Multi-Stage Pricing for Coordination of Thermostatically Controlled Loads: A Dynamic Stackelberg Game Approach

This paper focuses on multi-stage coordination for a population of thermostatically controlled loads (TCL). Each load maximizes the individual utility in response to an energy price, while the coordinator determines the price to maximize the social welfare subject to a peak energy constraint. The coordination problem is formulated as a dynamic Stackelberg game. The Stackelberg solution is derived using an indirect approach: we first obtain an upper bound of the Stackelberg game, then we prove that this upper bound is attainable. Sufficient conditions for the optimal solution are characterized. Based on these conditions, a pricing algorithm with guaranteed convergence is developed to compute the Stackelberg solution employing the monotonicity of the user responses. Our control strategy is validated using real market data and weather information, and realistic simulation results are presented to demonstrate the effectiveness of the proposed control framework.