Optimal Operation of a Multiple Reservoir System

OPTIMAL OPERATION OF A MULTIPLE RESERVOIR SYSTEM

GIANNIN1 FOUNDATION OF Miguel A Marirlo AGRICULTWAL ECONOMICS LtARY and 0,6 Hugo A Loaiciga it-'t< 2 -cN85

CALIFORNIA WATER RESOURCES CENTER University of California ontribution No. 193 ISSN 05754941 March 1985 This publication is furnished as a part of the information exchange program in water resources research and is published and distributed by the DIRECTOR'S OFFICE of the CALIFORNIA WATER RESOURCES CENTER. The Center sponsors projects in water resources and related research on the several campuses of the University of California with funds provided by the U.S. Department of Interior and from the State of California. Copies of this and other re- ports published by the Center may be obtained from:

WATER RESOURCES CENTER UNIVERSITY OF CALIFORNIA 2102 WICKSON HALL DAVIS, CA 95616 (916) 752-1544

Copies of Center publications may be examined at either location of the Water Resources Center Archives: 410 O'Brien Hall, Berkeley Campus (415) 642-2666 or 2081 Engineering 1, Los Angeles Campus (213) 825-7734. OPTIMAL OPERATION OF A MULTIPLE RESERVOIR SYSTEM

Miguel A. Mari& and Hugo A Loaiciga

Land, Air and Water Resources University of California Davis, California 95616

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. Contents of this publication do not necessarily reflect the views and policies of the U.S. Department of Interior nor does mention of trade names or commercial products constitute their endorsement or recommendation for use by the U.6. Government.

Contribution No. 193 February 1985 ISSN 05754941 TABLE OF CONTENTS

ACKNOWLEDGMENTS

ABSTRACT -iv-

NOTATION -v-

1. INTRODUCTION 1-1

2. DESCRIPTION OF THE SOLUTION ALGORITHM 2-1

3. THE NCVP SYSTEM: FUNCTIONS, OPERATIONAL CONSTRAINTS, PHYSICAL FEATURES, AND FLOW FORECASTING 3-1 3.1 Description of the NCVP System 3-1 Functions 3-1 Data Relevant to the Constraints of the System 3-3 Power Generation 3-4 Net Losses 3-6 Benefits Accruing from the Operation of the System 3-8 Modeling of Spillages 3-9 3.2 Streamflow Forecasting 3-13

4. DEVELOPMENT OF THE OPTIMIZATION MODEL 4-1 4.1 General Optimization Model 4-1 4.2 Simplified Linear Model 4-12 4.3 Simplified Quadratic Model 4-15

5. APPLICATIONS AND DISCUSSION OF RESULTS 5-1 5.1 Linear Model 5-1 5.2 Quadratic Model 5-8

6. SUMMARY AND CONCLUSIONS 6-1

REFERENCES 7-1 ACKNOWLEDGMENTS

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. Contents 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. We thank Mr. George Link, of the Central

Valley Operations Office, U.S. Bureau of Reclamation, Sacramento, for his valuable assistance in providing data and explanations with regard to the

operation of the California Central Valley Project.

111 ABSTRACT

This report presents a methodology to obtain optimal reservoir operation policies for a large-scale reservoir system. The model maximizes the system annual energy revenues while satisfying all other functions imposed on the operation of the reservoir network by an appropriate definition of

the constraint set. The model incorporates the stochasticity of river flows and keeps future operating schedules up-to-date with the actual realization of those random variables. It yields medium-term (one-year ahead) optimal

release policies that allow the planning of activities within the current

water year, with the possibility of updating preplanned activities to

account for uncertain events that affect the state of the system. The

solution approach is a sequential dynamic decomposition algorithm that keeps

computational requirements and dimensionality problems at low levels. The

model is applied to a nine-reservoir portion of the California Central

Valley Project and the results are compared with those from conventional

operation methods currently in use, showing that the use of the model can

improve the levels of energy production (about 30 percent increase) while

the optimal release policies meet satisfactorily all other functions of the

reservoir system. Sensitivity analysis is conducted to assess the optim-

ality of the solutions and several alternative formulations of the model are

developed and tested, the results showing the robustness of the optimal

policies to the choice of model. Innovative features of the reservoir

operation model presented in this research are: (1) development of a model

of minimum dimensionality; (2) treatment of spillage and penstock releases

as decision variables; (3) implementation of a computationally efficient and

numerically stable solution algorithm for nonpositive-definite quadratic

programming problems; and (4) adequate fulfillment of the multiple functions

of the reservoir system by an appropriate definition of the constraint set.

iv NOTATION

Scalar Variables, Coefficients, and Indices 3 average surface area of reservoir i during month t, in 10 acres. i a energy coefficient in the linear and quadratic energy rate equations.

a energy coefficient in the linearized energy rate equation of the

quadratic model (i = 1, 4, 6, 8). i b energy coefficient in the linear and quadratic energy rate equations. i b energy coefficient in the linearized energy rate equation of the

quadratic model (i = 1, 4, 6, 8). i c energy coefficient in the quadratic energy rate equations.

c : net loss rate at reservoir i during month t, in ft/month. t "1 c : coefficient in the spillway equation for dam i in month t. t c : coefficient in the net loss equation at reservoir i for month t. t De: water demand at demand point k for month t, in Kaf.

di: coefficient in net loss equation at dam i for month t.

i: coefficient in the spillway equation at dam i for month t. dt Et: total system energy produced during month t, in Mwh.

hi- water surface elevation at reservoir i for month t, in ft above mean t. sea level.

1: reservoir index (i = 1, 2,..., 9).

Ka: kiloacre (1 Ka = 103 acre = 4.047 x 106 m2) 6 3 Kaf: kiloacre-ft (1 Kaf = 1.233 x 10 m ).

k : constant term in the objective function of the quadratic model. t n: dimension of the storage vector, n = 9.

t: denotes beginning of month t, t = 1, 2,..., 13.

n: exponent in the spillway equation at dam i for month t.

tfs: unit energy price, in $/Mwh. Matrices

A: matrix of constraints in the linear model.

matrix of constraints in the generalized optimization model.

A : diagonal matrix in the continuity equation of the generalized t+1 optimization model; its components are given by 1 + c . t A : matrix of constraints in the quadratic model. t B: diagonal matrix of energy coefficients in the generalized optimiza-

tion model.

B : diagonal matrix in the continuity equation of the generalized t optimization model; its elements are given by 1 - c . t

B:B • matrix of parameters in the AR(1) streamflow model.

Bt: matrix estimate of the parameter matrix Bt. *t B : diagonal matrix of energy coefficients in the quadratic model; its 11 i . components are b = 1, 4, 6, 8. t' B : diagonal matrix of energy coefficients in the quadratic model for 22 reservoirs i = 2, 3, 5, 7, and 9.

matrix in the continuity equation of the generalized optimization -1 model, Ctil. = 11 At4.1.

C(i): crosscorrelation matrix between streamflow vectors yt and y_ —L-J D : matrix in the continuity equation for the generalized model, t -1 = I B . Dt 1 t D : diagonal matrix in the continuity equation of the quadratic model t ^1 ^4 ^6 ^8 with elements given by dt, dt, dt, dt, 0, 0, 0, 0, 0.

F: matrix in the continuity equation of the generalized optimization -1 model, F = r

H : matrix in the continuity equation for month t of the quadratic model, t+1 H = IA . t+1 1 t+1 1 H : Hessian matrix in the generalized optimization model. t H : Hessian matrix iii the quadratic model. t vi H : matrix in the continuity equation for month t-1 of the quadratic t ^ ^ -1 model, H = F A t 1 t ^ ^ H : matrix in the constraint set of the quadratic model, Ht = Ht + Dt_ i. t M : matrix in the continuity equation of the quadratic model for month t, t -1 M = 11 B. t ^ M : matrix in the continuity equation for month t of the quadratic model, t M = M + ND . t t t M : matrix in the constraint set of the quadratic model. t N: matrix in the continuity equation of the quadratic model.

covariance matrix of the streamflow vector yt. R:t R : matrix estimate of the covariance matrix R . t t F: matrix in the continuity equation of the generalized optimization -1 model, r = IFF2.

: topological matrices in the continuity equation of the generalized 112 model.

It: covariance of the white noise vector

2 : matrix estimate of the covariance matrix 2 t t

Vectors

a: vector of energy coefficients in the generalized optimization model. ^*(1) : vector of energy coefficients in the quadratic model, its components "i . are a = 1, 4, 6, and 8. t' : vector of energy coefficients in the quadratic model for reservoirs —t 2, 3, 5, 7, and 9.

: right-hand side vector in the constraint set of the linear model. 12-t 1 right-hand side vector in the constraint set of the generalized

optimization model.

: right-hand side vector in the constraint set of the quadratic model. 12-t : vector of coefficients in the quadratic model continuity equation for ^1 ^4 ^6 "8 month t whose components are given by ct, ct, ct, ct, 0, 0, 0, 0, 0. vi I k c : vector containing a linear combination of water demands at control —t point k during month t, in Kaf.

• vector of net losses during month t; its components are denoted by i . e in Kaf. t' : white-noise vector in the AR(1) streamflow model.

t : vector of coefficients in the objective function of the linear t model.

K: vector containing spillway equations for reservoirs 2, 3, 5, 7, and

9 in the continuity equation of the quadratic model.

k: vector of fixed storages for regulating reservoirs 2, 3, 5, 7,

and 9.

k': vector of constant storages whose components are x i = 3, 3, 5, 7, t' and 9 in the objective function of the quadratic model.

Et: vector of diversions/accretions in the generalized model set of

constraints; its components are denoted by Rt, in Kaf.

: vector of feasible spillage values, in Kaf.

r • vector of spillages; its components are denoted by ri in Kaf. —t. t' !t: vector of coefficients in the objective function of the generalized

optimization model.

at: vector of coefficients in the objective function of the quadratic

model.

: vector of feasible releases, in Kaf. t : vector of maximum releases, in Kaf.

: vector of minimum releases, in Kaf. t,min i : vector of penstock releases; its components are denoted by u in t' Kaf.

: vector of optimal penstock releases, in Kaf.

y : vector in the continuity equation of the generalized optimization t i model; its elements are given by yt - d - i = 1, 2,..., 9. t' viii vector of feasible penstock plus spillage releases, in Kaf.

: vector of streamflows minus diversions and/or accretions in the Ig-t continuity equation of the linear model; its components are given by

y - R i = 1, 2,.. t t' • , 9- vector in the continuity equation of the quadratic model,

vector in the constraint set of the quadratic model, Ht = Ht - 2t

: vector of feasible storages, in Kaf. Ict ?..i : vector of maximum storages, in Kaf. t,max )_..ct,min: vector of minimum storages, in Kaf. i : vector of storages; its components are denoted by x in Kaf. ?!t. t' (0) x : vector of storage values about which first-order Taylor expansions —t are performed in the spillway and quadratic energy equations; its i components are denoted by xt,o, in Kaf.

vector of optimal storages, in Kaf. i vector of monthly streamflows; its components are denoted by yt, in

Kaf. i i i : vector whose components z are equal to y - e - R. --t. t t t t i : vector of energy rates; its components are denoted by t in t t' Mwh/Kaf.

Acronyms

AR(1): Autoregressive Model of Order 1.

CC: Chance Constraint.

CCLP: Chance Constrained Linear Programming.

CVP: Central Valley Project.

DP: Dynamic Programming.

DDP: Differential Dynamic Programming.

ix DPSA: Dynamic Programming Successive Approximations.

IDPSA: Combination of SIDP and DPSA.

LP: Linear Programming.

LP-DP: Linear Programming - Dynamic Programming.

NCVP: Northern Portion of the Central Valley Project.

NLP: Nonlinear Programming.

POA: Progressive Optimality Algorithm.

PPO: Principle of Progressive Optimality.

PSD-QP: Positive Semidefinite Quadratic Programming

QP: Quadratic Programming.

SIDP: State Increment Dynamic Programming.

X CHAPTER 1

INTRODUCTION

Optimal operation of reservoir systems is of fundamental importance for the adequate functioning of regional economies and environments as well as for the well being of the population served by such reservoir developments.

Reservoirs provide a wide variety of indispensable services that affect every facet of modern society. Those services include the provision of water supply for human consumption, agricultural and industrial activities, hydropower generation, flood control protection, ecological and environmental enhancement, navigation, and recreation. Large-scale reservoir developments have made possible the economic growth and flourishment of entire regions and nations. In parallel with the increasing expansion of population centers and economic activity, the demands exerted on the water stored by the reservoirs has been augmenting steadily. With the size of the systems almost at a maximum possible due to water availability limitations — with the most suitable locations for development already harnessed — and with tighter budgetary constraints, it has become mandatory to operate the

reservoir systems in an efficient manner, so as to effectively and reliably provide their intended services to the public.

Efficient operation of reservoir systems, although desirable, is by no

means a trivial task. First, there are multiple components (reservoirs,

canals, river diversions, pumping plants, etc.) that must be operated jointly. Second, there exist many, usually conflicting, interests and

constraints that influence the management of the system. Third, there exist

uncontrollable and uncertain elements that determine the reservoirs'

operation such as streamflows. Fourth, conditions such as water demands,

institutional regulations, and even infrastructural elements of the system,

are continuously changing due to the inherent dynamic nature of society and

technology. 1-1 In the context of a much needed efficient management of reservoir systems, mathematical optimization models for reservoir operation become a valuable tool for improved planning of complex operational schedules for any

reservoir system. Several aspects make optimization models prone to suitable use in reservoir management: the usually well defined structure and links between the physical components of the system, the quantitative

nature of demands and constraints imposed on the system, and the advent of

sophisticated computational equipment.

Modeling of reservoir management constitutes a classical example of the

application of optimization theory to resource allocation. Reservoir

management modeling, however, cannot be accomplished without problems of its

own, among which predominantly figure its stochastic nature (streamflows,

demands, etc.), the large size of the models (dimensionality), and the

limitations of mathematical tools that force the modeler to compromise

between accurate system modeling and complexity of the resulting optimiza-

tion models.

There exists a substantial body of reservoir literature. A few studies

related to large-scale reservoir operation are worth mentioning herein due

to their relevance to the methods employed in this study. First, in the

deterministic environment, Hall et al. (1969) were among the first inves-

tigators to propose a deterministic model for the monthly operation of a

reservoir system. They used a version of Larson's (1968) state increment

dynamic programming (SIDP) to solve for the monthly releases of a two-

reservoir system. A historical low-flow record was used to determine the

annual firm energy.

Becker and Yeh (1974) used a linear programming-dynamic programming

(LP-DP) approach for the monthly operation of a five-reservoir portion of

the California Central Valley Project (CVP). The performance index con-

sisted of minimization of potential energy losses of water backed up in the

reservoirs. They argued that this type of criterion is justified by the 1-2 high relative value of firm energy in comparison with that of firm water.

Streamflows were assumed to be known for the one-year ahead planning horizon. By assuming that two of the reservoirs are kept at constant elevation throughout the year and a third reservoir is operated by a rule curve, the number of state variables was reduced to two, allowing an efficient implementation of the model. The model yielded a 35% energy production increase over contract levels. Fults et al. (1976) applied SIDP to a four-reservoir

system in northern California, in which a highly nonlinear objective func-

tion represented power production. Two different initial policies gave

different release policies, indicating the existence of multiple local

optima.

Yeh et al. (1978) employed a combination of SIDP and Larson and

Korsack's (1970) dynamic programming successive approximations (DPSA)

(called IDPSA, after an abbreviation for SIDP-DPSA) to find optimal hourly

operational criteria for the northern portion of the CVP system. They used

a three-level approach in which monthly schedules (Becker and Yeh, 1974)

were used as input to the daily model (Yeh et al., 1976) and the results of

the daily model served as a basis to the hourly model. In essence, the

system is decomposed into smaller subsystems and DPSA is used to advance the

solution towards a local optimum. Each separate subsystem optimization is

obtained by SIDP. The model also used an expanded approximate LP with a

subsequent adjustment technique to develop an initial feasible policy, which

is subsequently used to start the DPSA iterations.

Turgeon (1981) applied the principle of progessive optimality algo-

rithm (POA) (Howson and Sancho, 1975) to maximize energy production in a

system of four reservoirs in series. He considered deterministic inflows in

an hourly operation model which incorporated time delays in the continuity

equation.

1-3 Yeh and Becker (1982) presented a multiobjective technique for the operation of a portion of the CVP system. The technique uses the criterion of a decision maker as an input to choose among the multiple objectives served by the system. The constraint method of solution is used to solve the multiobjective programming problem.

Maria() and Mohammadi (1984) extended the monthly operation model of

Becker and Yeh (1974) to allow for maximization of both water releases and energy from the reservoir system. The proposed model is a combination of LP

(used for month-by-month optimization) and DP (used for annual optimization). The efficiency of the LP-DP solution algorithm is improved through the use of parametric LP (reduces computation time) and an iterative solution procedure (reduces computation time and storage requirements). Those efficiency measures allowed the use of minicomputers (Marino and Mohammadi,

1983b), which are suitable for frequent updating purposes. The use of the model was illustrated for Shasta and Folsom reservoirs (CVP). In addition,

Mohammadi and Marifio (1984a) reported on an efficient algorithm for the monthly operation of a multipurpose reservoir with a choice of objective functions. The choice of objective functions gives the reservoir operator flexibility to select the objective that would best satisfy the needs of the demand area. Finally, Mohammadi and Mariiio (1984b) presented a daily operation model for maximization of monthly water and energy output from a system of two parallel reservoirs. The daily model uses optimum monthly water and energy contract levels obtained from the monthly release policy.

The model was applied to Shasta and Folsom reservoirs.

In the stochastic environment, Turgeon (1980) presented applications of system decomposition (DPSA) and aggregation/decomposition for the weekly operation of six reservoir-hydroplants in a stochastic environment. The decomposition approach works only for systems composed of parallel subsystems of reservoirs in series along rivers. In the decomposition technique, 1-4 each subsystem is lumped into a composite equivalent and thus the (weekly) operation policies are developed for each composite subsystem as a whole.

The aggregation/decomposition scheme lumps all but one of the parallel subsystems into a composite unit and solves a two-state variable problem.

The process is repeated for all subsystems and then an adjustment is made to the solution of the aggregation phase to obtain a final solution. Results showed that the aggregation/decomposition method is superior to the decomposition technique for a six-reservoir system.

Simonovic and Marino (1980) extended Colorni and Fronza's (1976) monthly operation model to allow for two reliability constraints, one for flood protection and the other for drought protection. The solution procedure consisted of an LP optimization and a two-dimensional Fibonaccian search technique. The search technique was used to select the reliability levels, and the LP was used to determine the releases for those reliabilities and thus evaluate the objective function. Simonovic and Marifio (1981) also developed a methodology for the development of risk-loss functions in the reliability programming approach to a single reservoir. Flood and drought risk-loss functions were developed by using economic data. The reliability programming model was later applied to a three-reservoir system by Simonovic and Marifio (1982). Fixed reliability levels for each month in each reservoir were used to overcome computer storage requirements. Large memory requirements are needed for multireservoir systems; only local optimality can be guaranteed by the approach.

Marifio and Simonovic (1981) developed a two-step algorithm for the design of a multipurpose reservoir. The model was formulated as a chance constrained-linear programming problem (CCLP) which maximized downstream releases. The first step transforms the CC model into its deterministic equivalent through the use of an iterative convolution procedure. The second step finds the optimum size of the reservoir by solving the 1-5 deterministic LP developed in step one. The model allows the use of random

inflows and demands together with other deterministic demands.

Marifio and Mohammadi (1983a) improved the work of Simonovic and Maria°

(1980, 1981, 1982) by developing a new reliability programming approach for

the monthly operation of a single multipurpose reservoir. The modeluses

CCLP and DP and differs from other reliability programming approaches with

respect to the following three points. First, the development of risk-loss

functions is not necessary. Second, the reliability levels are not assumed

to be constant throughout the year, and are different from month to month.

This will eliminate the need for unnecessary extra releases in summer for

high flood reliabilities, and extreme low releases during winter for high

drought reliabilities. Third, the reliability of the hydroelectric energy

production is included in the model.

The brief review of reservoir operation models outlined above does not

pretend to have exhausted the list of innumerable reservoir models pre-

viously reported in the literature. Instead, emphasis has been given to

those reservoir operation models that are related with the methods and/or

the reservoir system employed in this study.

This report is devoted to the development and application of a large-

scale optimization model for the management of the northern portion of the

California Central Valley Project (NCVP). The NCVP is a multireservoir,

multipurpose system that constitutes the backbone of central and northern

California water supply development. The product of this research is a

model for computing monthly water release policies that would maximize the

annual system energy revenues while satisfying all constraints imposed on

the operation of the NCVP. Emphasis is given to the following items throughout the development of the model:

(i) The optimization model must be able to represent the physical features of the system as closely as possible and include the pertinent 1-6 links that exist between its different components. Those links are of hydraulic, hydrological, and operational nature. Links of operational nature refer to the effects that water releases from any reservoir have on the operation of any other reservoir.

(ii) The model must be tractable both numerically and in its implemen- tability. The number and nature of the computations as well as the computer storage requirements have been kept to a practical, manageable size, despite the fact that the model is aimed at large-scale problems. Also, the model has been developed to match the real operation conditions faced by the reservoir managers of the NCVP. The mathematical development of the model is kept at a general level (and subsequently tailored to the NCVP), so that the approach can be extended and applied to other systems with minor modifications.

(iii) The uncertainty of streamflows is handled in a simple way, and allows to keep the operating policy up to date with the actual evolution of the storages of the reservoir system.

(iv) Along with the task of incorporating streamflow stochasticity and maintaining an adequate resemblance of the real operating scenario, conceptual rigor has not been sacrificed. Included in the analysis are types of optimality achieved (local or global), computational and computer storage burdens, existence of multiple optimal solutions, and a discussion of the advantages and disadvantages of the proposed optimization model.

(v) Test of the model with a large-scale system (the NCVP) under different scenarios: below-average, average, and'above-average streamflow conditions.

The model is tailored to fit the planning modus operandi of the NCVP system managing agencies (i.e., the U.S. Bureau of Reclamation and the

California Department of Water Resources). At the beginning of each month, inflow forecasts are made by a streamflow forecasting model developed by the 1-7 authors. Those forecasts are then treated as deterministic inputs and a multistage deterministic problem is solved for the remaining of the reservoir planning horizon (i.e., one water year). The computed release policy is followed for the current period only. As inflow forecasts deviate from actual realizations, updated forecasts are computed and a revised future release policy is developed on the basis of the observed storages and updated inflow forecasts. Each multistage deterministic problem, in turn, is decomposed into a sequence of two-period quadratic problems that are solved one at a time, using linear programming or a numerically stable active set method. The decomposition of each multistage deterministic problem is done within the framework of the progressive optimality algorithm (P0A)(Howson and Sancho, 1975; Marifio and Loaiciga, 1985a, 1985b).

The multireservoir monthly management model developed in this study has the following innovative features: (1) the model has the minimum possible number of unknowns (dimensionality); (2) penstock releases and spillages are treated both as independent decision variables; (3) the multifunction nature of the application reservoir system is appropriately handled by maximizing system energy revenues — the only profit generating function — while satisfying other functions by an appropriate set of constraints on storages, penstock, and spillage variables; and (4) the solution algorithm for the resulting quadratic programming (QP) problem can handle the nonpositive definiteness of the Hessian matrix in a computationally efficient and numerically stable manner.

The remainder of the report is organized as follows. Chapter 2 describes the solution algorithm of the optimization model. Chapter 3 contains a description of the NCVP system, including data on constraints, power generation, and reservoir spillages necessary in the optimization model. Chapter 4 gives a development of the optimization model and several

1-8 alternative versions that arise from different assumptions on energy production, reservoir losses, and spillages. Chapter 5 provides the application of two alternative versions of the optimization model. Chapter 6 is a summary and final statement of conclusions and further research needs.

1-9 CHAPTER 2

DESCRIPTION OF THE SOLUTION ALGORITHM

This chapter contains a description of the solution algorithm adopted in this study, a modified version of the progressive optimality algorithm

(POA) that reduces memory and computational requirements.

First, let us summarize the overall optimization philosophy adopted in this study. The longest-term operation activities of the NCVP are planned for each water-year. On October 1, the managing staff estimates future streamflows for the next 12 months. Based on that forecast, a tentative

release policy is proposed for the 12-month period. Because actual streamflows deviate from their expected values and institutional and/or technical

conditions may vary from month to month, streamflow forecasts are updated at

the beginning of each month and the release policy is revised for the

remaining months of the year. The proposed optimization model of the NCVP

system is developed to fit this recurrent scheme for release policies. The

updating scheme takes into consideration the most recent streamflow informa-

tion and the actual system storage evolution.

The updating scheme utilized in this study consists of a sequential

solution of deterministic problems in which reforecasted flows enter in the

formulation of each new problem, and initial storages are set equal to the

actual values at every beginning of month. Such updating scheme has been

successfully applied in control engineering and is usually referred to as

the certainty-equivalence controller (CEC) or under some circumstances as

open-loop feedback controller (Bertsekas, 1976). The solution of each

(consecutive) problem is obtained by a deterministic optimal control method,

namely the POA. For this study it is sufficient to point out that the

adopted CEC is quasi-adaptive, i.e., if the value of the objective function

achieved by applying a completely deterministic control is denoted by J0, 1 the value obtained by the CEC as J , and that for the true (unknown) optimal 2-1 * * 1 0 control as J, then (under maximization) J .. J ..' J (Bertsekas, 1976, p. 199). This inequality indicates that the use of the CEC control scheme would result in a performance that is at least as good as if a purely deterministic control was used.

The POA solves a multistage dynamic problem as a sequence of two-stage optimization problems. It is based on the principle of progressive optimality (PPO) (Howson and Sancho, 1975) which states that: "The optimal path has the property that each pair of decision sets is optimal in relation to its initial and terminal values." The PPO is derived from Bellman's (1957) principle of optimality.

This chapter presents an extension of the work of Howson and Sancho

(1975) to the case in which there exists bounds on state and decision variables. Also discussed are some powerful programming innovations introduced by the authors in the POA that can be used to accelerate its convergence rate.

Before outlining the solution approach, several of the favorable features of the POA are summarized:

(1) The decision (i.e., releases) and state (i.e., storage) variables need not be discretized.

(2) The two-stage optimization problem (objective function and constraints) can be solved by any adequate algorithm, either of the DP or NLP families. Thus, the objective function and constraints can be of arbitrary form, although for convergence to a global optimum the problem must be concave (or convex for the case of minimization).

(3) The dynamics of the system (i.e., the continuity equation) need not be invertible for the application of the POA, although that implies the possible existence of multiple solutions. By invertibility is meant that decision (state) variables are related in a one-to-one unique manner to state (decision) variables. 2-2 (4) Time delays in the continuity equation can be incorporated easily.

(5) Convergence rates depend on the scheme utilized in the two-stage maximization. When a fast convergent algorithm like Fletcher's (1981) active set method is used, the POA convergence rate is better than linear and perhaps close to quadratic if a good initial policy is chosen, however, it has not been possible to establish quadratic convergence rigorously.

Actual applications of the POA (Chapter 5) have shown convergence in less than nine iterations, five being the average.

(6) Assuming that the active set method is used to solve the two-stage optimization problem, the computation effort to complete one iteration (from 3 3 time t = 1 to t = N) is an N and b(2m + n) N for invertible and noninvertible continuity equations, respectively, in which a and b are positive constants larger than one and independent of m and n, the respective dimen- sions of the decision and state variables, and N is the number of optimization periods (N = 12 in the applications of this work).

(7) Storage requirements are proportional to nN and (2m + n)N for invertible and noninvertible cases, respectively. Those requirements account for the storage of current state trajectories. Parameter (transition matrices in the continuity equation) storage requirements vary greatly from one application to another, but for most applications they are 2 2 proportional to n and (2m + n) for invertible and noninvertible cases, respectively.

It is shown in Chapter 4 that the multireservoir operation optimization model developed in this study has the following mathematical structure:

12 Maximize kis 2 F(x) (2.1) x t=1 —t subject to

(2.2) Vk't

2-3 Equation (2.1) represents the maximization of energy revenues for the entire

water year, in which Ft can be either a linear or a quadratic operator on

the storage variables x and tfs represents the unit price of energy. Equa-

tion (2.2) denotes a set of de = 1, L constraints on the storage

variables for each month t, t = 1, 2,..., 12. For the NCVP system,,xt is a

9 x 1 vector whose elements represent the storages at each of the reser-

voirs. By using the POA, problem (2.1)-(2.2) is decomposed into a series of

two-stage problems, namely,

Maximize IMF + F ()] (2.3) t-i(x) —t t x —t

subject to

f (2.4) de,t-1( t) b1

(2.5) fk,t(M bk,t

fixed (2.6)

Equation (2.3) denotes the maximization of the energy revenues during months

t-1 and t. Equations (2.4) and (2.5) represent the constraints on xt for

months t-1 and t. Equation (2.6) specifies that the storages at the

beginning of months t-1 and t+1 are fixed at their (k+l)th and kth iteration

values, respectively. To clarify how the sequential solution of the two-

stage problems is accomplished iteratively, the steps of the POA algorithm

are listed next:

(1) The initial and final states and are fixed. Set the ;i1 1\1+1 iteration counter k equal to one (k = 1). Find an initial feasible release ( policy { .k)1 and its corresponding state trajectory {4.k)1.

(2) Solve the problem given by eqs. (2.3)-(2.6) by using any convenient method. In Chapter 4 it is shown that for alternative formulations of

2-4 the optimization model, linear programming and quadratic programming algo- rithms are suitable for solving the two-stage problems.

Denote the solution obtained in step 2 by x Set (k+1)t. = (3) `--t Increase the time index by one (i.e., set t equal to t + 1) and go to step 2. Repeat steps 2 and 3 until a complete iteration is performed (t = 1 through t = N). This is the end of the kth iteration. (k+1) (k) 2 (4) Perform a convergence test, e.g., Ilu - < &, & 0, t 2 for all t. If the test is satisfied, stop; otherwise, increase the iteration index, k = k +1, set t = 1, and go to step 2, or stop if k has exceeded a specified limit. Notice that at the end of each iteration the new values k+1) (k+1) { are used to derive the corresponding control sequence {ut by means of the continuity equation. Figure 2.1 shows the advancing scheme for the POA. Notice that to solve, say, for the unknown storage at time t = 2, the values of the storage at time t = 1, xi, and of the storage at time t = 3, x3, are held fixed. The two-period problem involving months 1 and 2 * is solved for x yielding the solution which in turn becomes the —2' ?_c_.2, (k+1) (4) (k+l)th POA iteration value. Subsequently, x2 (= Lc ) and x are held 2 —4 fixed and the two-period problem involving months 2 and 3 is solved for x —3' etc.

Some remarks about the POA are necessary. First, it would appear that

the iterations progress from t = 1 to t = N, but this is not necessarily so.

The algorithm can be modified in such a way that successive local optimiza-

tions are made at some periods in which relatively higher improvements in

the objective function are observed. Once the rate of improvement falls

below some preset value (e.g., 5% improvement in the objective function with

respect to the previous iteration), the algorithm would advance towards

ending the iteration at t = N. Figures 2.2 and 2.3 show two different

advancing schemes. This flexibility of the POA, to be able to take full

advantage of localized conditions, accelerates the overall convergence rate 2-5 at little expense in programming complexity, and is an innovation introduced

in this study. Second, time lags in the continuity equation are incorpor-

ated in a straightforward manner. Consider reservoir i which is directly

connected to (downstream) reservoir j. Let d.. be the travel time between

reservoirs i and

j; x!, uit, and yi the respective state, decision, and streamflow variables for

reservoir j at time t; and u the decision for reservoir i at time t-d ij - d... The continuity equation for reservoir j can be then written as

i x,1 = xi - + u + yi + ei (2.7) tt-d.. ut 13

in which ej is the sum of diversions/accretions, net losses, and runoff into

reservoir j. Equations similar to (2.7) must be written for every reservoir

in the network and the POA must be programmed so that previous decisions

()are retrieved from storage to be used in the continuity equation,

eq. (2.7).

Convergence proofs to an optimal solution in a finite number of steps

can be derived straightforwardly for the POA for the constrained case,

following similar arguments to those used by Howson and Sancho (1975) for

the unconstrained case.

t=1 t=2 1=3 1=4 1=12 1=13 month 1 month 2 month 3 month 12

—13

(k+1)=x* —3 —3

Fig. 2.1 Advancing scheme for the POA. 2-6 5

X -7

(2)

X -7

1 (i) (I) v(1) (I) x x x (1) )SI x ?S-7 State Variable Fig. 2.2 Standard POA (To achieve state x(3).. (1). and a‹..3(1) yield —3 —x1 2) (1) (2) (2) (1) (ii) x( and yield x ; (iii) x and x5 yield )i4 (2) (2) (iv) and yield x(3). and (v) (3) and x4 yield )1 )i3

(5) X -2

(4 4) '-2 -3X( X ' -i

X(3)• X(3) X -I

-2 -3 ,

X X(I) X(I) x(1) X(I) X -3 -5 -6 -7 State Variable(, Fig. 2.3 Modified POA (To achieve state (i) xi and x"-) yield —2x(5)-• (1) (ii) )i(2)2 and x4 yield x(2)- (iii) x and 42) yieldeld x(3)- —3 ' —1 —2 ' (1) ((iv) ie and ?i5 yield x(3)- (v) x and )43) yield x( '4)- —1 —2 (1) (4) (5) (vi) 44) and 4 yield x(4)- and (vii) x and yield x ). —3 ' —1 )-S3 —2 This scheme should be used when significant improvements (e.g., 5%

improvement in the objective function with respect to the previous

iteration) arise from the two-stage problem involving periods 1

and 2). 2-7 CHAPTER 3

THE NCVP SYSTEM: FUNCTIONS, OPERATIONAL CONSTRAINTS,

PHYSICAL FEATURES, AND FLOW FORECASTING

This chapter contains a description of the NCVP system, its configura-

tion, functions, as well as flood control, water supply, environmental,

recreational, power generation, reservoir losses and spillage data. This

information is contained in Section 3.1. The description of the streamflow forecasting technique is briefly outlined in Section 3.2. The data and

other information developed in this chapter are used in the applications of

Chapter 5.

3.1 Description of the NCVP System

Functions

The system under analysis is composed of the following reservoirs:

Clair Engle, Lewiston, Whiskeytown, Shasta, Keswick, Natoma, Folsom,

New Melones, and Tullock (reservoirs 1-9, respectively). Figure 3.1 shows a

schematic representation of the Central Valley Project (CVP). The portion

of the system analyzed in this study (NCVP) is shown within the dashed

lines.

The NCVP is managed jointly by the U.S. Bureau of Reclamation (USBR)

and the California Department of Water Resources (DWR). It stores flood and

snowmelt waters and releases them at appropriate times to serve different

functions. The main purposes of the NCVP are: provision of water for

irrigation (I), municipal and industrial uses (MI), environmental control

and enhancement (E), fish and wildlife requirements (F), river navigation

(N), water quality control (WQ), flood regulation (FC), hydropower (HP),

recreation (R), and control of ocean intrusion and erosion. Fults and

Hancock (1972, 1974) and Madsen and Coleman (1974) presented a thorough

geographical, institutional, and historical description of the CVP. The

discussion herein will be centered on the nine reservoirs mentioned earlier 3-1 and, specifically, on the joint operation of those reservoirs. From now on,

the term system will refer to those nine reservoirs.

Table 3.1 shows basic data of the NCVP. The reservoirs of the NCVP

must operate jointly to perform the multiple functions enumerated in

Table 3.1. The system release policy is subject to physical and technical

constraints that arise from the capacity and technology of the facilities,

as well as institutional and environmental regulations. It is evident from

Table 3.1 that Shasta, Clair Engle, Folsom, and New Melones are the larger

reservoirs within the system. Lewiston, Whiskeytown, Keswick, Natoma, and

Tullock play an important role as regulating reservoirs. A regulating

reservoir maintains an adequate flow magnitude downstream of a larger reser-

voir and a stable hydraulic head for a downstream power plant.

Winter flows in the Trinity River are stored for later release from

Clair Engle Lake. Normally, water behind Trinity Dam (at Clair Engle Lake)

is released through the Trinity Power Plant and regulated downstream in

Lewiston Reservoir. The major portion of the water reaching Lewiston Dam is diverted to the Sacramento River watershed via the 11-mile-long Clear Creek

Tunnel. The remaining water is released to the Trinity River to support fishery. Water diverted through Clear Creek Tunnel exits through the

J. F. Carr Power Plant in Whiskeytown Reservoir. It is possible to release water from Whiskeytown Dam to Clear Creek or make diversions through the

Spring Creek Tunnel.

Sacramento River water is stored for later release from Shasta Dam.

Ordinarily, water is released from Shasta Reservoir through the Shasta Power

Plant and flows downstream to Keswick Reservoir. The inflow to Keswick includes releases from Shasta and Spring Creek Tunnel. Releases from

Keswick may be made through the Keswick Power Plant and flow in excess of the power plant penstock capacity are spilled to the Sacramento River.

3-2 Folsom Dam stores American River water and releases it under normal operation through its power house. Excess flows are spilled to the American

River. Part of those releases is diverted by the Folsom South Canal and the remainder goes into Lake Natoma, which acts as a regulating reservoir. As much water as possible is released through the Nimbus Power Plant at Lake

Natoma, with any excess water spilled to the American River. The American and Sacramento Rivers converge near the city of Sacramento and flow to the

Sacramento-San Joaquin Delta to serve several purposes.

New Melones Dam stores Stanislaus River flows to release them during the summer for agricultural use. It also releases water to maintain water quality standards in the San Joaquin River. Tullock Reservoir, upstream from New Melones, is primarily a regulating reservoir.

Data Relevant to the Constraints of the System

The optimization model requires a quantitative statement of the constraints on the operation of the system. Figure 3.2 shows a schematic representation of the NCVP and the points at which accretions/diversions exist. Table 3.2 contains information on flow requirements. There exist also regulations on maximum and minimum storages and releases that arise from flood control provisions, recreation, aesthetic concerns, power plant performance, etc. Those bounds on storages and releases are used as constraints in the optimization model. Storage-area-elevation data for all reservoirs in the system are available from the authors.

Flood control constraints are specified by diagrams prepared by the

Corps of Engineers. Figure 3.3 displays the flood control diagram for

New Melones reservoir. Similar diagrams exist for other reservoirs.

Power Generation

Power generation will be considered in the objective function of the optimization model. The amount of power generated by the system depends on 3-3 the effective hydraulic head at the intake of the turbine, the flow through the penstocks, and the efficiency of the turbines. Curves that relate the rate of energy production (in megawatt-hour per kiloacre-foot, Mwha 0 to reservoir storage (in Kaf) were developed from actual records of operation for the reservoirs since their power plants began operating. Those curves can be used to compute the energy generated in any period without having to include constraints on power production, which are nonlinear and usually create numerical and analytical difficulties. The rate of energy production vs. reservoir storage curves were tested with actual operation data for the

NCVP. In comparison with actual energy output, the error in the predicted energy generation was less than 2 percent.

Figure 3.4 shows the energy production vs. reservoir storage curve for

New Melones Power Plant. The energy vs. storage curves were approximated by linear and quadratic equations (for J. F. Carr and Nimbus power plants, the equations were strictly linear). Linear and quadratic equations are used in alternative optimization models in Chapter 4. The equations are given by:

Trinity (at Clair Engle Lake) ti = 221.0 + 0.0858 Xi (linear) (3.1a) t t 2 r = 94.6% 1 -4 -1 2 t = 133.0 + 0.228 xi - 0.468 • 10 (x ) (quadratic) (3.1b) t t 2 r = 99.3%

J. F. Carr t2 = 606.3 - 0.254 X3 (linear) (3.2) t t 2 r = 99.0%

Spring Creek t3 = 460.0 + 0.434 X3 (linear) (3.3a) t t 2 r = 98.0%

3-4 t•3 = 445.0 + 0.738 X3 - 1.10 • 10-3(X3)2 (quadratic) (3.3b) 2 r = 99.0%

Shasta t•4 = 234.0 + 0.0462 X4 (linear) (3.4a) 2 r = 94.8% t = 169.0 + 0.107 xt - 0.115 • 1o_4()2 (quadratic) (3.4b) t 2 r = 99.6%

Keswick 5 tt = 80.3 + 0.6 X5 (linear) (3.5) 2 r = 92.0%

Folsom -6 = 201.0 + 0.120 x (linear) (3.6a) t 2 r = 95.8% 10-3(T c6)2 = 171.0 + 0.265 X6 - 0.130 • (quadratic) (3.6b) t 2 r = 98.7%

Nimbus

= 26.3 + 0.8 X7 (linear) (3.7) 2 r = 91.0%

New Melones et = 268.0 + 0.123 x (linear) (3.8a) t 2 r = 98.0% -8 -4 -8 2 = 169.0 + 0.275 x - 0.479 • 10 (x t) (quadratic) (3.8h) t 2 r = 98.6%

Tullock 9 t• = 64.9 + 0.931 X9 (linear) (3.9a) 2 r = 99.4% 3-5 t•9 = 63.4 + 1.020 X9 - 1.37 • 10-3(X9)2 (quadratic) (3.9b) 2 r = 99.0%

1 In eq. (3.1), t is the energy rate in megawatt-hour/kiloacre-foot (Mwh/Kaf) t -1 for Trinity Dam (at Clair Engle Lake), x is the average reservoir storage t 2 in Kaf during a specified period, and r is the adjusted regression correlation coefficient. Other terms in eqs. (3.2)-(3.9) are defined similarly.

Net losses

Evaporation and direct rainfall input are considered for the larger reservoirs only, i.e., Clair Engle, Shasta, Folsom, and New Melones. His- torical records of operation for Trinity and New Melones were used to derive monthly coefficients of net loss rates (evaporation minus direct rainfall input). For Shasta and Folsom, those coefficients were derived from data provided by Hall et al. (1969). Table 3.3 gives net loss rates data for the reservoirs. The total net loss in any month t (et) in Kaf is expressed by

i = c Ai (3.10) et t t in which A is the average surface area of reservoir i in month t (kiloacre, t i Ka) and c is the net loss rate during month t (ft/month). It is possible to express eq. (3.10) as a function of average storage if an area-storage relation is available. Several linear functions that relate area (A) and storage (x) were obtained for the four reservoirs mentioned earlier:

Clair Engle -1 A•l = 3.33 + 0.0078 x (3.11) t 2 r = 97.0%

Shasta 4 -4 A• = 3.99 + 0.0061 x (3.12) t 2 r = 96.0% 3-6 Folsom 6 -6 A = 2.67 + 0.0094 x (3.13) t t 2 r = 95.0%

New Melones 8 -8 A = 2.91 + 0.0088 x (3.14) t t 2 r = 96.0% in which the area A is in Ka and the storage in Kaf. Equations (3.11)-(3.14) were developed from area-storage data.

By substituting eqs. (3.11)-(3.14) into their respective equivalents to eq. (3.10), the net loss for each month t can be expressed as a function of average storage (x) as follows:

Clair Engle e = 3.33 c + 0.0078 c x (3.15) t t t t

Shasta e = 3.99 c + 0.0061 c x (3.16) t t t t

Folsom e = 2.67 c + 0.0094 c x (3.17) t t t t

New Melones 8 8 8 -8 e = 2.91 c + 0.0088 c x (3.18) t t

Benefits Accruing from the Operation of the System

Due to the multiobjective nature of the NCVP operation, multiple benefits arise from the operation of the system. Flood control benefits arise from the reduced damage caused by floods that would otherwise occur without the project. As an example of the services provided to the public, Madsen and Coleman (1974) estimated that in 1970 the system averted flood damages for about $55 million (in 1970 dollars). 3-7 Irrigation benefits can be measured by the cost of providing an alter-

native source of water supply. The criterion of alternative cost can also

be applied to economic benefits accruing from hydropower, municipal and

industrial use, and navigation. Jaquette (1978) estimated the cost of

developing new reservoir water supply at $100 per acre-foot. The issue of

benefits computation is more complicated with regard to water -quality and

fisheries. The economics of reservoir operation is a topic that needs

further research.

For the purpose of this study, the performance of the system is

measured by the total energy revenues produced during a water year. As

shown in Chapter 5, it is rational to use energy revenues as a performance

criterion because a large amount of power generation is usually associated

with increased water deliveries for other purposes and with adequate flood

control storages. The satisfaction of all other reservoir functions, e.g.,

flood control, water supply, recreation, etc., is enforced by imposing an

appropriate set of constraints on the operation of the reservoir system.

Thus, the multiple functions of the NCVP system are handled in the optimiza-

tion model by obtaining a release schedule that maximizes energy revenues,

while providing adequate services for other purposes via constraints on

releases and storages.

Modeling of Spillages

In Chapter 4, it is shown that one of the alternative formulations of

the optimization model expresses the reservoir spillages (r denotes the t spillage from reservoir i) during month t as linear functions of storages.

Those equations are developed next.

The exponential interpolation of the spillway discharge tables yielded the following equations (flows are in cfs and elevations are in ft above mean sea level):

3-8 Trinity (at Clair Engle reservoir) 1 1 2370)1.29 r = 781 (h - (3.19) t t 2 r = 98.4%

Lewiston 2 2 0.626 r = 412 (h - 1871) (3.20) t t 2 r = 99.8%

Whiskeytown 3 3 1.52 r = 992 (h• - 1208) (3.21) 2 r = 98.7%

Shasta 4 4 1.56 r• = 314 (h• - 1039) (3.22) 2 r = 99.9%

Keswick 5 5 0.436 r• = 720 (h• - 547) (3.23) 2 r = 99.2%

Folsom 6 6 0.466 r = 242 (h• - 420) (3.24) t 2 r = 99.9%

Nimbus 7 7 0.317 r• = 437 (h - 110) (3.25) 2 r = 99.9%

New Melones 8 8 1.55 r = 420 (h - 1088) (3.26) t t 2 r = 99.6%

3-9 Tuliock 9 9 0.478 r = 750 (h - 495) (3.27) t 2 r = 95.0%

2 in which r is the adjusted regression correlation coefficient. Equa- tions (3.19)-(3.27) need to be (i) converted from cfs to acre-ft/month before they can be used in the development that follows and (ii) expressed in terms of storages, because the optimization is expressed in terms of storages rather than elevation.

From elevation-storage data, shapes of elevation vs. storage curves were analyzed to determine appropriate interpolation functions. The inter- val of interest is for the range of elevations above the spillway crest, otherwise the spillage would be zero, which means that only the shape of the elevation vs. storage at high stages is of concern. Fortunately, from the perspective of numerical simplicity, the plots were nearly straight lines for all but low elevations. Figure 3.5 depicts the elevation-storage curve for Clair Engle, which shows high nonlinearity for elevations below 2010 ft and an almost perfect linear curve everywhere else. Similar behavior was determined to exist in the other major reservoirs (Shasta, Folsom, and

New Melones) for which the elevation-storage curves are needed. A similar pattern holds for the smaller reservoirs, but for those the interest is centered at a single elevation because the storage is held constant at the maximum permissible storage as shown in Chapter 5 and there is no need for elevation-storage curves. The following linear functions were developed for the four major reservoirs:

Clair Engle 1 1 h = 2142 -I- 0.0971 x (3.28) t t 2 r = 97.2%

3-10 Shasta 4 4 h = 871 + 0.0444 x (3.29) 2 r = 99.3%

Folsom 6 6 h = 364 + 0.101 x (3.30) t t 2 r = 99.5%

New Melones 8 8 h = 860 + 0.0945 x (3.31) t 2 r = 99.5%

In eqs. (3.28)-(3.31), hit denotes the elevation at time t at reservoir i i = 1, 4, 6, 8. To complete the information regarding elevation vs. storage, elevations in ft. above mean sea level corresponding to constant storages at 2 3 Lewiston, Whiskeytown, Keswick, Natoma, and Tullock (x = x = 241, 14.7' t 5 7 9 x = x = 8.8, and xt = 57.0 Kaf, respectively) are 1901.1, 1210.0, 23.8' t 587.4, 125.1, and 501.6 ft, respectiveLy (from elevation-storage data). Upon substitution of eqs. (3.28)-(3.31) into eqs. (3.19), (3.22), (3.24), and

(3.26), respectively, and after a subsequent first-order Taylor expansion linearization, the following expressions are obtained:

Clair Engle 1 1 1 r c + d x (3.32) t t t t

In which

3-11 1 1 1 1/x0 [1( xt,0 xt+1 1 n1 1 1 -1 c = c g - d - x c g t 2 t,0 2

1 l ]1ag ,1 - d 0 dx t 1 x t,0

1 + x +1)) (4 1.29 781 [(2142 + 0.0971 o' _ 2370' 4 .(1.29)(781) = 2

1 xl + 1 [(2142 + 0.0971( " vj 2370 0'29 (0.0971) 6 (3.33) 2 t+1))

1 1 x 1 1-1[ 1 xt, + t+1 _ iT-11-1 1 d c g d 6 t 2

(3.34) 1 1 + 0.0971( xt+1 1 1(1.29)084(2142 + "0 2370]0.29(0.0971) 6 = 2

In eqs. (3.33) and (3.34), 61 = 1 if the reservoir elevation is above the 1 1 spillway crest and 6 = 0 otherwise; xt,0 is the value of storage about

which the linearization is made (the selection of x is explained in t,0 ,.. . i 1 1 Chapter 4). Similar equations of the form r = IC + d x were developed in t t t t an analogous manner for Shasta, Folsom, and New Melones reservoirs (i = 4,

6, 8, respectively) and are omitted to conserve space.

4.2 Streamflow Forecasting

Streamflows are modelled as a multivariate autoregressive first-order

process (AR(1)). The parameters of the AR(1) model, i.e., transition proba-

bilities and covariance matrices, are estimated via maximum likelihood and

expressed in analytical form. The advantage of the maximum likelihood

estimation technique over, say, least-squares estimation resides on the fact 3-12 that the maximum likelihood approach allows to perform statistical tests on the assumptions implied by the AR(1) model, namely: (1) on the order of the model; (2) on the time variance of the transition matrices; (3) on the non- stationarity of the covariance matrices; and (4) on the non-independence of streamflow realizations at different stations. Details of these tests are available from the authors. It was found (after conducting the statistical tests) that the following AR(1) model was adequate (see Anderson, 1978):

/V = B + (3.35) Yt t Yt-1 —t in which yt is a p-component vector with mean E(y) = 0 (in this study, yt

/V is a five-dimensional flow vector whose mean has been subtracted); Bt is a p x p matrix (the "transition" matrix); and gt is a sequence of independent random vectors with expected values E(et) = 0_ and covariance matrices

E(et itr) = I and independent of v . Let the covariance of y t' Yt-2" ' t T be E(yt yt) = R. It follows that

R = rg R I (3.36) t t t-1 a + t

If the observations are made for t = 1, 2,..., T (T = 12) and if yi and the

are normal, then the model for the observation period is specified by

R B2,..., ii , 2 ,...I . The maximum likelihood estimators (MLE) of 'i2, 1' 1 2- T — B3,..., B are I'

= C (1) C 1 1(0)(0), t = 2, 3,..., T (3.37) t t- '

The MLE of R I ... 2 are, respectively, 1' 2' , T , A R = C (0) (3.38) 1 1 and ^T 2 = C (0) - B C (0) B t = 2, 3,..., T (3.39) t t t t-1 t'

In eqs. (3.37)-(3.39), 3-13 N 1 T (3.40) C(J) = R 2 Ytu Yt-j,c/ ci1=

Clearly, C(j) is the sample cross-correlation matrix between the observa-

that knowledge of R 2 tions at period t and those at period t-j. Notice 1, t'

and B allows the computation of R by means of eq. (3.36)., t R2'.." T Application of the AR model to the five streams considered in the NCVP

I's. /... the parameters B (estimate of the transition matrices) and 2 yielded t t

(estimate of the noise covariance matrices), which are not presented to

conserve space. Prediction of future inflows having the last period reali-

zation of y (i.e., yt-1) is accomplished by using the following expression t recursively:

.04. E(yti.1 I yt) = Bt yt (3.41)

Inflow forecasts computed from eq. (3.41) converge to the historical means

after eight periods. Thus, the best way to use eq. (3.41) is by updating

the forecasts once a value of yt is observed, i.e., by changing the base

time yt in eq. (3.41) and considering only the most recently updated fore-

.... el. ,i. Also, parameters B 2 and R can be casts for operation planning. t' t' 1

modified as new realizations yt become available so that those estimates can

be kept up-to-date with the most recent information. The forecasted stream-

flows ranged within ±10% of the actual values, a satisfactory accuracy for

the purpose of the planning model to be developed in this study.

3-14 -----I I I I ....- 0 -C I '9cY 1 'g

o I I Clair Engle Res. ZIV 0 I I v) I c!".; Shosto I I Lewiston Res. '6- v Res. I Whiskeytown I—1 .. I Res. Spring Cr. c I 'i• P H -6 I • A • > - - Clear Cr. it .. I I -1 J.F.Corr Spring Cr. Tunnel PH. Keswick I Tunnel Res. .4,z I Folsom' 1---) P. H. I V.. Res. 1 I ...Wilkins SI. P. H. i L _ _ _ _ Nimbus I P.H., I I < L -c -S I DE LTA I Notomo Res. 0 OUTFLOW < < (n_ 1I r7Clif ton E 0 ,,v Court V A I 0 g I SWP Delta' r-i Tracy I East By 1-..-1 P.P. I Tullock New Melones - Aqueouct Res. Res. I ti I -6 Son Luis :513 c o 3 Stomsious R. 1 Res. 0 cy. 00 0 L _ _ _. _ _ _ _. _ ...... __J V(.) vO 0 0 -) P.G.P. "ic) c A O'Neill 0 '4) O'Neill P.G.P. v2 Foreboy P.H. = Power House PP. = Pumping ri Dos Amigos Plant L---1 P. P. P.G.P. = Pump-generating Plant E t.„ .

Fig. 3.1 Schematic representation of the Central Valley Project.

3-15 R.

1 Sacramento I Trinity PP 1 r4

J.F.Corre Spring Cr. 3 2 3 M R PP R PP

, CC 0 , CU Q) (.) -c o cu U5 U. V > 5 U6 Wilkins SI. <

Rio U7 Nimbus PP v R7 Sacramento R. <

R. / e9 e8 R13 8

Jooquin Stanislaus R.

Son Ye v R8

Fig. 3.2 Schematic representation of NCVP diversions, losses, releases, and

spills.

SEP OCT NOV DEC JAN 4.... FEB MAR APR MAY JUN JUL ...)"6 AUG 0 0 - —

(X..- - 2420 ••-o- 1088.0 msl) - i-- 100 - ZZ FLOOD CONTROL SPACE 2320 1079.9 00 200 - - U r-- 2220 Lij 1071.5 (feet, < 300 - CONDIT ONAL 0 0> SPACE - 2120

Fig. 3.3 New Melones flood control diagram (Source: Central Valley

Operations Office, U.S.B.R., Sacramento, CA). 3-16 3000 —

-- 2500 Spillway Elevation 0

2000

1500 STORAGE

1000 RESERVOIR 500

0 200 300 400 500 600 AVERAGE PLANT ENERGY OUTPUT (Kwhian

Fig. 3.4 New Melones power plant gross generation curve.

Spillway Crest Elevation = 2370 ft 2040 (g 2020

1a)- 2000 a) 4- 1980

0 1960 < 1940

1920

1900 I 1 1 I I I 1 1 I I I I 1 1 0 400 800 1200 1600 2000 2400 2800 STORAGE ( Kaf)

Fig. 3.5 Elevation vs. storage (Clair Engle reservoir). 3-17 Table 3.1. Basic NCVP Data.

Managing First Year Installed* Functions Reservoir Institution of Operation Capacity Capacity Served (Kaf) (Mw)

Shasta USBR 1944 4552 559 I, FC, HP, MI, WQ, N, R, F

Clair Engle . USBR 1960 2448 128 I, FC, HP, MI, WQ, N, R, F

Lewiston USBR 1962 14.7 - Regulation, HP, F, R

Whiskeytown USBR 1963 241 154 Regulation, I, HP, MI, R, F W (J.F. Carr) 1 190 co (Spring Cr.)

Keswick USBR 1948 23.8 90 Regulation, HP, R, F

Folsom USBR 1955 1010 198 I, FC, HP, MI, WQ, R, F

Natoma USBR 1955 8.8 15 Regulation, HP, R, F

New Melones USBR 1978 2600 383 I, WQ, FC, F, HP, R

Tullock Oakdale 1958 67 17 I, HP, Regulation Irr. Dist.

* As of July 1982. Table 3.2 NCVP Flow Requirements (in cfs).

Requirement Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Function

Trinity River min. 300 300 300 300 300 300 300 300 300 300 300 300 F diversion (R2) Trinity River max. 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 FC release (r2 + R2) Clear Creek min. 50 100 100 50 50 50 50 50 50 50 50 50 F diversion (R3) Accretion (+) or 1000 2000 3000 4000 7000 4000 1000 3000 4000 3500 100 MI, I diversion (-), (R5) (+) (+) (+) (+) (+) (-) (-) (-) (-) (-)

Wilkins Slough min. 5000 5000 5000 5000 5000 5000 5000 5000 5000 5000 5000 5000 N flow Folsom South Canal 50 50 50 50 50 50 50 50 50 50 50 50 MI, I min. diversion coo (R7) Folsom min. 30 30 30 30 30 30 30 30 30 30 30 30 MI o diversion (R6) Shasta max. 79000 79000 79000 79000 79000 79000 79000 79000 79000 79000 79000 79000 FC release (u4 + r4) Keswick max. 49000 49000 49000 49000 49000 49000 49000 49000 49000 49000 49000 49000 FC release (u5 + r5), Keswick min. 3250 3250 3250 3250 3250 3250 3250 3250 3250 3250 3250 3250 F release (u5 + r5) Lake Natoma min. 250 250 250 250 250 250 250 250 250 250 250 250 F release, dry year (u7 + r7) Lake Natoma min. 500 500 500 500 500 500 500 500 500 500 500 500 F release, wet year (u7 + r7) Sacramento min. 75 10 10 50 50 35 220 410 520 550 530 250 MI, I diversion (R10) Natoma desired 1250 1250 1250 1250 1250 1250 1250 1250 1250 1250 1250 1250 F release, (117 + r7) Table 3.2 (Continued)

Requirement Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Function

New Melones min. 336 0 0 0 0 420 1429 1681 1849 1849 1765 1765 I diversion (R8) New Melones min. 20 50 202 538 538 538 639 639 118 118 118 118 F release (u8 + r8) New Melones min. 168 101 34 0 0 34 101 68 168 168 168 168 WQ release, dry year (118 + r8) New Melones min. 136 68 0 0 0 0 101 55 136 136 136 136 WQ release, average year (u8 + r8) New Melones min. 68 0 0 _ 0 0 0 0 0 101 101 101 101 WQ release, wet year (118 + r8) Tracy accretion (+) 1980 895 35 940 670 168 1025 2030 3100 4500 4240 2800 MI, I w or diversion (-)(R11) (-) (-) (-) (0- (0- (+) (-) (-) (-) (-) (-) (-) 1 N) C.) Tracy pumping plant 4200 4200 4200 4200 4200 4600 4600 3000 3000 6200 6200 4600 MI, I diversion (R12) NCVP share of Delta 1220- 1640- 2200- 2480- 1480- 2200- 2840- 5360- 4160- 3200- 1360- 720- WQ, E requirement (R13) 1530 2050 2800 3100 1850 2750 3550 6700 5200 4000 1700 900

Trinity min. penstock 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 HP flow, total (u1) Lewiston min. penstock 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 HP flow, total (u2) Whiskeytown min. 1300 1300 1300 1300 1300 1300 1300 1300 1300 1300 1300 1300 HP penstock flow, total (u3) Shasta min. penstock 5000 5000 5000 5000 5000 5000 5000 5000 5000 5000 5000 5000 HP flow, total (u4) Keswick min. penstock 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 HP flow, total (u5) Folsom min. penstock 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 HP flow, total (u6) Table 3.2 (Continued)

Requirement Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Function

Natoma min. penstock 1000 1000 1000 ' 1000 1000 1000 1000 1000 1000 1000 1000 1000 HP flow, total (u7) New Melones min. 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 HP penstock flow, total (u8) Tullock min. penstock 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 HP W flow, total (u9) I N.) --I Notes:

(1) All flows are in cfs. , (2) A desired release need not be met under drought conditions.

(3) For the reach between Keswick and Natoma there is a stringent requirement: whatever flow exists as of October 15 (must be greater than 5000 cfs) at Wilkins Slough, it must be maintained at least at that level through December 31 to permit normal fish spawning.

(4) Data provided by the Central Valley Operations Office, USBR, Sacramento, CA. Table 3.3. Reservoir Net Rate Losses, ct (in ft/month).

Month Clair Engle Shasta Folsom New Melones

Jan 0.040 -0.020 -0.020 0.020

Feb 0.047 -0.001 -0.005 0.023

Mar 0.090 0 -0.001 0.050

Apr 0.215 0.002 0.001 0.008

W1 May 0.340 0.005 0.003 0.120 r,..) IN3 Jun 0.484 0.006 0.004 0.200

Jul 0.715 0.008 0.006 0.260

Aug 0.635 0.007 0.005 0.150

Sep 0.450 0.005 0.004 0.060

Oct 0.170 0.003 0.001 0.030

Nov 0.070 0 0 0.001

Dec 0.022 -0.003 -0.002 0 CHAPTER 4

DEVELOPMENT OF THE OPTIMIZATION MODEL

The mathematical structure of the optimization model is developed in this chapter. The model consists of an objective function, a set of constraints, and a solution algorithm. The objective function consists of maximizing annual energy revenues while satisfying flood control, water supply, environmental and recreational functions by appropriate definition of the constraint set. Chapter 2 described the details of the solution algorithm. Input data necessary to express the objective function and constraints as well as streamflow sequences were presented in Chapter 3.

Section 4.1 provides a general optimization model. Section 4.2 presents an alternative LP approximation to the general model. Section 4.3 develops an alternative quadratic version of the general optimization model.

4.1 General Optimization Model

The basic approach of the optimization model is as follows. At the beginning of a water year, the NCVP system managers announce a release policy, which is updated each month after actual flows and water demands are known. The random nature of inflows is handled by making statistical forecasts of flows for the remaining months of the current water year (see

Section 3.2). Those forecasts are updated each month to account for the most recent actual realization of river flows. Updated flow forecasts as well as actual reservoir storages are input to the model and a revised release policy is found for the remainder of the water year.

The following notation is used in this section: i = 9-dimensional vector whose components x are the beginning of month

storages at each reservoir i, i = 1, 2,..., 9 (see Fig. 3.2).

= 9-dimensional control or decision vector that represents water released Lit i through penstocks; its components are u. Decisions are made at the

beginning of month t. 4-1 = 9-dimensional spillage vector; its components are r . t Figure 4.1 shows the relationship between the time index t and the vectors

r st, Lit, and f.2 u .-1 1- 12 113 x x x x -12 -13 t = 1 t = 2 t = 3 t = 12 t = -13

period: 1 2 3 12

Figure 4.1. Relation between t and vectors xt, ut, and . Et

The first step in developing the optimization model for the operation of the NCVP system is to write the law of motion, or continuity equation, for the NCVP system (see Fig. 3.2):

=I + r + z (4.1) lit+l 1 Lit in which I denotes the n x n (n = 9) identity matrix and

-1 1 -1 1 -1 -1 1 1 -1 (4.2) 1

1 -1 1 1 -1

4111.1.•

_ 1 1 -1

1 r = 1 -1 2 (4.3) 1 1 -1 zt = yt - where yt, and Et are the forecast inflow, net loss,

and diversion vectors, respectively (see Fig. 3.2). Notice that 1 and 1 12 are nonsingular lower triangular matrices that considerably simplify the

numerical computations. The triangularity of r and arises from a proper 1 12 numbering of the network as done in Fig. 3.2. Equation (4.1) can be

rewritten as

+r r = x - x - z (4.4) 11 lit ——tt

Recall that e the net losses for reservoir i during month t, is given by t' [see eqs. (3.15)-(3.18)]

e = d + c + x ) (4.5) t t t (x+1 t

in which d and c are coefficients and (x +1 + x ) signifies the use of t t t t average storage. Substitution of eq. (4.5) into eq. (4.4) gives

(4.6)

in which A is a diagonal matrix whose (diagonal) elements are 1 + c t+1 t' i = 1, 2,..., 9, Bt is a diagonal matrix whose (diagonal) elements are

1 - ct,• and is the vector -/Lt

1 1 - d Yt t 2 2 -R - d t t 3 3 3 y - R - d t t t 4 4 y - d t t 5 V = - d (4.7) —t 6 6 6 y R - d t t t 7 7 -R - d t t 8 8 8 y - R - d t t t 9 - d 4-3 From eq. (4.6),

x =(1) x +P u +P r +F v (4.8) —t+1 t —t lt —t 2t —t t —t -1 -1 -1 -1 in which (I) = A F• = A F- and F = A . Equa- t+1 B•P1t' =At+1 1' P2t t+1 2' t t+1 tion (4.8) is the equation of continuity for the NCVP system (a linear equation). Vectors and 1_. represent deterministic control terms and F v Lit t t—t is a stochastic disturbance term. Notice that is a stochastic vector Yt because it includes the random inflows yt. In addition, it is evident from i eq. (4.7) that if water requirements (R ) were also considered random, then t i would include only d as a nonrandom term. It is emphasized that in this -\11. t study, the flows are forecast and then a deterministic problem is solved with the most recent forecast of flows used in the vector 17-t. Equations

(4.6) and (4.8) play a central role in the development that follows.

The objective function of the optimization model consists of maximization of the energy revenues generated during each year. The energy generated at reservoir i during month t (in Mwh) is

Ei = ui = [ai + bi (xi + x )]u (4.9) t tit t t t+1 t i i in which t is the energy rate given by eqs. (3.1)-(3.9) and u is the t t penstock release from reservoir i during month t. For the whole system, the total energy revenues generated during any month t can be expressed as

T E = Oa + B(x + x )] u (4.10) t —t —t+1 —t in which IP is the unit price per Mwh. Equations (4.9) and (4.10) are valid for the cases in which the energy rates given in eqs. (3.1)-(3.9) are either linear or quadratic, because the latter are linearized by a first-order

Taylor expansion, as discussed in Section 4.3. The vector a and matrix B are given numerical values in the two alternate versions of the model presented in Sections 4.2 and 4.3.

4-4 Recall that the POA maximizes a sequence of two-stage problems, i.e., maximize E + E for t = 2, 3,..., 12 subject to a set of constraints (see t-1 t Chapter 2). It follows from eq. (4.10) that

T T E - + E = tp[a + B(x + x )] u + + B(x + x )] u (4.11) t 1 t —t-1 —t —t-1 —t

It is clear from eq. (4.6) that

-1 -1 -1 -1 u = - F r r + r A x - r B x - r v —t 1 2 —t 1 t+1 —t+1 1 t —t 1 —t

= - r +C - D - F (4.12) t+1 ?.t+1 t Yt

A similar expression can be developed for . Substitution of those expres- -11t-1 sions for LI and into eq. (4.11) yields t ilt-1

T - Brr + x G x +k ) (4.13) t —1t- —t t —t t in which q•t. = - a r - xt+1 B r (4.14)

T T - a r-x 2•t = -t-1 Br (4.15)

T _ T T T T T -a D +x -vFB-x BD+aC Rt — t —t+1 t+1 B —t —t-1 t t

T T T T - x D B - v F B + B C (4.16) ?st-1 t

G = B C - B D t t t (4.17)

T T T k = a C x - a F v + x B C t — t+1 —t+1 — — t+1▪t+1

T T - B F v - a x - a F v •t+1 —t Dt1- —t-1 — —t-1

- • B D - B F (4.18) t-1 ict-1 17t-1

4-5 Notice that k is a constant because and are held fixed at every t ..t.-1 2s.t+1 two-stage maximization. Thus, kt plays no role in the two-stage solutions.

Let

...... , x —t r (4.19) —t r —t-1

...... e

that is, construct an augmented vector containing the variables for which

the optimization is to be made. Equation (4.13) can be now expressed as

T T T T 1 Et_i +E = 0[[gt, qt, Et] 2t + 2t Ht p_t + ktl (4.20)

in which

...... , T + G -Br -Br Gt t 1 T -r B 0 0 2 (4.21) T -r B 0 0

......

where 0 is a 9 x 9 null matrix. By dropping the constant term kt,

eq. (4.20) can be rewritten as

T 1 T E + E = 0[6 H 6 + s 6 1 (4.22) t-1 t —ttt tt in which T T T T -s-t = [RC qt' Rtl (4.23)

Thus, it has been demonstrated that the two-stage objective function is quadratic. It is worthwhile to recall that the continuity equation was used to develop eq. (4.22) and thus it will not appear as a constraint in the constraint set associated with eq. (4.22).

The final step to complete the formulation of the two-stage optimization problem is to include the constraint set associated with eq. (4.22).

4-6 Recall that Section 3.1 gives a description of the NCVP constraints that

arise from contractual agreements on water supply and recreational require-

ments, from flood control, environmental, and wildlife-fisheries water

demands, etc. All such constraints can be expressed in terms of the

augmented vector using the alternative expressions for the -9-t, continuity equation (i.e., eqs. (4.6), (4.8) and (4.12)) as follows:

Constraints on total releases for month t, u+rEW

(-1 + I) + C - D -Fv EW (4.24) t+1 ?_st+1 t —t —t

in which is a vector of feasible total release 141. values and I is a 9 x 9

identity matrix. Equation (4.24) can also be expressed in terms of the

augmented vector Ot as

C + (-1+I) - Fv E W (4.25) t+1 [-Dr —t —t —t

where 0 is a 9 x 9 null matrix.

Total release requirements at control point k (see Fig. 3.2), month t, kT impose additional constraints on total releases, c (u +r ) G Dee,

kT kT kT kT c (-1-+I)r + c C - c D x - c Fv E (4.26) —t —t —t t+1 —xt+1 Det

or in terms of

kT kT kT k c C + c [-D - c Fv E —t t+1 —t t De (4.27)

where is a vector expressing the C appropriate linear combination of

releases in month t, at control point k, and De is a feasible value for t month t, at control point k.

Constraints on penstock releases for month t, rt E Ut,

-F +C -D - Fv e U (4.28) t+1 14+1 t —t —t or in terms of 9-t' 4-7 -F 0) + C - F e (4.29) t t+1 )4+1 in which U is a vector of feasible penstock release values.

Constraints on spillages for month t) _rt E

[0 I E]et E (4.30 in which is a vector of feasible spillage values. II Constraints on storage values for month t E X ' ?.s.t —t'

[I 0 0]0t E Xt (4.31) in which is a vector of feasible storage values. The appropriate equa- ?it tions for month t-1 are obtained in a similar manner. All the constraints, eqs. (4.24)-(4.31), are linear. Constraints on power generation, which are intrinsically nonlinear, have been avoided by the way in which energy generation is computed (Section 3.1).

The two-stage optimization problem can be summarized as a linearly- constrained quadratic programming (QP) problem,

T 1 T maximize (WO H + s 0 ] (4.32)

subject to

1 Al 0 b (4.33)

[I 0 [I 0 fixed 0]2t-1' 0]2t+1 in which

4-8 kT - (-D -F+I-[+1 0) St kT -c (C 0 -F+I) -t t

D 1 t A (4.34) t ct 0 -r -ct 0 o o I I 0 0 -I o o

- C + F 141-t t+1 lit+1 Yt

W +D x + F v -t-1 t -t-1 -t-1 k kT T -De + c C c F v t -t t+1 -t k kT T -De - c D - c F v t-1 -t t-1 -t-1 -t-1

-C x + F v -t,max t+1 -t

-U +C x- F v -t,min t+1 -t (4.35) + D + F -t-1,max t-1 Yt-1

-U - D - F v -t-1,min t-1 -t-1

—t

4-9 where and are minimum and maximum penstock releases for Ilt,min Ilt,max month t, and and X are minimum and maximum storages for month t. —t,max Notice that eq. (4.33) lumps into one expression all the (linear) constraints on 2t Solution of eqs. (4.32) and (4.33) by the POA would yield the optimal * sequences {2t} and accordingly {xt}, {lid, and fEti. The vector

is of dimension 3 x 9 = 27. Thus, the 14 computational effort involved in methods such as Fletcher's (1981) active set method would be proportional to a number between a1(27)2 and a2(27)3, in which a and a are positive real 1 2 numbers independent of the dimension of (Gill and Murray, 1977). The 2t aforementioned computational burden is not limiting in a fast modern computer but it is still unpleasantly large. Storage requirements would be 2 proportional to (27) , a reasonably low number.

It is possible to reduce the complexity and dimensionality of problem

(4.32)-(4.33). This can be accomplished in two ways:

(1) By neglecting net losses and handling spillages in a special way.

The neglection of net losses is appropriate, and the appropriateness follows from examining matrices A and B in eq. (4.6). Their diagonal coeffi- t+1 t cients are given by 1 + c and 1 - c respectively, and are close to one t t' because the c values are in all cases less than 0.0044, t as can be verified by substituting the c values given in Table 3.3 into t eqs. (3.15)-(3.18).

Spillages, on the other hand, are set to zero whenever the penstock release

is below U When u = U the additional release necessary to —t,max —t —t,max' maintain feasiblity is set equal to the spillage rt. This eliminates the treatment of as a decision variable and reduces It the dimensionality from 3n = 27 to n = 9 in each two-stage problem (4.32)-(4.33).

(2) By using the linearized relationships between storages and releases developed in Chapter 3, see eq. (3.32), and by treating the regulating reservoirs (reservoirs 2, 3, 5, 7, and 9) as constant-storage 4-10 reservoirs. This would reduce the dimensionality from 27 to 4, as shown in

Section 4.3.

Simplification (1) leads to a linear programming (LP) model of dimensionality equal to 9. This alternative formulation is presented in Sec- tion 4.2. Simplification (2) leads to a quadratic programming (QP) model of dimensionality equal to 4 and is presented in Section 4.3.

4.2 Simplified Linear Model

By using the simplification (1) outlined above, the continuity equation can be written as

= +Fu + w (4.36) 1—t —t in which w = y - R It is possible to solve for u in eq. (4.36), and —t ——tst —t substitute u (and a similarly derived expression for Ilt-1) into eq. (4.11) —t and into the constraint equations (4.24), (4.26), (4.28), and (4.31) [constraint (4.30) on spillages is not written as part of the rows of the constraint matrix], to obtain the following LP model for the two-stage problems (dropping constant terms irrelevant to the optimization in the objective function):

T maximize 0 x —t —t (4.37) -xt subject to

A x b —t —t fixed (4.38) in which

T T T T T T T h = (x - x MBG - (BG) ] - (w + w )(BG) (4.39) —t —t-1 t+1 —t-1 —t

-1 G = F (4.40) 1

4-11 -G

-G

A = kT (4.41) c G -t kT - G

- U + Gx - G -t,min -t+1 14-t

- - G - G /4-1

- + Gw -t,max G:t+1 -t

+ G + G t-1,max b -t (4.42) k kT kT - De + c Gx - c Gw t -t -t+1 -t -t k kT kT - De - c Gx - c Gw t-1 -t -t-1 -t -t-1

- X -t,min

X -t,max

where I is a 9 x 9 identity matrix and B = diag(0.0429, 0.4500, 0.02170,

0.0231, 0.3000, 0.0600, 0.4000, 0.0615, 0.4660). Notice that eqs. (4.37) and (4.38) are expressed in terms of a 9-dimensional unknown vector, It.

Thus, the two fundamental assumptions discussed previously (neglect net losses and treat spillage as excess over penstock capacity) have reduced the computational burden considerably. Equations (4.37) and (4.38) will be solved sequentially by using the POA. Once the optimal sequence [ is )it

4-12 known, the optimal release policies {ut} can be readily obtained through the continuity equation.

In the POA, the beginning and ending storage vectors xl and ?s.13, respectively, are fixed. Vector x1 is specified at the beginning of period

1, and vector (= ?.sN+1) must be a value ranging from 1/2 to 2/3 of the ?--S13 capacities of the reservoirs. This range has been established as satisfactory from past experience. If desired, the value of can be updated 2sN+1 at every end of the month. This study adopted a value of 7/12 of reservoir capacity, which was not updated because it proved satisfactory.

The sequential solution procedure can be summarized as follows:

(1) The initial and final states and x13 are fixed. Subindex I can take ?.s.I values 1 through 11, depending on which month the future release policy is being computed. Forecast flows for the remaining (13-I) months, develop (or (k) adjust) an initial feasible state trajectory {xt } (see Chapter 5), and set the iteration counter k equal to 1 and t = I; (2) Set the time index t = t+1 and solve the LP two-stage problem (4.37)-(4.38); (3) Denote the solution of •;'; (k+1) _ * step 2 by Set — and go to step 2. Repeat steps 2 and 3 until x.—t a complete iteraton sweep is performed (t = N). This ends the kth iteration; (4) Perform a convergence test, e.g., is

[(4 .)( k+1) - (xi)t (k) 0.01 for all i and t (t = 12)? If yes, go to step 5. Otherwise, set k = k+1 (provided that a maximum iteration limit is not exceeded), t = I and go to step 2; and (5) Apply the optimal policy for current month I. Set

I = I+1 and go to step 1.

4.3 Simplified Quadratic Model

In this model, net losses are included, quadratic energy rate equations are used instead of linear ones, and releases are related through storages 4-13 by the relationships developed in Section 3.1, i.e., the continuity equation Ai l l is given by eq. (4.6) subject to the equations rit = c + d x derived in t t t Section 3.1. In addition, the regulating reservoirs (Lewiston, Whiskeytown,

Keswick, Natoma, and Tullock) are treated as constant-storage reservoirs,

with the constant storages equal to their maximum permissible values (these

values were given in Section 3.1, after eq. (3.31)). The application of the

simplified linear model of Section 4.2 shows (Chapter 6) that keeping the

regulating reservoirs at their maximum permissible level is in fact an

ootimal strategy. Since the regulating reservoirs are kept at constant

sLorage values, it is convenient to partition the storage vector as T (1)T T 1 4 6 8 T 2 3 x = (xt k ), where x(1)T = (x x x ) and k = (x x —t —t t' xt' t' t t' t' 5t. 7t. 9t. x x x ) = (14.7, 241.0, 23.8, 8.8, 57.0), all units in Kaf. Similar

partitions hold for the release and spillage vectors and -14_. All the 'It matrices in eq. (4.6) are partitioned and reordered conformally. If the

linear expressions relating storages and releases given in Section 3.1 are

substituted by the releases appearing in the continuity eq. (4.6), and after

a subsequent partition of the storage and release vectors, as discussed

above, the continuity equation becomes, after solving for

•••••••• r- t+1 (1) 1) H x(1) 11 -t+1

t+1 t+1 Mt t ,.(2) u(2) H H M w —t 21 22 21 22 —t

...alb 4....

that is

(1) x x(1) —t+1 —t = H - M -w Ht t+1 —t (4.43)

411.111.1

in which

-1 H = F A t+1 1 t+1 (4.44)

4-14 M = M + t t NDt (4.45)

M t = (4.46)

-1 N = F F (4.47) 1 2

D = diag(d d d d , 0, 0, 0, 0, 0) (4.48) t' t' t' t

c —t 1,;71 = F-11 + N (4.49) —L

T „4 c = (c c c c 0 0 0, 0, 0) (4.50) —t t' t' t' t' ' ' „. 1, where the c s and d s are given in Section 3.1; K is a 5 x 1 vector whose t t elements are given by eqs. (3.20), (3.21), (3.23), (3.25), and (3.27), which i are constant once the fixed elevations h (i = 2, 3, 5, 7, and 9) are sub- i stitnted in them (see values for the h 's in Section 3.1, after eq. (3.31)).

It is remarked that the values of x appearing in the equations for the t,0 „i c 's and d 's in Section 3.1 are automatically generated by the solution t t procedure outlined below. A similar equation to (4.42) can be developed for

by performing analogous steps to those leading to eqs. (4.42)-(4.45);

is expressed as Lit.-1 x(1)• -t x—t-1(1) —t-1 =H - M (4.51) t-1 t- -1 ,v)(2) —t-1 in which

-11 H = 1 A - ND (4.52) t t t-1

-1 M = 1 B (4.53) t-1 1 t-1

4-15 = lv +N (4.54) —t-1 1 —t-1

where Fl, At, N, Dt_i, B and K have been defined in Sec- t-1' 17t-1' tions 4.1-4.2. The expressions for ut and u are used to develop the

objective function and constraints of the two-stage problems below.

To set up the system energy production rate in matrix form, the

quadratic energy rates developed in Section 3.1 are linearized. By per-

forming a first-order Taylor series expansion about a guessed value x t,O, the energy generation rate becomes

t a + b x = 1, 4, 6, 8 (4.55) t t t t'

i in which the coefficients a and b depend on the initial guessed value x t t t,0 i i for x - x is automatically generated by the solution algorithm outlined t' t,0 ^ below. The expressions for a and b are given below. Clearly, no t t linearization is needed for linear energy rates, since the energy rate

eqs. (3.2), (3.5), and (3.7) are already in linear form. The quadratic

energy rates for the constant-storage reservoirs, (eqs. (3.3b) and (3.9b))

need not be linearized due to the constant value of the storage in reser-

voirs 3 and 9. The energy rate for month t for the entire NCYP system can

be expressed as

*t B 0 11 tt (4.56) 0 B 22 in which

„*(1)T Al A4 A6 A8 = (a a a a ) '2t t' t' t' t (4.57)

*(2)T a = (a a a a , a ) = (606.3, 445.0, 80.3, 26.3, 63.4) (4.58)

4-16 *t ,-1 -4 -6 -8 B = diag b , b , (4.59) 11 (b r. t t b)t

-3 3 B = diag (-0.254, 0.738 - 1.10 • 10 x 0.60, 0.80, 22 t'

-3 9 1.020 - 1.37 • 10 x ) t

*T 3 3 5 7 9 k = (x x x x x ) t' t' t' t' t 3 5 7 9 where the constant values for x x x and x were given in Section 3.1, t' t' t' t' after eq. (3.31). The double appearance of superscript 3 in eq. (4.61) 3 follows from the use of x in eqs. (3.2) and (3.3b). The values for a and t b appearing in eqs. (4.57) and (4.59), respectively, for i = 1, 4, 6, 8, t can be obtained from the following expressions,

i 2 x + x [x + x „i i 1_ i t0, t+1 i t,0 t+1 a = a b c t 2 2

i [bi c i 1 i + (xi + x )1 x (4.62) 2 2 t,0 t+1 t,0

i x + x i b i[ b = + c (4.63) t -2 2 t+1]

-4. in which ai, bi, and ci are given by (133.0, 0.228, -0.468 • 10 ), (169.0,

0.107, -0.115 • iO 4), (171.0, 0.265, -0.130 • 10 ), and (169.0, 0.275, -4 i i 1 -0.479 • 10 ) for i = 1, 4, 6, and 8, respectively. The a , b , c values are taken from eqs. (3.1b), (3.4b), (36b), and (3.8b). An expression i similar to eq. (4.56) can be obtained for tt...1 by replacing x with xi t+1 t-1 in eqs. (4.62)-(4.63).

The objective function for the two-stage QP problems is readily available by using (4.43) and (4.51) to represent and i.e., u 14-1'

4-17 T ) max E + E = tfs u + u = (fsik + q x t -t-1 -t

1 (1)T * (1) + H x J (4.64) t

in which

, * *(2)T *T t+1 (1) t+1 t -(2) k = + k B [H 1.5. - M y_ t 22 21 1L-ct+1 1122 22

t t-1 (1) t-1 „(2) *(1)T (1) + k - x - M k - w + (H H22 - M21 -t-1 22 - -t-1 At 11 )4+1

_ -(1)w ) _ *(1)T t-1 (1) (m w ) (4.65) —t —t-1 11 Lct-1 t-1

*T *(1)T t *(2)T *T (m t Ht = - M - ( + at At 11 A 22' ` 21 21'

x(1)T *t t+1 T -(1)T *t ,,*(1)T A t (B H ) - w B + a H -t+1 11 11 -t 11 -t-1 11

(1)T *t-1 t-1 T ,.(1)T *t-1 (B M ) - B (4.66) -t-1 11 11 11

*t-1 At *t At H = 2(B H - B M ) t 11 11 11 11 (4.67)

*(1) ,,*(2) *t where , a B 8 and k have been defined in 11' 22' eqs. (4.57)- (1) (1) (4.61), and x and x are fixed in the two-stage problems. Notice that (1) eq. (4.64) is written in terms of only, i.e., the solution ?_c_t to the (1) two-stage problems is in terms of the non-regulating storage vector xt

only, of dimension four in this application. Had the formulation of the

two-stage problems been expressed in terms of x r and u the total

number of unknowns would have been 27 for each two-stage QP problem, as in

eqs. (4.32)-(4.33).

4-18 The two-stage QP problem is fully specified by subjecting storages, penstock releases, spillages, and total releases to a set of (linear) constraints. Constraints play two important roles: (0 enforce feasibility due to physical and/or technical features in the system, and (ii) guarantee that functions other than power generation are adequately fulfilled, by introduc- ing suitable constraints on penstock releases, spillages, and storages so as to satisfy contractual agreements and regulations related to flood control, wildlife and fisheries requirements, water quality, etc. By writing the constraints (see eqs. (4.24)-(4.31)) in terms of the releases and u 1"-it —t-1 (see eqs. (4.43) and (4.51)) and of the storages, the two-stage maximization problem becomes,

*T (1) 1 x(1)T /I* x(1)1 maximize Oqt xt + • (4.68) 2 — t t —t x(1) —t subject to

A* x(1) < b* (4.69) x(1) x(1) fixed in which

4-19

,C111111111•11=M111

(1) A -^t )it+l W + M - H + t+1 -t k ...... `=:--- -t -t-1 W - H + M + -t 2 t s2t-1 k

(4.71) x(1) k kT ^t kT -t-1 kT :: - e - c M k + c H - c w t -t 2 - -t .t+1 -t - _k r...r.. ==, x(1) k kT t kT -t-1 - e + c ii k - c kT 1 M - c w k

where t t M = [M M ] =D -M (4.72) 1 2 t t

A t ^t ^ H = [H H ] = H + D (4.73) t 1 2 t t-1

2 w = - (4.74) -t —t

4-21 and other terms have been previously defined in Sections 4.1-4.2.

The sequential solution kocedure to obtain monthly release schedules

can be summarized as follows (we drop the superscript (1) on storages for

notational clarity):

(1) The initial and final states and x are fixed. The subindex I 13 can take values 1 through 11, depending on the month for which .the future

release policy is being computed. Forecast flows for the remaining 13-I

months within the current water year and develop an initial feasible state (k), trajectory ixt 1, in which the time index is initialized at t = I, the

counter k for the sweep iterations (from t = I to t = 12) is set equal to

one, and the counter 2 for the iterations within each two-stage problem is

set equal to zero.

(2) Construct the QP problem given by (4.68)-(4.69), in which lineari- (k) (k) zations are made about x (x = for 2 = 0 only). —t —t (3) Solve the QP problem and denote the solution by xt. If x does not —t satisfy a convergence test, then set 2 = 2 + 1, set xt —, and go to (k+1) step 2, otherwise, set = increase the time index by one, set )4 = 0, and go to step 2. Repeat steps 2 and 3 until a complete iteration

sweep is performed (t = I to t = 12). This ends the kth iteration.

(4) Perform a convergence test for t = I, 1+1,...,12. If convergence

is attained, go to step 5. Otherwise, set k = k+1, 2 = 0, and go to step 2.

(5) Apply the optimal computed policy for current month I. At the beginning of next month, set I = I+1 and go to step 1.

A few remarks concerning the solution method are warranted. (i) In step 1, the fixed state xi is specified. It is equal to the beginning of month storages for the nonregulating reservoirs. The final state x is —13 also fixed. From previous operational experience, a value of ranging ?..13 from 1/2 to 2/3 of reservoir capacity was found to be appropriate (the value can be updated every month if deemed convenient). In this study, a value of

4-22 7/12 of reservoir capacity was adopted. (ii) Initial policies are -)!13 determined by a trial-and-error procedure (see Chapter 5). (iii) The bulk

of the computations resides in step 3 of the solution procedure of the QP problem. In effect, the existence of an efficient, stable way to carry out

step 3 practically implies the successful computation of the release

policies.

The solution algorithm for the QP problems adopted in this study is a

numerically stable active set method for quadratic programming problems.

The active set method can handle quadratic problems in which the Hessian

matrix is indefinite (i.e., has positive and negative eigenvalues), as is

the case in this work. The details of the active set method can be found in

Fletcher (1981).

4-23 CHAPTER 5

APPLICATIONS AND DISCUSSION OF RESULTS

The application of the simplified linear and quadratic models developed in Sections 4.2 and 4.3 is presented in this chapter. Sections 5.1 and 5.2 discuss the results obtained from the simplified linear and quadratic models, respectively. The models are simply referred to as the linear and quadratic models henceforth.

5.1 Linear Model

Initial operating policies for the NCVP system were developed by using a trial-and-error procedure that considers some heuristic criteria used by

NCVP managers to set up their release policies. In essence, desired reservoir storages at the end of the water year are selected and a feasible

(initial) release policy that achieves those targets is chosen. As the system operation progresses through the year, actual flow conditions may lead to a revision of the ending storages selected initially. The overall philosophy is that reservoir storages must be kept high at the beginning of the dry season (usually, May) to meet increasing agricultural and Delta water requirements during the summer. Also, the operation during the rainy season (November-March) is conservative in the sense that a substantial flood storage volume is allocated to store eventual large-runoff events.

Clearly, there is a trade-off between the desire to maintain the reservoir levels below some specified elevation during the rainy season and the desire to have as large a storage volume as possible at the beginning of the dry season. A general rule would be to maintain reservoir storages at the maximum permissible levels during the rainy season and to make large releases during the dry season. Interestingly, because most power installations in the NCVP are of the high-head type, a greater generation of power will not result from the largest releases but from some optimal reservoir

5-1 elevation associated with moderate releases. The largest releases would

drive reservoir levels below the range at which turbines can operate

efficiently.

The optimization model was tested with different streamflow conditions:

below-average inflow volumes (1975-76, total inflow of 6,057 Kaf), ,average

inflow conditions (1974-75 and 1979-80, total inflows of 12,198 and

13,936 Kaf, respectively), and above-average conditions (1973-74, total

inflow of 20,146 Kaf). This was done to determine if different initial

policies under various streamflow scenarios result in distinct optimal

policies. Two initial policies were developed for 1974-75, 1975-76, and

1979-80 while a single initial policy was considered for 1973-74. Due to

space limitations, initial policies are not included herein. A complete set

of initial policies (policies I and II) for the various years under consideration is available from the authors. Development of the initial policies indicated that for below-average streamflow conditions there is little opportunity for optimizing the operation of the system, because prevailing low inflows barely meet the system's demands by releasing flows near their minimum permissible values. For both average and above-average streamflow conditions, there is a larger feasible region and the gains from the optimization model can be significant. During the winter months of an extremely wet year such as 1973-74, the initial policy is nearly optimal because the reservoirs are at near capacity during those months and total releases are set equal to maximum permissible flows. In those circumstances, the optimization model allows the determination of the best feasible release policy that simultaneously minimizes the spillage and maximizes the power generation and revenues accruing from it. Because the reservoirs are at a high stage after the winter, substantial improvements in energy generation can be obtained during the subsequent summer months.

Revenues are obtained from energy figures by multiplying by the appropriate 5-2 unit price (P, which was set to $50 per Mwh (1980 constant dollars) in this

application.

Some of the initial policies were refined so as to make them near

possible optimal releases whereas others were deliberately set to be poor

(but feasible) initial estimates. This was done to estimate the number of

iterations and CPU time needed by the POA to reach optimality. Inital

policies I for average (1974-75, 1979-80) and below-average (1975-76) inflow

years were carefully refined, attempting to be near their respective optimal

policies. in those cases, convergence to the optimum was attained in six to

eight iterations. In contrast, initial policies ll for average-inflow years

1974-75 and 1979-80 were purposely developed to be far from good initial

policies. That was accomplished by releasing heavily during wet (winter)

months to maintain a year-round low head and a corresponding decrease in

power generation. That is also suboptimal from the standpoint of agricul-

tural and Delta requirements, because those demands are low in the winter

and thus larger than necessary flows will be of no use. This strategy

forces summer releases to be at minimum permissible levels, when an addi-

tional acre-foot of water during this season has a greater marginal value

than in the rainy season. Those deliberately-poor initial policies resulted

in an increase in the number of iterations needed to attain convergence,

ranging now from eight to ten iterations. Table 5.1 summarizes the required

iterations and CPU times for the specified initial policies and inflow

conditions. It is evident that the CPU time increases as the inflow condi-

tions vary from below-average to average. That is because, for below-

average flow conditions, the feasible region becomes so tight that there is

no freedom to optimize any policy. Any feasible initial policy will be very

close to an optimal release policy. As flow volumes increase to average-

flow conditions, there is a corresponding increase in CPU time. Notice that

5-3 policies II for average-flow years 1974-75 and 1979-80 (which were deliber-

ately chosen to be inferior to their counterparts, policies 1) also required

more CPU time. For extremely wet conditions such as water year 1973-74, the

feasible region becomes very tight during the winter and that implies a

reduction in CPU time as shown in Table 5.1.

Optimal state trajectories (i.e., end of month storages) and their

corresponding release policies were obtained by applying the POA to the

initial policies. Tables 5.2 and 5.3 show the optimal strategies for

average-flow conditions (1979-80) corresponding to the first initial policy

(policy I). Table 5.3 also shows the energy produced by the optimal policy

and the total annual energy revenues as well as the monthly water deliveries

to the Delta. For Clair Engle, Shasta, and New Melones reservoirs, the

optimal state policies (end-of-month storages and releases) resulting from

initial policy I were different from those resulting from initial policy II.

For Folsom reservoir, the optimal end-of-month storages and release policies

were the same for initial policies I and IT. For the remaining five smaller

reservoirs, the end-of-month storages were the same for initial policies I

and II, but their release policies were different. The results also showed

that the value of the objective function of the model (total energy revenues

for the year) is practically the same, approximately $400 million, for

release policies I and II, with a difference of less than 0.17% due essen-

tially to the effect of roundoff in the convergence test. This implies that

there are multiple ways of achieving the optimum performance index.

It was found that an optimal release policy is achieved by releasing

less water than the maximum possible penstock capacity. Because hydropower

production depends on the storage level (the larger the head, the greater

the energy production for a given discharge), an optimal release policy is a

feasible trade-off point between a high head and a small release and a low head and a large release. Such a tradeoff point is the optimal solution 5-4 given by the POA. Because the power installations in the NCVP are of the high-head type, except Nimbus (at Natoma) and Keswick, the trade-off point is shifted towards a relatively high head with a moderate discharge.

Optimal release policies for average-flow year 1974-75 essentially led to the same findings for 1979-80, except that due to the slightly lower annual inflow, no spillage occurred and the annual system energy also decreased. For water year 1975-76 with below-average streamflows, initial policies I and II were near optimal policies. Because of tight feasible- region conditions, the benefits from running the optimization model were marginal. Initial policies I and II yielded the same optimal state and release policies. The gain in energy production and revenues accruing from it (as obtained from the model) associated with both initial policies I and II was about 1 %. For above-average flow year 1973-74, with almost twice as much inflow as in 1974-75 or 1979-80, substantial spillage occurred. Also, the higher storage levels and greater releases that occurred in this year resulted in an increased total energy production and revenues. Figure 5.1 shows the relationship between total annual energy and total annual inflow, obtained from the values of the objective function computed for the water years under consideration. Notice that revenues are obtained by multiplying the energy figures by the appropriate unit price tp.

The energy vs. volume of inflow curve, a fairly straight curve, is applicable to the range of inflow volumes depicted in Fig. 5.1.

It was found from the computed results (e.g., Table 5.2) that the optimal state trajectories for the smaller reservoirs were to keep them full all year. That stems from the ratio of the capacities of the major reservoirs to their corresponding downstream regulating reservoirs. The largest capacity ratio of the system is 241/2448 = 10%, corresponding to Clair Engle and Whiskeytown reservoirs. When a capacity ratio becomes less than the largest capacity ratio of the system, all th state variables corresponding 5-5 to downstream, smaller, regulating reservoirs can be treated as constant and

equal to the maximum capacity of the regulating reservoirs. Those nodes in

the network can thus be treated as transmission points only. The number of

state variables in the NCVP system would be reduced from nine to four:

Clair Engle, Shasta, Folsom, and New Melones. This led to the modified

quadratic model of Section 4.3, whose application is given in Section 5.2.

Care must be taken in reformulating the model in terms of the reduced number

of state variables because the constraints that hold for the operation of

the smaller reservoirs must still be satisfied. For example, if constraints

representing penstock and spillage capacities are not observed, releases

from Shasta reservoir could cause overtopping of Keswick reservoir.

Table 5.4 summarizes the energy production levels obtained for water

year 1979-80 by using the optimization model and actual operation schedules.

The ratio of actual energy (Ea) and maximized energy (Em) varied from 29 %

at New Melones power plant to 72 % at Shasta power plant. Those ratios

should be interpreted as an indication of the potential that exists to

improve energy generation levels. For example, at New Melones, legal

battles of environmental origin kept the reservoir from being filled completely, and also the power plants were in complete halt during three months, which affected the actual power production adversely. The actual

realizable increase is definitely lower but not estimable with the available information.

Further insight into the differences between actual operation policies and those resulting from the optimization model can be gained from Fig. 5.2, which shows actual and optimal state trajectories (for policies I and II,

1979-80) for Shasta reservoir. It is evident that substantially smaller storages are maintained from November to February in the optimal policies.

That is accomplished by releasing large volumes of water through the penstocks, resulting in greater available flood control storages than in the 5-6 actual operation. Thus, the level of energy generation during November-

February is higher with the optimal trajectory because the releases are routed through the penstocks at a larger magnitude relative to the actual operation. Also, when the high inflows of January-April occur, the actual operation follows the flood control regulations by spilling large volumes of water because the empty volume in the flood control pool is not as large as that attained with the optimal state trajectories. In March-June, the optimal state trajectories maintain higher storage elevations than in the actual operation. That also results in increased energy production because energy is linearly dependent on storages. The lower storages during March-

June in the actual operation are due to water spillages that drive the reservoir level to lower stages. Those spillages reflect the conservative- ness of the actual operation policy. Because they bypass the power plant, those spillages do not generate energy. In contrast, the reliance of the optimal trajectories on greater penstock outflows and smaller spillages reflects (i) the foreknowledge of future inflows (within a certain range of error) that arises from streamflow forecast and (ii) the knowledge that for a given release, the higher the storage level the higher the energy generation rate. During July-September, there is a steady drawdown of the reservoir storage level in the actual and optimal policies, reflecting increased demands for water and energy during the summer. Because the optimal state trajectories start at a higher elevation in July and end at a slightly lower level in September than the actual policy, the rate of water release during this period is higher for the optimal policies. That results in a greater generation of energy for the optimal policies in the summer as is evident in

Table 5.4. Optimal policies I and II follow a similar pattern throughout the year and result in the same total energy production. The actual state trajectory shows high peaks in January and February that are due to short- term floods that raise the reservoir level for a few days. Those floods are 5-7 partly spilled and do not contribute to energy generation at the reservoir power plant. Those short-term flood events are not well captured within the monthly period framework, resulting in an overestimation of energy production of approximately 2% during such high-inflow months. These findings are also applicable to the other major reservoirs.

Benefits of the optimization model can also be measured in terms of increased water deliveries to downstream users. For example, the Delta requires a delivery of 3,850 Kaf of water per year. Optimal release policies indicated a total annual release of 12,627 Kaf (for 1979-80), more than three times the required amount. For May-August, when most agricultural activities take place, additional water could be supplied for leaching and crop-growing purposes. The Delta requirements for May-August are about

2,698 Kaf. For the same period, optimal releases indicated that 4,813 Kaf were delivered in 1979-80. This suggests the possibility of a conjunctive use of surface water and groundwater reservoirs. Also, with increased deliveries, cultivated areas could be expanded or better leaching of salts might be achieved, resulting in an expanded economic output. Fish spawning, water quality, and navigation would also benefit from increased water deliveries.

5.2 Quadratic Model

By using the method outlined in the preceding section, release policies for the NCVP were computed for 1979-80, a water year with average inflow conditions (total yearly inflow equaled 13,936 Kaf). After deriving two initial release policies, the model was run to determine if both policies yielded the same performance, as measured by the total annual system energy revenues generated. Derivation of the initial policies (I and II) was accomplished by a trial-and-error procedure on the basis of past operation experience and with the assistance of the NCVP managing staff, as explained

5-8 in Section 5.1. Tables 5.5 and 5.6 show optimal releases corresponding to initial policies I and II, respectively. These tables also show that substantial spillages occur in the regulating reservoirs (Lewiston, Keswick,

Natoma, and Tullock; at Whiskeytown, spillages are slightly greater than the downstream water requirements of 3 Kaf/month). The major reservoirs pass most of their total release through penstocks, with the exception of (high- inflow) March. Optimal policies I and II in Tables 5.5-5.6 are clearly different except for the subsystem New Melones-Tullock where initial policies I and II yielded the same optimal release and state sequences

(state or storage sequences have been omitted to conserve space). Both solutions I and II yielded the same volume of Delta releases as specified in

Tables 5.5 and 5.6 (annual total Delta release = 14,697 Kaf). The total annual energy production is almost the same for policies I and II, 6 6 7.764 x 10 and 7.772 x 10 Mwh, respectively. For all practical purposes, it can be claimed that the two alternative optimal policies produce a com- parable performance as measured by energy production, which determines the revenues by multiplying total energy by the unit factor 41, as explained in

Section 5.1. Table 5.7 summarizes the results obtained from the quadratic and linear models and the actual operations. The linear model results in larger Delta releases (14,773 Kaf) than those obtained with the quadratic model (14,697 Kaf, for both policies I and II) and also in larger annual 6 energy production (8.077 x 106 Mwh as compared to 7.764 x 10 and 6 7.772 x 10 Mwh for the two optimal policies of the quadratic model) and revenues ($400 and $387 million, in 1980 constant dollars, for the linear and quadratic models, respectively).

Figure 5.3 shows the state trajectories at Shasta for the different models. It is evident that quadratic policies I and II follow a pattern similar to the linear policy but, overall, maintain a lower storage elevation. That is explained by the fact that when spillages are functions of 5-9 storage, there is a penalty for achieving higher levels because the spilled

(non-energy producing) water increases exponentially with the differential of reservoir elevation minus spillway crest elevation. It can be expected that penstock releases will increase (in the quadratic model) to keep reservoir levels from reaching such high elevations. Because energy production T is linear in the penstock release (recall that E = t ), it would follow t —t u—t that the quadratic model is more likely to generate more energy than the linear model, however, it was stated earlier that the linear model resulted in a greater energy production level than the quadratic model. The resolu- tion of the contradiction established by this argument and the observed results (which indicate more energy from the linear model) lies in the fact that energy production is a quadratic function of storages, through the energy rates, and that offsets the effect of the higher penstock release, for in the linear case the storages are greater. In the more realistic quadratic model, the trade-off between higher elevations and smaller releases is more complex than in the linear case. It can be observed in

Table 5.7 that values of (actual over maximized annual energy ratios) Ea/Em are higher for the quadratic model than for the linear model. The overall

E/E ratio for policy I of the quadratic model is 5.2/7.764 = 0.67, a m slightly larger than the 0.64 obtained with the linear model, implying that a potential increase of up to 27 % over energy actually produced could be achieved by using release policies from the quadratic model. A 27 % 6 increase will be about 1.4 x 10 Mwh per year with average inflow conditions. The comments in Section 5.1 explaining that energy increases must be seen only as a potential upper bound apply equally well in this section.

The similarity of the state trajectories shown in Fig. 5.3 for the linear and quadratic models can be explained by noticing that high inflow forecasts result in a drawdown of reservoirs in December, mainly by routing

5-10 large volumes of water through penstocks. Reservoir elevations are relatively steady throughout the winter so that the trade-off between elevation and discharge is optimal in the sense that for given conditions, the total energy would be maximized. The volume of water released during the summer

(4,967 Kaf in May-August), obtained from the quadratic model policy

(Table 5.5), is larger than the agricultural requirements (2,698 Kaf in

May-August). This points to the feasibility of extending agricultural activities in the Sacramento-San Joaquin valleys. Finally, at the expense of a moderate increase in the complexity of the quadratic model, both in its formulation and solution, it appears that the quadratic model should be preferred over the linear model due to the closer representation to the actual system that it commands.

INFLOW TO RESERVOIRS TOTALS

YEAR NEW OAELONESISHASTAi FOLSOMICLAIR ENGLEt wHISKEYTOWN T iNFLOw1 ENERGY 1 1973-74 1498 10 796 4408 ' 2672 771 20146 11.14 1974-75 1206 6405 2786 1408 394 12198 7.35 1975-76 470 3611 1142 695 139 60571 4 53 1979-80 1734 6415 3972 1471 344 139361 8 01 _ INFLOWS IN 103 ACRE FEET ENERGY IN 106 MWM

Mwh) 12

6 co (10 10 NCVP 8 0/ FOR

0/ ENERGY 4

2 ANNUAL

0 1 t 1 1 1 I I 1 0 2 4 6 8 10 12 14 16 18 20 22 TOTAL ANNUAL TOTAL INFLOW (Kaf)

Figure 5.1 Total annual energy vs. total annual inflow for the NCVP.

5-11 Figure Figure RESERVOIR STORAGE (Kaf ) 4000 3000 2500 4500 300 RESERVOIR STORAGE (Kaf) 4600 4200 3400 3000 3800 5.3 5.2 • OCT OCT Operation Operation 1 ---

NOV NOV I

Optimization, Optimization, Nei,..• Actual DEC DEC ../ of of Shasta I Shasta JAN JAN /.? /V 1 FEB Policy Policy reservoir FEB reservoir —•••—•••

—• — — MAR — 1 5-12 — II MAR I . — TIME ...N. —

TIME Optimal Optimal Optimal Optimal Actual APR 1 (water (water APR state MAY state state state state MAY • year year policy policy, policy, JUN policy, • policy, • • 'N . 1979-80) 1979-80). JUN I, • It, JUL II, I, N. • Linear * N Linear Full ‘-• Full % \:\ S. JUL \ S. AUG S. S.

model • . model model S. •• S. • S. \ \.\\\-% model S. • . SEP • AUG \\*.• • ... •.. • \\. • • %. : • \•,. •... % sc • 4600 3800 4200 IN ... SEP -...... :. ...‘ • ... %.. \ \ ••% • — — *, J Table 5.1 Number of Iterations to Attain Convergence and CPU Time

Requirements.

Inflow Number of Iterations Burroughs B7800 Condition Policy to Attain Convergence CPU Time (min.)

Average I 6 6.01 (1974-75) II 9 8.94

Average I 8 8.51 (1979-80) II 10 10.32

Below Average I 3 2.98 (1975-76) II . 3 2.79

Above Average I 8 7.28 (1973-74)

5-13 Table 5.2 Optimal State Trajectory Corresponding to Initial Policy I, 1979-80.

Month Clair Engle Lewiston Whiskeytown Shasta Keswick Folsom Natoma New Melones Tullock Oct 1632.0 14.7 241.0 3035.0 23.8 673.0 8.8 1600.0 60.0 Nov 1577.2 14.7 241.0 3008.0 23.8 567.6 8.8 1462.0 57.0 Dec 1547.8 14.7 241.0 3046.0 23.8 364.7 8.8 1337.0 57.0 Jan 1543.7 14.7 178.1 2729.0 23.8 206.6 8.8 1185.0 57.0 Feb 1690.8 14.7 183.4 3184.0 23.8 733.5 8.8 1253.0 57.0 cn Mar 1907.6 1 14.7 201.7 3800.0 23.8 985.1 8.8 1303.0 60.0 -, Apr 1974.7 14.7 241.0 4 3900.0 23.8 1000.0 8.8 1230.0 61.0 May 2080.4 14.7 241.0 4168.0 23.8 1010.0 8.8 1098.0 67.0 Jun 2072.7 14.7 241.0 4294.0 23.8 1010.0 8.8 1068.0 67.0 Jul 1969.1 14.7 241.0 4134.0 23.8 1010.0 8.8 992.0 67.0 Aug 1808.5 14.7 241.0 3691.0 23.8 992.5 8.8 800.0 67.0 Sep 1637.1 14.7 241.0 3187.0 23.8 798.6 8.8 536.0 67.0 Oct 1542.6 14.7 241.0 2622.0 22.8 624.8 8.8 323.0 57.0 6 3 Storages are in Kaf (1 Kaf = 1.233 x 10 m ). Table 5.3 Optimal Energy Production, Release Policy, and Delta Releases Corresponding to 1979-80. Policy I,

Clair Engle Lewiston Whiskeytown Shasta Keswick Month Energy Release Energy Release Energy Release Energy Release Energy Release Oct 33356.7 93.0 44117.3 75.0 43473.7 77.0 112078.0 300.0 35664.2 377.0 Nov 38985.9 109.8 53999.5 91.8 53523.5 94.8 112154.2 300.0 37348.1 394.8 Dec 31578.7 89.3 41940.8 71.3 77793.4 141.2 243955.3 664.0 76171.9 805.2 Jan 32126.6 89.3 41940.8 71.3 41460.3 77.0 291284.0 786.0 81639.8 863.0 Feb 33520.7 89.3 41940.8 71.3 41854.6 77.0 310729.7 786.0 81639.8 863.0 Mar 34608.3 89.3 41940.8 71.3 42817.1 77.0 323729.8 786.0 81639.8 863.0 Apr 35941.7 91.0 41940.8 73.0 54201.0 96.0 126111.2 300.0 37461.6 396.0 May 83426.1 209.0 112351.8 191.0 116871.0 207.0 128841.7 300.0 47962.2 507.0 Jun 82428.2 209.0 107057.9 182.0 105014.5 186.0 184764.0 431.0 58368.2 617.0 Jul 80059.3 209.0 105293.2 179.0 103885.3 184.0 281620.3 679.0 81639.8 863.0 Aug 66386.9 180.0 88234.5 150.0 86947.5 154.0 278553.2 709.0 81639.8 863.0 Sep 3-6455.7 102.0 44117.3 75.0 43473.7 77.0 289027.5 785.0 81639.8 863.0 Total 588874.9 764875.5 811315.6 2682848.9 782815.0 Table 5.3 (Continued)

Folsom Natoma New Melones Tullock Delta Month Energy Release Energy Release Energy Release Energy Release Release

Oct 53544.8 194.0 6373.6 191.4 71641.1 157.0 7165.3 60.0 628.4 Nov 77852.2 303.0 9990.0 300.0 70422.2 160.0 7081.4 60.0 754.8 Dec 71289.2 303.0 9990.0 300.0 85889.9 203.0 12156.5 103.0 1208.2 Jan 118406.8 460.0 9990.0 300.0 87766.8 210.0 12982.6 110.0 1273.0 Feb 139893.4 460.0 9990.0 300.0 90566.3 213.0 13136.4 110.0 1273.0 Mar 147248.8 460.0 9990.0 300.0 89417.5 211.0 13341.5 110.0 1273.0 Apr 126260.2 392.6 7745.6 232.6 88813.2 216.0 13700.3 110.0 C.71 738.6 ...._,I May 131038.7 406.7 8215.1 246.7 84253.9 210.0 14007.8 110.0 863.7 ol Jun 76361.4 237.0 7792.2 234.0 82884.9 210.0 14007.8 110.0 961.0 Jul 63362.9 197.3 6470.2 194.3 79423.7 210.0 14007.8 110.0 1167.3 Aug 93465.2 303.0 9990.0 300.0 70032.8 200.0 12734.4 100.0 1263.0 Sep 86780.4 303.0 9990.0 300.0 48124.3 150.0 7361.0 60.0 1223.0

Total 1185503.9 106526.7 949236.5 141683.0 12627.0

Energy is in megawatt-hour (Mwh) and releases in kiloacre-foot (Kaf; 1 Kaf = 1.233 x 106m3). Delta releases are the sum of Keswick, Natoma, and Tullock releases. In addition to the above penstock releases, spillages are 160 Kaf at Lake Natoma during Jan-Apr and 100 Kaf at Tullock during Oct-Sep. Total annual energy equals 8,013,680 Mwh; this implies a total revenue of about $400 million for a unit price tis = $50/Mwh, in 1980 constant dollars. Table 5.4 Actual (Ea) and Maximized (Em) Energy Production (in 103 Mwh) Corresponding to Policy II, 1979-80.

Trinity Power Plant Judge Francis Carr Spring Cree., Shasta Keswick Folsom Nimbus Power Plant New Melones ' at Clair Engle Power Plant Power Plant Power Plant Power Plant Power Plant at Lake Natoma Power Plant Month Actual Max. Actual Max. Actual Max, Actual Max. Actual Max. Actual Max. Actual Max. Actual Max.

Oct 30.4 73.9 36.8 112.4 42.3 109.0 76.5 112.1 19.0 46.6 37.7 53.5 4.6 6.4 * 71.6 Nov 9.6 71.2 4.8 112.4 19.5 109.5 89.6 126.2 20.7 50.3 41.3 77.9 4.7 10.0 * 70.4 Dec 17.1 61.4 15.4 98.5 19.2 98.5 111.1 251.3 23.2 81.6 37.2 71.3 4.7 10.0 * 68.1 Jan 6.2 29.7 * 41.9 23.9 77.2 237.4 266.9 47.6 81.6 107.0 118.4 8.7 10.0 19.6 81.9 Feb 17.7 31.1 3.4 41.9 55.3 41.8 209.9 310.7 39.2 81.6 84.3 139.9 6.7 10.0 23.6 92.1 Mar 73.4 32.2 78.2 41.9 99.5 42.8 228.3 323.7 49.9 81.6 130.0 147.2 10.6 10.0 47.8 90.9 cn Apr 44.9 32.9 53.5 41.9 54.7 53.2 112.9 126.1 29.0 37.3 99.8 126.3 9.7 7.7 55.8 90.4 I May 21.2 33.7 21.2 41.9 18.6 49.3 164.4 ...._, 128.8 33.7 36.6 82.0 131.0 9.1 8.2 39.5 85.8 Jun 51.5 75.3 55.2 101.2 59.6 99.4 212.9 129.5 47.3 45.0 66.6 76.4 7.6 7.8 23.3 84.4 Jul 54.3 39.4 57.2 44.7 57.2 45.7 237.9 307.5 52.2 77.0 84.4 63.4 9.7 6.5 38.5 80.9 Aug 75.7 37.6 87.2 42.9 85.9 43.5 165.7 310.2 43.6 81.6 35.6 93.5 4.3 10.0 22.1 74.9 Sep 71.1 36.5 84.7 44.2 88.6 43.5 86.8 289.0 29.3 81.6 40.5 86.8 4.6 10.0 9.0 64.8

Total 473.1 817.5 497.6 765.9 624.3 813.7 1933.4 2682.2 434.7 782.8 846.4 1185.5 85.0 106.5 279.2 956.5

E/Ea m 0.58 0.65 0.77 0.72 0.56 0.71 0.80 0.29

* Power plant not in operation.

To obtain revenues, multiply the energy production by the unit price LP; in this study, 41 = $50/Mwh in 1980 constant dollars. Table 5.5 Optimal Release Policy, 1979-80 (Policy I)

Clair Engle Lewiston Whiskeytown Shasta Keswick Month Spill Penstock Spill Penstock Spill Penstock Spill Penstock Spill Penstock

Oct 99 26 73 5 77 300 50 327 Nov 102 26 76 5 87 300 50 337 Dec 89 26 63 5 76 664 50 690 Jan 89 26 63 5 105 786 50 841 Feb 89 26 63 5 168 786 110 844 7/ Mar 91 26 65 5 121 11 875 220 787 -6,-, Apr 92 26 66 5 93 78 394 50 515 May 209 26 183 5 192 56 495 50 693 Jun 209 26 183 5 190 14 436 50 590 Jul 209 26 183 5 186 680 50 816 Aug 180 26 154 5 157 600 50 707 Sep 100 26 74 5 78 353 50 381 Table 5.5 (Continued)

Folsom Natoma New Melones Tullock Delta total Month Spill Penstock Spill Penstock Spill Penstock Spill Penstock release

Oct 20 170 19 168 157 55 102 721 Nov 300 19 278 159 55 104 843 Dec 300 19 278 199 110 89 1236 Jan 476 184 289 218 110 108 1582 Feb 70 406 184 289 213 110 103 1640 Mar 156 344 184 313 205 110 95 1709 Apr 45 305 62 285 217 110 107 1129 May 45 355 124 273 213 110 103 1353 01 1 Jun 23 277 33 264 211 110 101 1148 --1QD Jul 22 222 19 222 210 110 100 1317 Aug 18 177 19 173 200 110 90 1149 Sep 14 277 19 269 150 55 95 869

6 3 Releases are in kiloacre-feet (Kaf). 1 Kaf = 1.233 x 10 m . Total annual Delta releases = 14,697 Kaf.

Entries are the optimal implemented spillage and penstock releases at the beginning of each month. 6 Total NCVP annual energy production corresponding to the optimal release policy = 7.764 • 10 Mwh.

Annual revenues equal $387 million Op = $50/Mwh, in 1980 constant dollars). Table 5.6 Optimal Release Policy, 1979-80 (Policy II)

Clair Engle Lewiston Whiskeytown Shasta Keswick Month Spill Penstock Spill Penstock Spill Penstock Spill Penstock Spill Penstock

Oct 190 26 164 5 168 300 25 443 Nov 190 26 164 5 175 338 50 463 Dec 170 26 144 5 157 688 50 795 Jan 100 26 74 5 116 728 50 794 Feb 100 26 74 5 179 812 220 771 Mar 100 26 74 5 130 876 220 786 Apr 9.5 26 69 5 96 78 370 50 494 May 170 26 144 5 153 57 319 50 479 (.711 N) Jun 110 26 84 5 91 35 465 50 541 CD Jul 125 26 99 5 102 700 50 752 Aug 109 26 84 5 87 602 25 664 Sep 99 26 73 5 77 460 25 512 Table 5.6 (Continued)

Folsom Natoma New Melones Tullock Delta total Month Spill Penstock Spill Penstock Spill Penstock Spill Penstock release

Oct 18 204 19 200 157 55 102 844 Nov 290 19 268 159 55 104 959 Dec 295 19 273 199 110 89 1336 Jan 476 184 289 218 110 108 1535 Feb 52 424 184 289 213 110 103 1677 Mar 150 310 184 273 205 110 95 1668 Apr 79 381 62 395 217 110 107 1218 cn May 45 275 125 192 213 110 103 1059 1 Jun 22 268 30 257 211 110 101 1089 ....JN) Jul 22 278 19 278 210 110 100 1309 Aug 19 161 19 158 200 110 90 1066 Sep 15 238 19 231 150 55 95 937

6 3 Releases are in kiloacre-feet (Kaf). 1 Kaf = 1.233 x 10 m . Total annual Delta releases = 14,697 Kaf.

Entries are the optimal implemented spillage and penstock releases at the beginning of each month.

Total NCVP annual energy production corresponding to the optimal release policy = 7.772 • 106 Mwh.

Annual revenues equal $387 million GIs = $50/Mwh, in 1980 constant dollars). Table 5.7 Actual and Maximized Energy Production (in 103 Mwh) for 1979-80

Trinity Power Plant Judge Francis Carr Spring Creek at Clair Engle Power Plant Power Plant Month Actual Linear Quad. I Quad. II Actual Linear Quad. I Quad. II Actual Linear Quad. I Quad. II

Oct 30.4 73.9 37.4 71.1 36.8 112.4 39.8 98.8 42.3 111.2 43.0 93.9 Nov 9.6 71.2 39.4 68.9 4.8 112.4 43.1 98.8 19.5 115.2 49.7 97.8 Dec 17.1 61.4 33.3 60.0 15.4 98.5 34.3 86.8 19.2 101.3 42.5 87.8 Jan 6.2 29.7 33.8 35.6 -A- 41.9 34.3 44.6 23.9 95.5 58.7 64.8 Feb 17.7 31.1 34.9 37.2 3.4 41.9 34.3 44.6 55.3 87.0 93.9 100.1 Mar 73.4 32.2 36.3 38.2 78.2 41.9 35.4 44.6 99.5 50.0 67.6 72.7 Apr 44.9 32.9 37.0 36.8 53.5 41.9 36.0 41.6 54.7 56.6 52.0 53.7 cri May 21.2 33.7 84.6 66.6 21.2 41.9 99.8 86.8 18.6 46.5 107.3 85.5 1 N.) Jun 51.5 75.3 84.2 43.1 55.2 101.2 99.8 50.6 59.6 102.2 106.2 50.9 N) Jul 54.3 39.4 82.9 48.8 57.2 44.7 99.8 59.7 57.2 45.7 104.0 57.2 Aug 75.7 37.6 69.6 41.9 87.2 42.9 84.0 50.6 85.9 44.0 87.8 48.1 Sep 71.1 36.5 38.5 37.3 84.7 44.2 41.4 44.0 88.6 45.7 44.7 43.0

Total 473.1 555.1 611.8 585.7 497.6 765.9 681.9 751.4 623.3 901.0 857.5 855.2

Ea/Em 0.58 0.77 0.81 0.65 0.73 0.66 0.69 0.73 0.73 Table 5.7 (Continued)

Shasta Power Plant Keswick Power Plant Folsom Power Plant Month Actual Linear Quad. I Quad. II Actual Linear Quad. I Quad. II Actual Linear Quad. I Quad. II

Oct 76.5 112.1 116.2 116.2 19.0 47.0 30.9 41.9 37.7 53.5 48.6 57.9 Nov 89.6 126.2 116.2 130.7 20.7 51.3 32.1 43.8 41.3 77.9 80.1 76.3 Dec 111.1 251.3 254.5 253.9 23.2 81.6 65.3. 75.2 1 37.2 71.3 71.3 69.0 Jan 237.4 266.9 302.5 279.2 47.6 81.6 79.5 75.1 107.0 118.4 127.2 126.0 Feb 209.9 310.7 316.3 326.4 39.2 81.6 79.8 72.9 84.3 139.9 122.7 127.8 Mar 228.3 323.7 358.3 358.3 49.9 81.6 74.4 74.3 130.0 147.2 104.8 94.7 Apr 112.9 126.1 162.0 151.8 29.0 37.3 48.7 46.7 99.8 126.3 93.2 116.2 May 164.4 128.8 203.8 131.4 33.7 36.2 65.5 45.3 82.0 131.0 108.6 84.0 (xi Jun 212.9 129.5 178.3 190.6 47.3 45.5 55.8 51.2 66.6 76.4 84.7 81.9 1 ,r .) Jul 237.9 307.5 273.1 281.0 52.2 77.0 77.2 71.1 84.4 63.4 65.6 84.4 w Aug 165.7 310.2 232.8 232.2 43.6 81.6 66.9 62.7 35.6 93.5 53.3 48.1 Sep 86.8 289.0 131.8 172.9 29.3 81.6 36.0 48.4 40.5 86.8 83.5 69.4

Total 1933.4 2682.2 2646.0 2625.6 434.7 784.5 712.2 708.7 846.4 1185.5 1045.5 1035.8

E/Ea m 0.72 0.73 0.74 0.55 0.61 0.61 0.71 0.81 0.82 .. Table 5.7 (Continued)

Nimbus Power Plant New Melones at Lake Natoma Power Plant Month Actual Linear Quad. I Quad. II Actual Linear Quad. I Quad. II

Oct 4.6 6.4 5.6 6.7 * 71.6 75.0 75.0 Nov 4.7 10.0 9.3 8.9 -A- 70.4 73.2 73.2 Dec 4.7 10.0 9.3 9.1 * 68.1 87.6 87.6 Jan 8.7 10.0 9.6 9.6 19.6 81.9 94.4 94.4 Feb 6.7 10.0 9.6 9.6 23.6 92.1 94.1 94.1 Mar 10.6 10.0 10.4 9.1 47.8 90.9 90.3 90.3 Apr 9.7 7.7 9.5 13.2 55.8 90.4 92.1 92.1 May 9.1 8.2 9.1 6.4 39.5 85.8 87.5 87.5 Jun 7.6 7.8 8.8 8.6 23.3 84.4 84.6 84.6 Jul 9.7 6.5 7.4 9.3 38.5 80.9 79.1 79.1 01 Aug 4.3 10.0 5.8 5.3 22.1 74.9 66.3 66.3 1 rN) Sep 4.6 10.0 9.0 7.7 9.0 64.8 41.7 41.7 -P Total 84.9 106.5 103.4 103.5 279.2 956.5 966.0 966.0

Ea/Em 0.80 0.82 0.82 0.29 0.29 0.29

Power plant not in operation. E = actual energy production, a E = maximized energy production. Quad I and II denote quadratic model m using initial policies I and II, respectively. To obtain energy revenues, multiply energy figures by the unit price t[s (ts = $50/Mwh, in 1980 constant dollars). CHAPTER 6

SUMMARY AND CONCLUSIONS

Alternative optimization models have been developed to obtain reservoir operation policies. Two different models have been used to find optimal release policies for the NCVP system. Several conclusions can be drawn from this study:

(1) It is possible to increase the annual energy production of the system for below-average, average, and above-average inflow conditions. For a sample year of average inflow conditions, an upper bound to the potential increase of annual energy production was found to be of the order of 6 10 Mwh, from both the linear and quadratic models. This potential increase is relative to the energy production levels achieved by the NCVP system using heuristic policies.

(2) Delta and agricultural water deliveries can be increased by adopting the optimal release policies. For a year of average inflow conditions, the water released from the system exceeded the agricultural demands by a factor of 1.6 (for both models). That suggests the possibility of increasing irrigated areas, providing better leaching of agricultural fields, and improving conjunctive management of surface water and groundwater reservoirs.

(3) Much of the improvement achieved by the optimal operation policies developed in this study relative to the actual implemented operation schedules is due to an accurate river inflow forecasting technique and an integrative analysis, intrinsic in the optimization model, that allows to represent all the links and constraints that act simultaneously and interac- tively within the system. Clearly, this integral conceptualization cannot be achieved by a heuristic approach based solely on experience.

6-1 (4) It is difficult to establish a direct comparison between the

optimal and actual implemented operation policies. This is due to the

following reasons: (i) There is a significant amount of idle time in the

power plants of the system that is caused by breakdowns and maintenance.

Those operation halts occur randomly during any season of the year. In

addition, those idle periods are difficult to consider in any optimization

model because it is not known when they will occur, how long they will last,

and what repercussion they will have in the integrated energy network;

(ii) there are legal and institutional regulations that are highly variable

which affect the directives of the NCVP management staff; and (iii) the

managers of the system consider many intangible effects that cannot be

properly considered with a mathematical model. This is especially true for

establishing flood-control regulations, where a conservative attitude

prevails with regard to flood management.

(5) The improved performance reported by the use of the optimization

models should be viewed as an upper bound to the possible gains that could

be derived from the use of mathematical models. The more knowledgeable the

system managers become with reservoir optimization models, the closer the performance of the system will be to the optimal operation policies obtained

under the conditions assumed by the models. Clearly, the use of mathematical models and the better understanding that emanates from their use should result in a feedback to the models, with their probable reformulation and modification that would bring closer the unpleasant difficulties of any

real-world system and the sophistications inherent to any mathematical

model. With regard to the two different models, linear and quadratic, the

optimal policies proved to be robust to the choice of model, i.e., the

results obtained from both models were relatively close. However, the quadratic model gives a better representation of the physical features of the system, and its implementation is worthwhile because that model can be 6-2 expected to provide more reliable results under a wider range of conditions than the linear model.

The experience gained in this investigation has made evident several areas related to reservoir operation that deserve further study:

(1) Reliable forecasts of river inflows. Conceptual hydrologic and statistical methods that would allow to predict accurate streamflows to the system are perhaps the most needed element in reservoir operation planning.

Development of techniques for river-flow forecasting is a major task due to the size of the basins and the difficulties in modeling the hydrologic elements that interact to determine the volume of runoff feeding the reservoirs. For short-term events, the use of satellite information and raingage networks offers a possibility to improve reservoir management. Statistical forecasting techniques for real-time prediction of floods also may prove useful in developing flood-control strategies.

(2) Interrelationships that exist between the functions served by the reservoir system and the economic environment in which the system is imbedded. The state of the economy determines to a great extent the demands of water for agricultural activities and the energy requirements for industrial uses. Price schedules of water releases for agricultural uses also determine to some extent the quantities of water requested by the agricultural sector.

(3) Conjunctive use of surface water and groundwater reservoirs.

Improved management of surface water reservoirs leads to a greater recharge potential at groundwater basins. Thus, excess, releases over contractual levels could be recharged to groundwater reservoirs rather than discharged into the ocean.

(4) Implementation and installation of software that would automatize the operation of reservoir systems. This is a possible way to perform better control decisions for routing flood events by integrating inflow 6-3 forecasts, control decision making, and execution of control policies into a unique, coordinated operation.

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