Distributed Nash Equilibrium Seeking Algorithm Under Switching Topologies And Its Application

In recent years,some efficient techniques from noncooperative game theory have been wide-ly applied to solve the problems of designing mobile sensor networks,resource allocation,energy management and energy trading for smart grids,etc.As an important research problem of nonco-operative game,distributed Nash equilibrium seeking has been paid more attention by researchers in various fields.The most of the existing works about distributed Nash equilibrium seeking were performed on the basis of the players' communication topologies being(strongly)connected.In practice,however potential random link failures or attacks on communication links may damage the connectivity of the communication network and lead to the switching characteristics of the underlying communication topology of the players.Considering the stochastic switching of com-munication network of players in multiple disconnected communication topology candidates,this paper investigates the distributed Nash equilibrium seeking problem under switching topologies and applies the proposed distributed Nash equilibrium seeking algorithm to the energy trading among microgrids.The main works are as follows:Firstly,for the case of the actions of players being not subject to constraints and the communi-cation topology of the players being assumed to be Markovian switching,a distributed discrete-time Nash equilibrium seeking algorithm is proposed.Specifically,the players utilize a switched consensus protocol to estimate the actions of all players,based upon the local estimates,each player employs a gradient-like play to update its action.Under the condition that the union of the communication topology candidates is undirected and connected,it is proved that all players' actions will converge to the Nash equilibrium in the sense of mean square under the proposed distributed algorithm.The reliability and validity of the theoretical results are verified by numerical simulations.Secondly,consider the case that the players' actions are subject to local constraint sets,a distributed Nash equilibrium seeking algorithm is designed by combining the gradient-like projection algorithm and the switched consensus protocol.It is shown that the players' actions converge to arbitrarily small neighborhood of the Nash equilibrium in the sense of mean square by adjusting the algorithm parameters.Moreover,it is proved that the Nash equilibrium is mean-square stable if the Nash equilibrium lies in the interior of the constraint sets.The reliability and validity of the theoretical results are verified by numerical simulations.Thirdly,the developed distributed Nash equilibrium seeking algorithm is applied to solve the energy game among multiple microgrids.The energy game model is established where the microgrids are classified into sellers with surplus energy and buyers with insufficient energy and each buyer is modelled as a rational player using a purchase price as its action.The energy market uses the principle of proportional distribution to allocate the excess energy from sellers according to the purchase price provided by buyers.Thus,the payoff functions of players are determined and the game model is established.The distributed Nash equilibrium seeking algorithm is applied to the microgrid energy game to determine the equilibrium purchase price of players.The reliability and validity of the theoretical results are verified by numerical simulations.Finally,we conclude the whole works and prospect the future research work.