We propose a stochastic control problem to manage cooperatively heterogeneous Thermostatically Controlled Loads (TCLs: water heaters, heat pumps...) to promote power balance in electricity networks. We develop a method to solve this problem with a decentralized architecture, in order to respect privacy of individual users and to reduce both the telecommunications and the computational burden compared to the setting of an omniscient  central planner. This paradigm is called federated learning in the machine learning community,hence the name federated stochastic control problem. The optimality conditions are expressed in the form of a high-dimensional Forward-Backward Stochastic Differential Equation (FBSDE), which is decomposed into smaller FBSDEs, which fully characterize the Nash equilibrium of a stochastic Stackelberg differential game. In this game, a coordinator (the leader) aims at controlling the aggregate behavior of the  population, by sending appropriate signals, and agents (the followers) respond to this signal by optimizing their storage system locally. A mean-fi eld-type approximation is proposed to circumvent telecommunication
constraints and privacy issues. Convergence results and error bounds depending on the size of the population are obtained for this approximation. A numerical illustration is provided to show the interest of the control scheme and to exhibit the convergence of the approximation. An implementation which answers practical industrial challenges to deploy such a scheme is presented.

- The seminar is supported by the Labex (Laboratory of excellence) of Institut Louis Bachelier -