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