Skip to main content

Advertisement

Springer Nature Link
Log in

Multi-Resolution Large Population Stochastic Differential Games and Their Application to Demand Response Management in the Smart Grid

  • Published:
Dynamic Games and Applications Aims and scope Submit manuscript

Abstract

Dynamic demand response (DR) management is becoming an integral part of power system and market operational practice. Motivated by the smart grid DR management problem, we propose a multi-resolution stochastic differential game-theoretic framework to model the players’ intra-group and inter-group interactions in a large population regime. We study the game in both risk-neutral and risk-sensitive settings, and provide closed-form solutions for symmetric mean-field responses in the case of homogeneous group populations, and characterize the symmetric mean-field Nash equilibrium using the Hamilton–Jacobi–Bellman (HJB) equation together with the Fokker–Planck–Kolmogorov (FPK) equation. Finally, we apply the framework to the smart grid DR management problem and illustrate with a numerical example.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

Notes

  1. Allowing the demand to take negative values captures the possibility of the scenario that users can act as a power supplier in the smart grid, that is, they can generate their own energy from renewable sources. Users can sell their excess energy when energy generated exceeds the load of the users while they can also buy energy from the market if their load exceeds self-generated energy. Hence, \(d_{i}^{k}\) here captures this effect and users would want to sell excess energy when \(d_{i}^{k}\) becomes negative. A detailed discussion of this can be found in [27].

References

  1. Ipakchi A, Albuyeh F (2009) Grid of the future. IEEE Power Energy Mag 7(2):52–62

    Article  Google Scholar 

  2. Garrity TF (2009) Getting smart. IEEE Power Energy Mag 6(2):38–45

    Article  MathSciNet  Google Scholar 

  3. Gellings CW (2009) The smart grid: enabling energy efficiency and demand response, vol 21. CRC Press, Boca Raton

    Google Scholar 

  4. Han Z, Hossain E, Poor V (eds) (2012) Smart grid communications and networking. Cambridge University Press, Cambridge

    Google Scholar 

  5. The smart grid: an introduction. Department of Energy, Office of Electricity Delivery and Energy Reliability. http://energy.gov/node/235057

  6. Cruz JB Jr., Tan X (2009) Dynamic noncooperative game models for deregulated electricity markets. Nova Science, New York

    Google Scholar 

  7. Medina J, Muller N, Roytelman I (2010) Demand response and distribution grid operations: opportunities and challenges. IEEE Trans Smart Grid 1(2):193–198

    Article  Google Scholar 

  8. Strbac G (2008) Demand side management: benefit and challenges. Energy Policy 36:4419–4426

    Article  Google Scholar 

  9. Kirschen DS (2003) Demand-side view of electricity markets. IEEE Trans Power Syst 18(2):520–527

    Article  Google Scholar 

  10. Albadi MH, El-Saadany EF (2007) Demand response in electricity markets: an overview. In: IEEE power engineering society general meeting, pp 1–5

    Google Scholar 

  11. Tembine H, Zhu Q, Başar T (2011) Risk-sensitive mean-field stochastic differential games. In: Proceedings of IFAC world congress, Milan, Italy, Aug 28–Sept 2, 2011

    Google Scholar 

  12. Zhu Q, Hamidou T, Başar T (2011) Hybrid risk-sensitive mean-field stochastic differential games with application to molecular biology. In: Proceedings of 50th IEEE conference on decision and control and European control conference, Orlando, FL, Dec 12–15, 2011

    Google Scholar 

  13. Başar T, Olsder GJ (1998) In: Dynamic noncooperative game theory. SIAM classics in applied mathematics, 2nd ed

    Google Scholar 

  14. Lasry J-M, Lions P-L (2007) Mean field games. Jpn J Math 2:229–260

    Article  MathSciNet  MATH  Google Scholar 

  15. Zhu Q, Başar T (2010) Price of anarchy and price of information in linear quadratic differential games. In: Proceedings of American control conference (ACC), Baltimore, MD, June 30–July 2, 2010, pp 782–787

    Google Scholar 

  16. Başar T, Zhu Q (2011) Prices of anarchy, information, and cooperation in differential games. Dyn Games Appl 1(1):50–73

    Article  MathSciNet  MATH  Google Scholar 

  17. Zhu Q, Tembine H, Başar T (2009) Evolutionary game for hybrid additive white Gaussian noise multiple access control. In: Proceedings of IEEE Globecom, Honolulu, HI, Dec 2009, 2009, pp 1–6

    Google Scholar 

  18. Zhu Q, Tembine H, Başar T (2009) A constrained evolutionary Gaussian multiple access channel game. In: Proceedings of international conference on game theory for networks (GameNets), Istanbul, Turkey, May 13–15, 2009, pp 403–410

    Google Scholar 

  19. Başar T, Srikant R (2002) A Stackelberg network game with a large number of followers. J Optim Theory Appl 115(3):479–490

    Article  MathSciNet  MATH  Google Scholar 

  20. Coolen ACC (2005) The mathematical theory of minority games: statistical mechanics of interacting agents. Oxford University Press, Oxford

    MATH  Google Scholar 

  21. Kasbekar GS, Altman E, Sarkar S (2009) A hierarchical spatial game over licensed resources. In: Proceedings of international conference on game theory for networks (GameNets), Istanbul, Turkey, May 13–15, 2009, pp 70–79

    Google Scholar 

  22. Parpas P, Webster M (2012) A stochastic multiscale model for electricity generation capacity expansion. Preprint, available on http://www.optimization-online.org, accessed on Feb 17

  23. Altman E, Boulogne T, El Azouzi R, Jiménez T, Wynter L (2006) A survey on networking games in telecommunications. Comput Oper Res 33:286–311

    Article  MathSciNet  MATH  Google Scholar 

  24. Başar T, Bernhard P (1991) H-infinity optimal control and related minimax design problems: a dynamic game approach. Birkhäuser, Boston

    Google Scholar 

  25. Dolbow J, Khaleel MA, Mitchell J (2004) Multiscale mathematics initiative: a roadmap. Technical report PNNL-14966, Pacific Northwestern National Laboratory, Dec 2004

  26. Manshaei MH, Zhu Q, Alpcan T, Başar T, Hubaux J-P (2013) Game theory meets network security and privacy. ACM Comput Surv 45(3), to appear

  27. Zhu Q, Zhang J, Sauer P, Dominguez-Garcia A, Başar T (2012) A game-theoretic framework for control of distributed renewable-based energy resources in smart grids. In: Proceedings of 2012 American control conference (ACC), Montreal, Canada, June 27–29, 2012

    Google Scholar 

  28. Couillet R, Perlaza S, Tembine H, Debbah M (2012) A mean field game analysis of electric vehicles in the smart grid. In: Proceedings of 1st INFOCOM workshop on communications and control for sustainable energy systems: green networking and smart grids, Orlando, FL, March 30, 2012

    Google Scholar 

  29. Zhu Q, Han Z, Başar T (2012) A differential game approach to distributed demand side management in smart grid. In: Proceedings IEEE international conference on communications (ICC)—selected areas in communications symposium (ICC’12 SAC), Ottawa, Canada, June 10–15, 2012

    Google Scholar 

  30. Zhu Q, Yuan Z, Song JB, Han Z, Başar T (2010) Dynamic interference minimization routing game for on-demand cognitive pilot channel. In: Proceedings of IEEE global communication conference (GLOBECOM), Miami, FL

    Google Scholar 

  31. Zhu Q, Song JB, Başar T (2011) Dynamic secure routing game in distributed cognitive radio networks. In: Proceedings of IEEE global communications conference (GLOBECOM), Houston, TX, Dec 5–9, 2011

    Google Scholar 

  32. Tanabe Y (2006) The propagation of chaos for interacting individuals in a large population. Math Soc Sci 51:125–152

    Article  MathSciNet  MATH  Google Scholar 

  33. Tembine H, Zhu Q, Başar T Risk-sensitive mean field games. arXiv:1210.2806, IEEE Trans Autom Control, under review

  34. Tembine H (2011) Mean field stochastic games: convergence, Q/H-learning and optimality. In: Proceedings of American control conference (ACC), San Francisco, CA, pp 2413–2428

    Google Scholar 

  35. Yaïche H, Mazumdar RR, Rosenberg C (2000) A game theoretic framework for bandwidth allocation and pricing in broadband networks. IEEE/ACM Trans Netw 8(5):667–678

    Article  Google Scholar 

  36. Kelly FP, Maulloo AK, Tan DKH (1998) Rate control in communication networks: shadow prices, proportional fairness and stability. J Oper Res Soc 49:237–252

    MATH  Google Scholar 

  37. von Meier A (2006) Electric power systems: a conceptual introduction. Wiley/IEEE Press, New York

    Book  Google Scholar 

Download references

Acknowledgements

This is an expanded revised version of a paper that was presented at the International Conference on Network Games, Control, and Optimization (NetGCOOP 2011), Oct. 12–14, 2011; Paris, France.

The research was partially supported by an AFSOR MURI Grant (FA9550-10-1-0573).

Author information

Authors and Affiliations

  1. Coordinated Science Laboratory and the Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, 1308 West Main Street, Urbana, IL, 61801, USA

    Quanyan Zhu & Tamer Başar

Authors
  1. Quanyan Zhu
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Tamer Başar
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Quanyan Zhu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhu, Q., Başar, T. Multi-Resolution Large Population Stochastic Differential Games and Their Application to Demand Response Management in the Smart Grid. Dyn Games Appl 3, 68–88 (2013). https://doi.org/10.1007/s13235-013-0072-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13235-013-0072-0

Keywords