Socially Optimal Energy Usage via Adaptive Pricing

Using price signals to coordinate the electricity consumption of a group of users has been studied extensively. Typically, a system operator broadcasts a price, and users optimizes their own actions subject to the price and internal cost functions. A central challenge is the operator's lack of knowledge of the users, since users may not want to share private information. In addition, learning algorithms are being increasingly used to load control, and users maybe unable to provide their costs in analytical form. In this paper, we develop a two time-scale incentive mechanism that alternately updates between the users and a system operator. The system operator selects a price, and the users optimize their consumption. Based on the consumption, a new price is then computed by the system operator. As long as the users can optimize their own consumption for a given price, the operator does not need to know or attempt to learn any private information of the users. We show that under a wide range of assumptions, this iterative process converges to the social welfare solution. In particular, the cost of the users need not be strictly convex and its consumption can be the output of a learning algorithm.