Fuzzy Optimization and Decision Making

, Volume 15, Issue 1, pp 103–128 | Cite as

A unit commitment-based fuzzy bilevel electricity trading model under load uncertainty



In this study, we establish a bilevel electricity trading model where fuzzy set theory is applied to address future load uncertainty, system reliability as well as human imprecise knowledge. From the literature, there have been some studies focused on this bilevel problem while few of them consider future load uncertainty and unit commitment optimization which handles the collaboration of generation units. Then, our study makes the following contributions: First, the future load uncertainty is characterized by fuzzy set theory, as the various factors that affect the load forecasting are often assessed with some non-statistical uncertainties. Second, the generation costs are obtained by solving complicated unit commitment problems, rather than approximate calculations used in existing studies. Third, this model copes with the optimizations of both the generation companies and the market operator, where the unexpected load risk is particularly analyzed by using fuzzy value-at-risk as a quantitative risk measurement. Forth, a mechanism to encourage the convergence of the bilevel model is proposed based on fuzzy maxmin approach, and a bilevel particle swarm optimization algorithm is developed to solve the problem in a proper runtime. To illustrate the effectiveness of this research, we provide a test system-based numerical example and discuss about the experimental results according to the principle of social welfare maximization. Finally, we also compare the model and algorithm with conventional methods.


Bilevel programming Fuzzy set theory Electricity trading  Unit commitment Fuzzy value-at-risk Particle swarm optimization 

List of symbols



Index of generation company


Number of total generation companies


Index of each generation unit


Index of each scheduling period


Number of total scheduling periods


Number of units in company \(m\)


Empirical lower bound of \(\textit{AP}_{t}^{m}\)


Cold/hot star-up cost of unit \(j_{m}\)


Cost function coefficients of unit \(j_{m}\)


Maximal generation capability of \(j_{m}\)


Minimal generation constraint of \(j_{m}\)


Maximal capability of company \(m\)


Minimal ‘on’ hours of unit \(j_{m}\)


Minimal ‘off’ hours of unit \(j_{m}\)


Number of hours unit \(j_{m}\) is required to be on at the start of the planning period


Number of hours unit \(j_{m}\) is required to be off at the start of the planning period


Minimal value of \(U_{j_{m}}\) and T


Minimal value of \(D_{j_{m}}\) and T


Maximal downward ramp rates of \(j_{m}\)


Maximal upward ramp rates of \(j_{m}\)


Forecasted fuzzy load of period \(t\)


Lower bound of \(\widetilde{L}_{t}\)


Upper bound of \(\widetilde{L}_{t}\)


Estimated fuzzy target profit of GC \(m\)


Estimated fuzzy target cost of MO


Reservation budget of a MO



Cost function of unit \(j_{m}\) with output \(G_{t}^{j_{m}}\)

\(F^{\prime }_{m}\)

Cost function of company \(m\)


Upper level objective function


Lower level objective function



Average bidding of company \(m\) in \(t\)


Higher payment for unexpected load


Real generation of unit \(j_{m}\) in period \(t\)


On/off (1/0) state of unit \(j_{m}\) in period \(t\)


Startup action at time \(t\) of generator \(j_{m}\)


Shutdown action at time \(t\) of generator \(j_{m}\)


Generation of company \(m\) in period \(t\)


Total generation of all companies in \(t\)


Unexpected load of period \(t\)


Spinning reserve of \(m\) in period \(t\)


Unexpected load cost of period \(t\)


Unified market clearing price of period \(t\)



Supported by the Fundamental Research Funds for the Central Universities (No. 2062014286).


  1. 1.
    Bianco, L., Caramia, M., & Giordani, S. (2009). A bilevel flow model for hazmat transportation network design. Transportation Research Part C, 17(2), 175–196.CrossRefGoogle Scholar
  2. 2.
    Blanco, R. F., Arroyo, J. M., & Alguacil, N. (2012). A unified bilevel programming framework for price-based market clearing under marginal pricing. IEEE Transactions on Power Systems, 27(1), 1446–1456.Google Scholar
  3. 3.
    Bracken, J., & McGill, J. (1973). Mathematical programs with optimization problems in the constraints. Operations Research, 21(1), 37–44.CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Candler, W., Fortuny-Amat, J., & McCafl, B. (1981). The potential role of multilevel programming in agricultural economics. American Journal of Agricultural Economics, 63(6), 521–531.CrossRefGoogle Scholar
  5. 5.
    Carrion, M., Arroyo, J. M., & Conejo, A. J. (2009). A bilevel stochastic programming approach for retailer futures market trading. IEEE Transactions on Power Systems, 24(3), 1446–1456.CrossRefGoogle Scholar
  6. 6.
    Christie, R. D., Wollenberg, B. F., & Wangensteen, I. (2000). Transmission management in the deregulated environment. Proceedings of the IEEE, 88(2), 170–195.CrossRefGoogle Scholar
  7. 7.
    Colson, B., Marcotte, P., & Savard, G. (2007). An overview of bilevel optimization. Annals of Operations Research, 153(1), 235–256.CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Duffie, D., & Pan, J. (1997). An overview of value-at-risk. Journal of Derivatives, 4(3), 7–49.CrossRefGoogle Scholar
  9. 9.
    Hong, T., & Wang, P. (2014). Fuzzy interaction regression for short term load forecasting. Fuzzy Optimization and Decision Making, 13(1), 91–103.CrossRefMathSciNetGoogle Scholar
  10. 10.
    Juste, K., Kita, H., & Tanaka, E. (1999). An evolutionary programming solution to the unit commitment problem. IEEE Transactions on Power Systems, 14(4), 1452–1459.CrossRefGoogle Scholar
  11. 11.
    Kennedy, J., & Eberhart, R. (1995). Particle swarm optimization, In: Proceedings of the 1995 IEEE international conference on neual networks, IV, pp. 1942–1948.Google Scholar
  12. 12.
    Lan, Y. F., Zhao, R. Q., & Tang, W. S. (2011). A bilevel fuzzy principal-agent model for optimal nonlinear taxation problems. Fuzzy Optimization and Decision Making, 10(3), 211–232.CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    Liu, B., & Liu, Y. K. (2002). Expected value of fuzzy variable and fuzzy expected value models. IEEE Transactions on Fuzzy Systems, 10(4), 445–450.CrossRefGoogle Scholar
  14. 14.
    Ostrowski, J., Anjos, M. F., & Vannelli, A. (2012). Tight mixed integer linear programming formulations for the unit commitment problem. IEEE Transactions on Power Systems, 27(1), 39–46.CrossRefGoogle Scholar
  15. 15.
    Peng, J. (2013). Risk metrics of loss function for uncertain system. Fuzzy Optimization and Decision Making, 12(1), 53–64.CrossRefGoogle Scholar
  16. 16.
    Saber, A. Y., Senjyu, T., Miyagi, T., Urasaki, N., & Funabashi, T. (2006). Fuzzy unit commitment scheduling using absolutely stochastic simulated annealing. IEEE Transactions on Power Systems, 21(2), 955–964.CrossRefGoogle Scholar
  17. 17.
    Takriti, S., Birge, J. R., & Long, E. (1996). A stochastic model for the unit commitment problem. IEEE Transactions on Power Systems, 11(2), 1497–1508.CrossRefGoogle Scholar
  18. 18.
    Wang, B., Li, Y., & Watada, J. (2011). Re-scheduling of unit commitment based on customers’ fuzzy requirements for power reliability. IEICE Transactions on Information and Systems, E94–D(7), 1378–1385.Google Scholar
  19. 19.
    Wang, B., Li, Y., & Watada, J. (2013). Supply reliability and generation cost analysis due to load forecast uncertainty in unit commitment problems. IEEE Transactions on Power Systems, 28(3), 2242–2252.CrossRefGoogle Scholar
  20. 20.
    Wang, B., Wang, S., & Watada, J. (2011). Fuzzy portfolio selection models with value-at-risk. IEEE Transactions on Fuzzy Systems, 19(4), 758–769.CrossRefGoogle Scholar
  21. 21.
    Wang, S., Watada, J., & Pedrycz, W. (2009). Value-at-risk-based two-satge fuzzy facility location problems. IEEE Transactions on Industrial Informatics, 5(4), 465–482.CrossRefGoogle Scholar
  22. 22.
    Wu, X. L., & Liu, Y. K. (2012). Optimizing fuzzy portfolio selection problems by parametric quadratic programming. Fuzzy Optimization and Decision Making, 11(4), 411–449.CrossRefMathSciNetMATHGoogle Scholar
  23. 23.
    Yuan, X., Nie, H., Su, A., Wang, L., & Yuan, Y. (2009). An improved binary particle swarm optimization for unit commitment problem. Expert Systems with Applications, 36(4), 8049–8055.CrossRefGoogle Scholar
  24. 24.
    Zhang, G., Zhang, G., Gao, Y., & Lu, Jie. (2011). Competitive strategic bidding optimization in electricity markets using bilevel programming and swarm technique. IEEE Transactions on Industrial Electronics, 58(6), 2138–2146.CrossRefGoogle Scholar
  25. 25.
    Zimmermann, H. J. (1978). Fuzzy programming and LP with several objective functions. Fuzzy Sets and Systems, 1(1), 45–55.CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.School of Management and EngineeringNanjing UniversityNanjingChina
  2. 2.Graduate School of Information, Production and SystemsWaseda UniversityKitakyushuJapan

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