Regime-switching constrained viscosity solutions approach for controlling dam-reservoir systems

TL;DR

This paper develops a mathematical and numerical framework for optimizing dam-reservoir operations under stochastic regime-switching inflow conditions, balancing operational and environmental objectives.

Contribution

It introduces a regime-switching constrained viscosity solutions approach to solve the nonlinear optimality equations for dam-reservoir control problems.

Findings

Explicit solution for the optimality equation under certain conditions.

Numerical scheme applied to real dam-reservoir system.

Operational policies adapted to environmental concerns.

Abstract

A new stochastic control problem of a dam-reservoir system installed in a river is analyzed both mathematically and numerically. Water balance dynamics of the reservoir are piece-wise deterministic and are driven by a stochastic regime-switching inflow process. The system is controlled to balance among the operation purpose and the internal and downstream environmental conditions. Finding the optimal operation policy of the system reduces to solving an optimality equation with a discontinuous Hamiltonian, which is a system of nonlinear degenerate parabolic (or hyperbolic) equations. We show that the optimality equation has at most one constrained viscosity solution and find the solution explicitly under certain conditions. The model is applied to numerical computation of the operation policy of an existing dam-reservoir system using a high-order finite difference scheme. The computational results can suggest how the operation policy should be adapted according to the environmental concerns of the river.