Optimal price signal generation for indirect control of flexible energy demand under uncertainty

As more and more energy is generated from intermittent sources, such as wind and solar, it becomes increasingly difficult to balance energy supply and demand. Flexible energy consumption, also referred to as demand response, is a cost-effective way to mitigate this growing challenge. In this paper, we propose an approach that generates optimal price signals for indirect control of the flexible energy consumption. With a stochastic and dynamic demand-response model, called the flexibility function, we generate the price signals by solving a stochastic open-loop optimal control problem (OCP). Using numerical examples, we demonstrate the improved performance of the generated price signals compared to those obtained by solving a deterministic OCP. Furthermore, we compare with the performance of an optimal stationary feedback law obtained by solving the Hamilton-Jacobi-Bellman equation. Based on the numerical results, we conclude that the performance is improved by accounting for uncertainty when generating the price signals. Furthermore, the improvement increases with the level of uncertainty in the demand response model.