Report

Economics of Electric Vehicle Charging - A Game Theoretic Approach Nan Cheng Smart Grid & VANETs Joint Group Meeting 2012.2.13 IEEE Trans. on Smart Grid, Vol. 3, No. 4, Dec. 2012 1 Roadmap Introduction System model Non-cooperative generalized Stackelberg game Proposed solution and algorithm Adaption to time-varying conditions Numerical analysis 2 Introduction Challenges of PEVs Optimal charging strategies Efficient V2G communications Managing energy exchange PEV charging may double the average load Simultaneous charging may lead to interruption Little has been done to capture the interactions between PEVs and the grid. 3 Contribution Framework to analyze the interactions between SG and PEV groups (PEVGs) Decision making process of both SG and PEVGs Leader-follower Stackelberg game. PEVGs choose the amount that they need to charge; Grid optimizes the price to maximize its revenue. Existence of generalized Stackelberg equilibrium (GSE) is proved A distributed algorithm to achieve the GSE Adapt to a time-varying environment 4 System Model (1) A power system Grid: Serves the primary customers Sets an appropriate price, and sells the surplus to the secondary customers Primary customers : Houses, industries, offices Secondary customers : PEVGs (PEVs in a parking lot) Smart energy manager (SEM) Charging period is divided into time slots (5 mins~0.5 h) 5 System Model (2) In each time slot, Totally N PEVGs Each PEVG requests energy Maximum energy that can sell: Demand constraint: The price of per unit energy: 6 System Model (3) Model the interaction based on the demand constraint PEVGs strategically choose demand to optimize their satisfaction level Grid sets up price 7 to maximize its revenue Game Formulation (1) Grid (leader) and PVEGs (follower) make the decision – Stackelberg game is utilized – for multi-level decision making Game formulation: 8 are players is demand set, with : Utility function of PEVG n : Utility function for the SG Game Formulation (2) Utility function of PEVG n We have the following: It is considered: 9 Game Formulation (3) Utility function for the SG The objective of PVEG n is It is a jointly convex generalized Nash equilibrium problem (GNEP) 10 Game Formulation (4) Among PEVGs – GNEP Between grid and PEVGs – Stackelberg game So, we have generalized Stackelberg game (GSG) with generalized Stackelberg equilibrium (GSE) Definition of GSE: and 11 Existence of GSE (1) Variational Equilibrium (VE) is a social optimal GNE, i.e., it is a GNE that maximizes Theorem: A social optimal VE exists in the GNEP. Proof (in brief): KKT conditions of PVEG n: 12 Existence of GSE (2) Reformulation of GNEP: variational inequality (VI) KKT conditions [1]: 13 [1] F. Facchinei and C. Kanzow, “Generalized Nash equilibrium problems,” 4OR, vol. 5, pp. 173–210, Mar. 2007. Existence of GSE (3) Jacobian of F is positive definite. Thus F is strictly monotone, and the GNEP has one unique global VE [1]. VE+SG optimally sets its price = GSE 14 [1] F. Facchinei and C. Kanzow, “Generalized Nash equilibrium problems,” 4OR, vol. 5, pp. 173–210, Mar. 2007. Algorithm (1) How to solve the VI? Solodov and Svaiter (S-S) hyperplane projection method [2]: Obtain GNE for fixed price: 15 [2] M. V. Solodov and B. F. Svaiter, “A new projection method for variational inequality problems,” SIAM J. Control Optim., vol. 37, pp. 765–776, 1999. Algorithm (2) SG sets the price: Maximize price to maximize revenue: 16 Time-Varying Conditions Number of vehicles in a PEVG Available energy In each t, the grid estimates the amount of energy to sell PEVGs constitute VE in t, and send the demands to the grid. Team optimal solution in the discrete time game. 17 Simulation Parameters PVEG->1000 PEVs 22 kWh->100 miles Battery capacity : 35 MWh~65 MWh Total available energy : 99 MWh Initial price : 17 USD per MWh Satisfaction parameter : range [1,2] 18 Numerical Results (1) Demand v.s. number of iterations 19 Numerical Results (2) Utility v.s. number of iterations 20 Numerical Results (3) 21 v.s. number of iterations Numerical Results (4) Price v.s. number of iterations 22 Numerical Results (5) Price v.s. number of PEVGs 23 Numerical Results (6) Average demand v.s. number of vehicles 24 Numerical Results (7) Average utility v.s. number of PEVGs 25