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.