  In this paper, we investigate a practical demand side management scenario
where the selfish consumers compete to minimize their individual energy cost
through scheduling their future energy consumption profiles. We propose an
instantaneous load billing scheme to effectively convince the consumers to
shift their peak-time consumption and to fairly charge the consumers for their
energy consumption. For the considered DSM scenario, an aggregative game is
first formulated to model the strategic behaviors of the selfish consumers. By
resorting to the variational inequality theory, we analyze the conditions for
the existence and uniqueness of the Nash equilibrium (NE) of the formulated
game. Subsequently, for the scenario where there is a central unit calculating
and sending the real-time aggregated load to all consumers, we develop a one
timescale distributed iterative proximal-point algorithm with provable
convergence to achieve the NE of the formulated game. Finally, considering the
alternative situation where the central unit does not exist, but the consumers
are connected and they would like to share their estimated information with
others, we present a distributed agreement-based algorithm, by which the
consumers can achieve the NE of the formulated game through exchanging
information with their immediate neighbors.
