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Optimal Operation of a Multiple Reservoir System

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Optimal Operation of a Multiple Reservoir System / I /I r, / OPTIMAL OPERATION OF A MULTIPLE RESERVOIR SYSTEM by Miguel A. Marino Principal Investigator and Hugo A. Loaiciga Research Assistant Land, Air and Water Resources University of California, VI .TEr~R'::SO - r. ~S-- C~\ I f"R P ;,(, ... f ••.5 .I) "..~, Office of the Director CALIFORNIA WATER RESOURCES CENTER University of California DaVis, California 95616 - The research leading to this report was supported in part by the United States Department of the Interior, under the Annual Cooperative Program of Public Law 95-467, Project No. A-088-CAL, and by the University of California Water Resources Center, Project UCAL-WRC-W-617. Cont'ent.s of this publication do not necessarily reflect the views and poliCies of the Office of Water Policy, U.S. Department of the Interior, nor does mention of trade names or commercial products constitute their endorsement or recommendation for use by the U.S. Government. TECHNICAL COMPLETION REPORT September 1983 TABLE OF CONTENTS LIST OF FIGURES v LIST OF TABLES vii ACKNOWLEDGMENTS xii ABSTRACT xiii 1. INTRODUCTION 1 2. TERMINOLOGY 6 2.1 Control or Decision Variable 6 2.2 State Variable 6 2.3 Constraints 6 2.4 Feasible Region 6 2.5 Objective Function 7 2.6 Convex Set 7 2.7 Optimization Model 7 2.8 Global and Local Optima 7 2.9 Initial Policy 8 2.10 Optimal Policy 8 ·2.11 Multiple Optimal Solutions 8 3. REVIEW OF RESERVOIR OPERATION MODELS 9 3.1 Deterministic Models 9 3.2 Stochastic Models 16 3.3 Discussion 21 4. REVIEW OF SOLUTION PROCEDURES 25 4.1 Formulation of the Problem 25 4.2 Discrete Dynamic Programming 28 -ii- 4.3 State Increment Dynamic Programming 31 4.4 Dynamic Programming Successive Approximations 33 4.5 Differential Dynamic Programming 37 4.6 Linear Quadratic Gaussian Method 40 4.7 Nonlinear Programming 45 4.8 Progressive Optimality Algorithm 46 4.9 Convergence Proofs of the POA for Bounded States and Decisions 55 5. OPTIMAL OPERATION POLICIES FOR THE NCVP 59 5.1 Description of the NCVP System 59 Data Relevant to the Constraints of the System 75 Benefits Accruing from the Operation of the System 98 5.2 Streamflow Forecasting Technique 99 5.3 Optimization Model for the NCVP 109 Simiplified Linear Model 126 Simplified Quadratic Model 1 131 Simplified Quadratic Model 2 133 5.4 Selection of Initial Operation Policy 138 6. DISCUSSION OF RESULTS 141 6.1 Initial Policies 141 6.2 Optimal Operation Policies 160 7. SYSTEM-DEPENDENT FEATURES: AND EXTENSION OF THE 'OPTIMIZATION MODEL 193 7.1 Modeling Regulating Reservoirs 193 7.2 Simiplified Linear Model 208 -iii- 7.3 Simplified Quadratic Modell 213 7.4 Simplified Quadratic Model 2 217 7.5 Nonlinearity in the Energy Rate Equation 220 7.6 Nonlinearities in the Constraints 223 7.7 A Full Model of Minimum Dimensionality 228 7.8 Application of General Nonlinear Model 241 Energy Generation Rates 242 Spillage Discharge Equations 247 Initial Policies 259 Solution Algorithm 259 Discussion of Results 268 8. SUMMARY AND CONCLUSIONS 286 REFERENCES 290 -iv- LIST OF FIGURES Fig. 4.1 Schematic representation of a trial trajectory f.~~O)}, the corridor boundaries {!~O) ± a}, and a substitute trial , f (1), traJectory t!t j. Fig. 4.2 Standard progressive optimality algorithm (paA) flow diagram. Fig. 4.3 Standard paA (To achieve state !~3): (L) ~1 and ~;l) yield (2) (1'1') (2) d (1) i e Ld (2) (",) (2) d (1) Le l d ~2 j !2 an!4 y~e !3 ; ~~1!3 an!S Yle 3 ~~2) (iv) ~1 and ~;2) yield ~;3); and (v) !i ) and !~2) yield ~j3)). (1) Fig. 4.4 Modified paA s (To achieve state !i ): (i) 3f1 and ~3 yield (2) (2) C') (2) and xC!) l eLd (2) and yield ~2 ; 11 ~2 -4 Y1e ?5.3 ; (iii) !1 !3 (3 (1) (3) x ). C) (3) and yield x(3). (v) and yield -2 ' ~v ~2 !S -3 ' !1 ~3 x(4). (4 (1) x(4). (4) (vi) x ) and yield and (vii) !1 and -2 ' -2 E6 -3 ' !3 s yield !i )). This scheme should be used when significant improvements (e.g., 10% improvement in the objective function with respect to the previous iteration) arise from the two- stage problem involving periods 1 and 2). Fig. 5.1 Schematic representation of Central Valley Project. Fig. 5.2 Shasta flood control diagram. Fig. 5.3 Folsom flood control diagram. Fig, 5.4 New Melones flood control diagram. Fig. 5.5 Tullock flood control diagram. Fig. 5.6 Schematic representation of NCVP diversions, losses, releases, and spills. Fig. 5.7 Trinity power plant (at Clair Engle Lake) gross generation curve. -v- Fig. 5.8 Judge Francis Carr power plant gross generation curve. Fig. 5.9 Spring Creek power plant gross generation curve. Fig. 5.10 Shasta power plant gross generation curve. Fig. 5.11 Folsom power plant gross generation curve. Fig. 5.12 New Melones power plant gross generation curve. Fig. 5.13 Tullock power plant gross generation curve. Fig. 5.14 Relation between t and vectors !t' ~t' and !t' Fig. 5.15 NCVP monthly optimization flow chart. Fig. 6.1 Typical energy vs. release curve (developed by using Folsom reservoir data). Fig. 6.2 Total annual energy vs , total annual inflow for the NCVP. Fig. 6.3 Operation of Shasta reservoir (water year 1979-80). Fig. 6.4 Operation of Folsom reservoir (water year 1979-80). Fig. 7.1 Elevation vs. storage (Clair Engle reservoir). Fig. 7.2 Operation of Shasta reservoir (water year 1979-80). Fig. 7.3 Operation of Folsom reservoir (water year 1979-80). Fig. A.1 Spillway rating curve (New Melones Dam). -vi- LIST OF TABLES Table 5.1 Basic NCVP Data. Table 5.2 Capacities of Tunnels and Penstocks. Table 5.3. Flood Control Provisions for Trinity Dam at Clair Engle Reservoir. Table 5.4 NCVP Flow Requirements. Table 5.5 Reservoir Capacity Allocations. Table 5.6 Maximum Reservoir Storages. Table 5.7 Minimum Reservoir Storages. Table 5.8 Maximum Reservoir Releases, Excluding Spills, that are Limited to Penstock Capacity. Table 5.9 Minimum Reservoir Releases. Table 5.10 Reservoir Net Rate Losses. Table 5.11 Transition and Noise Covariance Matrices for NCVP Monthly Inflows. Table 5.12 Actual and Forecast Monthly Inflows for Year with Below- Average Inflows, October 1975 - September 1976. Table 5.13 Actual and Forecast Monthly Inflows for Year with Average Inflows, October 1974 - September 1975. Table 5.14 Actual and Forecast Monthly Inflows for Year with Above- Average Inflows, October 1979 - September 1980. Table 5.15 Versions of the Optimization Model. Table 6.1 Initial Storage Policy, 1974-75 (Policy I). Table 6.2 Initial Release Policy, 1974-75 (Policy I) . Table 6.3 Initial Storage Policy, 1974-75 (Policy II) . -vii- Table 6.4 Initial Release Policy, 1974-75 (Policy II). Table 6.5 Initial Storage Policy, 1979-80 (Policy I) . Table 6.6 Initial Release Policy, 1979-80 (Policy I). Table 6.7 Initial Storage Policy, 1979-80 (Policy II) . Table 6.8 Initial Release POlicy, 1979-80 (Policy II) . Table 6.9 Initial Storage Policy, 1975-76 (Policy I). Table 6.10 Initial Release Policy, 1975-76 (Policy I) . Table 6.11 Initial Storage Policy, 1975-76 (Policy II) . Table 6.12 Initial Release Policy, 1975-76 (Policy II) . Table 6.13 Initial Storage Policy, 1973-74 (Policy I). Table 6.14 Initial Release Policy, 1973-74 (Policy I). Table 6.15 Number of Iterations to Attain Convergence and CPU Time Requirements. Table 6.16 Optimal State Trajectory Corresponding to Initial Policy I, 1974-75. Table 6.17 Optimal Energy Production, Release Policy, and Delta Releases Corresponding to Initial Policy I, 1974-75. Table 6.18 Optimal State Trajectory Corresponding to Initial Policy II, 1974-75. Table 6.19 Optimal Energy Production, Release Policy, and Delta Releases Corresponding to Initial Policy II, 1974-75. Table 6.20 Optimal State Trajectory Corresponding to Initial Policy I, 1979-80. Table 6.21 Optimal Energy Production, Release Policy, and Delta Releases Corresponding to Initial Policy I, 1979-80. Table 6.22 Optimal State Trajectory Corresponding to Initial Policy II, 1979-80. -viii- Table 6.23 Optimal Energy Production, Release Policy, and Delta Releases Corresponding to Initial Policy II, 1979-80. Table 6.24 Optimal State Trajectory Corresponding to Initial Policies I and II, 1975-76. Table 6.25 Optimal Energy Production, Release Policy, and Delta Releases Corresponding to Initial Policies I and II, 1975-76. Table 6.26 Optimal State Trajectory, 1973-74. Table 6.27 Optimal Energy Production, Release Policy, and Delta Releases, 1973-74. Table 6.28 Actual and Maximized Energy Production for Water Year 1979-80. Table 7.1 Initial Storage Policy, 1979-80 (Policy I). Table 7.2 Initial Release Policy, 1979-80 (Policy I). Table 7.3 Initial Storage Policy, 1979-80 (Policy II). Table 7.4 Initial Release Policy, 1979-80 (Policy II). Table 7.5 Optimal State Trajectory, 1979-80 (Policy I). Table 7.6 Optimal Release Policy, 1979-80 (Policy I). Table 7.7 Optimal Energy Production, Penstock Release, and Delta Releases Corresponding to Initial Policy I, 1979-80. Table 7.8 Optimal State Trajectory, 1979-80 (Policy II). Table 7.9 Optimal Release Policy, 1979-80 (Policy II). Table 7.10 Optimal Energy Production, Penstock Release, and Delta Releases Corresponding to Initial Policy II, 1979-80. Table 7.11 Actual and Maximized Energy Production for 1979-80. -ix- Table A.l Capacities of Clair Engle Reservoir. Table A.2 Capacities of Lewiston Reservoir. Table A.3 Capacities of Whiskeytown Reservoir.
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