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Optimal Control of Hydro-Electric Power Plants with Uncontrolled Spillways

Maria do Rosario de Pinho, Maria Margarida A. Ferreira, Georgi Smirnov

TL;DR

This work tackles profit-maximizing control of cascaded hydroelectric plants with reversible turbines and uncontrollable spillways, where spillways activate at reservoir capacity and thus create state-discontinuities that defy classical optimality conditions. The authors introduce an exponential-penalty sequence of standard optimal control problems and apply the Pontryagin Maximum Principle to each, then pass to the limit to obtain necessary conditions for the original boundary-constrained system. They establish convergence results showing that solutions to the penalized problems converge to a solution of the original problem, and they derive a BV-adjoint framework with a structured maximum principle, including complementary slackness for spillways. An illustrative two-plant example demonstrates how the conditions identify the optimal strategy and underscores the potential for numerical schemes based on solving tractable approximations for larger cascades.

Abstract

In this paper, we study an optimal control problem for a cascade of hydroelectric power plants with reversible turbines and uncontrolled spillways. The system dynamics are governed by a linear control model subject to path constraints. The aim is to maximize the power production profit while respecting operational restrictions on reservoir water levels. The challenge is the presence of uncontrollable spillways: their discontinuous nature and the fact that they are activated at the state boundary prevent the application of known necessary conditions of optimality. To overcome this, we derive necessary conditions by approximating the original problem through a sequence of standard optimal control problems using exponential penalty functions. The applicability of resulting conditions are illustrated by an example.

Optimal Control of Hydro-Electric Power Plants with Uncontrolled Spillways

TL;DR

This work tackles profit-maximizing control of cascaded hydroelectric plants with reversible turbines and uncontrollable spillways, where spillways activate at reservoir capacity and thus create state-discontinuities that defy classical optimality conditions. The authors introduce an exponential-penalty sequence of standard optimal control problems and apply the Pontryagin Maximum Principle to each, then pass to the limit to obtain necessary conditions for the original boundary-constrained system. They establish convergence results showing that solutions to the penalized problems converge to a solution of the original problem, and they derive a BV-adjoint framework with a structured maximum principle, including complementary slackness for spillways. An illustrative two-plant example demonstrates how the conditions identify the optimal strategy and underscores the potential for numerical schemes based on solving tractable approximations for larger cascades.

Abstract

In this paper, we study an optimal control problem for a cascade of hydroelectric power plants with reversible turbines and uncontrolled spillways. The system dynamics are governed by a linear control model subject to path constraints. The aim is to maximize the power production profit while respecting operational restrictions on reservoir water levels. The challenge is the presence of uncontrollable spillways: their discontinuous nature and the fact that they are activated at the state boundary prevent the application of known necessary conditions of optimality. To overcome this, we derive necessary conditions by approximating the original problem through a sequence of standard optimal control problems using exponential penalty functions. The applicability of resulting conditions are illustrated by an example.
Paper Structure (7 sections, 4 theorems, 137 equations, 2 figures)

This paper contains 7 sections, 4 theorems, 137 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation and problem statement
  3. Notation
  4. Problem statement
  5. Preliminary results
  6. Necessary conditions of optimality
  7. Conclusions

Key Result

Lemma 1

Consider a sequence $u^{\gamma}$ converging in the weak-* topology of $L_{\infty}([0,T],R^{\cal I})$ to some control $\tilde{u}\in [u^m,u^M]$, a sequence $V^{\gamma,0}\leq V^M$ converging to $V^0\in R^{\cal I}$ and the Cauchy problems where $s^{\gamma}_i=\gamma^i e^{\gamma^i(V_i^{\gamma}-V_i^M)}$, $i=1,\ldots,{\cal I}$. (Here and in what follows $\gamma^i$ stands for $\gamma$ to the $i$-th power

Figures (2)

  • Figure 1: Cascade of five hydro-power plants. Three of them, $2$, $3$ and $4$, marked with double sided arrows, have reversible turbines.
  • Figure 2: Cascade of two hydro-power stations. One of them, $2$, marked with double sided arrows, has reversible turbine.

Theorems & Definitions (4)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 4