  This paper presents the equilibrium analysis of a game composed of
heterogeneous electric vehicles (EVs) and a power distribution system operator
(DSO) as the players, and charging station operators (CSOs) and a
transportation network operator (TNO) as coordinators. Each EV tries to pick a
charging station as its destination and a route to get there at the same time.
However, the traffic and electrical load congestion on the roads and charging
stations lead to the interdependencies between the optimal decisions of EVs.
CSOs and the TNO need to apply some tolling to control such congestion. On the
other hand, the pricing at charging stations depends on real-time
distributional locational marginal pricing, which is determined by the DSO
after solving the optimal power flow over the power distribution network. This
paper also takes into account the local and the coupling/infrastructure
constraints of EVs, transportation and distribution networks. This problem is
modeled as a generalized aggregative game, and then a decentralized learning
method is proposed to obtain an equilibrium point of the game, which is known
as variational generalized Wardrop equilibrium. The existence of such an
equilibrium point and the convergence of the proposed algorithm to it are
proven. We undertake numerical studies on the Savannah city model and the IEEE
33-bus distribution network and investigate the impact of various
characteristics on demand and prices.
