Stochastic games for the smart grid energy management with prospect prosumers

In this paper, the problem of the smart grid energy management under stochastic dynamics is investigated. In the considered model, at the demand side, it is assumed that customers can act as prosumers who own renewable energy sources and can both produce and consume energy. Due to the coupling between the prosumers' decisions and the stochastic nature of renewable energy, the interaction among prosumers is formulated as a stochastic game, in which each prosumer seeks to maximize its payoff, in terms of revenues, by controlling its energy consumption and demand. In particular, the subjective behavior of prosumers is explicitly reflected into their payoff functions using the prospect theory, a powerful framework that allows modeling real-life human choices, rather than objective, user-agnostic decisions, as normative models do. For this prospect-based stochastic game, it is shown that there always exists a stationary Nash equilibrium where the prosumers' trading policies in the equilibrium are independent of the time and their histories of the play. Moreover, to obtain one of such equilibrium policies, a novel distributed algorithm with no information sharing among prosumers is proposed and shown to converge to an \epsilon-Nash equilibrium in which each prosumer is able to achieve its optimal payoff in an equilibrium up to a small additive error \epsilon. On the other hand, at the supply side, the interaction between the utility company and the prosumers is formulated as an online optimization problem in which the utility company's goal is to learn its optimal energy allocation rules. For this case, it is shown that such an optimization problem admits a no-regret algorithm meaning that regardless of the actual outcome of the game among the prosumers, the utility company can follow a strategy that mitigates its allocation costs as if it knew the entire demand market a priori. Simulation results justify the convergence of the proposed algorithms and present new insights toward more efficient energy management in the smart grids.