A Quantilized Mean Field Game Approach To Energy Pricing With Application To Fleets Of Plug-In Electric Vehicles

We consider the problem of designing the price of electricity by an energy provider to a pool of homogeneous loads. The energy provider is risk sensitive and considers that its energy production cost at any particular time is related to the instantaneous maximum excursion of the random aggregate demand of the loads. A statistical measure of this excursion is the α- quantile of the distribution of the individual electricity demands of the loads, or equivalently the value d α at risk α, of the electricity demand per vehicle. The price is assumed to be a known and possibly time varying function of d α . The loads are associated with individual price sensitive costs. For a very large number of loads, in particular a large fleet of electric vehicles, this results in a mean field game (MFG). The existence of an MFG equilibrium associated with a price trajectory, and the epsilon- Nash property of the resulting limiting control laws, are established in this work.