Hierarchical Mean-Field Type Control of Price Dynamics for Electricity in Smart Grid

This paper solves a mean-field type hierarchical optimal control problem in electricity market. The authors consider n − 1 prosumers and one producer. The ith prosumer, for 1 < i < n, is a leader of the (i − 1)th prosumer and is a follower of the (i + 1)th prosumer. The first player (agent) is the follower at the bottom whereas the nth is the leader at the top. The problem is described by a linear jump-diffusion system of conditional mean-field type, where the conditioning is with respect to common noise, and a quadratic cost functional involving, the square of the conditional expectation of the controls of the agents. The authors provide a semi-explicit solution of the corresponding mean-field-type hierarchical control problem with common noise. Finally, the authors illustrate the obtained result via a numerical example with two different scenarios.