We analyse deterministic aggregative games, with large but finite number of players, that are subject to both local and coupling constraints. Firstly, we derive sufficient conditions for the existence of a generalized Nash equilibrium, by using the theory of variational inequalities together with the specific structure of the objective functions and constraints. Secondly, we present a coordination scheme, belonging to the class of asymmetric projection algorithms, and we prove that it converges R-linearly to a generalized Nash equilibrium. To this end, we extend the available results on asymmetric projection algorithms to our setting. Finally, we show that the proposed scheme can be implemented in a decentralized fashion and it is suitable for the analysis of large populations. Our theoretical results are applied to the problem of charging a fleet of plug-in electric vehicles, in the presence of capacity constraints coupling the individual demands.

We consider the framework of aggregative games, in which the cost function of each agent depends on his own strategy and on the average population strategy. As first contribution, we investigate the relations between the concepts of Nash and Wardrop equilibria. By exploiting a characterization of the two equilibria as solutions of variational inequalities, we bound their distance with a decreasing function of the population size. As second contribution, we propose two decentralized algorithms that converge to such equilibria and are capable of coping with constraints coupling the strategies of different agents. Finally, we study the applications of charging of electric vehicles and of route choice on a road network.

We consider quasi-aggregative games for large populations of heterogeneous agents, whose interaction is determined by an underlying communication network. Specifically, each agent minimizes a quadratic cost function, which depends on its own strategy and on a convex combination of the strategies of its neighbors, and is subject to heterogeneous convex constraints. We suggest two distributed algorithms that can be implemented to steer the best responses of the rational agents to a Nash equilibrium configuration. The convergence of these schemes is guaranteed under different sufficient conditions depending on the matrices defining the agents' cost functions and on the communication network

We consider the framework of average aggregative games, where the cost function of each agent depends on his own strategy and on the average population strategy. We focus on the case in which the agents are coupled not only via their cost functions, but also via constraints coupling their strategies. We propose a distributed algorithm that achieves an almost-Nash equilibrium by requiring only local communications of the agents, as specified by a sparse communication network. The proof of convergence of the algorithm relies on the auxiliary class of network aggregative games and exploits a novel result of parametric convergence of variational inequalities, which is applicable beyond the context of games. We apply our theoretical findings to a multi-market Cournot game with transportation costs and maximum market capacity.

Abstract-This paper focuses on reactive power flow and voltage stability in electrical grids. We provide novel analytical understanding of the solutions to the classic nonlinear polynomial equations describing the decoupled reactive power flow. As of today, solutions to these equations can be found only via numerical methods. Yet an analytical understanding would enable rigorous design of future electrical grids. This paper has two main contributions. First, for sufficiently-high reference voltages, we guarantee the existence of a high-voltage solution for the reactive power flow equations and provide its approximate analytical expression. The approximation error is bounded in terms of network topology and parameters. Second, we consider a recently-proposed droop control strategy for voltage stabilization in a microgrid equipped with inverters. For sufficiently-high reference voltages, we prove the existence and the exponential stability of a high-voltage fixed point of the closed-loop dynamics. We provide an approximate expression for this fixed point and find the limiting value of the approximation error for high reference voltages. Finally, we validate the accuracy of our approximations through numerical simulation of the IEEE 37 standard test case.