Skip to main content

Advertisement

SpringerLink
Log in

Particle Swarm Optimization for Computing Nash and Stackelberg Equilibria in Energy Markets

  • Original Research
  • Published:
SN Operations Research Forum Aims and scope Submit manuscript

Abstract

Interactions among stakeholders in deregulated markets lead to complex interdependent optimization problems. The present study is motivated by load control programs in energy markets and more precisely by using the power supply interruption as a tool for reducing consumers’ demand voluntarily, also known as voluntary load curtailment programs. The problem is formulated as a Stackelberg game, specifically, as a bilevel optimization problem that belongs to the mathematical programs with equilibrium constraints. In this game, a player that acts as leader determines the actions of the players that act as followers and play a Nash game among them through a subsidy program. The corresponding equilibria need to be found and the presence of nonconvex functions makes the use of metaheuristic algorithms attractive. An extension of particle swarm optimization is proposed for solving such problems based on the unified particle swarm optimization that is a variation of the plain particle swarm optimization algorithm. The proposed algorithm is tested by solving some examples of the formulated games in order to study its efficiency and the interactions between the stakeholders of the market.

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

Fig. 1
Fig. 2

Similar content being viewed by others

Abbreviations

ISO:

Independent system operator

KKT:

Karush-Kuhn-Tucker

LICQ:

Linear independence constraint qualification

MPEC:

Mathematical program with equilibrium constraints

PSO:

Particle swarm optimization

UPSO:

Unified particle swarm optimization

VLC:

Voluntary load curtailment

References

  1. Gabriel SA, Conejo AJ, Fuller JD, Hobbs BF, Ruiz C (2013) Complementarity modeling in energy markets. International Series in Operations Research and Management Science 180. Springer Science & Business Media, New York

  2. Marzband M, Azarinejadian F, Savaghebi M, Guerrero JM (2017) An optimal energy management system for islanded microgrids based on multi-period artificial bee colony combined with Markov chain. IEEE Syst J 11(3):1712–1722

    Article  Google Scholar 

  3. Marzband M, Parhizi N, Savaghebi M, Guerrero JM (2016) Distributed smart decision-making for a multimicrogrid system based on a hierarchical interactive architecture. IEEE T Energy Conver 31(2):637–648

    Article  Google Scholar 

  4. Marzband M, Yousefnejad E, Sumper A, Domínguez-García JL (2016) Real time experimental implementation of optimum energy management system in standalone microgrid by using multi-layer ant colony optimization. Int J Electr Power Energy Syst 75:265–274

    Article  Google Scholar 

  5. Parsopoulos KE, Vrahatis MN (2010) Particle swarm optimization and intelligence: advances and applications. Information Science Publishing (IGI Global), Hershey

  6. US Department of Energy (2006) Benefits of demand response in electricity markets and recommendations for achieving them: report to US Congress pursuant to section 1252 of the Energy Policy Act of 2005. US Department of Energy, Washington DC

  7. Federal Energy Regulatory Commission (2006) Assessment of demand response and advanced metering, staff report docket number AD-06-2-00. Federal Energy Regulatory Commission, Washington DC

  8. Walawalkar R, Fernands S, Thakur N, Chevva KR (2010) Evolution and current status of demand response (DR) in electricity markets: insights from PJM and NYISO. Energy 35(4):1553–1560

    Article  Google Scholar 

  9. Cappers P, Goldman C, Kathan D (2010) Demand response in US electricity markets: empirical evidence. Energy 35(4):1526–1535

    Article  Google Scholar 

  10. Marzband M, Ghadimi M, Sumper A, Domínguez-García JL (2014) Experimental validation of a real-time energy management system using multi-period gravitational search algorithm for microgrids in islanded mode. Appl Energy 128:164–174

    Article  Google Scholar 

  11. Marzband M, Parhizi N, Adabi J (2016) Optimal energy management for stand-alone microgrids based on multi-period imperialist competition algorithm considering uncertainties: experimental validation. Int T Electr Energy 26(6):1358–1372

    Google Scholar 

  12. Marzband M, Sumper A, Domínguez-García JL, Gumara-Ferret R (2013) Experimental validation of a real time energy management system for microgrids in islanded mode using a local day-ahead electricity market and MINLP. Energy Convers Manag 76:314–322

    Article  Google Scholar 

  13. Marzband M, Moghaddam MM, Akorede MF, Khomeyrani G (2016) Adaptive load shedding scheme for frequency stability enhancement in microgrids. Electr Power Syst Res 140:78–86

    Article  Google Scholar 

  14. Larsen ER, Osorio S, van Ackere A (2017) A framework to evaluate security of supply in the electricity sector. Renew Sust Energ Rev 79:646–655

    Article  Google Scholar 

  15. Soliman HM, Leon-Garcia A (2014) Game-theoretic demand-side management with storage devices for the future smart grid. IEEE T Smart Grid 5(3):1475–1485

    Article  Google Scholar 

  16. Su W-C, Huang AQ (2014) A game theoretic framework for a next-generation retail electricity market with high penetration of distributed residential electricity suppliers. Appl Energy 119:341–350

    Article  Google Scholar 

  17. Papavassilopoulos GP (1980) Algorithms for leader-follower games. In: Proceedings of the 18th Annual Allerton Conference on Communication Control and Computing, pp 851–859

  18. Papavassilopoulos GP (1982) Algorithms for static Stackelberg games with linear costs and polyhedra constraints. In: Proceedings of the 21st IEEE Conference on Decision and Control, vol 21, pp 647–652

  19. Bialas WF, Karwan MH (1980) Multilevel optimization: a mathematical programming perspective. In: Proceedings of the 19th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes, vol 19, pp 761–765

  20. Luo Z-Q, Pang J-S, Ralph D (1996) Mathematical programs with equilibrium constraints. Cambridge University Press

  21. Marzband M, Ardeshiri RR, Moafi M, Uppal H (2017) Distributed generation for economic benefit maximization through coalition formation–based game theory concept. Int T Electr Energy 27:e2313

    Google Scholar 

  22. Andreou GT, Bouhouras AS, Milioudis AN, Labridis DP (2017) Energy efficiency in urban electrical grids through consumer networking. In: Computers and Operations Research Series, vol 8, Network Design and Optimization for Smart Cities, Gakis K, Pardalos PM (eds), World Scientific, Chapter 2, pp. 32–52

  23. Marzband M, Javadi M, Domínguez-García JL, Moghaddam MM (2016) Non-cooperative game theory based energy management systems for energy district in the retail market considering DER uncertainties. IET Gener Transm Distrib 10(12):2999–3009

    Article  Google Scholar 

  24. Bard JF (2013) Practical bilevel optimization: algorithms and applications. In: Nonconvex optimization and its applications series, vol 30, Springer

  25. Dempe S (2010) Foundations of bilevel programming. In: Nonconvex optimization and its applications series, vol 61, Springer

  26. McCarl BA et al (2013) McCarl expanded GAMS user guide, GAMS release 24.2.1. GAMS development corporation, Washington, DC, USA

  27. Harker PT (1991) Generalized Nash games and quasi-variational inequalities. Eur J Oper Res 54(1):81–94

    Article  Google Scholar 

  28. Facchinei F, Kanzow C (2007) Generalized Nash equilibrium problems. 4OR 5(3):173–210

  29. Bertsekas DP (1999) Nonlinear programming. Athena scientific, Belmont

    Google Scholar 

  30. Luenberger DG, Ye Y-Y (2016) Linear and nonlinear programming. In: International series in operations research and management science series, vol 228, Springer

  31. Wachsmuth G (2013) On LICQ and the uniqueness of Lagrange multipliers. Oper Res Lett 41(1):78–80

    Article  Google Scholar 

  32. Fortuny-Amat J, McCarl B (1981) A representation and economic interpretation of a two-level programming problem. J Oper Res Soc 32(9):783–792

    Article  Google Scholar 

  33. Schaefer TJ (1978) The complexity of satisfiability problems. In: Proceedings of the tenth annual ACM Symposium on Theory of Computing (STOC), pp 216–226

  34. Kennedy J, Eberhart R (1995) Particle swarm optimization. In: Proceedings of IEEE International Conference on Neural Networks IV, pp1942–1948

  35. Clerc M, Kennedy J (2002) The particle swarm – explosion, stability, and convergence in a multidimensional complex space. IEEE T Evol Comput 6(1):58–73

    Article  Google Scholar 

  36. Parsopoulos KE, Vrahatis MN (2007) Parameter selection and adaptation in unified particle swarm optimization. Math Comput Model 46(1–2):198–213

    Article  Google Scholar 

  37. Selvakumar AI, Thanushkodi K (2007) A new particle swarm optimization solution to nonconvex economic dispatch problems. IEEE T Power Syst 22:42–51

    Article  Google Scholar 

  38. Yang JM, Chen YP, Horng JT, Kao CY (1997) Applying family competition to evolution strategies for constrained optimization. In: Angeline PJ, Reynolds RG. McDonnell JR, Eberhart R (eds), Evolutionary Programming VI. (EP 1997). Lecture notes in computer science 1213:201–211, Springer

  39. Parsopoulos KE, Vrahatis MN (2002) Recent approaches to global optimization problems through particle swarm optimization. Nat Comput 1(2–3):235–306

    Article  Google Scholar 

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

    Article  Google Scholar 

  41. GAMS–The Solver Manuals (2020) GAMS Release 30.1.10. GAMS Development Corporation, Washington, DC, USA

  42. Parsopoulos KE, Vrahatis MN (2004) On the computation of all global minimizers through particle swarm optimization. IEEE T Evol Comput 8(3):211–224

    Article  Google Scholar 

  43. Pavlidis NG, Parsopoulos KE, Vrahatis MN (2005) Computing Nash equilibria through computational intelligence methods. J Comput Appl Math 175(1):113–136

    Article  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the anonymous reviewers for their helpful comments.

Author information

Authors and Affiliations

  1. Computational Intelligence Laboratory (CILab), Department of Mathematics, University of Patras, GR-26110, Patras, Greece

    Michael N. Vrahatis

  2. School of Electrical and Computer Engineering, National Technical University of Athens, GR-15780, Athens, Greece

    Panagiotis Kontogiorgos & George P. Papavassilopoulos

Authors
  1. Michael N. Vrahatis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Panagiotis Kontogiorgos
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. George P. Papavassilopoulos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael N. Vrahatis.

Ethics declarations

Conflict of Interest

The authors declare that they have no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Vrahatis, M.N., Kontogiorgos, P. & Papavassilopoulos, G.P. Particle Swarm Optimization for Computing Nash and Stackelberg Equilibria in Energy Markets. SN Oper. Res. Forum 1, 20 (2020). https://doi.org/10.1007/s43069-020-00021-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s43069-020-00021-4

Keywords

Search

Navigation