A Novel Mean Field Game-Based Strategy for Charging Electric Vehicles in Solar Powered Parking Lots

We develop a strategy, with concepts from Mean Field Games (MFG), to coordinate the charging of a large population of battery electric vehicles (BEVs) in a parking lot powered by solar energy and managed by an aggregator. A yearly parking fee is charged for each BEV irrespective of the amount of energy extracted. The goal is to share the energy available so as to minimize the standard deviation (STD) of the state of charge (SOC) of batteries when the BEVs are leaving the parking lot, while maintaining some fairness and decentralization criteria. The MFG charging laws correspond to the Nash equilibrium induced by quadratic cost functions based on an inverse Nash equilibrium concept and designed to favor the batteries with the lower SOCs upon arrival. While the MFG charging laws are strictly decentralized, they guarantee that a mean of instantaneous charging powers to the BEVs follows a trajectory based on the solar energy forecast for the day. That day ahead forecast is broadcasted to the BEVs which then gauge the necessary SOC upon leaving their home. We illustrate the advantages of the MFG strategy for the case of a typical sunny day and a typical cloudy day when compared to more straightforward strategies: first come first full/serve and equal sharing. The behavior of the charging strategies is contrasted under conditions of random arrivals and random departures of the BEVs in the parking lot.

A probabilistic numerical method for a class of mean field games

The Mean Field Games PDEs system is at the heart of the Mean Field Games theory initiated by [J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. I–le cas stationnaire, C. R. Math. 343 (2006) 619–625; J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. II–horizon fini et contrôle optimal, C. R. Math. 343 (2006) 679–684; J.-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math. 2 (2007) 229–260] which constitutes a seminal contribution to the modeling and analysis of games with a large number of players. We propose here a numerical method of resolution of such systems based on the construction of a discrete mean field game where the controlled state-variable is a Markov chain approximating the controlled stochastic differential equation [H. Kushner and P. G. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time, Stochastic Modeling and Applied Probability, Vol. 24 (Springer Science & Business Media, 2013)]. In particular, existence and uniqueness properties of the discrete MFG are investigated with convergence results under adequate assumptions.