Real-time pricing leading to optimal operation under distributed decision makings

This paper investigates the optimal regulation problem with the steady-state constraints under distributed decision makings, where each agent is allowed to determine its own optimal set-point according to an individual profit. On the other hand, the utility, which corresponds to an individual public commission, tries to realize a socially optimal solution that fulfills steady-state constraints. In order to align the individual decision making of each agent with the socially optimal solution, the utility is allowed to provide additional price, which corresponds to tax or subsidy for the agent. This paper proposes a real-time pricing strategy of the utility. The proposed pricing strategy is heuristic, but we show that the resulting closed-loop system is stable, if the step size parameter in the real-time pricing strategy is sufficiently small, and realizes tracking to the socially optimal solution.