Optimal Control of Linked Reservoirs

Abstract

Control rules for linked reservoirs meeting a common demand are evaluated for a two storage case. Draw‐off can be made from either of the reservoirs directly to supply, and transfers are permitted from the smaller to the larger reservoirs. Dynamic programming is effective in selecting the optimal control rules, for any stage of reservoir contents, given a defined objective of operation. The objective is expressed in monetary terms, relating to transmission, purification, or shortage costs, which are to be minimized in the long term. The case considered allows monthly inflows to the reservoirs to be treated as random variates; first order serial correlation of inflows is expressed by using ‘high’ or ‘low’ inflow distributions, according to whether the previous month's inflow was above or below its mean. Present worth factors, switch‐on costs, and costs of shortage that vary nonlinearly with total deficit can all be brought into the reckoning. The paper includes a numerical example of the dynamic programming calculation for a system of a finite surface reservoir and a full aquifer, the latter having limited pumpage. Also, the flow diagram for a computer program is given, which incorporates inflow, draw‐off, storage volumes, and operating costs as general parameters. Using this program, the convergence to optimal control rules has been obtained, for the most part within 5 years of iteration. Given the optimal control rules for an assumed reservoir system, it becomes possible to form transition matrices of contents, by an adaptation of Gould's method. The steady‐state solutions of the matrices show probabilities of each reservoir's contents in the long term. These lead to a long‐term operating cost for consideration at the design stage of the system.

IMPLEMENTATION OF AN OPTIMIZATION MODEL FOR OPERATION OF A METROPOLITAN RESERVOIR SYSTEM1

“…Because of the complexity of the objective function, forward dynamic programming was used to carry out the optimization. Dynamic programming (DP) has been extensively used (e.g., Buras, 1966;Butcher, 1971;Hall, et ul., 1968Hall, et ul., , 1969Schweig and Cole, 1968;Tobabi and Mobasheri, 1973) in reservoir operation optimization. Its logic is well known (Bellman and Dreyfus, 1962;Buras, 1972;Hall and Dracup, 1970;Larson, 1968) and need not be detailed here.…”

COMPUTATIONAL IMPROVEMENT FOR STOCHASTIC DYNAMIC PROGRAMMING MODELS OF URBAN WATER SUPPLY RESERVOIRS1

“…• Using coarse discretization for state variables (Schweig and Cole, 1968). This study used only two states to describe the inflow state variable.…”

Optimization of a Class of River Aeration Problems by the Use of Multivariable Distributed Parameter Control Theory

“…Mathematical programing methods applied to water pollution problems, as evidenced by recent publications in the field [Hall 1965;Liebman, 1966;Davidson and Bradshaw, 1967;Ortolano and Thomas, 1968;Revelle et al, 1968;Schweig and Cole, 1968], have been highly successful, especially in cases where hypothetic models were used. In the majority of the applications mentioned, linear programing, dynamic programing, or Pontryagin's Maximum Principle methods were demonstrated using steady state Streeter-Phelps system equations at some stage in the optimization study.…”