Towards control of dam and reservoir systems with forward–backward stochastic differential equations driven by clustered jumps

Abstract

AbstractWe deal with a new maximum principle‐based stochastic control model for river management through operating a dam and reservoir system. The model is based on coupled forward–backward stochastic differential equations (FBSDEs) derived from jump‐driven streamflow dynamics and reservoir water balance. A continuous‐time branching process with immigration driven by a tempered stable subordinator efficiently describes clustered inflow streamflow dynamics. This is a completely new attempt in hydrology and control engineering. Applying a stochastic maximum principle to the dynamics based on an objective functional for designing cost‐efficient control of dam and reservoir systems leads to the FBSDEs as a system of optimality equations. The FBSDEs under a linear‐quadratic ansatz lead to a tractable model, while they are solved numerically in the other cases using a least‐squares Monte‐Carlo method. Optimal controls are found in the former, while only sub‐optimal ones are computable in the latter due to a hard state constraint. Model parameters are successfully identified from a real data of a river in Japan having a dam and reservoir system. We also show that the linear‐quadratic case can capture the real operation data of the system with underestimation of the outflow discharge. More complex cases with a realistic time horizon are analyzed numerically to investigate impacts of considering the environmental flows and seasonal operational purposes. Key challenges towards more sophisticated modeling and analysis with jump‐driven FBSDEs are discussed as well.