Skip to main content
SpringerLink
Log in

Distributed Nash Equilibrium Seeking for Aggregative Games via Derivative Feedback

  • Regular Papers
  • Control Theory and Applications
  • Published:
International Journal of Control, Automation and Systems Aims and scope Submit manuscript

Abstract

In this paper, we investigate a continuous-time distributed Nash equilibrium seeking algorithm for a class of aggregative games, with application to the real-time pricing demand response. To seek the Nash equilibrium via local communication among neighbors, by combining projected gradient dynamics and consensus tracking dynamics, we propose a novel distributed algorithm for the players. We prove the convergence of the distributed algorithm via a constructed Lyapunov function and the variational inequality technique, and show an illustrative simulation related to the energy consumption control in smart grids.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. K. Jensen, “Aggregative games and best-reply potentials,” Economic Theory, vol. 43, no. 1, pp. 45–66, 2010.

    Article  MathSciNet  Google Scholar 

  2. R. Cornes, “Aggregative environmental games,” Environmental & Resource Economics, vol. 63, no. 2, pp. 339–365, 2016.

    Article  Google Scholar 

  3. H. Yin, U. V. Shanbhag, and P. G. Mehta, “Nash equilibrium problems with scaled congestion costs and shared constraints,” IEEE Transactionson on Automatic Control, vol. 56, no. 7, pp. 1702–1708, 2011.

    Article  MathSciNet  Google Scholar 

  4. L. Chen, N. Li, S. H. Low, and J. C. Doyle, “Two market models for demand response in power networks,” Proceedings of the 1st IEEE conference on Smart Grid Communications, Gaithersburg, MD, USA, pp. 397–402, Oct. 2010.

  5. C. Wu, H. Mohsenian-Rad, and J. Huang, “Vehicle-to-aggregator interaction game,” IEEE Transactions on Smart Grid, vol. 3, no. 1, pp. 434–442, 2012.

    Article  Google Scholar 

  6. Z. Ma, D. S. Callaway, and I. A. Hiskens, “Decentralized charging control of large populations of plug-in electric vehicles,” IEEE Transactions on Control Systems Technology, vol. 21, no. 1, pp. 67–78, 2013.

    Article  Google Scholar 

  7. K. Ma, G. Hu, and C. J. Spanos, “Distributed energy consumption control via real-time pricing feedback in smart grid,” IEEE Transactions on Control Systems Technology, vol. 22, no. 5, pp. 1907–1914, 2014.

    Article  Google Scholar 

  8. D. Paccagnan, B. Gentile, F. Parise, M. Kamgarpour, and J. Lygeros, “Distributed computation of generalized Nash equilibria in quadratic aggregative games with affine coupling constraints,” Proceedings of 55th IEEE conference on decision and control, Las Vegas, USA, pp. 6123–6283, Dec. 2016.

  9. S. Grammatico, “Dynamic control of agents playing aggregative games with coupling constraints,” IEEE Transactionson on Automatic Control, vol. 62, no. 9, pp. 4537–4548, 2017.

    Article  MathSciNet  Google Scholar 

  10. S. S. Kia, J. Cortés, and S. Martínez, “Distributed convex optimization via continuous-time coordination algorithms with discrete-time communication,” Automatica, vol. 55, pp. 254–264, 2015.

    Article  MathSciNet  Google Scholar 

  11. J. Liu, W. Chen, and H. Dai, “Sampled-data based distributed convex optimization with event-triggered communication,” International Journal of Control, Automation and Systems, vol. 14, no. 6, pp. 1421–1429, 2016.

    Article  Google Scholar 

  12. X. Wang, Y. Hong, P. Yi, H. Ji, and Y. Kang, “Distributed optimization design of continuous-time multiagent systems with unknown-frequency disturbances,” IEEE Transactions on Cybernetics, vol. 47, no. 8, pp. 2058–2066, 2017.

    Article  Google Scholar 

  13. F. Parise, B. Gentile, S. Grammatico, and J. Lygeros, “Network aggregative games: Distributed convergence to Nash equilibria,” Proceedings of 54th IEEE conference on decision and control, Osaka, Japan, pp. 2295–2300, Dec. 2015.

  14. J. Koshal, A. Nedić, and U. V. Shanbhag, “Distributed algorithms for aggregative games on graphs,” Operations Research, vol. 63, no. 3, pp. 680–704, 2016.

    Article  MathSciNet  Google Scholar 

  15. S. Liang, P. Yi, and Y. Hong, “Distributed Nash equilibrium seeking for aggregative games with coupled constraints,” Automatica, vol. 85, pp. 179–185, 2017.

    Article  MathSciNet  Google Scholar 

  16. M. Ye and G. Hu, “Game design and analysis for price-based demand response: an aggregate game approach,” IEEE Transactions on Cybernetics, vol. 47, no. 3, pp. 720–730, 2017.

    Article  Google Scholar 

  17. B. Gharesifard, T. Basar, and A. D. Dominguez-Garcia, “Price-based coordinated aggregation of networked distributed energy resources,” IEEE Transactions on Automatic Control, vol. 61, no. 10, pp. 2936–2946, 2016.

    Article  MathSciNet  Google Scholar 

  18. C. Godsil and G. Royle, Algebraic Graph Theory, Springer-Verlag, New York, 2001.

    Book  Google Scholar 

  19. B. Li, H.-Y. Yang, Z.-Q. Chen, and Z.-X. Liu, “Distributed containment control of multi-agent systems with general linear dynamics and time-delays,” International Journal of Control, Automation and Systems, vol. 16, no. 6, pp. 2718–2726, 2018.

    Article  Google Scholar 

  20. J.-B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms I: Fundamentals, Springer-Verlag, Berlin Heidelberg, 1993.

    Book  Google Scholar 

  21. F. Facchinei and J. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer-Verlag, New York, 2003.

    MATH  Google Scholar 

  22. M. Fukushima, “Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems,” Mathematical Programming, vol. 53, pp. 99–110, 1992.

    Article  MathSciNet  Google Scholar 

  23. J.-l. Zhang, D.-l. Qi, and M. Yu, “A game theoretic approach for the distributed control of multi-agent systems under directed and time-varying topology,” International Journal of Control, Automation and Systems, vol. 12, no. 4, pp. 749–758, 2014.

    Article  Google Scholar 

  24. Y. Yuan and F. Sun, “Data fusion-based resilient control system under dos attacks: A game theoretic approach,” International Journal of Control, Automation and Systems, vol. 13, no. 3, pp. 513–520, 2015.

    Article  Google Scholar 

  25. J. S. Vardakas, N. Zorba, and C. V. Verikoukis, “A survey on demand response programs in smart grids: pricing methods and optimization algorithms,” IEEE Communications Surveys & Tutorials, vol. 17, no. 1, pp. 152–178, 2015.

    Article  Google Scholar 

  26. M. H. Albadi and E. El-Saadany, “A summary of demand response in electricity markets,” Electric Power Systems Research, vol. 78, no. 11, pp. 1989–1996, 2008.

    Article  Google Scholar 

  27. L. Chen, N. Li, L. Jiang, and S. H. Low, “Optimal demand response: Problem formulation and deterministic case,” in Control and Optimization Methods for Electric Smart Grids, pp. 63–85, Springer, New York, 2012.

    Chapter  Google Scholar 

  28. L. Gelazanskas and K. A. Gamage, “Demand side management in smart grid: a review and proposals for future direction,” Sustainable Cities and Society, vol. 11, pp. 22–30, 2014.

    Article  Google Scholar 

  29. J. S. Shamma and G. Arslan, “Dynamic fictitious play, dynamic gradient play, and distributed convergence to Nash equilibria,” IEEE Transactions on Automatic Control, vol. 50, no. 3, pp. 312–327, 2005.

    Article  MathSciNet  Google Scholar 

  30. X. Zeng, P. Yi, Y. Hong, and L. Xie, “Continuous-time distributed algorithms for extended monotropic optimization problems,” 2016, arXiv:1608.01167.

  31. H. K. Khalil, Nonlinear Systems, 3rd ed, Prentice Hall, Upper Saddle River, 2002.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departement of Automation, University of Science and Technology of China, Hefei, 230027, China

    Yawei Zhang & Haibo Ji

  2. Key Laboratory of Knowledge Automation for Industrial Processes of Ministry of Education, School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing, 100083, China

    Shu Liang

Authors
  1. Yawei Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shu Liang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Haibo Ji
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yawei Zhang.

Additional information

Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Recommended by Associate Editor Shun-ichi Azuma under the direction of Editor Yoshito Ohta. This work is supported by National Natural Science Foundation of China (61873029, 61273090) and Beijing Natural Science Foundation (4192068).

Yawei Zhang received his B.E. degree in automatic control from the Nanjing Normal University in 2014. He is currently a Ph.D student in the Department of Automation, University of Science and Technology of China. His research interests include distributed Nash game and distributed optimization.

Shu Liang received his B.E. degree in automatic control and a Ph.D. degree in engineering from the University of Science and Technology of China, Hefei, China, in 2010 and 2015, respectively. He was a postdoctoral fellow at Academy of Mathematics and Systems Science, Chinese Academy of Sciences from 2015 to 2017, and was a visiting scholar at Wayne State University from 2017 to 2018. He is currently a lecturer in the School of Automation and Electrical Engineering, University of Science and Technology Beijing, China. His research interests include distributed optimizations and fractional order systems.

Haibo Ji was born in Anhui, China, in 1964. He received his B.Eng. degree and Ph.D. degree in Mechanical Engineering from Zhejiang University and Peking University, in 1984 and 1990, respectively. He is currently a Professor in Department of Automation, University of Science and Technology of China. His research interests include nonlinear control and its applications.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, Y., Liang, S. & Ji, H. Distributed Nash Equilibrium Seeking for Aggregative Games via Derivative Feedback. Int. J. Control Autom. Syst. 18, 1075–1082 (2020). https://doi.org/10.1007/s12555-019-0011-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12555-019-0011-y

Keywords

Search

Navigation