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

We 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.