Nash Equilibrium among Mobile Energy Storage Systems Game for Load Restoration of Faulted Microgrids

Mobile energy storage systems (MESSs) represent a proactive approach to load restoration of faulted microgrids. Existing studies mainly focus on minimizing the overall cost of the MESS fleet but fail to guarantee that self-interested MESSs are willing to follow the optimal social cost solution. Hence, this article allows MESSs to make independent decisions and formulates a nonconvex game. In particular, we incorporate the maximum tolerable service waiting time and construct a potential function to prove that the selfish actions of MESSs converge to a Nash equilibrium. We derive a small upper bound on the price of anarchy (PoA), demonstrating that granting MESSs the autonomy to make self-interested Nash decisions does not significantly increase the overall social cost. Moreover, we model subjective behaviors under uncertain grid power availability, accounting for both loss-sensitive and gain-seeking tendencies. Simulation results show that for different MESS fleet sizes and battery anxiety extents, Nash equilibria are always achieved, with all PoA values below the theoretical bound. Higher MESS numbers or battery anxiety extents improve load restoration performance. Under uncertain surplus energy, gain-seeking MESSs behave more aggressively and earn higher average profits.