We address a class of Nash games with nonconvex coupling constraints for which we define a novel notion of local equilibrium, here named local generalized Nash equilibrium (LGNE). Our first technical contribution is to show the stability in the game theoretic sense of these equilibria on a specific local subset of the original feasible set. Remarkably, we show that the proposed notion of local equilibrium can be equivalently formulated as the solution of a quasi-variational inequality with equal Lagrange multipliers. Next, under the additional proximal smoothness assumption of the coupled feasible set, we define conditions for the existence and local uniqueness of a LGNE. To compute such an equilibrium, we propose two discrete-time dynamics, or fixed-point iterations implemented in a centralized fashion. Our third technical contribution is to prove convergence under (strongly) monotone assumptions on the pseudo- gradient mapping of the game and proximal smoothness of the coupled feasible set. Finally, we apply our theoretical results to a noncooperative version of the optimal power flow control problem.

In this paper, we consider a smart grid where users behave selfishly, aiming at minimizing cost in the presence of uncertain wind power availability. We adopt a demand side management (DSM) model, where active users (so-called prosumers) have both private generation and local storage availability. These prosumers participate to the DSM strategy by updating their energy schedule, seeking to minimize their local cost, given their local preferences and the global grid constraints. The energy price is defined as a function of the aggregate load and the wind power availability. We model the resulting problem as a non-cooperative Nash game and propose a semi-decentralized algorithm to compute an equilibrium. To cope with the uncertainty in the wind power, we adopt a rolling-horizon approach, and in addition we use a stochastic optimization technique. We generate several wind power production scenarios from a defined probability density function (PDF), determining an approximate stochastic cost function. Simulations results on a real dataset show that the proposed approach generates lower individual costs compared to a standard expected value approach.