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Economics of Electric Vehicle Charging - A Game Theoretic Approach

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. Roadmap. Introduction System model Non-cooperative generalized Stackelberg game

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Economics of Electric Vehicle Charging - A Game Theoretic Approach

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  1. 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

  2. Roadmap • Introduction • System model • Non-cooperative generalized Stackelberg game • Proposed solution and algorithm • Adaption to time-varying conditions • Numerical analysis

  3. 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.

  4. 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

  5. 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)

  6. 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:

  7. 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

  8. 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

  9. Game Formulation (2) • Utility function of PEVG n • We have the following: • It is considered:

  10. Game Formulation (3) • Utility function for the SG • The objective of PVEG n is • It is a jointly convexgeneralizedNash equilibrium problem (GNEP)

  11. Game Formulation (4) • Among PEVGs – GNEP • Between grid and PEVGs – Stackelberg game • So, we have generalized Stackelberggame (GSG) with generalized Stackelbergequilibrium (GSE) • Definition of GSE: and

  12. 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:

  13. 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. 173–210, Mar. 2007.

  14. 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. 173–210, Mar. 2007.

  15. Algorithm (1) • How to solve the VI? • Solodov and Svaiter (S-S) hyperplane projection method [2]: • Obtain • GNE for fixed price: [2] M. V. Solodov and B. F. Svaiter, “A new projection method for variationalinequality problems,” SIAM J. Control Optim., vol. 37, pp. 765–776, 1999.

  16. Algorithm (2) • SG sets the price: • Maximize price to maximize revenue:

  17. 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.

  18. 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]

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

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

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

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

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

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

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

  26. Thanks!

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