Optimal control of energy flexibilities in an uncertain context.

Summary In this thesis, we use tools from stochastic optimal control and stochastic and convex optimization to develop mechanisms to drive energy storage systems to manage the production uncertainty of intermittent energy sources (solar and wind).First, we introduce a mechanism in which a consumer commits to follow a consumption profile on the grid, and then controls its storage systems to follow this profile in real time. We model this situation by a mean-field control problem, for which we obtain theoretical and numerical results. Then, we introduce a problem of controlling a large number of thermal storage units subject to a common noise and providing services to the network. We show that this control problem can be replaced by a stochastic differential Stackelberg problem. This allows a decentralized control scheme with performance guarantees, while preserving the privacy of the consumers' data and limiting the telecommunication requirements. Next, we develop a Newton method for stochastic control problems. We show that the Newton step can be computed by solving Stochastic Retrograde Differential Equations, then we propose an appropriate linear search method, and prove the global convergence of the obtained Newton method in a suitable space. Its numerical performance is illustrated on a problem of controlling a large number of batteries providing services to the network. Finally, we study the extension of the "Alternating Current Optimal Power Flow" problem to the stochastic multistage case in order to control an electrical network equipped with storage systems. For this problem, we give realistic and verifiable a priori conditions guaranteeing the absence of relaxation jumps, as well as an a posteriori bound on the latter. In the broader framework of non-convex multistage problems with a generic structure, we also establish a priori bounds on the duality jump, based on results related to the Shapley-Folkman Theorem.