  We consider a stylized model for a power network with distributed local power
generation and storage. This system is modeled as network connection a large
number of nodes, where each node is characterized by a local electricity
consumption, has a local electricity production (e.g. photovoltaic panels), and
manages a local storage device. Depending on its instantaneous consumption and
production rates as well as its storage management decision, each node may
either buy or sell electricity, impacting the electricity spot price. The
objective at each node is to minimize energy and storage costs by optimally
controlling the storage device. In a non-cooperative game setting, we are led
to the analysis of a non-zero sum stochastic game with $N$ players where the
interaction takes place through the spot price mechanism. For an infinite
number of agents, our model corresponds to an Extended Mean-Field Game (EMFG).
In a linear quadratic setting, we obtain and explicit solution to the EMFG, we
show that it provides an approximate Nash-equilibrium for $N$-player game, and
we compare this solution to the optimal strategy of a central planner.
