  In this paper, the problem of grid-to-vehicle energy exchange between a smart
grid and plug-in electric vehicle groups (PEVGs) is studied using a
noncooperative Stackelberg game. In this game, on the one hand, the smart grid
that acts as a leader, needs to decide on its price so as to optimize its
revenue while ensuring the PEVGs' participation. On the other hand, the PEVGs,
which act as followers, need to decide on their charging strategies so as to
optimize a tradeoff between the benefit from battery charging and the
associated cost. Using variational inequalities, it is shown that the proposed
game possesses a socially optimal Stackelberg equilibrium in which the grid
optimizes its price while the PEVGs choose their equilibrium strategies. A
distributed algorithm that enables the PEVGs and the smart grid to reach this
equilibrium is proposed and assessed by extensive simulations. Further, the
model is extended to a time-varying case that can incorporate and handle slowly
varying environments.
