### Economics of Electric Vehicle Charging - A Game

```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
 Introduction
 System model
 Non-cooperative generalized Stackelberg game
 Proposed solution and algorithm
 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
 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)

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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
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