State constrained stochastic optimal control of a PV system with battery storage via Fokker–Planck and Hamilton-Jacobi-Bellman equations

With the growing global emphasis on sustainability and the implementation of contemporary environmental policies, photovoltaic (PV) generation is playing an increasingly important role in modern power systems, while its intrinsic variability poses challenges for real-time operation and electricity market participation. This paper proposes a continuous-time stochastic optimal control framework for the joint optimization of real-time battery management and day-ahead market bidding of PV plants with energy storage. Solar irradiance, electricity prices, and battery dynamics are modeled through stochastic differential equations (SDEs), leading to a constrained stochastic control problem characterized by a coupled Hamilton-Jacobi-Bellman (HJB) and Fokker-Planck (FP) formulation. To mitigate the associated computational burden, a dimension-reduction strategy is introduced by decomposing the state space into controllable and uncontrollable components, yielding lower-dimensional optimality systems while preserving the continuous-time Dynamic Programming structure. Numerical results show that the reduced formulations achieve substantial computational savings, enabling real-time applicability without significant loss of performance. The proposed methodology is benchmarked against two rule-based strategies and a stochastic Model Predictive Control (MPC) approach, highlighting a favorable trade-off in terms of economic performance, computational efficiency, and suitability for day-ahead market participation.