A singular stochastic control problem for hydropower generation in renewable energy markets

We consider a singular stochastic control problem for hydroelectric power production in an energy market where the electricity spot market prices dynamics is described by a Vasicek's process, allowing also for negative prices. We propose a hydroelectric production system that can react in two different modes when it is convenient to produce energy through an instantaneous release of water. We endow the system with the possibility of producing ``less efficiently'' when negative prices appear in the market, but it is still preferable to produce instantaneously rather than waiting for positive prices. We defined a novel optimization problem whose performance functional exhibits a state-dependent instantaneous marginal revenue whose sign is directly affected by the sign of the prices dynamics. We aim to maximize such functional among all the admissible control policies into the class of the adapted stochastic processes whose paths are not necessarily absolutely continuous with respect to the Lebesgue's measure but only non-decreasing, left-continuous and with finite right limits (càglàd). We prove the Verification Theorem, allowing to characterize the value function of our singular stochastic control problem among the solutions of the associated Hamilton-Jacobi-Bellman equation which turns to be a variational inequality with state-dependent gradient constraint. The Verification Theorem sheds light on the structure of the optimal control which turns to be a purely discontinuous process that, at the first time of action, exerts all the available fuel with a single instantaneous jump. Under some assumptions on the characteristics of our hydropower production model, we identify the value function of the optimal control problem in terms of the optimal reward function of an associated family of optimal stopping problems. We identified a unique positive boundary, separating the action and inaction regions and we showed that the optimal strategy consists in completely discharge the water reservoir as soon as the price dynamics reaches values grater or equal such optimal threshold. We highlight the difficulties that arise when the aforementioned assumptions are replaced by other alternative hypotheses. In this more challenging context, we provide some intuitions on the tricky structure of the action and inaction regions as well as on the nature of the candidate optimal control policy.