A Potential Game Framework for Charging PHEVs in Smart Grid Shahab Bahrami and Vincent W.S. Wong Department of Electrical and Computer Engineering The University of British Columbia, Vancouver, Canada email: {bahramis, vincentw}@ece.ubc.ca Abstract—Due to the proliferation of plug-in hybrid electric vehicles (PHEVs), the peak load in the power grid is expected to increase in future. The peak load can be reduced by implementing appropriate load scheduling schemes using advanced metering infrastructure (AMI) and smart chargers. In this paper, we formulate the charging problem of PHEVs as a potential game to jointly optimize the cost of the utility company and payoff of the customers. The potential game approach enables us to study the existence and uniqueness of the pure strategy Nash equilibrium and to design a polynomial time distributed algorithm to achieve that equilibrium. It also enables us to define a Lyapunov function to show that the Nash equilibrium is globally asymptotically stable, i.e., the proposed distributed algorithm converges to the Nash equilibrium from any arbitrary initial conditions. To evaluate the efficiency of our proposed algorithm, we compare its running time with an algorithm based on the customers’ best response. Keywords: Charging PHEVs, potential game, Lyapunov function. I. I NTRODUCTION When a large number of plug-in hybrid electric vehicles (PHEVs) are integrated into the grid, the aggregate charging demand can further increase the existing peak load demand. It may also introduce new peaks to the daily load profile. To de- crease the cost of generating electricity, utility companies need to motivate customers to shift the charging load of PHEVs from peak to off-peak periods [1]. In smart grid, advanced metering infrastructure (AMI) and smart chargers enable utility companies to provide customers with the price data [2]. They also enable utility companies to deploy automated charging scheduling approach to mitigate the charging impact of PHEVs on the load profile [3]. In the literature, there are several studies that address the charging problem of PHEVs. Wu et al. in [4] proposed a game theoretic framework to model the interaction between the aggregator and electric vehicles (EVs) to achieve optimal frequency regulation in the power grid. Shi et al. in [5] modeled vehicle-to-grid (V2G) control problem as a Markov decision process. They proposed an algorithm that can adapt to the unknown pricing information and make optimal control decisions. Wang et al. [6] designed an optimal V2G aggregator to control the charging and frequency regulation processes of a group of PHEVs. Couillet et al. in [7] formulated a competitive interaction between PHEVs in a Cournot power market. They modeled the competition market as a mean field game and introduced a set of differential equations to model the actions of the vehicles at the mean field equilibrium. Fan in [8] applied the congestion pricing concept used in communication networks and distributed systems to PHEVs charging and demand response problems in power networks. Nguyen et al. in [9] proposed centralized and decentralized optimization models for PHEVs charging. The centralized approach aims to minimize the Euclidean distance between the instantaneous load demand and the average demand. In the decentralized approach, a distributed algorithm is proposed in which users independently determine their charging schedules. The PHEVs charging problem has been studied in several works [4]–[9]. The key challenge in this paper is to propose a scheduling algorithm to tackle the overload associated with the charging demand of PHEVs. In our system model, each PHEV is equipped with a smart charger that can collect the market clearing price data. We take into account the interaction between the charging strategies of PHEVs. Hence, we use non- cooperative game theoretic approach to model the problem. The contributions of this paper are as follows: • We propose a novel charging approach for PHEVs. We show that the charging problem is an ordinal potential game with strictly concave potential function. We prove that the proposed potential game has a unique pure strategy Nash equilibrium, and we develop a distributed algorithm to determine that equilibrium. • We model the game as a dynamical system. We show that there exists a Lyapunov function for this system, and the Nash equilibrium is globally asymptotically stable. • The proposed scheduling approach is simulated on the system with 1000 households and 800 PHEVs. Simu- lations show that the proposed approach decreases the peak load, and increases the payoff of the customers. We compare the proposed algorithm with an algorithm based on the best response of the vehicle owners. We show that the running time of our charging algorithm increases linearly with the number of PHEVs. The rest of this paper is organized as follows. In Section II, the charging problem of PHEVs is formulated as a potential game. The existence, uniqueness and stability of the Nash equilibrium are proven, and a distributed algorithm is proposed to determine the equilibrium. In Section III, simulation results are presented, and the performance of the proposed algorithm 978-1-4673-7788-1/15/$31.00 c 2015 IEEE