c. 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 opera- tion 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.