Demand response management in the smart grid in a large population regime

In this paper, we introduce a hierarchical system model that captures the decision making processes involved in a network of multiple providers and a large number of consumers in the smart grid, incorporating multiple processes from power generation to market activities and to power consumption. We establish a Stackelberg game between providers and end users, where the providers behave as leaders maximizing their profit and end users act as the followers maximizing their individual welfare. We obtain closed-form expressions for the Stackelberg equilibrium of the game and prove that a unique equilibrium solution exists. In the large population regime, we show that a higher number of providers help to improve profits for the providers. This is inline with the goal of facilitating multiple distributed power generation units, one of the main design considerations in the smart grid. We further prove that there exist a unique number of providers that maximize their profits, and develop an iterative and distributed algorithm to obtain it. Finally, we provide numerical examples to illustrate the solutions and to corroborate the results.