A Discontinuous Hamiltonian Approach for Operating a Dam-Reservoir System in a River

We present a new optimal control model for designing operation rules of a dam-reservoir system in a river. Operation of the system is assumed to be optimized so that the operation purpose of the system and downstream aquatic ecosystem are reasonably balanced. Starting from the water balance dynamics of a reservoir, we show that finding the optimal operation policy of the system ultimately reduces to solving a Hamilton-Jacobi-Bellman (HJB) equation having a discontinuous Hamiltonian. This discontinuity comes from a physical constraint in modeling the water balance dynamics and a functional shape of the performance index to be optimized. We obtain an exact continuous viscosity solution to the HJB equation. We also present numerical schemes for discretization of the HJB equation, which are the local Lax-Friedrichs scheme and its Weighted Essentially Non-Oscillatory (WENO) counterpart. Both schemes can approximate the exact solution, the latter being more accurate and efficient. A computational example of a hypothetical dam-reservoir system receiving a seasonal inflow discharge is finally demonstrated.