Economics of Electric Vehicle Charging - A Game Theoretic Approach

Nan ChengSmart Grid & VANETs Joint Group Meeting2012.2.13Economics of Electric Vehicle Charging - A Game Theoretic ApproachIEEE Trans. on Smart Grid, Vol. 3, No. 4, Dec. 20121RoadmapIntroductionSystem modelNon-cooperative generalized Stackelberg gameProposed solution and algorithmAdaption to time-varying conditionsNumerical analysis2IntroductionChallenges of PEVsOptimal charging strategiesEfficient V2G communicationsManaging energy exchangePEV charging may double the average loadSimultaneous charging may lead to interruptionLittle has been done to capture the interactions between PEVs and the grid.

3ContributionFramework to analyze the interactions between SG and PEV groups (PEVGs)Decision making process of both SG and PEVGsLeader-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 provedA distributed algorithm to achieve the GSEAdapt to a time-varying environment4System Model (1)A power systemGrid: Serves the primary customersSets an appropriate price, and sells the surplus to the secondary customersPrimary customers :Houses, industries, officesSecondary customers :PEVGs (PEVs in a parking lot)Smart energy manager (SEM)Charging period is divided into time slots (5 mins~0.5 h)5System Model (2)In each time slot,Totally N PEVGsEach PEVG requests energy Maximum energy that can sell:Demand constraint:

The price of per unit energy:

6System Model (3)Model the interaction based on the demand constraint

PEVGs strategically choose demand to optimize their satisfaction level

Grid sets up price to maximize its revenue

7Game Formulation (1)Grid (leader) and PVEGs (follower) make the decision Stackelberg game is utilized for multi-level decision makingGame formulation:

are players is demand set, with : Utility function of PEVG n : Utility function for the SG

8Game Formulation (2)Utility function of PEVG n

We have the following:

It is considered:

9Game Formulation (3)Utility function for the SG

The objective of PVEG n is

It is a jointly convex generalized Nash equilibrium problem (GNEP)

10Game Formulation (4)Among PEVGs GNEPBetween grid and PEVGs Stackelberg gameSo, we have generalized Stackelberg game (GSG) with generalized Stackelberg equilibrium (GSE)Definition of GSE:

and

11Existence 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:

12Existence of GSE (2)Reformulation of GNEP: variational inequality (VI)

KKT conditions [1]:

[1] F. Facchinei and C. Kanzow, Generalized Nash equilibrium problems, 4OR, vol. 5, pp. 173210, Mar. 2007.13Existence 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

[1] F. Facchinei and C. Kanzow, Generalized Nash equilibrium problems, 4OR, vol. 5, pp. 173210, Mar. 2007.14Algorithm (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. 765776, 1999.Algorithm (2)SG sets the price:

Maximize price to maximize revenue:

16Time-Varying Conditions17Number of vehicles in a PEVGAvailable energy

In each t, the grid estimates the amount of energy to sellPEVGs constitute VE in t, and send the demands to the grid.Team optimal solution in the discrete time game.

Simulation Parameters18PVEG->1000 PEVs22 kWh->100 milesBattery capacity : 35 MWh~65 MWhTotal available energy : 99 MWhInitial price : 17 USD per MWhSatisfaction parameter : range [1,2]

Numerical Results (1)19Demand v.s. number of iterations

Numerical Results (2)20Utility v.s. number of iterations

Numerical Results (3)21 v.s. number of iterations

Numerical Results (4)22Price v.s. number of iterations

Numerical Results (5)23Price v.s. number of PEVGs

Numerical Results (6)24Average demand v.s. number of vehicles

Numerical Results (7)25Average utility v.s. number of PEVGs

Thanks!