Decentralized Control via Dynamic Stochastic Prices: The Independent System Operator Problem

A smart grid connects several electricity consumers/producers, e.g., wind/solar/storage farms, fossil-fuel plants, industrial/commercial loads, or load-serving aggregators, all modeled as stochastic dynamical systems. In each time period, each consumes/supplies some electrical energy. Each such agent's utility is the benefit accrued from its consumption or the negative of its generation cost. The social welfare, the sum of all these utilities, is the total benefit accrued from all consumption minus the total cost of generation. The independent system operator is charged with maximizing the social welfare subject to total generation equalling consumption in each time period, but without the agents revealing their system states, dynamic models, utility functions, or uncertainties. It has to announce prices after interacting with agents via bid-price interactions. This paper examines the case where the agents respond in a compliant price-taking manner. It is shown that there is an iterative bid-price interaction, where agents respond to price announcements by complying to the requirement to announce their optimal responses according to their true stochastic model, or, in the case where the agents are linear quadratic Gaussian (LQG) systems, according to deterministic versions of their true stochastic models, that leads to the same global maximum value of social welfare attainable if all agents had pooled their information. In the important LQG case, the bid-price iteration is dramatically simple, exchanging only real-valued vectors of future prices and consumptions/generations at each time step. Agents need not even know of the existence of other agents. DC power flow equations can also be incorporated. The results may be of broader interest vis-a-vis general equilibrium theory of economics for stochastic dynamic agents.