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We consider an energy system with $n$ consumers who are linked by a DemandSide Management (DSM) contract, i.e. they agreed to diminish, at random times,their aggregated power consumption by a predefined volume during a predefinedduration. Their failure to deliver the service is penalised via the differencebetween the sum of the $n$ power consumptions and the contracted target. We areled to analyse a non-zero sum stochastic game with $n$ players, where theinteraction takes place through a cost which involves a delay induced by theduration included in the DSM contract. When $n \to \infty$, we obtain aMean-Field Game (MFG) with random jump time penalty and interaction on thecontrol. We prove a stochastic maximum principle in this context, which allowsto compare the MFG solution to the optimal strategy of a central planner. In alinear quadratic setting we obtain an semi-explicit solution through a systemof decoupled forward-backward stochastic differential equations with jumps,involving a Riccati Backward SDE with jumps. We show that it provides anapproximate Nash equilibrium for the original $n$-player game for $n$ large.Finally, we propose a numerical algorithm to compute the MFG equilibrium andpresent several numerical experiments.

MFG model with a long-lived penalty at random jump times: application to demand side management for electricity contracts