Optimal control of a large dam, taking into account the water costs [New Edition]

TL;DR

This paper investigates optimal water release strategies for large dams considering water costs that depend on water levels, analyzing how cost structures influence long-term expense minimization.

Contribution

It introduces a model incorporating water costs into dam control, proving existence and uniqueness of optimal solutions, especially for linear cost structures.

Findings

Optimal control parameters are derived for large dam models.

Water cost structure significantly impacts the optimal solution.

Existence and uniqueness of solutions are established under natural assumptions.

Abstract

This paper studies large dam models where the difference between lower and upper levels, $L$, is assumed to be large. Passage across the levels leads to damage, and the damage costs of crossing the lower or upper level are proportional to the large parameter $L$. Input stream of water is described by compound Poisson process, and the water cost depends upon current level of water in the dam. The aim of the paper is to choose the parameters of output stream (specifically defined in the paper) minimizing the long-run expenses. The particular problem, where input stream is ordinary Poisson and water costs are not taken into account, has been studied in [Abramov, \emph{J. Appl. Prob.}, 44 (2007), 249-258]. The present paper addresses the question \textit{How does the structure of water costs affect the optimal solution?} Under natural assumptions we prove an existence and uniqueness of a solution and study the case of linear structure of the costs.