Stable real-time pricing and scheduling for serving opportunistic users with deferrable loads

In this paper, we address real-time pricing and control of opportunistic consumers with deferrable demands that are motivated by the envisioned smart electrical grid. In the smart grid, demand-side flexibilities from deferrable loads enable consumers to respond to real-time electricity prices for their own economic benefit. However, the aggregate load created by many such economically-driven consumers can be highly time-varying, which can cause significant fluctuations in both the electricity demand and the real-time price, and ultimately incur additional generation and capacity costs for the suppliers. In this paper, we propose a distributed pricing and load scheduling algorithm that alleviates such undesirable fluctuations and high volatility. In particular, we formulate the pricing and scheduling problem as an optimization problem with proximal terms that incur penalty for rapid consumption changes over time. Through a continuous-time approximation, we show how the overall system evolves towards the optimal operating regime under the proposed algorithm. In the limiting operating regime, even though the individual consumption of each consumer may still exhibit oscillatory behavior, the aggregate load and the real-time price at any time stay arbitrarily close to a stable operating point. Our design also enables third-party intermediaries to adjust the change of load penalty to balance the payments made by the consumers to the suppliers, thereby achieving market clearance.