In this article, we investigate the competitive interaction between electrical vehicles or hybrid oil-electricity vehicles in a Cournot market consisting of electricity transactions to or from an underlying electricity distribution network. We provide a mean field game formulation for this competition, and introduce the set of fundamental differential equations ruling the behavior of the vehicles at the system equilibrium, namely the mean field equilibrium. This framework allows for a consistent analysis of the evolution of the sale-and-purchase price of electricity as well as of the instantaneous total demand. Simulations precisely quantify those parameters and suggest that following the charge and discharge policy at the equilibrium allows for a significant reduction of the daily electricity peak demand.