Mean field game based control of dispersed energy storage devices with constrained inputs

In this paper, we study the mean field control of a large population of electric space heaters with linear dynamics and saturation constraints on the inputs. The mean field model is described by the fixed point of a system of coupled partial differential equations. A numerical algorithm is proposed to find this fixed point, which takes advantage of the special form of the individual control problem. We derive a decentralized control mechanism based on the mean field equilibrium solution under which the mean temperature of the population follows a set target temperature, while controls for each device are generated locally and attempt to keep individual temperature deviations small. We illustrate the results using numerical simulations, and compare the solutions obtained to the case where control inputs are unconstrained.