Charging Control of Electric Vehicles in Smart Grid: a Stackelberg Differential Game Based Approach

In this paper, we investigate the charging control problems of the electrical vehicles in smart grid, where electricity transactions exist between the aggregation and the electrical vehicles. We use the Stackelberg differential game to formulate the charging/discharging interactions between the aggregation and the electrical vehicles, where the dynamic behavior of the energy levels of the aggregation and the electrical vehicles are formulated as a differential game, and the charging/discharging interactions between the aggregation and the electrical vehicles are formulated as a Stackelberg game. The aggregation acts as the leader, and control the electricity price for to maximize the payoff. The electrical vehicles acts as the followers, and control their charging/discharging power to minimize the energy cost. Two kinds of equilibrium solutions to the Stackelberg differential game will be analyzed, which are the open loop equilibriums and the feedback equilibriums, respectively. The obtained equilibrium solutions can be considered as the optimal control for the aggregations and the electrical vehicles. Numerical simulations and results show the effectiveness and advantages of the proposed algorithms.