Stochastic Reservoir Calculations

TL;DR

This paper analyzes the stationary distribution of reservoir storage levels under stochastic inflow modeled by Gamma distributions, focusing on probabilities of depletion and spillage, and explores optimal outflow rates to minimize these risks.

Contribution

It clarifies detailed calculations of reservoir behavior with Gamma inflows and identifies optimal outflow to reduce depletion and spillage probabilities.

Findings

Derived explicit formulas for depletion and spillage probabilities.

Identified outflow rates that minimize these probabilities.

Provided detailed examples illustrating the calculations.

Abstract

Prabhu (1958) obtained the stationary distribution of storage level $Z_{t}$ in a reservoir of finite volume $v$, given an inflow $X_{t}$ and an outflow $Y_{t}$. Time $t$ is assumed to be discrete, $X_{t} \sim$ Gamma$(p,\mu)$ are independent and $p$ is a positive integer. The mean inflow is $p/\mu$; the target outflow is $m$ (constant). We attempt to clarify intricate details, often omitted in the literature, by working through several examples. Of special interest are the probabilities of depletion ($Z_{t}=0$) and spillage ($Z_{t}=v$). For prescribed {$v,p,\mu$}, what value of $m$ minimizes both of these?