Nan Cheng Smart Grid & VANETs Joint Group Meeting 2012.2.13 Economics of Electric Vehicle Charging -...
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- Slide 1
- Nan Cheng Smart Grid & VANETs Joint Group Meeting 2012.2.13
Economics of Electric Vehicle Charging - A Game Theoretic Approach
IEEE Trans. on Smart Grid, Vol. 3, No. 4, Dec. 2012 1
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- Roadmap Introduction System model Non-cooperative generalized
Stackelberg game Proposed solution and algorithm Adaption to
time-varying conditions Numerical analysis 2
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- 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
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- 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
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- 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
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- 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
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- System 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
7
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- Game Formulation (1) Grid (leader) and PVEGs (follower) make
the decision Stackelberg game is utilized for multi-level decision
making Game formulation: are players is demand set, with : Utility
function of PEVG n : Utility function for the SG 8
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- Game Formulation (2) Utility function of PEVG n We have the
following: It is considered: 9
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- 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
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- 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
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- 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
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- Existence 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. 13
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- 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 [1] F. Facchinei and C.
Kanzow, Generalized Nash equilibrium problems, 4OR, vol. 5, pp.
173210, Mar. 2007. 14
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- 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. 765776, 1999.
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- Algorithm (2) SG sets the price: Maximize price to maximize
revenue: 16
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- Time-Varying Conditions 17 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.
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- Simulation Parameters 18 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]
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- Numerical Results (1) 19 Demand v.s. number of iterations
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- Numerical Results (2) 20 Utility v.s. number of iterations
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- Numerical Results (3) 21 v.s. number of iterations
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- Numerical Results (4) 22 Price v.s. number of iterations
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- Numerical Results (5) 23 Price v.s. number of PEVGs
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- Numerical Results (6) 24 Average demand v.s. number of
vehicles
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- Numerical Results (7) 25 Average utility v.s. number of
PEVGs
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