  In this paper, an aggregate game approach is proposed for the modeling and
analysis of energy consumption control in smart grid. Since the electricity
user's cost function depends on the aggregate load, which is unknown to the end
users, an aggregate load estimator is employed to estimate it. Based on the
communication among the users about their estimations on the aggregate load,
Nash equilibrium seeking strategies are proposed for the electricity users. By
using singular perturbation analysis and Lyapunov stability analysis, a local
convergence result to the Nash equilibrium is presented for the energy
consumption game that may have multiple Nash equilibria. For the energy
consumption game with a unique Nash equilibrium, it is shown that the players'
strategies converge to the Nash equilibrium non-locally. More specially, if the
unique Nash equilibrium is an inner Nash equilibrium, then the convergence rate
can be quantified. Energy consumption game with stubborn players is also
investigated. Convergence to the best response strategies for the rational
players is ensured. Numerical examples are provided to verify the effectiveness
of the proposed methods.
