Robust strategy synthesis for probabilistic systems applied to risk-limiting renewable-energy pricing

We address the problem of synthesizing control strategies for Ellipsoidal Markov Decision Processes (EMDP), i.e., MDPs whose transition probabilities are expressed using ellipsoidal uncertainty sets. The synthesized strategy aims to maximize the total expected reward of the EMDP, constrained to a specification expressed in Probabilistic Computation Tree Logic (PCTL). We prove that the EMDP strategy synthesis problem for the fragment of PCTL disabling operators with a finite time bound is NP-complete and propose a novel sound and complete algorithm to solve it. We apply these results to the problem of synthesizing optimal energy pricing and dispatch strategies in smart grids that integrate renewable sources of energy. We use rewards to maximize the profit of the network operator and a PCTL specification to constrain the risk of power unbalance and guarantee quality-of-service for the users. The EMDP model used to represent the decision-making scenario was trained with measured data and quantitatively captures the uncertainty in the prediction of energy generation. An experimental comparison shows the effectiveness of our method with respect to previous approaches presented in the literature.