 10 Day Period
Figure 6.23a Comparison of computed reservoir levels in Type Four model for Tarbela Reservoir
Optimal Releases under Type Four Model, Verification Case Tarbela Reservoir, Minimizing Irrig Shortfalls. Stationary policies, Simulation with Forecasted Flow
3.5
)
F 3 A
M 2.5
( 1985-86 e s 1986-87
a 2
l e 1987-88 e 1.5 1989-90 Irrig.Demand l R 1 i m a
t 0.5 p O 0 Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep 10 Day Period
Figure 6.23b Comparison of computed reservoir releases in Type Four model for Tarbela Reservoir
Chapter Seven 245 Stochastic Network Flow Programming
Chapter 7 STOCHASTIC NETWORK FLOW PROGRAMMING (Application to Indus River System)
7.1 Background
Stochastic Network Flow Programming (SNFP) model has been developed and applied for the optimization of the Indus River System. A 10 day time step is adopted for the working model. Network Flow Programming model uses advantages of the incorporation of the results obtained from SDP model for the optimization. A two stage mixed optimization procedure improves the operation of the system with minimum execution time and considering the nonlinear objectives at reservoir nodes and incorporating the uncertainty explicitly into the NFP model which is basically deterministic.
7.2 Suggested Approach
1. Select a 10 day time step which is a traditional time step for the actual and WAPDA operation criteria for the Indus Basin. 2. Represent the Indus Basin Irrigation System (IBIS) as a capacitated network in which nodes are reservoirs, local inflow locations, diversion locations and system input locations where flow or flow limits need to be specified. 3. Connect the nodes with the arcs which represent river and canal reaches, siphons or any other water transportation facility in the system. 4. Define the maximum and minimum capacities of the river and canal reaches of the Indus River System with the help of known data. 5. With the aid of step 2 to 4 construct the NFP model of the Indus River System. (see Figure 4.17) 6. From the first stage of the stochastic optimization of reservoir operation model in chapter 6, incorporate the stochastic operating policies at the nodes in the network where reservoirs are located. The stochasticity is explicitly included in the optimization and the model becomes Stochastic Network Flow Programming (SNFP) model. (see Figure 3.6)
Chapter Seven 246 Stochastic Network Flow Programming
7. Calibrate the model with 10 day historic data. 8. Verify the model with the forecasted inflows from section 5.16.4 and compare the results with the actual historic operation. 9. Illustrate the applicability and limitations of such methodology.
7.3 Problem Formulation for System Network Operation
The model is formulated as a SNFP (Stochastic Network Flow Programming) model based on the Out of Kilter algorithm (as described in detail in chapter 3) and operating policies derived from SDP. It is a multi period network model for the optimum operation of the large scale Indus River System. The inflow points, reservoirs, or diversion points are nodes and rivers and canals as arcs in a single period. This network is connected to the network in the next period by carry over arcs like reservoir storage. This formulation gives results in the form of a long chain of single period networks connected by storage carry over arcs. The model considers long term inflow data and optimizes the system operation as per demand in each time step. Formulation of the model requires an objective function. One of the simplest criteria can be the minimization of the total system cost.
7.3.1 The Objective Function NFP minimizes the total cost of flow through the arcs. It is therefore a minimum cost circulation problem. The rate of flow through the arcs can be regulated by proper assignment of cost to various arcs. The model can be written mathematically (in a primal problem form) as: (Phillips and Garcia-Diaz, 1981)
N N Min z = ∑ ∑ cij * fij (7.1) i=1 j=1 Where c is the cost from arc i t j, f is the flow from arc i to j and N is the number of nodes. The dual formulation of the objective function and other details has been defined in detail in chapter 3.
Chapter Seven 247 Stochastic Network Flow Programming
7.3.2 Constraints Following are the constraints associated with NFP model in (7.1)
∑ fij - ∑ fji = 0 (7.2) i j and Lij < fij < Uij (7.3)
fij ≥ 0 (7.4)
- fij ≤ - Lij (7.5)
Where Lij and Uij are the lower and upper bounds respectively on fij.
7.3.3 Node-Arc Representation The NFP model contains nodes and arcs. These represent the physical system of reservoirs, rivers, link canals, barrages and irrigation areas. Nodes in the OKA (Out of Kilter Algorithm) are the control points. These are used to represent reservoirs, inflow location diversion location (like barrages), system input locations and any other locations where flows or flow limits need to be specified. Arcs are the links that connect the nodes. These represent river and canal reaches, pipeline or tunnels, the water transportation facilities within the system. The nodes and arcs representation for Indus River System are shown in the computer output Table 7.3. For example we can see from the Table that arcs 57 through 87 represent the demand arcs of various canals in the system.
7.3.4 Strategy of Out of Kilter Algorithm The operating policy problem is the allocation and flow of water. The strategy of the algorithm is to determine the optimal allocation and flow of water through the system network.
7.3.5 Structure of the Program The flow chart in Figure 3.6 shows the basic structure of the program. The program is to solve a dual problem of minimum cost circulation. Possible state of an arc is determined and forces to be in kilter with the help of clearly defined steps as described in chapter 3 (Ford and Fulkerson, 1961). Scanned nodes and the flow augmenting paths are identified with the help of labeling procedure. The five principal steps of the OKA are described in chapter 3.
Chapter Seven 248 Stochastic Network Flow Programming
7.4 Application of the Methodology to Indus River System
Indus River basin is the main political and population center in Pakistan. The proposed optimization procedure has been applied to the Indus River multiple reservoir system. Water and Power Development Authority is responsible for the operation of the system. Their operation methodology is mainly based on the six month ahead operation criteria of the three reservoirs, computed from the historic data and the traditional methods.
The Indus River basin reservoirs and diversions are represented schematically in Figure 4.17. Few actual diversion structures and control points are integrated together in the given diagram to reduce the number of variables in the network optimization model.
• The proposed Stochastic Network Flow Programming model (SNFP) was first calibrated and compared to the actual operation record for the simulation period. In this case historic inflows of Chenab at Marala and Kabul at Nowshera has been used. • The SNFP model was verified by running the model with forecasted streamflows of Chenab and Kabul and comparing the results with actual operation.
7.4.1 The Operating Policy Problem The operational policy problem can be sketched in the form of a network flow problem as shown in Figure 7.1. In a water resources problem the OKA (Out of Kilter Algorithm) is repeatedly applied to solve the operating policy problem of the canal network. It optimizes the flow variables of the original network whose configuration is physically known. The simulation starts from the first 10 day period of the first year and OKA is applied. Then it is preceded to the next 10 day period of the first year and so on. The optimization procedure stops when simulation is completed for all the 10 day periods of all the years selected for the simulation as given below:
1 2 3 36 1 2 m-2 m-1 m t1 , t 1 , t 1 , . . . , t 1 , t 2 , t 2 , . . . , t n , t n , t n
Where m is counter for 10 day period (varies between 1 and 36) and n is the number of years of the simulation period.
Chapter Seven 249 Stochastic Network Flow Programming
10 day period 1 No. of year 1
10 day period 36 No of year = N
Figure 7.1 Stochastic network flow model is repeatedly applied for each 10 day period in each year
7.4.2 Calibration of the SNFP model For the modeling purpose, Indus Basin is shown in the form of nodes and arcs. (see Figure 4.17). Arcs are the links that connect the nodes. These represent river and canal reaches,
Chapter Seven 250 Stochastic Network Flow Programming
siphons, or the water transportation facilities within the system. Each arc is defined by three parameters
1. Upper bound of flow 2. Lower bound of flow 3. The cost per unit of flow
The upper bounds are the canal capacities and river reach capacities of the system. The lower bound is the minimum flow capacity of the reach. It is set equal to zero in most of the reaches. These bounds are obtained from the physical data as discussed in chapter 4. Since the OKA is a minimal cost network flow problem, water will be allocated first to those arcs with the lowest cost parameters. In other words, those arcs with the lowest cost parameters have the highest priority of allocation. Therefore the cost parameter will be called the priority factor because the term cost parameter may be misleading since the cost parameter is used not as an economic indicator but as a driving tool to specify relative priorities of allocation within the system (Chung et al. 1989).
The SNFP allocates water sequentially in each 10 day period, starting at the most upstream demand point and proceeding downstream. At time of short supply the downstream demands suffer shortage first, regardless of any intention. With the SNFP model, a system operator is given the ability to define the relative priorities of various demands. For the diversion canals from where the water is to be supplied for irrigation, the lower bound is set equal to zero and the upper bound is set equal to the demand from that canal. 10 day demand of the canals in the system belongs to the Water Accord demand.
The input to the system are stochastic releases from Tarbela and Mangla reservoir, inflow of Chenab at Marala, Kabul at Nowshera, Gomal, Ravi at Balloki, Sutlej at Sulaimanki. The flows from Ravi, Sutlej and Gomal are comparatively negligible as discussed in chapter 4. In the calibration case 10 day historic flows of these rivers have been used. A 10-year simulation period is selected and SNFP is run with the priority factors same for all the demand arcs. The symbols representing various diversion canals in the system are explained below. These symbols have been used in the computer output (see Table 7.1 and 7.3)
Chapter Seven 251 Stochastic Network Flow Programming
Symbol Node Description TarbelaR Release from Tarbela reservoir ManglaR Release from Mangla reservoir Thal Thal canal P/CRBC Paharpur/CRBC DGKhan Dera Ghazi Khan canal from Taunsa Barrage MZAF Muzafar Garh canal from Taunsa Barrage Patfee Pat Feeder + Desert + Beghari canals from Guddu Barrage Ghotki Ghotki feeder from Guddu Barrage Rohri Rohri + Khanpur E + Khanpur W + Nara from Sukkar Barrage NW/Rice NW + Rice + Dadu from Sukkar Barrage Fuleli Fuleli + Pinyari + Lined channel from Ghulam Muhammad (Kotri) Barrage Kalri Kalri from Ghulam Muhammad (Kotri) Barrage UJC Upper Jhelum canal LJC Lower Jhelum canal from Rasul Barrage Rang Rangpur from Trimmu Barrage TSHavl TS + Haveli Internal from Trimmu Barrage Punjnd Punjnad + Abbasian from Punjnad Barrage MRI Marala Ravi Internal UCI Upper Chenab Internal LCULGU Lower Chenab Upper + Lower Chenab Lower + Gugera Branch from Khanki Udepal Upper Depalpur from CBD Central Bari Doab LBD Lower Bari Doab from Balloki Ldepal Lower Depalpur from Balloki Sidnai Sidnai canal from Sidnai Barrage Esadiq East Sadiqia from Sulemanki Barrage Fordwh Fordwah from Sulemanki Barrage UPakpt Upper Pakpattan from BRBD link Qaimpr Qaimpur + Upper Bahawal from Islam Barrage UMails Upper Mailsi from Islam Barrage LBahaw Lower Bahawal from SMB LPakpt Lower Pakpattan from + Lower Mailsi from SMB
It is important to note that few actual diversion canals are integrated together in the model (as shown in the above list) to reduce the number of variables in the network optimization model. The model was run first with the known capacities of each canal and diversion structure, the stochastic releases from the reservoirs and historic inflows of Chenab and Kabul. In this run the priority factor was same for each demand point. The water was sequentially allocated starting at the most upstream demand point and proceeding downstream in each time period. If the priority factor is same, the downstream demands suffer shortage first, regardless of any intention, if there is short supply in the system inputs.
Chapter Seven 252 Stochastic Network Flow Programming
This happens in the last seven canals (Esadiq through LPakpt) in above list when the system is calibrated with same priority factor. The shortage of water supply was observed in Rabi season only in these canals during the 10 year simulation period. Enough supply was available in Kharif season and there was no remarkable shortage during the model run.
In the next run the priority factors are altered and adjusted for the best results. Best results were achieved when the model is calibrated for the priority factor 3 to all demand points except the last seven canals in the above list. The priority factor for these canals was given 4. The 10 day 10 year simulation results for the period 1985-86 to 1994-95 were performed. The 10 day results of each demand point are summed to get annual summary. It is presented in Table 7.1. The computer output of the first 10 day period of first year of the simulation was presented in Table 7.2. The results are compared with the actual demand. It is shown in Figure 7.2. Results indicated that with Water Accord demands there are some shortages in LBD, LJC and TSHavil canals. These shortages are mostly in Rabi seasons. However these were the best possible results with the given inputs, given canal capacities and known inflows. These results may be improved providing some more storages and higher capacities in some canals in the system and utilizing some of the outflows which are going to the sea.
Comparison of Actual and Computed Canal Allocations Calibration of Stochastic Network Flow Model
25.0
1985-86 1986-87 20.0 1987-88 AF)
M 1988-89 ( s
n 15.0 1989-90 o i t a
c 1990-91 o l 10.0 1991-92 Al l
a 1992-93 u 1993-94 Ann 5.0 1994-95 Actual 0.0 i r li I I n d q a e r F vl al al ai al D ki D C i C e p l Pt n e a l aw wh gpr ails GU A CE ha BC k ot ep ep F m dn nj ad n LJ Th Se t UC i UJ MR ah Ka LB rd Fu CB M a Rohri RI Si UL B Gh CR Pu MZ Pa To R ES U LD TSH Qa LPakPt UD Fo UPa L C P/ DGK NW L Name of Canal
Figure 7.2 Comparison of actual and computed canal allocations during model calibration
Chapter Seven 253 Stochastic Network Flow Programming
Observed and computed flows in Rohri Canal 1994-95
40
35
30 ) s f 25 Observed
1000 x c 20 (
e Computed g r 15 ha c s i
D 10
5
0 l t r r g c n v p b Ju Ju Jan Oc Se Fe Ap Au De Ma No May 10 Day Period
Figure 7.3 Comparison of observed and computed discharges in Rohri Canal during model calibration
Chapter Seven 254 Stochastic Network Flow Programming
Table 7.1 Summary results for calibration of stochastic network flow programming model, simulation period 1985-95.
Chapter Seven 255 Stochastic Network Flow Programming
Table 7.2 Sample result of calibration of SNFP model 10 day time period 10 year simulation for 1985-1995 (values in 1000 x cfs) Dual Arc I J HI LO Flow Cost Location of Arc Cost 1 66 1 11381 11381 11381 0 0 Indus [ Unregulated flows ] 2 66 16 3755 3755 3755 0 0 Jhelum [ unregulated flows ] 3 66 32 2779 2779 2779 0 0 Chenab 4 66 4 1400 1400 1400 0 0 Kabul 5 66 9 100 100 100 0 0 Gomal 6 66 39 310 310 310 0 0 Ravi 7 66 48 200 200 200 0 0 Sutluj 8 66 1 100 0 100 0 0 Tarbela 9 66 16 100 0 100 0 0 Mangla 10 1 2 80000 0 11481 0 0 Tarbela to d/s [ Regulated flows 11 2 4 80000 0 6265 0 0 By pass to power generation 12 4 5 80000 0 12881 0 0 Indus to Jinnah 13 5 7 80000 0 12221 0 0 Jinnah to chasma 14 7 23 2170 0 1307 0 0 CJ link 15 7 9 82000 0 10784 0 0 Gomal Confluence on Indus 16 9 10 82000 0 10884 0 0 Indus to Taunsa 17 10 28 1200 0 1100 0 0 TP link 18 10 13 85500 0 8314 0 0 Taunsa to Guddu 19 13 58 85500 0 7485 0 0 Guddu to sukkur 20 58 61 85500 0 3025 0 0 Sukkur to Ghulam Muhammad Ghulam Muhammad to Sink node 21 61 65 85500 0 1126 0 0 /Sea 22 16 17 30000 0 3855 0 0 Mangla to d/s [ Regulated flows ] 23 17 19 30000 0 2371 0 0 By pass to power generation 24 19 34 780 0 0 0 0 UJ link 25 19 21 30000 0 3565 0 0 Rasul 26 21 36 1900 0 1307 0 0 RQ link 27 21 23 35000 0 1728 0 0 Confluence with CJ link 28 23 24 35000 0 3035 0 0 Confluence with chenab 29 24 25 35000 0 3035 0 0 Trimmu 30 25 46 1100 0 1100 0 0 TS link with Sidnai Barrage 31 25 27 40000 0 1285 0 0 with Ravi 32 27 28 40000 0 1285 0 0 TP link with Jehlum/ravi 33 28 29 45000 0 2385 0 0 with Sutluj 34 29 30 55000 0 2385 0 0 Punjnad 35 30 13 55000 0 1305 0 0 Punjnad to Guddu 36 32 34 30000 0 1262 0 0 Marala to UJ link confluence + Kha 37 32 39 2200 0 0 0 0 MR link with Ravi 38 32 40 310 0 310 0 0 BRBD link from Marala to Ravi 39 32 43 780 0 638 0 0 Upper Chenab (UC) link with Ravi 40 34 36 45000 0 143 0 0 Qadirabad + RQ link 41 36 44 1450 0 1450 0 0 QB link from Chenab to Balloki 42 36 24 55000 0 0 0 0 Chenab / Jhelum confluence 43 39 40 32000 0 310 0 0 Ravi from Origin to BRBD link 44 40 43 32000 0 300 0 0 Ravi from BRBD to UC link
Chapter Seven 256 Stochastic Network Flow Programming
Dual Arc I J HI LO Flow Cost Location of Arc Cost 45 43 44 32000 0 938 0 0 Ravi from UC to Balloki 46 44 46 32000 0 250 0 0 Ravi Balloki to Sidnai Barrage 47 46 27 32000 0 0 0 0 Ravi Sidnai to Jhelum (confluence) 48 46 55 1500 0 940 0 0 SMB Link Sidhnai to Mailsi Siphon 49 44 48 2500 0 1078 0 0 BS link I & II with Sutlej 50 48 52 32000 0 70 0 0 Sutlej at Islam Barrage 51 52 55 32000 0 0 0 0 Sutlej at SMB link 52 55 29 70000 0 0 0 0 Sutlej above Punjand 53 2 3 5216 0 5216 0 0 Tarbela power house [Energy Flow 54 3 4 5216 0 5216 -2 -3 T 55 17 18 1484 0 1484 0 0 Mangla Power house [Energy Flow 56 18 19 1484 0 1484 -2 -3 M 57 5 6 660 0 660 -2 -3 Thal [ Tarbela command Irrigation 58 7 8 130 0 130 -2 -3 Paharpur/CRBC 59 10 12 760 0 760 -2 -3 DG Khan 60 10 11 710 0 710 -2 -3 Muzafarghar 61 13 14 1554 0 1554 -2 -3 Pat feeder/Desert/Beghari 62 13 15 580 0 580 -2 -3 Ghotki 63 58 60 2710 0 2710 -2 -3 Khairpur West/Rohri/Nara/Khairpur 64 58 59 1750 0 1750 -2 -3 N.W/Rice/Dadu 65 61 63 1130 0 1130 -2 -3 Fuleli/Pinyari/Lined channel 66 61 62 669 0 669 -2 -3 Kalri 67 61 64 100 0 100 -1 -2 To sea Upper Jhelum [Mangla command 68 19 20 290 0 290 -2 -3 Irrig 69 21 22 530 0 530 -2 -3 Lower Jhelum [ do Irrig 70 25 26 150 0 150 -2 -3 Rangpur 71 25 38 500 0 500 -2 -3 TS link + Haveli 72 30 31 1080 0 1080 -2 -3 Punjnad/Abassian MR Internal [Chenab command 73 32 33 100 0 100 -2 -3 Irrig 74 32 37 469 0 469 -2 -3 Upper Cenab Internal (UCI) From Khanki to L.C. upper & 75 34 35 1119 0 1119 -2 -3 Lower 76 40 41 170 0 170 -2 -3 U.Depalpur from BRBD link 77 40 42 150 0 150 -2 -3 Central Bari Doab from BRBD link Lower Bari Doab from Balloki 78 44 45 700 0 700 -2 -3 Barra 79 44 67 360 0 360 -2 -3 Lower Depalpur from BS-I link 80 46 47 410 0 410 -2 -3 Sidnai from Sidnai Barrage 81 48 49 469 0 469 -3 -4 East Sadqia from Sulemanki 82 48 50 270 0 270 -3 -4 Fordwah from Sulemanki 83 48 51 469 0 469 -3 -4 Upper Pakpattan from Sulemanki Qaimpur+Upper Bahawal from 84 52 53 70 0 70 -3 -4 Islam B 85 52 54 0 0 0 -3 -4 Upper Mailsi from Islam Barrage Lower bahawal from SMB Link 86 55 56 360 0 360 -3 -4
Chapter Seven 257 Stochastic Network Flow Programming
Dual Arc I J HI LO Flow Cost Location of Arc Cost Lower Pakpattan + Lower Mailsi 87 55 57 580 0 580 -3 -4 from 88 6 65 4000 0 660 0 0 Thal [ Tarbela command Exit flows 89 8 65 4000 0 130 0 0 Paharpur/CRBC 90 12 65 4000 0 760 0 0 DG Khan 91 11 65 4000 0 710 0 0 Muzafarghar 92 14 65 4000 0 1554 0 0 Pat feeder/Desert/Beghari 93 15 65 4000 0 580 0 0 Ghotki Khairpur 94 60 65 4000 0 2710 0 0 West/Rohri/Nara/Khairpur 95 59 65 4000 0 1750 0 0 N.W/Rice/Dadu 96 63 65 4000 0 1130 0 0 Fuleli/Pinyari/Lined channel 97 62 65 4000 0 669 0 0 Kalri 98 64 65 90000 0 100 0 0 to sea Upper Jhelum [Mangla command 99 20 65 4000 0 290 0 0 Exit 100 22 65 4000 0 530 0 0 Lower Jhelum [ do Exit 101 26 65 4000 0 150 0 0 Rangpur 102 38 65 4000 0 500 0 0 TS link + Haveli 103 31 65 4000 0 1080 0 0 Punjnad/Abassian MR Internal [ Chenab command 104 33 65 4000 0 100 0 0 Exit 105 37 65 4000 0 469 0 0 Upper Cenab Internal (UCI) From Khanki to L.C. upper & 106 35 65 4000 0 1119 0 0 Lower 107 41 65 4000 0 170 0 0 U.Depalpur from BRBD link 108 42 65 4000 0 150 0 0 Central Bari Doab from BRBD link Lower Bari Doab from Balloki 109 45 65 4000 0 700 0 0 Barra 110 67 65 4000 0 360 0 0 Lower Depalpur from BS-I link 111 47 65 4000 0 410 0 0 Sidnai from Sidnai Barrage 112 49 65 4000 0 469 0 0 East Sadqia from Sulemanki 113 50 65 4000 0 270 0 0 Fordwah from Sulemanki 114 51 65 4000 0 469 0 0 Upper Pakpattan from Sulemanki Qaimpur+Upper Bahawal from 115 53 65 4000 0 70 0 0 Islam b 116 54 65 4000 0 0 0 0 Upper Mailsi from Islam Barrage 117 56 65 4000 0 360 0 0 Lower bahawal from SMB Link Lower Pakpattan + Lower Mailsi 118 57 65 4000 0 580 0 0 fro 119 65 66 1E+08 0 20125 0 0 Sink to source node Total Project Cost -53516
Chapter Seven 258 Stochastic Network Flow Programming
7.4.3 Verification of the SNFP model After calibrating the model all the priority factors and the canal capacities are fixed. In the verification case the current inflows of the system are unknown. At the reservoir locations the stochastic operating policies are incorporated. It does no require the current inflows as discussed in chapter 6. These policies require previous period inflows. The flows from Chenab and Kabul entering into the system requires forecast because current inflows are unknown. Forecast models for these rivers have been developed in chapter 5. The forecasted inflows are therefore incorporated in the SNFP model at these locations. Although a major part of the inputs into the system are from the reservoirs, therefore any forecast error at Chenab and Kabul may be small when considering the system as a whole. The flows entering from Ravi, Sutlej and Gomal are negligible as compared to the quantities of other inputs. However a mean inflow of these minor rivers is considered as inputs to the system. Forecast models for these minor rivers do not improve the system performance due to the negligible inflows.
After the data is setup for verification case, the model is simulated for the 10 year period with a time step of 10 day. These results of each demand point are summed to get annual summary and presented in Table 7.3. The computer output of the first 10 day period of first year of the simulation was presented in Table 7.4. The results are compared with the actual demand (Figure 7.4). The shortages are found in Rabi season in some of the canals. The results indicated that with Water Accord demands there are some shortages in Rabi season in LBD, LJC and TSHavil canals. No remarkable shortage was found in Kharif seasons.
7.5 Improved Strategy
• The capability to provide different relative priorities to the different demand points. • The complete system is under control and can be viewed like a physical view of the system. • The incorporation of SDP operating policies replaces the deterministic problem to a stochastic one. • The results of the network optimization model indicated that the model may be used as an efficient tool for the 6 month ahead criteria of the Indus reservoir system
Chapter Seven 259 Stochastic Network Flow Programming
7.6 Comments
A comparative study was made between the model performance and actual operation data for the Indus River System. In all cases, model performance was superior to that of the historical operational records. It reveals through case study that networks flow optimization with out of kilter algorithm may be one of the best technique for the system excluding the nodes in the network where reservoirs are located. These specific nodes are optimized with the stochastic dynamic programming.
Comparison of Actual and Computed Canal Allocations 1994-95 to 2003-04 Validation of of Stochastic Network Flow Model
25.0 1994-95 20.0 1995-96 )
F 1996-97 A
M 1997-98 ( s
n 15.0 1998-99 o ti a
c 1999-00 llo 10.0 2000-01 l A
a 2001-02 u n
n 2002-03 A 5.0 2003-04 Actual 0.0 l l i i i i r r t t li I n q a r h U r e F al a a v al k D i C C p l ils a P P nd e t h e le a n d a w G A CE h BC k k o m ep F ep u d a nj ngp S LJ Th d a i UCI H t a UJ L MR LBD Ka ahaw r CB F M a Ro u D RI D Si B Gh P CR MZ Pa To R ES U L LP TS Qa / U Fo UP W L P DGK LCU N Name of Canal
Figure 7.4 Comparison of actual and computed canal allocations during model validation 1995-96 to 2003-04
Chapter Seven 260 Stochastic Network Flow Programming
Table 7.3 Summary results for validation of stochastic network flow programming model, simulation period 1994-95 to 2003-04.
Chapter Seven 261 Stochastic Network Flow Programming
Table 7.4 Sample result of validation of SNFP model 10 day time period 10 year simulation for 1995-96-2003-04 (values in 1000 x cfs) Dual Arc I J HI LO Flow Cost Location of Arc Cost 1 66 1 11280 11280 11280 0 0 Indus [ Unregulated flows ] 2 66 16 4255 4255 4255 0 0 Jhelum [ unregulated flows ] 3 66 32 1800 1800 1800 0 0 Chenab 4 66 4 839 839 839 0 0 Kabul 5 66 9 100 100 100 0 0 Gomal 6 66 39 310 310 310 0 0 Ravi 7 66 48 200 200 200 0 0 Sutluj 8 66 1 100 0 100 0 0 Tarbela [ Reservoir Initial conditions ] 9 66 16 100 0 100 0 0 Mangla 10 1 2 80000 0 11380 0 0 Tarbela to d/s [ Regulated flows ] 11 2 4 80000 0 6164 0 0 By pass to power generation 12 4 5 80000 0 12219 0 0 Indus to Jinnah 13 5 7 80000 0 11559 0 0 Jinnah to chasma 14 7 23 2170 0 2170 0 0 CJ link 15 7 9 82000 0 9259 0 0 Gomal Confluence on Indus 16 9 10 82000 0 9359 0 0 Indus to Taunsa 17 10 28 1200 0 1160 0 0 TP link 18 10 13 85500 0 6729 0 0 Taunsa to Guddu 19 13 58 85500 0 6400 0 0 Guddu to sukkur 20 58 61 85500 0 1940 0 0 Sukkur to Ghulam Muhammad 21 61 65 85500 0 41 0 0 Ghulam Muhammad to Sink node /Sea 22 16 17 30000 0 4355 0 0 Mangla to d/s [ Regulated flows ] 23 17 19 30000 0 2871 0 0 By pass to power generation 24 19 34 780 0 780 0 0 UJ link 25 19 21 30000 0 3285 0 0 Rasul 26 21 36 1900 0 1450 0 0 RQ link 27 21 23 35000 0 1305 0 0 Confluence with CJ link 28 23 24 35000 0 3475 0 0 Confluence with chenab 29 24 25 35000 0 3475 0 0 Trimmu 30 25 46 1100 0 1100 0 0 TS link with Sidnai Barrage 31 25 27 40000 0 1725 0 0 with Ravi 32 27 28 40000 0 1725 0 0 TP link with Jehlum/ravi 33 28 29 45000 0 2885 0 0 with Sutluj 34 29 30 55000 0 2885 0 0 Punjnad 35 30 13 55000 0 1805 0 0 Punjnad to Guddu 36 32 34 30000 0 339 0 0 Marala to UJ link confluence + Khanki 37 32 39 2200 0 0 0 0 MR link with Ravi 38 32 40 310 0 112 0 0 BRBD link from Marala to Ravi 39 32 43 780 0 780 0 0 Upper Chenab (UC) link with Ravi 40 34 36 45000 0 0 0 0 Qadirabad + RQ link 41 36 44 1450 0 1450 0 0 QB link from Chenab to Balloki 42 36 24 55000 0 0 0 0 Chenab / Jhelum confluence 43 39 40 32000 0 310 0 0 Ravi from Origin to BRBD link
Chapter Seven 262 Stochastic Network Flow Programming
Dual Arc I J HI LO Flow Cost Location of Arc Cost 44 40 43 32000 0 102 0 0 Ravi from BRBD to UC link 45 43 44 32000 0 882 0 0 Ravi from UC to Balloki 46 44 46 32000 0 194 0 0 Ravi Balloki to Sidnai Barrage 47 46 27 32000 0 0 0 0 Ravi Sidnai to Jhelum (confluence) 48 46 55 1500 0 940 0 0 SMB Link Sidhnai to Mailsi Siphon 49 44 48 2500 0 1078 0 0 BS link I & II with Sutlej 50 48 52 32000 0 70 0 0 Sutlej at Islam Barrage 51 52 55 32000 0 0 0 0 Sutlej at SMB link 52 55 29 70000 0 0 0 0 Sutlej above Punjand 53 2 3 5216 0 5216 0 0 Tarbela power house [ Energy Flows ] 54 3 4 5216 0 5216 -2 -3 T 55 17 18 1484 0 1484 0 0 Mangla Power house [ Energy Flows ] 56 18 19 1484 0 1484 -2 -3 M Thal [ Tarbela command Irrigation 57 5 6 660 0 660 -2 -3 flows ] 58 7 8 130 0 130 -2 -3 Paharpur/CRBC 59 10 12 760 0 760 -2 -3 DG Khan 60 10 11 710 0 710 -2 -3 Muzafarghar 61 13 14 1554 0 1554 -2 -3 Pat feeder/Desert/Beghari 62 13 15 580 0 580 -2 -3 Ghotki Khairpur West/Rohri/Nara/Khairpur 63 58 60 2710 0 2710 -2 -3 East 64 58 59 1750 0 1750 -2 -3 N.W/Rice/Dadu 65 61 63 1130 0 1130 -2 -3 Fuleli/Pinyari/Lined channel 66 61 62 669 0 669 -2 -3 Kalri 67 61 64 100 0 100 -1 -2 To sea Upper Jhelum Mangla command 68 19 20 290 0 290 -2 -3 Irrigation flows Lower Jhelum [ do Irrigation 69 21 22 530 0 530 -2 -3 flows ] 70 25 26 150 0 150 -2 -3 Rangpur 71 25 38 500 0 500 -2 -3 TS link + Haveli 72 30 31 1080 0 1080 -2 -3 Punjnad/Abassian MR Internal [ Chenab command 73 32 33 100 0 100 -2 -3 Irrigation Flows ] 74 32 37 469 0 469 -2 -3 Upper Cenab Internal (UCI) From Khanki to L.C. upper & Lower + 75 34 35 1119 0 1119 -2 -3 Gugera 76 40 41 170 0 170 -2 -3 U.Depalpur from BRBD link 77 40 42 150 0 150 -2 -3 Central Bari Doab from BRBD link 78 44 45 700 0 700 -2 -3 Lower Bari Doab from Balloki Barrage 79 44 67 360 0 360 -2 -3 Lower Depalpur from BS-I link 80 46 47 410 0 354 -2 -3 Sidnai from Sidnai Barrage 81 48 49 469 0 469 -3 -4 East Sadqia from Sulemanki 82 48 50 270 0 270 -3 -4 Fordwah from Sulemanki 83 48 51 469 0 469 -3 -4 Upper Pakpattan from Sulemanki
Chapter Seven 263 Stochastic Network Flow Programming
Dual Arc I J HI LO Flow Cost Location of Arc Cost Qaimpur+Upper Bahawal from Islam 84 52 53 70 0 70 -3 -4 barrage 85 52 54 0 0 0 -3 -4 Upper Mailsi from Islam Barrage Lower bahawal from SMB Link 86 55 56 360 0 360 -3 -4
Lower Pakpattan + Lower Mailsi from 87 55 57 580 0 580 -3 -4 SMB Link 88 6 65 4000 0 660 0 0 Thal [ Tarbela command Exit flows ] 89 8 65 4000 0 130 0 0 Paharpur/CRBC 90 12 65 4000 0 760 0 0 DG Khan 91 11 65 4000 0 710 0 0 Muzafarghar 92 14 65 4000 0 1554 0 0 Pat feeder/Desert/Beghari 93 15 65 4000 0 580 0 0 Ghotki Khairpur West/Rohri/Nara/Khairpur 94 60 65 4000 0 2710 0 0 East 95 59 65 4000 0 1750 0 0 N.W/Rice/Dadu 96 63 65 4000 0 1130 0 0 Fuleli/Pinyari/Lined channel 97 62 65 4000 0 669 0 0 Kalri 98 64 65 90000 0 100 0 0 to sea Upper Jhelum [Mangla command Exit 99 20 65 4000 0 290 0 0 flows ] 100 22 65 4000 0 530 0 0 Lower Jhelum [ do Exit flows ] 101 26 65 4000 0 150 0 0 Rangpur 102 38 65 4000 0 500 0 0 TS link + Haveli 103 31 65 4000 0 1080 0 0 Punjnad/Abassian MR Internal [ Chenab command Exit 104 33 65 4000 0 100 0 0 Flows ] 105 37 65 4000 0 469 0 0 Upper Cenab Internal (UCI) From Khanki to L.C. upper & Lower + 106 35 65 4000 0 1119 0 0 Gugera 107 41 65 4000 0 170 0 0 U.Depalpur from BRBD link 108 42 65 4000 0 150 0 0 Central Bari Doab from BRBD link 109 45 65 4000 0 700 0 0 Lower Bari Doab from Balloki Barrage 110 67 65 4000 0 360 0 0 Lower Depalpur from BS-I link 111 47 65 4000 0 354 0 0 Sidnai from Sidnai Barrage 112 49 65 4000 0 469 0 0 East Sadqia from Sulemanki 113 50 65 4000 0 270 0 0 Fordwah from Sulemanki 114 51 65 4000 0 469 0 0 Upper Pakpattan from Sulemanki Qaimpur+Upper Bahawal from Islam 115 53 65 4000 0 70 0 0 barrage 116 54 65 4000 0 0 0 0 Upper Mailsi from Islam Barrage 117 56 65 4000 0 360 0 0 Lower bahawal from SMB Link Lower Pakpattan + Lower Mailsi from 118 57 65 4000 0 580 0 0 SMB Link 119 65 66 1E+08 0 18984 0 0 Sink to source node
Chapter Eight 264 Conjunctive Operation of Muliple Reservoirs Simulation
Chapter 8
CONJUNCTIVE OPERATION OF MULTIPLE RESERVOIRS SIMULATION
8.1 Background
Due to wide temporal variation of rainfall the agriculture in the Indus Plains is highly dependent on irrigation water supplies. With a large arable land base of 79 million acres, only about 35 million acres are canal commanded. Pakistan still has the potential of bringing several million acres of virgin land under irrigation. An important impediment in the way of this development is insufficient control over flood waters of the rivers. With virtually no limit on availability of land, it is unfortunate to allow large quantities of Indus water go waste to the sea. It is true that Indus Basin Irrigation System (IBIS) suffers severe water shortages including distributional inequity in critical crop demand periods. On the other hand, it is equally true that there is very large quantum of flood flows still being wasted to sea due to inadequate control over rivers. The real physical aspect is that about ¼ of Indus River System average annual flow is still being wasted to sea during the flood season of about three months. In post-Tarbela (1976-2003) about an annual average of 35.2 MAF (43.5 BCM) escaped to the sea. Out of this a sizeable volume of water can be effectively controlled and utilized to bring about prosperity to millions. The schematic diagram of the Indus Basin Irrigation System (IBIS), which feeds water to the existing irrigated area is shown in Figure 4.2. The major rivers contributing to the system are the Indus, the Chenab and the Jhelum. Rabi inflows which are 15 % to 20 % of the annual inflows are fully utilized, whereas the surplus flows of the flood period, normally from mid- June to mid-September are available for further development. The water supplies to the IBIS are presently regulated at Tarbela, Mangla and Chashma reservoirs. The irrigation water requirements of the Indus River System at Kalabagh/Chashma are estimated as 97.7 MAF (120.6 BCM) in year 2010-11. These water requirements cannot be met from the existing reservoirs. Mangla dam raisings with a useable storage of 2.9 MAF (3.6 BCM), Akhori dam with a useable storage of over 6.75 MAF (8.32 BCM), Basha Diamer dam with a useful storage of 7.3 MAF (9.1 BCM) and Kalabagh dam with useable storage of 6.1 MAF (7.5
Chapter Eight 265 Conjunctive Operation of Muliple Reservoirs Simulation
BCM) are planned to be added to the infrastructure to augment the regulation of the river Indus for irrigation water supplies and hydropower generation (WAPDA and Techno Consult, 2005). Tarbela, Mangla and Chashma reservoirs are losing their capacities progressively due to siltation. They have lost their gross capacities by 3.14 MAF (3.88 BCM), 1.18 MAF (1.46 BCM) and 0.37 MAF (0.46 BCM) by the year 2003, respectively. Thus the total gross storage which was originally 18.37 MAF (22.69 BCM) has been reduced to 13.68 MAF (16.89 BCM), which is about 26 % of their original capacity. Therefore, water resources development and its optimal management need an important position in the national investment planning. WAPDA, being responsible for the development of water and power, presented a Vision 2025 Programme for development of water resources and hydropower of the country. This programme includes Akhori dam, which has been upgraded recently in priority by the Government of Pakistan.
Akhori Dam is one of the off-channel storage projects located between the Jhelum and Indus Rivers and is planned to be implemented to develop additional surface water storage capacity in the IBIS. It will be a 128 m high dam located on Nandna Kas. The reservoir to be fed from surplus flows of Tarbela reservoir will have a reservoir surface area of 228 Km2 (88 sq. miles) at its maximum pool level. Tarbela-Akhori lined conveyance channel with its intake in the Siran pocket of Tarbela reservoir will be used to fill the Akhori reservoir. The reservoir will not submerge any of the major infrastructures like Motorway, G.T. Road, Railway Line, or town etc. The project would have a power potential of 600 MW and generate energy of around 2189 GWh per year.
8.2 Indus multi-reservoir system for conjunctive operation study
Conjunctive operation study was based on the simultaneous operation of existing and future reservoirs. In the Indus Basin Basha, Akhori and Kalabagh are the future reservoirs (WAPDA and Techno Consult, 2005). Schematic of Indus multi-reservoir system for conjunctive operation study is shown in Figure 8.1.
Chapter Eight 266 Conjunctive Operation of Muliple Reservoirs Simulation
Indus Jhelum
Basha Dam Power house Mangla Dam
Tarbela dam Chenab Kabul Akhori dam Ravi+Sutlej
Kalabagh dam Legend Punjnad Existing reservoirs Future reservoirs To sea Power house
Figure 8.1 Schematic of Indus multi-reservoir system for conjunctive operation study
Recently, new Government in year 2008 decided not to construct proposed Kalabagh dam due to political conflicts between provinces. Under the light of this decision, Kalabagh dam was given least priority while considering the possible cases for conjunctive operation study. The cases investigated are summarized in Table 8.1.
Chapter Eight 267 Conjunctive Operation of Muliple Reservoirs Simulation
Table 8.1 Possible cases for conjunctive operation
No. Case Year Remarks
1 T+M 2012 Do Nothing
2 T+M+B 2015 Basha comes in operation in year 2015
Basha comes in operation in year 2015 and 3 T+M+B+A 2020 Akhori in year 2020
Basha comes in operation in year 2015 and 4 T+M+B+A+K 2030 Akhori in year 2020 and Kalabagh in year 2030 T =Tarbela, M=Mangla, B=Basha, A=Akhori, K=Kalabagh A brief description of the future reservoirs on Indus River is given below: Basha Dam Basha Dam is proposed on the Indus River, located about 40 km downstream of Chilas town. (Figure 8.2). Its live capacity will be about 6.4 MAF (7.9 BCM). It will not only regulate the supplies of irrigation water but also to generate 4500 MW of hydro-power. It is expected to be commissioned by the year 2015. Area-elevation-capacity curves and other necessary data required for present research were collected from WAPDA. (WAPDA and NEAC Consultants (2004))
Figure 8.2 Aview of Basha Diamer damsite on Indus River (WAPDA and Diamer Basha Consultant, 2007).
Chapter Eight 268 Conjunctive Operation of Muliple Reservoirs Simulation
Akhori Dam Akhori dam is an off channel storage of Tarbela dam and located on a small stream called Nandna Kas. It will be a 420 feet (128 m) high dam. A 37.5 Km long Tarbela-Akhori conveyance canal will be used to get surplus water from Tarbela Reservoir at intake level 1492 ft AMSL. Out of total storage of 7.6 MAF, live capacity of the dam will be about 7 MAF. The project would have a power potential of about 600 MW and generate energy of 2100 GWh per year. Area-elevation-capacity curves and other necessary data required for present research were collected from WAPDA. (WAPDA and Techno Consult, 2005)
Kalabagh Dam Kalabagh dam is located on Indus River, about 120 miles (193 km) downstream of Tarbela and 16 miles (26 km) upsteam of Jinnah Barrage. Live capacity of the dam will be 6.1 MAF out of total storage volume 7.9 MAF. Initially, installed capacity for power generation will be 2776 MW. It will produce 11,423 GWh of energy per year. Area-elevation-capacity curves and other necessary data were collected from WAPDA. (WAPDA and Kalabagh Consultants (1988)
8.3 Model Calibration for Conjunctive Operation of Multi Reservoirs
SDP rules derived in chapter 6 were used to operate the system. In a first step existing reservoirs Mangla and Tarbela were operated with the help of historic inflows on 10 daily basis. Model performance was evaluated with the help of comparison between historic and model releases and between historic and model power generation. Figure 8.3 presents reservoir levels and power generated at Tarbela. Figure 8.4 shows reservoir levels during simulation period. Table 8.2 shows summary results for Tarbela reservoir. Comparison between historic and model operation is shown in Chapter 9. Results indicated that model shows better performance as compared to historic operation.
Chapter Eight 269 Conjunctive Operation of Muliple Reservoirs Simulation
Figure 8.3 Reservoir elevations and power generated during simulation period 1962-2004
Figure 8.4 Reservoir levels during operation at Tarbela during simulation period 1962-2004
Chapter Eight 270 Conjunctive Operation of Muliple Reservoirs Simulation
Figure 8.5 Releases from the dam during simulation period 1962-2004
Figure 8.6 Average water levels in the reservoir for period 1962-2004
Chapter Eight 271 Conjunctive Operation of Muliple Reservoirs Simulation
Table 8.2 Summary results of reservoir simulation at Tarbela (calibration case) ------Year Inflow Demand Release Shortage % Shortage Evaporation Spill Energy (KAF) (KAF) (KAF) (KAF) (KAF) (KAF) ( GWH) ------1962- 63 66700.00 58097.89 51388.60 6709.29 11.55 266.37 16856.15 16945.990 1963- 64 67154.00 58097.89 50376.30 7721.59 13.29 250.26 16047.63 15978.992 1964- 65 55887.00 58097.89 50082.10 8015.79 13.80 250.82 5882.30 15895.594 1965- 66 66295.00 58097.89 49834.10 8263.79 14.22 258.86 16423.07 16055.493 1966- 67 70245.00 58097.89 49171.93 8925.96 15.36 262.95 20333.39 15967.623 1967- 68 65280.00 58097.89 48770.27 9327.63 16.06 259.92 16281.95 15752.607 1968- 69 64969.00 58097.89 49775.90 8321.99 14.32 257.59 15751.62 15968.126 1969- 70 62384.00 58097.89 48975.60 9122.29 15.70 257.59 12296.32 16030.468 1970- 71 59371.00 58097.89 47308.60 10789.29 18.57 261.13 12092.65 15488.894 1971- 72 57924.00 58097.89 48362.60 9735.29 16.76 253.82 9734.61 15732.402 1972- 73 82410.00 58097.89 49378.90 8719.00 15.01 288.81 31029.99 16939.051 1973- 74 54827.00 58097.89 48120.34 9977.56 17.17 237.38 7921.46 14809.565 1974- 75 60890.00 58097.89 48055.60 10042.29 17.29 253.32 12331.63 15489.595 1975- 76 60150.00 58097.89 48856.60 9241.29 15.91 257.85 11180.43 15880.468 1976- 77 63844.00 58097.89 48308.30 9789.59 16.85 254.91 15117.77 15464.448 1977- 78 74059.00 58097.89 48264.93 9832.96 16.92 276.87 25406.55 16147.282 1978- 79 63044.00 58097.89 49136.93 8960.96 15.42 260.24 14159.79 15958.134 1979- 80 60002.00 58097.89 47665.93 10431.96 17.96 266.20 11828.88 15800.934 1980- 81 63444.00 58097.89 47176.60 10921.29 18.80 280.72 16034.69 15990.771 1981- 82 51833.00 58097.89 45660.60 12437.29 21.41 229.07 6131.53 13802.472 1982- 83 61956.00 58097.89 48575.30 9522.59 16.39 242.86 12577.25 15280.249 1983- 84 70758.00 58097.89 49263.60 8834.29 15.21 270.70 21713.26 16337.189 1984- 85 57483.00 58097.89 47520.63 10577.26 18.21 235.98 10264.09 14669.004 1985- 86 61967.00 58097.89 46494.30 11603.59 19.97 249.87 14379.57 14678.614 1986- 87 60789.00 58097.89 50231.90 7865.99 13.54 248.04 10305.90 15962.812 1987- 88 79909.00 58097.89 50925.40 7172.49 12.35 293.37 28294.19 17523.363 1988- 89 62410.00 58097.89 49747.60 8350.29 14.37 264.32 12614.22 16302.290 1989- 90 76667.00 58097.89 50890.60 7207.29 12.41 288.66 24979.84 17517.594 1990- 91 69883.00 58097.89 52090.90 6006.99 10.34 277.94 17686.46 17599.051 1991- 92 64163.00 58097.89 50743.60 7354.29 12.66 262.14 13575.83 16634.914 1992- 93 53250.00 58097.89 50153.25 7944.64 13.67 249.26 2722.22 16155.276 1993- 94 78575.00 58097.89 49708.60 8389.29 14.44 273.04 29123.54 16452.787 1994- 95 60200.00 58097.89 48857.27 9240.63 15.91 253.74 10908.67 15534.194 1995- 96 65643.00 58097.89 49608.60 8489.29 14.61 258.86 15400.75 16045.466 1996- 97 52769.00 58097.89 47473.40 10624.50 18.29 228.37 5321.89 14279.122 1997- 98 65132.00 58097.89 50511.90 7585.99 13.06 266.45 14300.15 16617.195 1998- 99 75723.00 58097.89 50680.90 7416.99 12.77 290.41 23776.08 17423.320 1999- 0 59851.00 58097.89 51173.10 6924.79 11.92 261.08 9324.89 16770.014 2000- 1 53647.00 58097.89 48091.10 10006.79 17.22 246.29 5798.28 15229.073 2001- 2 57915.00 58097.89 48118.60 9979.29 17.18 257.80 9349.55 15704.016 2002- 3 68460.00 58097.89 49700.90 8396.99 14.45 281.73 17990.60 16891.729 2003- 4 50702.00 58097.89 48984.90 9112.99 15.69 246.97 1861.85 15588.140
Mean 63775.33 58097.90 49148.03 8949.87 15.40 260.30 14407.42 15983.198 ------No of Years when no Shortage Occur = 0 No of Years when Shortage Occur = 42 No of Years when more than 5% Shortage Occur = 42 No of Years when more than 10% Shortage Occur = 42 ------
In the second reservoir Mangla dam, model performance was evaluated with the help of comparison between historic and model operation (See Chapter 9). Figure 8.7 presents reservoir levels and power generated at Mangla dam. Figure 8.8 shows reservoir levels during simulation period. Table 8.3 shows summary results for Mangla reservoir operation.
Chapter Eight 272 Conjunctive Operation of Muliple Reservoirs Simulation
Figure 8.7 Reservoir elevations and power generated at Mangla during simulation period 1922-2004
Figure 8.8 Reservoir levels at Mangla during simulation period 1922-2004
Chapter Eight 273 Conjunctive Operation of Muliple Reservoirs Simulation
Figure 8.9 Optimal releases from the dam during simulation period 1922-2004
Figure 8.10 Average reservoir levels at Mangla during simulation 1922-2004
Chapter Eight 274 Conjunctive Operation of Muliple Reservoirs Simulation
Table 8.3 Summary results of reservoir simulation at Mangla (calibration case) ------Year Inflow Demand Release Shortage % Shortage Evaporation Spill Energy (KAF) (KAF) (KAF) (KAF) (KAF) (KAF) ( GWH) ------1922- 23 23513.00 30390.00 24138.13 6251.88 20.57 199.99 235.71 6826.775 1923- 24 27467.00 30390.00 23797.07 6592.93 21.69 198.68 3220.68 6795.490 1924- 25 20774.00 30390.00 20746.42 9643.58 31.73 191.92 131.78 5882.184 1925- 26 22745.00 30390.00 21862.82 8527.18 28.06 197.55 378.75 6273.197 1926- 27 19541.00 30390.00 19016.17 11373.83 37.43 190.24 645.20 5424.324 1927- 28 26276.00 30390.00 23912.18 6477.82 21.32 208.91 1459.55 6898.835 1928- 29 21248.00 30390.00 19499.23 10890.77 35.84 191.53 1731.23 5465.914 1929- 30 28912.00 30390.00 27484.50 2905.50 9.56 218.94 1688.44 7973.798 1930- 31 25251.00 30390.00 22882.95 7507.05 24.70 196.69 2004.99 6512.219 1931- 32 22286.00 30390.00 21753.48 8636.52 28.42 194.22 639.49 6190.780 1932- 33 26025.00 30390.00 22471.23 7918.77 26.06 198.95 2870.82 6491.768 1933- 34 18026.00 30390.00 17982.90 12407.10 40.83 187.37 331.37 5104.441 1934- 35 23893.00 30390.00 22418.43 7971.57 26.23 198.65 1156.83 6409.659 1935- 36 25431.00 30390.00 23789.60 6600.40 21.72 214.61 1329.92 6897.384 1936- 37 20661.00 30390.00 20683.09 9706.91 31.94 195.79 0.00 5793.313 1937- 38 23434.00 30390.00 22627.31 7762.69 25.54 208.57 635.91 6474.953 1938- 39 23900.00 30390.00 23222.34 7167.66 23.59 204.96 286.27 6633.678 1939- 40 16629.00 30390.00 16575.46 13814.54 45.46 187.36 0.00 4643.204 1940- 41 16972.00 30390.00 16297.83 14092.17 46.37 186.10 0.00 4526.699 1941- 42 25806.00 30390.00 25597.63 4792.37 15.77 202.45 160.41 7275.030 1942- 43 25267.00 30390.00 24340.07 6049.93 19.91 200.37 795.91 6926.651 1943- 44 19036.00 30390.00 18499.87 11890.13 39.13 188.62 491.04 5173.281 1944- 45 21535.00 30390.00 20772.33 9617.67 31.65 194.55 474.36 5889.994 1945- 46 14961.00 30390.00 15119.63 15270.37 50.25 184.69 0.00 4171.310 1946- 47 15864.00 30390.00 15314.13 15075.87 49.61 185.36 0.00 4211.504 1947- 48 28041.00 30390.00 24948.50 5441.50 17.91 212.79 2882.78 7237.609 1948- 49 24918.00 30390.00 24154.58 6235.42 20.52 212.06 718.70 6964.582 1949- 50 30304.00 30390.00 24653.77 5736.23 18.88 210.98 4897.38 7120.065 1950- 51 21292.00 30390.00 21638.93 8751.07 28.80 195.57 0.00 6166.671 1951- 52 20490.00 30390.00 20289.66 10100.34 33.24 191.64 0.00 5688.595 1952- 53 18545.00 30390.00 18136.96 12253.04 40.32 188.67 0.00 5066.303 1953- 54 26671.00 30390.00 25730.20 4659.80 15.33 219.94 394.41 7472.568 1954- 55 19341.00 30390.00 19014.47 11375.53 37.43 190.15 514.02 5412.880 1955- 56 23660.00 30390.00 22127.19 8262.81 27.19 196.35 1357.90 6286.146 1956- 57 30806.00 30390.00 26370.93 4019.07 13.22 215.32 4089.59 7652.027 1957- 58 26919.00 30390.00 25018.07 5371.93 17.68 205.70 1614.03 7203.760 1960- 61 17501.00 30390.00 17163.37 13226.63 43.52 187.16 0.00 4819.164 1961- 62 15464.00 30390.00 15380.73 15009.27 49.39 183.46 0.00 4242.833 1962- 63 21893.00 30390.00 21848.77 8541.23 28.11 199.95 0.00 6241.623 1963- 64 23647.00 30390.00 22011.30 8378.70 27.57 194.66 1385.61 6251.865 1964- 65 26673.00 30390.00 23340.23 7049.77 23.20 209.80 3236.50 6742.099 1965- 66 21805.00 30390.00 21084.38 9305.62 30.62 193.04 0.00 5963.797 1966- 67 23938.00 30390.00 24184.83 6205.17 20.42 199.98 0.00 6874.899 1967- 68 21847.00 30390.00 21632.89 8757.11 28.82 193.20 77.90 6047.441 1968- 69 25394.00 30390.00 24011.20 6378.80 20.99 205.58 1155.18 6888.171 1969- 70 16481.00 30390.00 16070.37 14319.63 47.12 186.44 0.00 4466.901 1970- 71 13266.00 30390.00 13470.10 16919.90 55.68 180.12 0.00 3706.234 1971- 72 21253.00 30390.00 20629.83 9760.17 32.12 193.73 20.77 5865.953 1972- 73 29452.00 30390.00 27258.20 3131.80 10.31 220.66 2060.03 7943.282 1973- 74 17192.00 30390.00 17394.64 12995.36 42.76 187.46 0.00 4810.536 1974- 75 24018.00 30390.00 22135.21 8254.79 27.16 194.01 985.40 6306.080 1975- 76 25875.00 30390.00 23695.73 6694.27 22.03 196.12 2218.97 6695.896 1976- 77 18754.00 30390.00 18709.36 11680.64 38.44 189.74 0.00 5262.878 1977- 78 25000.00 30390.00 23956.80 6433.20 21.17 200.73 703.73 6831.731 1978- 79 20599.00 30390.00 20569.84 9820.16 32.31 191.76 0.00 5723.039 1979- 80 23154.00 30390.00 22759.33 7630.67 25.11 195.38 0.00 6372.081 1980- 81 24308.00 30390.00 24287.85 6102.15 20.08 214.11 196.47 6982.472 1981- 82 20086.00 30390.00 19785.52 10604.48 34.89 192.84 73.22 5530.924 1982- 83 28643.00 30390.00 26123.17 4266.83 14.04 213.95 2012.64 7586.276 1983- 84 19436.00 30390.00 19179.30 11210.70 36.89 191.64 124.45 5402.186 1984- 85 15376.00 30390.00 15416.30 14973.70 49.27 184.49 30.25 4285.366 1985- 86 26514.00 30390.00 24376.10 6013.90 19.79 210.99 1742.55 7016.128 1986- 87 29014.00 30390.00 26314.00 4076.00 13.41 220.56 2458.35 7660.936 1987- 88 26641.00 30390.00 24070.02 6319.98 20.80 203.55 2279.23 6863.327 1988- 89 22720.00 30390.00 21878.22 8511.78 28.01 195.83 747.87 6210.262 1989- 90 26987.00 30390.00 25110.57 5279.43 17.37 220.79 1555.20 7298.023 1990- 91 33303.00 30390.00 26218.54 4171.46 13.73 239.13 6662.03 7810.578 1991- 92 31648.00 30390.00 26698.03 3691.97 12.15 221.71 4189.25 7831.549 1992- 93 26046.00 30390.00 25697.20 4692.80 15.44 211.76 908.95 7381.109 1993- 94 25462.00 30390.00 23864.53 6525.47 21.47 200.18 1387.53 6839.315 1994- 95 28331.00 30390.00 25354.13 5035.87 16.57 208.60 2796.50 7276.484 1995- 96 31734.00 30390.00 26032.60 4357.40 14.34 220.56 5365.68 7584.898 1996- 97 21676.00 30390.00 20183.90 10206.10 33.58 191.88 1257.55 5688.277 1997- 98 26037.00 30390.00 25108.10 5281.90 17.38 225.05 949.03 7316.650 1998- 99 15362.00 30390.00 15266.50 15123.50 49.76 183.58 0.00 4150.373 1999- 0 13934.00 30390.00 13694.80 16695.20 54.94 179.71 0.00 3730.053 2000- 1 10925.00 30390.00 10995.83 19394.17 63.82 171.80 0.00 2965.808 2001- 2 16513.00 30390.00 16009.08 14380.92 47.32 185.81 0.00 4373.522 2002- 3 22762.00 30390.00 22428.31 7961.69 26.20 207.00 2.08 6376.936 2003- 4 16458.00 30390.00 16601.00 13789.00 45.37 185.15 0.00 4506.328 Mean 22833.40 30390.00 21564.93 8825.07 29.04 199.71 1084.05 6142.324 ------
Chapter Eight 275 Conjunctive Operation of Muliple Reservoirs Simulation
No of Years when no Shortage Occur = 0 No of Years when Shortage Occur = 82 No of Years when more than 5% Shortage Occur = 82 No of Years when more than 10% Shortage Occur = 80 ------
8.4 Model Validation for Conjunctive Operation of Multi Reservoirs
Model validation was carried out with the help of re-running the model with another inflow data set. Thomas Feiring model (Clarke (1973)) was used to generate the data for another 40 years period for 10 daily basis.
Figure 8.11 Reservoir elevations and power generated at Tarbela during simulation with forecasted flows for 42 year period (validation case)
Chapter Eight 276 Conjunctive Operation of Muliple Reservoirs Simulation
Figure 8.12 Reservoir levels at Tarbela during simulation with forecasted flows for 42 year period (validation case)
Figure 8.13 Optimal releases at Tarbela during simulation with forecasted flows for 42 year period (validation case)
Chapter Eight 277 Conjunctive Operation of Muliple Reservoirs Simulation
Figure 8.14 Average reservoir water levels at Tarbela during simulation with forecasted flows for 42 year period (validation case)
Table 8.4 Summary results of reservoir simulation at Tarbela (validation case) ------Year Inflow Demand Release Shortage % Shortage Evaporation Spill Energy (KAF) (KAF) (KAF) (KAF) (KAF) (KAF) ( GWH) ------2005- 6 70118.52 58097.89 51167.84 6930.05 11.93 273.02 20409.88 16968.092 2006- 7 50079.77 58097.89 46982.24 11115.66 19.13 228.62 2801.47 14273.823 2007- 8 58369.05 58097.89 48145.13 9952.76 17.13 236.29 9624.02 14927.496 2008- 9 58262.38 58097.89 51383.34 6714.55 11.56 255.03 7263.82 16527.465 2009- 10 57658.63 58097.89 49344.64 8753.25 15.07 257.95 7554.15 16180.983 2010- 11 70421.78 58097.89 51771.21 6326.68 10.89 278.06 18396.68 17358.572 2011- 12 60944.80 58097.89 49333.17 8764.73 15.09 254.36 11348.90 15948.809 2012- 13 67329.32 58097.89 49025.48 9072.41 15.62 265.96 18163.26 15964.115 2013- 14 56228.27 58097.89 47562.82 10535.07 18.13 246.46 8385.67 15188.458 2014- 15 80639.33 58097.89 48993.79 9104.11 15.67 282.26 31492.25 16646.971 2015- 16 61454.02 58097.89 47635.86 10462.04 18.01 244.53 13732.85 14740.324 2016- 17 69144.19 58097.89 50080.75 8017.14 13.80 274.29 18902.17 16761.498 2017- 18 64315.32 58097.89 47874.93 10222.96 17.60 269.27 16717.11 15658.875 2018- 19 57480.69 58097.89 46551.50 11546.40 19.87 242.05 9747.02 14491.521 2019- 20 60051.02 58097.89 50243.21 7854.68 13.52 266.04 9959.48 16579.012 2020- 21 49490.05 58097.89 42175.75 15922.15 27.41 226.28 6660.85 12957.108 2021- 22 64813.61 58097.89 49359.91 8737.98 15.04 261.32 15972.57 16077.237 2022- 23 61342.25 58097.89 48807.21 9290.68 15.99 244.86 11036.01 15288.330 2023- 24 73410.99 58097.89 51690.17 6407.73 11.03 268.11 22287.08 16900.311 2024- 25 71976.02 58097.89 48366.49 9731.41 16.75 275.64 23078.86 16126.363 2025- 26 64712.25 58097.89 49640.87 8457.03 14.56 264.86 14398.86 16362.146 2026- 27 58623.42 58097.89 49513.32 8584.57 14.78 240.97 9043.93 15487.608 2027- 28 55822.44 58097.89 48418.29 9679.61 16.66 243.27 7452.17 15307.602 2028- 29 62132.42 58097.89 48967.86 9130.03 15.71 272.74 12822.35 16376.109 2029- 30 69702.38 58097.89 49160.50 8937.39 15.38 278.77 20401.76 16479.947 2030- 31 71189.98 58097.89 49441.83 8656.06 14.90 262.75 21641.54 16034.420 2031- 32 50512.20 58097.89 46982.38 11115.52 19.13 229.92 3267.87 14292.981 2032- 33 74433.37 58097.89 50525.84 7572.05 13.03 278.60 23516.04 16952.545 2033- 34 59445.55 58097.89 48842.52 9255.37 15.93 266.95 10829.64 16228.230 2034- 35 60176.50 58097.89 46137.42 11960.48 20.59 248.63 12582.10 14713.670 2035- 36 64149.25 58097.89 49577.39 8520.50 14.67 261.66 14810.00 16241.659 2036- 37 67073.50 58097.89 49737.98 8359.92 14.39 256.95 17287.95 15923.772 2037- 38 64231.43 58097.89 50277.54 7820.36 13.46 264.12 13286.42 16511.137 2038- 39 70000.41 58097.89 51839.34 6258.56 10.77 269.77 17304.21 17202.307 2039- 40 61635.39 58097.89 49276.83 8821.06 15.18 274.47 13431.79 16571.631 2040- 41 55578.96 58097.89 45688.97 12408.93 21.36 239.73 9366.16 14152.530 2041- 42 57174.59 58097.89 47147.72 10950.18 18.85 255.65 9985.72 14982.409 2042- 43 60499.35 58097.89 46149.64 11948.26 20.57 273.95 13832.76 15468.639 2043- 44 71359.54 58097.89 49730.66 8367.23 14.40 277.14 21188.01 16626.426 2044- 45 56090.95 58097.89 49958.33 8139.57 14.01 249.43 6129.62 15919.590
Mean 62951.84 58097.89 48837.77 9260.13 15.94 259.02 13902.83 15835.016 ------
Chapter Eight 278 Conjunctive Operation of Muliple Reservoirs Simulation
No of Years when no Shortage Occur = 0 No of Years when Shortage Occur = 40 No of Years when more than 5% Shortage Occur = 40 No of Years when more than 10% Shortage Occur = 40 ------At Mangla dam, model performance was evaluated with the help of comparison between historic and model operation (See Chapter 9). Figure 8.7 presents reservoir levels and power generated at Mangla dam. Figure 8.8 shows reservoir levels during simulation period. Table 8.3 shows summary results for Mangla reservoir operation.
Figure 8.15 Reservoir elevations and power generated at Mangla during simulation with forecasted flows for 82 year period (validation case)
Chapter Eight 279 Conjunctive Operation of Muliple Reservoirs Simulation
Figure 8.16 Reservoir levels at Mangla during simulation with forecasted flows for 82 year period (validation case)
Figure 8.17 Optimal Releases from Mangla dam during simulation with forecasted flows for 82 year period (validation case)
Chapter Eight 280 Conjunctive Operation of Muliple Reservoirs Simulation
Figure 8.18 Average water levels in Mangla reservoir during simulation with forecasted flows for 82 year period (validation case)
Table 8.5 Summary results of reservoir simulation at Tarbela (validation case) ------Year Inflow Demand Release Shortage % Shortage Evaporation Spill Energy (KAF) (KAF) (KAF) (KAF) (KAF) (KAF) ( GWH) ------2005- 6 23513.00 30390.00 24138.13 6251.88 20.57 199.99 235.71 6826.775 2006- 7 27467.00 30390.00 23797.07 6592.93 21.69 198.68 3220.68 6795.490 2007- 8 20774.00 30390.00 20746.42 9643.58 31.73 191.92 131.78 5882.184 2008- 9 22745.00 30390.00 21862.82 8527.18 28.06 197.55 378.75 6273.197 2009- 10 19541.00 30390.00 19016.17 11373.83 37.43 190.24 645.20 5424.324 2010- 11 26276.00 30390.00 23912.18 6477.82 21.32 208.91 1459.55 6898.835 2011- 12 21248.00 30390.00 19499.23 10890.77 35.84 191.53 1731.23 5465.914 2012- 13 28912.00 30390.00 27484.50 2905.50 9.56 218.94 1688.44 7973.798 2013- 14 25251.00 30390.00 22882.95 7507.05 24.70 196.69 2004.99 6512.219 2014- 15 22286.00 30390.00 21753.48 8636.52 28.42 194.22 639.49 6190.780 2015- 16 26025.00 30390.00 22471.23 7918.77 26.06 198.95 2870.82 6491.768 2016- 17 18026.00 30390.00 17982.90 12407.10 40.83 187.37 331.37 5104.441 2017- 18 23893.00 30390.00 22418.43 7971.57 26.23 198.65 1156.83 6409.659 2018- 19 25431.00 30390.00 23789.60 6600.40 21.72 214.61 1329.92 6897.384 2019- 20 20661.00 30390.00 20683.09 9706.91 31.94 195.79 0.00 5793.313 2020- 21 23434.00 30390.00 22627.31 7762.69 25.54 208.57 635.91 6474.953 2021- 22 23900.00 30390.00 23222.34 7167.66 23.59 204.96 286.27 6633.678 2022- 23 16629.00 30390.00 16575.46 13814.54 45.46 187.36 0.00 4643.204 2023- 24 16972.00 30390.00 16297.83 14092.17 46.37 186.10 0.00 4526.699 2024- 25 25806.00 30390.00 25597.63 4792.37 15.77 202.45 160.41 7275.030 2025- 26 25267.00 30390.00 24340.07 6049.93 19.91 200.37 795.91 6926.651 2026- 27 19036.00 30390.00 18499.87 11890.13 39.13 188.62 491.04 5173.281 2027- 28 21535.00 30390.00 20772.33 9617.67 31.65 194.55 474.36 5889.994 2028- 29 14961.00 30390.00 15119.63 15270.37 50.25 184.69 0.00 4171.310 2029- 30 15864.00 30390.00 15314.13 15075.87 49.61 185.36 0.00 4211.504 2030- 31 28041.00 30390.00 24948.50 5441.50 17.91 212.79 2882.78 7237.609 2031- 32 24918.00 30390.00 24154.58 6235.42 20.52 212.06 718.70 6964.582 2032- 33 30304.00 30390.00 24653.77 5736.23 18.88 210.98 4897.38 7120.065 2033- 34 21292.00 30390.00 21638.93 8751.07 28.80 195.57 0.00 6166.671 2034- 35 20490.00 30390.00 20289.66 10100.34 33.24 191.64 0.00 5688.595 2035- 36 18545.00 30390.00 18136.96 12253.04 40.32 188.67 0.00 5066.303 2036- 37 26671.00 30390.00 25730.20 4659.80 15.33 219.94 394.41 7472.568 2037- 38 19341.00 30390.00 19014.47 11375.53 37.43 190.15 514.02 5412.880 2038- 39 23660.00 30390.00 22127.19 8262.81 27.19 196.35 1357.90 6286.146 2039- 40 30806.00 30390.00 26370.93 4019.07 13.22 215.32 4089.59 7652.027
Chapter Eight 281 Conjunctive Operation of Muliple Reservoirs Simulation
Continue Table 8.5……… ------Year Inflow Demand Release Shortage % Shortage Evaporation Spill Energy (KAF) (KAF) (KAF) (KAF) (KAF) (KAF) ( GWH) ------2040- 41 26919.00 30390.00 25018.07 5371.93 17.68 205.70 1614.03 7203.760 2041- 42 33749.00 30390.00 27819.00 2571.00 8.46 234.19 5172.03 8273.539 2042- 43 19028.00 30390.00 19616.52 10773.48 35.45 192.29 0.00 5535.188 2043- 44 17501.00 30390.00 17163.37 13226.63 43.52 187.16 0.00 4819.164 2044- 45 15464.00 30390.00 15380.73 15009.27 49.39 183.46 0.00 4242.833 2045- 46 21893.00 30390.00 21848.77 8541.23 28.11 199.95 0.00 6241.623 2046- 47 23647.00 30390.00 22011.30 8378.70 27.57 194.66 1385.61 6251.865 2047- 48 26673.00 30390.00 23340.23 7049.77 23.20 209.80 3236.50 6742.099 2048- 49 21805.00 30390.00 21084.38 9305.62 30.62 193.04 0.00 5963.797 2049- 50 23938.00 30390.00 24184.83 6205.17 20.42 199.98 0.00 6874.899 2050- 51 21847.00 30390.00 21632.89 8757.11 28.82 193.20 77.90 6047.441 2051- 52 25394.00 30390.00 24011.20 6378.80 20.99 205.58 1155.18 6888.171 2052- 53 16481.00 30390.00 16070.37 14319.63 47.12 186.44 0.00 4466.901 2053- 54 13266.00 30390.00 13470.10 16919.90 55.68 180.12 0.00 3706.234 2054- 55 21253.00 30390.00 20629.83 9760.17 32.12 193.73 20.77 5865.953 2055- 56 29452.00 30390.00 27258.20 3131.80 10.31 220.66 2060.03 7943.282 2056- 57 17192.00 30390.00 17394.64 12995.36 42.76 187.46 0.00 4810.536 2057- 58 24018.00 30390.00 22135.21 8254.79 27.16 194.01 985.40 6306.080 2058- 59 25875.00 30390.00 23695.73 6694.27 22.03 196.12 2218.97 6695.896 2059- 60 18754.00 30390.00 18709.36 11680.64 38.44 189.74 0.00 5262.878 2060- 61 25000.00 30390.00 23956.80 6433.20 21.17 200.73 703.73 6831.731 2061- 62 20599.00 30390.00 20569.84 9820.16 32.31 191.76 0.00 5723.039 2062- 63 23154.00 30390.00 22759.33 7630.67 25.11 195.38 0.00 6372.081 2063- 64 24308.00 30390.00 24287.85 6102.15 20.08 214.11 196.47 6982.472 2064- 65 20086.00 30390.00 19785.52 10604.48 34.89 192.84 73.22 5530.924 2065- 66 28643.00 30390.00 26123.17 4266.83 14.04 213.95 2012.64 7586.276 2066- 67 19436.00 30390.00 19179.30 11210.70 36.89 191.64 124.45 5402.186 2067- 68 15376.00 30390.00 15416.30 14973.70 49.27 184.49 30.25 4285.366 2068- 69 26514.00 30390.00 24376.10 6013.90 19.79 210.99 1742.55 7016.128 2069- 70 29014.00 30390.00 26314.00 4076.00 13.41 220.56 2458.35 7660.936 2070- 71 26641.00 30390.00 24070.02 6319.98 20.80 203.55 2279.23 6863.327 2071- 72 22720.00 30390.00 21878.22 8511.78 28.01 195.83 747.87 6210.262 2072- 73 26987.00 30390.00 25110.57 5279.43 17.37 220.79 1555.20 7298.023 2073- 74 33303.00 30390.00 26218.54 4171.46 13.73 239.13 6662.03 7810.578 2074- 75 31648.00 30390.00 26698.03 3691.97 12.15 221.71 4189.25 7831.549 2075- 76 26046.00 30390.00 25697.20 4692.80 15.44 211.76 908.95 7381.109 2076- 77 25462.00 30390.00 23864.53 6525.47 21.47 200.18 1387.53 6839.315 2077- 78 28331.00 30390.00 25354.13 5035.87 16.57 208.60 2796.50 7276.484 2078- 79 31734.00 30390.00 26032.60 4357.40 14.34 220.56 5365.68 7584.898 2079- 80 21676.00 30390.00 20183.90 10206.10 33.58 191.88 1257.55 5688.277 2080- 81 26037.00 30390.00 25108.10 5281.90 17.38 225.05 949.03 7316.650 2081- 82 15362.00 30390.00 15266.50 15123.50 49.76 183.58 0.00 4150.373 2082- 83 13934.00 30390.00 13694.80 16695.20 54.94 179.71 0.00 3730.053 2083- 84 10925.00 30390.00 10995.83 19394.17 63.82 171.80 0.00 2965.808 2084- 85 16513.00 30390.00 16009.08 14380.92 47.32 185.81 0.00 4373.522 2085- 86 22762.00 30390.00 22428.31 7961.69 26.20 207.00 2.08 6376.936 2086- 87 16458.00 30390.00 16601.00 13789.00 45.37 185.15 0.00 4506.328
Mean 22833.40 30390.00 21564.93 8825.07 29.04 199.71 1084.05 6142.324 ------No of Years when no Shortage Occur = 0 No of Years when Shortage Occur = 82 No of Years when more than 5% Shortage Occur = 82 No of Years when more than 10% Shortage Occur = 80 ------8.5 Model Prediction for Conjunctive Operation of Multi-reservoir Simulation with Future Reservoirs
One of the main purposes of a modelling study is prediction for future conditions once the model is calibrated. Conjunctive operation of multi-reservoirs for future reservoir was carried to find out best possible scenarios for new reservoirs and to find water available for new reservoirs after meeting demands of the existing reservoirs (See Figure 8.1). In addition the
Chapter Eight 282 Conjunctive Operation of Muliple Reservoirs Simulation
second objective was to estimate the downstream Kotri releases to check sea water intrusion after conjunctive operation of reservoirs. Refer to Table 8.1, four cases were investigated. Case 1 in Table 8.1 belongs to existing condition and it is described in preceding sections.
Case 2: TMB 2015
In this case, Basha dam comes to operation in 2015. These results showed that at Tarbela the dam was capable of regulating only 50.39 MAF of water to meet the downstream requirement. Table 8.6 shows summary results of conjunctive operation in this case. The balance annual amount that will flow to sea are 33.77 MAF which is quite higher than the recommended release of 10 MAF to check sea water intrusion. The results showed that demand met at Kalabagh without Kalabagh dam but with Basha dam was of the order of 73.54 MAF.
Table 8.6 Summary of mean annual results of 10-daily conjunctive operation, case TMB: 2015 Reservoir Optimal Water Spillage Energy Released (MAF) Generated (MAF) (GWh)
Tarbela 50.39 12.64 17954
Mangla Raising 21.15 0.146 6659
Basha 37.37 13.09 19776
Kotri Below 33.77
Case 3: TMBA 2020
In this case, Basha dam comes to operation in year 2015 and Akhori in year 2020. These results showed that at Tarbela the dam was capable of regulating only 45.54 MAF. Table 8.7 shows the summary results of conjunctive operation in this case. The results showed that demand met at Kalabagh without Kalabagh dam but with Basha and Akhori dams was of the order of 73.4 MAF. The balance annual amount that will flow to sea are 33.8 MAF. The graphs showing water levels in reservoirs during optimal operation for the case is placed in Appendix-B.
Chapter Eight 283 Conjunctive Operation of Muliple Reservoirs Simulation
Table 8.7 Summary of mean annual results of 10-daily conjunctive operation, case TMBA: 2020 Reservoir Optimal Water Spillage Energy Released (MAF) Generated (MAF) (GWh)
Tarbela 45.54 7.53 16801
Mangla Raising 19.98 2.61 6170
Basha 37.13 13.33 19634
Akhori 7.965 1.89 2635
Kotri Below 33.80
Case 4: TMBAK 2030
In this case, Basha dam comes to operation in year 2015 and Akhori in year 2020 and Kalabagh in year 2030. These results showed that at Tarbela the dam was capable of regulating only 36.93 MAF. Table 8.8 shows the summary results of conjunctive operation in this case. The results showed that demand met at Kalabagh with Basha, Akhori and Kalabagh dams was of the order of 76.29 MAF. The balance annual amount that will flow to sea are 30.52 MAF. Table 8.8 Summary of mean annual results of 10-daily conjunctive operation, case TMBAK: 2030 Reservoir Optimal Water Spillage Energy Released (MAF) Generated (MAF) (GWh)
Tarbela 36.93 11.22 14005
Mangla Raising 20.33 2.24 6500
Basha 35.28 15.16 19072
Akhori 7.965 6.76 2770
Kalabagh 76.29 11.54 11232
Kotri Below 30.52
Chapter Nine 284 Results and Discussion
Chapter 9 RESULTS AND DISCUSSION
9.1 General
The results of the study are compiled in three parts. The first part describes the results obtained from the reservoir operation optimization model which deals with the derivation of the operating policies for the reservoirs. Second part includes the results gathered from the optimum operation of the network flow model. These results are compared with the actual historic data records of the system and their performance is evaluated. Third part presents the results from conjunctive operation of multiple reservoirs for future scenarios.
9.2 Reservoir Operation Model
It has been shown in chapter 6 that the reservoir operation model could be used to explore various regulating policies to determine their effects on overall optimization.
9.2.1 Results and Discussion Four Types of DP models have been applied to the Indus multireservoir system for the derivation of operating rules for reservoir operation.
• DDP Type One uses 10 day mean inflow of the historic data to derive optimal policies. The decision in this case is optimal target releases which is function of the initial storage and current inflow.
• SDP Type-Two uses serially independent inflow probabilities and Marcov chains of storage states to derive the policies in terms of optimal target releases which is a function of initial storage and current or forested inflow.
• SDP Type-Three uses inflow transition probability (P(Qt+1/Qt) and optimal policies are derived in terms of final optimal storage which is a function of the initial storage and current or forecasted inflow.
Chapter Nine 285 Results and Discussion
• SDP Type-Four uses inflow transition probability P(Qt/Qt-1) and operating policies are derived in terms of optimal target release which is a function of initial storage and previous period inflow Qt-1.
• Alternate objective functions and mixture of various objective functions have been used in each model type (from item 1 to 4 above) to analyze the performance of the reservoirs under different objectives.
• To verify their performance and to identify which model type is superior for the test case, actual operation of the reservoirs employing derived policies is carried out simulating the reservoirs with 10 year of the historic or forecasted data.
Mangla Reservoir The results obtained in chapter 6 are summarized in Table 9.1 for the comparative evaluation.
Table 9.1 Summary results of Reservoir Operation Model, Mangla Reservoir.
Model Type Type 1 Type 2 Type 3 Type 4 OF1 OF2 OF3 OF2 OF1 OF2 OF3 OF1 OF2 OF3 Empty 5 1 0 0 6 0 0 9 0 0 Full 39 75 46 43 2 13 11 6 31 11 p(failure) 0.014 0.003 0.000 0.000 0.017 0.000 0.000 0.025 0.000 0.000 p(full) 0.108 0.208 0.128 0.119 0.006 0.036 0.031 0.017 0.086 0.031 I(deficit) 6.08 8.27 6.87 6.52 4.82 7.17 6.30 5.28 7.71 6.99 P(deficit) 2198 1947 1948 2350 2470 1807 1951 2610 2073 2218 Energy 6179 6407 6420 6028 5914 6548 6418 5766 6285 6145
Description of sybmols used in this Table OF = Objective Function Type OF1 = Minimizing Irrigation shortfalls OF2 = Minimizing Irrigation and power generation shortfalls OF3 = Minimizing Irrigation, Power generation shortfalls and flood protection.
Empty = No.of Times Reservoir is Empty Full = No.of times Reservoir is Full p(failure) = Prob.of Failure p(full) = Prob.of Reservoir Full I(deficit) = Average Annual Irrigation shortfall P(deficit) = Average Annual Power shortfall
Chapter Nine 286 Results and Discussion
Energy = Average Annual Energy Generation (MKWH) ------
i) Analysis of Objective Function Type • If we consider the Objective function (OF1) minimizing irrigation shortfalls ignoring other objectives, the irrigation deficits are comparatively low viz 4.82 and 5.28 in SDP Type-Three and SDP Type-Four respectively. But in both the cases the reservoir suffers with probability of failure. The energy generated is comparatively low because the head in the reservoir is less in this case. From Figure 6.13 and 6.16 (in chapter 6), it can be seen that the reservoir levels do not follow the design rule curve and most of the time the reservoir level is at lower stage.
• If we consider objective function (OF2) minimizing irrigation and power generation shortfalls, the energy generation is improved but irrigation deficits are comparatively higher. Further it can be seen from the Figure 6.12 and 6.15 (in chapter 6) that the reservoir levels does not follow the design rule curve and reservoir content is mostly at higher side. The probability of spill is high loosing more water through spillage.
• Objective Function, 'OF3' deals with minimizing irrigation and energy generation shortfalls and ensure for flood protection following the design rule curve. In this case the probability of failure in all model type is zero, the probability of spill is 0.1279, 0.031 and 0.031 in model Type-One, Type-Three and Type-Four respectively. Average annual irrigation deficits are 6.87, 6.30 and 6.99. Average annual energy generation is 6420, 6418, 6145 MKWH in model Type-One, Type-Three, Type-Four respectively. This policy follows the design rule for flood protection (see Figure 6.9, 6.11 and 6.14 in Chapter 6). The lower allocation of irrigation water supplies and energy generation is marginal as compared to the other objective function types.
Based on the above discussion, it reveals that Objective Function Type 'OF3' may be better than others for the Mangla Reservoir and adopted in this study for further work. ii) Analysis of the Model Type
Chapter Nine 287 Results and Discussion
• Although Type-One model performs well during simulation of the reservoir operation producing 6420 MKWH of energy and average annual irrigation deficits are 6.87 MAF. The probability of reservoir full/spill is about 0.1278. It means adopting this rule chances of spillage is more. Further to simulate the reservoir operation, current inflow is required which is unknown. A forecast inflow model can be incorporated in place of current inflow. But perfect forecast cannot be made due to the random errors.
• Type-Two produces less irrigation shortfalls, 6.52 MAF as compared to Type-One but it produces less energy generation (average annual 6028 MKWH) during the simulation of the reservoir operation. Further the probability of spill is again comparatively high (about 0.119). For applying this operating policy, current inflow is needed which is unknown. like in Type-One, a forecast model can be constructed but perfect forecast may not be made due to random errors in Jhelum River inflows.
• Type-Three is one of the best model for Mangla Reservoir if flood control is on priority and storage has to be controlled. On th average, it produces 6418 MKWH of energy per year. Irrigation shortfall was 6.30 MAF and probability of spill is only 0.031. The probability of failure is zero. Current inflow or forecast inflow is required to use this operating rule. Type-Three model seems better for flood control in which storage has to be controlled. The results showed that SDP Type-Three model is superior than all other types, if perfect forecast of inflows is available.
• Type-Four produces average annual energy generation of 6145 MKWH, average annual irrigation shortfalls of 6.99, probability of spill 0.031 and probability of failure is zero. For the operation of reservoir, it requires previous period inflow and initial storage. Both the variables are known and therefore no forecasting is required.
• Comparing with SDP Type-Three, the irrigation shortfalls are about 9% higher and average annual energy generation is about 4 percent lower in SDP Type-Four. The probability of spill is same in both the model types.
• The difference in irrigation shortfalls and average annual energy production is marginal in SDP Type-Three and Type-Four.
Chapter Nine 288 Results and Discussion
• In reality, given the limitation of forecasting in SDP Type-Three, SDP Type-Four may be better for irrigation, energy production where the release is a target.
iii) Evaluation of the Reservoir Operation Model To check the advantage of using the reservoir operation model in real-time operations, it is necessary to make a comparison with the current mode of operation over an extended period of time (for example, 10 years), since the basis of the model is statistical. Table 9.2 and 9.3 show comparison of the model performance and actual WAPDA operation over the period 1985-86 to 1994-95 in terms of irrigation and energy production. The historic and model performance in terms of irrigation and energy production is shown in Figure 9.1 and 9.2. Table 9.4 and Figure 9.3 present the comparison of total benefits achieved from water and power with historic and model reservoir operation. Table 9.5 and Figure 9.4 shows comparison with water wasted through spillage with historic and model operation. There is an indication of superior performance by these models with historic data records. The technique described in this work appears promising.
As discussed in the conclusion of section 9.2.1(a)ii above, SDP model Type Four with ‘OF3’ appears to be best fit for Mangla reservoir and adopted for the further work. The evaluation of model Type Four was made with historic data records in Figure 9.5 through 9.6.
Tarbela Reservoir Like in Mangla reservoir the reservoir operation model was continued to run with new data set for Tarbela reservoir under various model types and objective types. The results obtained in chapter 6 are summarized in Table 9.6 for the comparative evaluation.
Chapter Nine 289 Results and Discussion
Table 9.2 Annual hydropower generated under various model types. (MKWh) Mangla Reservoir
Table 9.3 Annual irrigation releases under various model types. (MAF) Mangla Reservoir
Chapter Nine 290 Results and Discussion
Table 9.4 Benefits from water and power under various model types. (Rs. Million) Mangla Reservoir
Table 9.5 Water wasted through spillage under various model types. (MAF) Mangla Reservoir
Chapter Nine 291 Results and Discussion
Mangla Reservoir
8000
7000
) H 6000 W Historic K
M 5000 Type 1 OF 1 ( Type 1 OF 2 g y
r 4000 TYpe 1 OF 3
n e Type II OF 2 E
3000 Type III OF 1 l Type IV OF 1 u a
n 2000 n
A 1000
0 1985-86 1986-87 1987-88 1988-89 1989-90 1990-91 1991-92 1992-93 1993-94 1994-95 Simulation Year
Figure 9.1 Comparison of historic and model hydropower generation
Mangla Reservoir
35
30 Historic
Type I OF 1 ) 25 F
A Type 1 OF 2
M TYpe 1 OF 3
e ( 20 s
a Type II OF 2 e l e Type III OF 1
R 15 al
u Type IV OF 1 n n 10 A
5
0 1985-86 1986-87 1987-88 1988-89 1989-90 1990-91 1991-92 1992-93 1993-94 1994-95 Simulation Year
Figure 9.2 Comparison of historic and model releases from the reservoir
Chapter Nine 292 Results and Discussion
Mangla Reservoir
9000 .
)
s 8000 n io 7000 Mill s R
( Historic
its 6000 f
e Type I OF 1 n e B
r 5000 Type 1 OF 2 e w
o TYpe 1 OF 3 P
d 4000 n Type II OF 2 a r e
t Type III OF 1 a 3000
W Type IV OF 1 d te
la 2000 u m u c c 1000 A
0 1985-86 1986-87 1987-88 1988-89 1989-90 1990-91 1991-92 1992-93 1993-94 1994-95 Simulation Year
Figure 9.3 Comparison of historic and model water and power benefits
Mangla Reservoir
8.00
7.00 Historic . ) F 6.00 Type I OF 1 A M (
e Type 1 OF 2 5.00
illag TYpe 1 OF 3 p S h
g 4.00 Type II OF 2 u o r Type III OF 1 th d
e 3.00 t s Type IV OF 1 a w
r te
a 2.00 W
1.00
0.00 1985-86 1986-87 1987-88 1988-89 1989-90 1990-91 1991-92 1992-93 1993-94 1994-95 Simulation Year
Figure 9.4 Comparison of historic and model water wasted through spillage
Chapter Nine 293 Results and Discussion
Mangla Reservoir
30
25
)
F 20 A M
e ( Historic eas
l 15 e Type IV OF3 R nual
n 10 A
5
0 1980-81 1981-82 1982-83 1983-84 Simulation Year
Figure 9.5 Releases from Mangla reservoir, historic vs SDP Type Four (New data set)
Mangla Reservoir
9.0
8.0
7.0
. 6.0 ) F 5.0 A Historic 4.0 Type IV OF 3 ill (M p S 3.0
2.0
1.0
0.0 1974-75 1975-76 1976-77 1977-78 1978-79 1979-80 1980-81 1981-82 1982-83 1983-84 Simulation Year
Figure 9.6 Water wasted through spillage, historic vs SDP Type Four (New data set)
Chapter Nine 294 Results and Discussion
Table 9.6 Summary results of Reservoir Operation Model, Tarbela Reservoir.
Model Type Type-One Type-Three Type-Four OF3 OF1 OF2 OF3 OF1 OF2 OF3 Empty 7 44 2 0 12 0 0 Full 30 3 2 7 2 35 8 p(failure) 0.019 0.122 0.006 0.000 0.033 0.000 0.000 p(full) 0.083 0.008 0.006 0.019 0.006 0.097 0.022 I(deficit) 13.59 11.46 14.76 12.78 14.26 20.86 16.39 P(deficit) 1313 2877 462 1456 3246 1199 1798 Energy 15422 14075 16070 15525 13928 14928 15026
Description of symbols used in this Table
OF = Objective Function Type OF1 = Minimizing Irrigation shortfalls OF2 = Minimizing Irrigation and power generation shortfalls OF3 = Minimizing Irrigation, Power generation shortfalls and flood protection. Empty = No.of Times Reservoir is Empty Full = No.of times Reservoir is Full p(failure) = Prob.of Failure p(full) = Prob.of Reservoir Full I(deficit) = Average Annual Irrigation shortfall P(deficit) = Average Annual Power shortfall Energy = Average Annual Energy Generation (MKWH) ------i) Analysis of Objective Function Type • In case of Tarbela reservoir, for Objective function (OF1) (minimizing irrigation shortfalls) the irrigation deficits are comparatively low, 11.46 and 14.26 in SDP Type-Three and SDP Type-Four respectively. But in both the cases the reservoir suffers with probability of failure which is 0.122 and 0.033 in Type Three and Type Four respectively. The energy generated is comparatively low (14075 and 13928 MKWH in Type Three and Type Four respectively). It is due to the low head in the reservoir. From Figure 6.20 and 6.23 (in chapter 6), it can be seen that the reservoir levels do not follow the design rule curve and most of the time the reservoir level is at lower stage.
Chapter Nine 295 Results and Discussion
• In case of objective function (OF2) minimizing irrigation and power generation shortfalls, the energy generation is improved but irrigation deficits are comparatively higher. Further it can be seen from the Figure 6.19 and 6.22 (in chapter 6) that the reservoir levels does not follow the design rule curve and reservoir content is mostly at higher side. The probability of spill is high loosing more water through spillage.
• In case of Objective Function, ‘OF3’ which deals with minimizing irrigation and energy generation shortfalls and ensure for flood protection following the design rule curve,. the probability of failure in model Type Three and Four is zero. But in Type One it is 0.019. The probability of spill is 0.0.083, 0.019 and 0.022 in model Type- One, Type-Three and Type-Four respectively. Average annual irrigation deficits are 13.59, 12.78 and 16.39. Average annual energy generation is 15422, 15525, 15026 MKWH in model Type-One, Type-Three, Type-Four respectively. This policy follows the design rule for flood protection (see Figure 6.17, 6.18 and 6.21 in Chapter 6). The lower allocation of irrigation water supplies and energy generation is marginal as compared to the other objective function types.
• Based on the above discussion, it reveals that Objective Function Type 'OF3' may be better than others for the Tarbela Reservoir and adopted in this study for further work. ii. Analysis of the Model Type
• Type-One model performs well for hydropower generation producing 15422 MKWH of energy and average annual irrigation deficits are 13.59 MAF. The probability of reservoir full/spill is about 0.083 which is comparatively high. Adopting this rule chances of spillage is more. Further to simulate the reservoir operation, current inflow is required which is unknown. A forecast inflow model can be incorporated in place of current inflow. But perfect forecast cannot be made due to the random errors.
• Stationary results could not be obtained by Type-Two model as the iterations did not converge to a steady state conditions. Therefore results from model Type Two are not satisfactory and not included for further analysis.
• On the basis of simulation statistics, Type-Three proves best model for Tarbela reservoir if flood control is on priority and storage has to be controlled. On the
Chapter Nine 296 Results and Discussion
average, it produces 15525 MKWH of energy per year. Irrigation shortfall was 12.78 MAF and probability of spill is only 0.019. The probability of failure is zero. Current inflow or forecast inflow is required to use this operating rule. Type-Three model seems better for flood control in which storage has to be controlled. The results showed that SDP Type-Three model is superior than all other types, if perfect forecast of inflows is available.
• In case of Tarbela reservoir, Type-Four produces average annual energy generation of 15026 MKWH, average annual irrigation shortfalls of 16.39, probability of spill 0.022 and probability of failure is zero. For the operation of reservoir, it requires previous period inflow and initial storage. Both the variables are known and therefore no forecasting is required.
Conclusion
• Comparing with SDP Type-Three, the irrigation shortfalls are about 22 percent higher and average annual energy generation is about 3 percent lower in SDP Type-Four. The difference between probabilities of spill is negligible in both the model types.
• The difference in irrigation shortfalls and average annual energy production is marginal in SDP Type-Three and Type-Four.
• In reality, given the limitation of forecasting in SDP Type-Three, SDP Type-Four may be better for irrigation, energy production where the release is a target.
iii) Evaluation of the Reservoir Operation Model
Evaluation of the reservoir operation model was made by comparing the model results with historic operation Table 9.7 through 9.8 shows comparison of the model performance and actual WAPDA operation over the period 1985-86 to 1994-95 in terms of irrigation and energy production. Table 9.9 presents the comparison of total benefits achieved from water and power with historic and model reservoir operation. In Table 9.10 comparison with water wasted through spillage with historic and model operation is presented. These results indicated that model performance is considerably superior than historic operation in case of Tarbela reservoir.
Chapter Nine 297 Results and Discussion
Due to the reasons explained in the conclusion of section 10.2.1(b)ii above, SDP model Type Four with ‘OF3’ were preferred for Tarbela reservoir and adopted for the further work. The evaluation of model Type Four was made with operation of a new data set 1974-75 to 1983- 84 and comparing with historic operation.
Final Remarks
The SDP Type-Four model produces to derive operating policies for reservoir operation of the system with Objective Function 'OF3' may be adopted due to the reasons described above.
9.3 Network Flow Model
9.3.1 Results and Discussions Network Flow model applied to the Indus River System in chapter 7 takes advantages of the stochastic inputs of optimal releases at reservoir nodes and optimize the system in a better way. The model formulation allows to use network linear programming as a sub problem that receives the optimal releases as inputs which are stochastically derived by the SDP model.
The proposed two-stage mixed optimization methodology provides
i. Incorporation of stochastic component in the Network Flow model which was considered to be deterministic in the previous work found in the literature. ii. Incorporation of nonlinear convex loss functions of reservoir operation in an explicit way in the Network Flow model which was considered to be using linear objective function and constraints in the previous work found in the literature.
Chapter Nine 298 Results and Discussion
Table 9.7 Annual hydropower generated under various model types. (MKWh) Tarbela Reservoir
Table 9.8 Annual irrigation releases under various model types. (MAF) Tarbela Reservoir
Chapter Nine 299 Results and Discussion
Table 9.9 Benefits from water and power under various model types. (Rs. Million) Tarbela Reservoir
Table 9.10 Water wasted through spillage under various model types. (MAF) Tarbela Reservoir
Chapter Nine 300 Results and Discussion
The model was calibrated for the Indus River System in chapter 7. The model system represent whole the Indus River System containing 119 reaches and 67 nodes. The system was operated with 10 day time step for a 10 year of period (1985-86 to 1994-95). At this stage historic data of Chenab at Marala and Kabul at Nowshera was used. The Operating release policies at reservoir nodes are a function of initial storage and previous period inflows which are known. The calibration results are presented in Figure 7.3. In calibration case, the channel capacities of different reaches are checked. The results were checked for any infeasible condition in the system. The calibration was carried out with different runs of the model and priority factors for different demand nodes are fixed.
In the verification case the unknown inflows are predicted with the help of forecast models already developed in chapter 5. The 10 day results are obtained from the model (see Figure 7.3 and Table 7.3). These results indicated optimal canal allocation in the system. From these results, it is observed that sufficient amount of water is available in Kharif (Apr-Aug) to meet the Water Accord requirement. In Rabi (Oct-Mar) few canals suffer water shortage due to non availability of water in certain 10 days. These shortages are prominent in October and November. If the system inflows in these months are high, the irrigation shortfalls are low. Therefore these shortages are due to resource limitation in Rabi period. The resource limitation in Rabi can be improved providing some more storage and storing the surplus water of Kharif. This water can then be used in Rabi. In Table 9.11, a comparison was made between the model performance (of verification case) and the historic operation of the Indus River System. Out of the 10 years of operation period, the model performance is better than historic operation in 8 years.
Chapter Nine 301 Results and Discussion
Table 9.11 Comparison of SNFP model performance with historic operation [annual canal withdrawals (MAF)] Year Historic Model (Oct-Sep) Operation Operation 1985-86 102.1 105.5 1986-87 107.1 106.1 1987-88 101.8 106.4 1988-89 101.1 103.2 1989-90 103.1 106.6 1990-91 109.1 107.6 1991-92 105.9 106.9 1992-93 97.8 106.9 1993-94 110.5 102.7 1994-95 89.7 104.3
9.3.2 Case Study Results In chapter 7, the model has been used for developing a plan of operation for the Indus River System. It was converted into an operational model and being used as a tool in the 10 day operation of the system. In this case study the model is set up for a new data set of 10 years. This data set is for the period 1974-75 to 1983-84. A 10 day time step is selected. The model parameters, obtained from the calibration case were used. Summary results are presented in Table 9.12. It is observed that LBD, LJC and Sidhnai canals are the main canals which suffer shortage of water in October and November when the available inflows are limited. A comparison is made between the model performance and the historic operation in Table 9.13. It can be seen that the model performance is better than the historic operation in each year.
Chapter Nine 302 Results and Discussion
Table 9.12 Verification of network flow model
Chapter Nine 303 Results and Discussion
Table 9.13 Comparison of SNFP model performance with historic operation [Annual canal withdrawals (MAF)] Year Historic Model (Oct-Sep) Operation Operation 1974-75 84.1 103.6 1975-76 92.0 102.6 1976-77 100.6 102.8 1977-78 95.6 105.1 1978-79 102.0 104.5 1979-80 103.5 105.2 1980-81 102.1 104.1 1981-82 99.3 100.5 1982-83 95.7 102.9 1983-84 101.1 99.8
• In these investigations, the model has proved to be effective in assessing the impact of alternative policies of operation in the Indus River System.
• The model has been used as an operational tool and shows considerable promise in the area.
• The model was developed with sufficient flexibility to cater to any arbitrary configuration of the network so that it would be quickly accepted as an effective tool for planning purposes.
9.4 Conjunctive Operation of Mutilple Reservoirs for future Scenarios
It has been shown in chapter 8 that the conjunctive reservoir operation model could be used to invstigate future planning and new locations for the dams in the system. It may also be used to determine their effects on overall system.
Among several combinations of cases comprising different dams to be build on Indus River, three cases vs TMB, TMBA and TMBAK for conjucntive operation were investigated depending upon their more favourable conditions. Comparison of summary results for mean annual water released and energy generated from 10-daily conjunctive operation is presented
Chapter Nine 304 Results and Discussion in Table 9.14 and 9.15. Case TMBAK seems to be better comparing quantative results for optimal water released and maximum energy generated.
Table 9.14 Comparison of Results (mean annual water released from 10-daily conjunctive operation) Reservoir Optimal Water Released (MAF)
Case: TMB: 2015 TMBA:2020 TMBAK: 2030
Tarbela 50.39 45.54 36.93
Mangla Raising 21.15 19.98 20.33
Basha 37.37 37.13 35.28
Akhori - 7.965 7.965
Kalabagh - - 76.29
Kotri Below 33.77 33.80 30.52
Table 9.15 Comparison of Results (mean annual energy generated from 10-daily conjunctive operation) Reservoir Energy Generated (GWh)
Case: TMB: 2015 TMBA:2020 TMBAK: 2030
Tarbela 17954 16801 14005
Mangla Raising 6659 6170 6500
Basha 19776 19634 19072
Akhori - 2635 2770
Kalabagh - - 11232
Chapter Ten 305 Conclusions and Suggestions for Future Work
Chapter 10 CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK
10.1. Conclusions
Following conclusions are drawn from this study 1. A two stage mix optimization procedure is adopted for the stochastic optimization of the Indus River System for 10 day operation.
2. The first stage of the proposed procedure cycles through three programs, a tpm computation algorithm, a DDP-SDP model and a simulation program. In DDP-SDP program, four model types and three objective types are investigated for multiresevoir system. Simulation program was used for the validation of the each policy derived through this cycle. The accumulation of these programs is called 10 day reservoir operation model of the multireservoir Indus River System.
3. The second stage of proposed procedure uses advantages of the stochastic optimal policies derived in item 2 above with a Network Flow programming model constructed for the Indus River System for 10 day operation.
4. The models in item 2 and 3 are calibrated for the Indus River System.
5. The results obtained from Item 4 are verified with the help of simulating the system with the next data set and comparing the results with actual historic data record.
6. The results indicated that such type of methodology is effective for the test case. The technique described in this work appears promising.
10.1.1 Theoretical Development
1. Optimization of reservoir operation is undertaken with Markov decision process and found effective for the test case.
2. A two stage frame work consisting of a steady state SDP 10 day reservoir operation model followed by a Network Flow model is developed for the optimization of the multireservoir multiobjective water resource Indus River System.
Chapter Ten 306 Conclusions and Suggestions for Future Work
3. The steady state model which describes the convergent nature of the prospective future operations is regarded as a periodic Marcov decision process and is optimized with the successive approximation method.
4. This result is in tern used as an interim step for the optimal operation of the complete Indus River System.
5. The stochastic behavior of the inputs and nonlinear objectives in the linear programming model is incorporated in this manner.
6. The computational effort required for optimization is significantly reduced compared to conventional procedures.
7. Three SDP-Types and one DDP-Type models are proposed in this study and their solution is obtained. Various model types in SDP/DDP formulation may produce different results in different reservoir conditions and different hydrologic regime. The model types are therefore system specific. For the Indus Reservoir System SDP Type- Four may be better than others.
8. Alternate multiobjective functions are proposed and analyzed in this study. Taking one or two objectives and ignoring other or considering all the objectives to optimize, produces different results in different model types. Especially the results are significantly different in terms of storage contents in the reservoir during simulation. The alternate means of analysis indicated that considering the minimization of irrigation shortfalls and power production and keeping the levels in the reservoirs for flood protection benefits may be the bestfit for the Indus Reservoir System.
9. The proposed procedure is capable of introducing any means of inflow forecasting into the optimization procedure, if SDP Type Three, Type Two and DDP Type One are used.
10. The results indicated that this procedure may be useful as the basis for the stochastic optimization of the multireservoir systems.
Chapter Ten 307 Conclusions and Suggestions for Future Work
11. The methodology developed provides a viable way of applying stochastic optimization into deterministic optimization under multireservoir, multiobjective water resource system with 10 day operation under uncertainty.
10.1.2 Practical Development 1. The stochastic inflow process of Indus at Tarbela and Jhelum at Mangla is shown as a Markov decision process.
2. Bestfit operating rules under uncertainty for the Indus reservoirs have been derived.
3. Optimum operation of reservoir for irrigation water supplies, hydropower production and flood protection is presented.
4. Optimal allocation of water resources in the canal network of Indus River System is identified.
5. Resource limitations at various locations are identified.
6. It is shown that if the derived operating policies are adopted for the system the benefits would be considerably higher than those experienced from the historic operation by WAPDA.
7. The hydropower production would be higher than the historic power generation.
8. The complete model may be used as a tool and a guide for the optimum 10 day operation of the Indus River System.
9. It may also be used for future planning for water resources development with the help of conjunctive operation of multiple reservoirs.
10.2. Recommendations
1. The results from the proposed modeling package can be used as guidelines for the operation of the Indus River System.
2. The proposed model is not capable of operating the system for daily or hourly operation during flood season.
Chapter Ten 308 Conclusions and Suggestions for Future Work
3. However the 10 day operating criteria may be used as boundary conditions (beginning and ending storage and 10 day releases) for a daily or even hourly operation model.
4. It is recommended to update the modeling package for daily and hourly operation for flood season.
5. The model is basically a hydrologic and system analysis optimization procedure for the optimization of the surface water facility of the Indus River System. It is not capable of providing any agricultural (e.g. crop type etc.), groundwater or water logging information in the system. It is recommended that the model may be updated to include such type of facility if needed in future.
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