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 meanfield-type hierarchical control problem with common noise.Finally,the authors illustrate the obtained result via a numerical example with two different scenarios.