Recent Progress in Modeling Unit Commitment Problems

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 62)

Abstract

The unit commitment problem is a fundamental problem in the operation of power systems. The purpose of unit commitment is to minimize the system-wide cost of power generation by finding an optimal power production schedule for each generator while ensuring that demand is met and that the system operates safely and reliably. This problem can be formulated as a mixed-integer nonlinear optimization problem, and finding global optimal solutions is important not only because of the significant operational costs but also because in competitive market environments, different near-optimal solutions can produce considerably different financial settlements. At the same time, the time available to solve the problem is a hard constraint in practice. Hence unit commitment is an important and challenging optimization problem. This article provides an introduction to the basic problem from the point of view of optimization, summarizes several related modeling developments in the recent literature while providing some possible directions for future research, and concludes with a brief mention of extensions to the basic problem that have great practical importance and are driving much current research.

Keywords

Optimization Unit commitment Mixed-integer linear programming 

References

  1. 1.
    Anjos, M.F., Lasserre, J.B. (eds.): Handbook on semidefinite, conic and polynomial optimization. In: International Series in Operations Research and Management Science. Springer, New York (2012)CrossRefGoogle Scholar
  2. 2.
    Arroyo, J.M., Conejo, A.J.: Optimal response of a thermal unit to an electricity spot market. IEEE Trans. Power Syst. 15(3), 1098–1104 (2000)CrossRefGoogle Scholar
  3. 3.
    Arroyo, J.M., Conejo, A.J.: Modeling of start-up and shut-down power trajectories of thermal units. IEEE Trans. Power Syst. 19(3), 1562–1568 (2004)CrossRefGoogle Scholar
  4. 4.
    Baldwin, C.J., Dale, K.M., Dittrich, R.F.: A study of the economic shutdown of generating units in daily dispatch. Power apparatus and systems, part III. Trans. Am. Inst. Electrical Engineers 78(4), 1272–1282 (1959)Google Scholar
  5. 5.
    Bautista, G., Anjos, M.F., Vannelli, A.: Formulation of oligopolistic competition in AC power networks: an NLP approach. IEEE Trans. Power Syst. 22(1), 105–115 (2007)CrossRefGoogle Scholar
  6. 6.
    Bautista, G., Anjos, M.F., Vannelli, A.: Modeling market power in electricity markets: is the devil only in the details? Electricity J 20(1), 82–92 (2007)CrossRefGoogle Scholar
  7. 7.
    Bertsimas, D., Litvinov, E., Sun, X.A., Zhao, J., Zheng, T.: Adaptive robust optimization for the security constrained unit commitment problem. IEEE Trans. Power Syst. 28(1), 52–63 (2013)CrossRefGoogle Scholar
  8. 8.
    Bhattacharya, K., Bollen, M.H.J., Daalder, J.E.: Operation of Restructured Power Systems. Springer, New York (2001)CrossRefGoogle Scholar
  9. 9.
    Billinton, R., Allan, R.N.: Reliability Evaluation of Power Systems, 2nd edn. Plenum Press, New York (1984)CrossRefGoogle Scholar
  10. 10.
    Borghetti, A., Frangioni, A., Lacalandra, F., Nucci, C.A., Pelacchi, P.: Using of a cost-based unit commitment algorithm to assist bidding strategy decisions. In: Power Tech Conference Proceedings, vol. 2 (2003)Google Scholar
  11. 11.
    Braun, A.: Anlagen- und Strukturoptimierung von 110-kV-Netzen. PhD thesis, RWTH Aachen University (2001)Google Scholar
  12. 12.
    Carrion, M., Arroyo, J.M.: A computationally efficient mixed-integer linear formulation for the thermal unit commitment problem. IEEE Trans. Power Syst. 21(3), 1371–1378 (2006)CrossRefGoogle Scholar
  13. 13.
    Coffrin, C., Van Hentenryck, P.: A linear-programming approximation of ac power flows. Technical report, CoRR, abs/1206.3614 (2012)Google Scholar
  14. 14.
    Costanzo, G.T., Zhu, G., Anjos, M.F., Savard, G.: A system architecture for autonomous demand side load management in smart buildings. IEEE Trans. Smart Grid 3(4), 2157–2165 (2012)CrossRefGoogle Scholar
  15. 15.
    Dennis, Jr, J.E., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Number 16 in Classics in Applied Mathematics. SIAM (1996)Google Scholar
  16. 16.
    Frangioni, A., Gentile, C.: Perspective cuts for a class of convex 0–1 mixed integer programs. Math. Program. 106(2), 225–236 (2006)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Frangioni, A., Gentile, C., Lacalandra, F.: Tighter approximated MILP formulations for unit commitment problems. IEEE Trans. Power Syst. 24(1), 105–113 (2009)CrossRefGoogle Scholar
  18. 18.
    Fu, Y., Shahidehpour, M., Li, Z.: Security-constrained unit commitment with AC constraints*. IEEE Trans. Power Syst. 20(3), 1538–1550 (2005) [*Corrected version of 20(2), 1001–1013]Google Scholar
  19. 19.
    Fu, Y., Shahidehpour, M., Li, Z.: AC contingency dispatch based on security-constrained unit commitment. IEEE Trans. Power Syst. 21(2), 897–908 (2006)CrossRefGoogle Scholar
  20. 20.
    Hedman, K.W., Ferris, M.C., O’Neill, R.P., Fisher, E.B., Oren, S.S.: Co-optimization of generation unit commitment and transmission switching with N-1 reliability. IEEE Trans. Power Syst. 25(2), 1052–1063 (2010)CrossRefGoogle Scholar
  21. 21.
    Hobbs, B.F., Rothkopf, M.H., O’Neill, R.P., Chao, H.-P. (eds.): The next generation of electric power unit commitment models. In: International Series in Operations Research & Management Science. Kluwer Academic Publishers, Norwell (2001)Google Scholar
  22. 22.
    Jeroslow, R.: Trivial integer programs unsolvable by branch-and-bound. Math. Program. 6, 105–109 (1974)CrossRefMATHGoogle Scholar
  23. 23.
    Jiang, R., Wang, J., Guan, Y.: Robust unit commitment with wind power and pumped storage hydro. IEEE Trans. Power Syst. 27(2), 800–810 (2012)CrossRefGoogle Scholar
  24. 24.
    Kaibel, V., Loos, A.: Branched polyhedral systems. In: IPCO 2010: The Fourteenth Conference on Integer Programming and Combinatorial Optimization, vol. 6080 of Lecture Notes in Computer Science, pp. 177–190. Springer (2010)Google Scholar
  25. 25.
    Kaibel, V., Pfetsch, M.E.: Packing and partitioning orbitopes. Math. Program. 114, 1–36 (2008)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Kaibel, V., Peinhardt, M., Pfetsch, M.E.: Orbitopal fixing. Discrete Optim. 8(4), 595–610 (2011)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Khodaei, A., Shahidehpour, M.: Transmission switching in security-constrained unit commitment. IEEE Trans. Power Syst. 25(4), 1937–1945 (2010)CrossRefGoogle Scholar
  28. 28.
    Koster, A.M.C.A., Lemkens, S.: Designing AC power grids using integer linear programming. In: Pahl, J., Reiners, T., Voß, S. (eds.) Network Optimization, vol. 6701 of Lecture Notes in Computer Science, pp. 478–483 (2011)Google Scholar
  29. 29.
    Lavaei, J., Low, S.H.: Zero duality gap in optimal power flow problem. IEEE Trans. Power Syst. 27(1), 92–107 (2012)CrossRefGoogle Scholar
  30. 30.
    Lee, J., Leung, J., Margot, F.: Min-up/min-down polytopes. Discrete Optim. 1(1), 77–85 (2004)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Lesieutre, B.C., Molzahn, D.K., Borden, A.R., DeMarco, C.L.: Examining the limits of the application of semidefinite programming to power flow problems. In: Proceedings of Communication, Control, and Computing (Allerton), pp. 1492–1499 (2011)Google Scholar
  32. 32.
    Liu, C., Shahidehpour, M., Li, Z., Fotuhi-Firuzabad, M.: Component and mode models for the short-term scheduling of combined-cycle units. IEEE Trans. Power Syst. 24(2), 976–990 (2009)CrossRefGoogle Scholar
  33. 33.
    Lotfjou, A., Shahidehpour, M., Fu, Y., Li, Z.: Security-constrained unit commitment with AC/DC transmission systems. IEEE Trans. Power Syst. 25(1) 531–542 (2010)CrossRefGoogle Scholar
  34. 34.
    Margot, F.: Pruning by isomorphism in branch-and-cut. Math. Program. 94, 71–90 (2002)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Margot, F.: Exploiting orbits in symmetric ILP. Math. Program. Series B 98, 3–21 (2003)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Mazadi, M., Rosehart, W.D., Malik, O.P., Aguado, J.A.: Modified chance-constrained optimization applied to the generation expansion problem. IEEE Trans. Power Syst. 24(3) 1635–1636 (2009)CrossRefGoogle Scholar
  37. 37.
    Meibom, P., Barth, R., Hasche, B., Brand, H., Weber, C., O’Malley, M.: Stochastic optimization model to study the operational impacts of high wind penetrations in ireland. IEEE Trans. Power Syst. 26(3), 1367–1379 (2011)CrossRefGoogle Scholar
  38. 38.
    Morales-España, G., Latorre, J.M., Ramos, A.: Tight and compact MILP formulation of start-up and shut-down ramping in unit commitment. IEEE Trans. Power Syst. 28(2), 1288–1296 (2013)CrossRefGoogle Scholar
  39. 39.
    Moser, A.: Langfristig optimale Struktur und Betriebsmittelwahl für 110-kV-Überlandnetze. PhD thesis, RWTH Aachen University (1995)Google Scholar
  40. 40.
    Ostrowski, J., Anjos, M.F., Vannelli, A.: Modified orbital branching with applications to orbitopes and to unit commitment. Cahier du GERAD G-2012-61, GERAD, Montreal, QC, Canada (2012)Google Scholar
  41. 41.
    Ostrowski, J., Anjos, M.F., Vannelli, A.: Tight mixed integer linear programming formulations for the unit commitment problem. IEEE Trans. Power Syst. 27(1), 39–46 (2012)CrossRefGoogle Scholar
  42. 42.
    Ostrowski, J., Linderoth, J., Rossi, F., Smriglio, S.: Orbital branching. Math. Program. 126(1), 147–178 (2009)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Ott, A.L.: Evolution of computing requirements in the PJM market: past and future. In: Power and Energy Society General Meeting, 2010 IEEE, pp. 1 –4 (2010)Google Scholar
  44. 44.
    Ozturk, U.A., Mazumdar, M., Norman, B.A.: A solution to the stochastic unit commitment problem using chance constrained programming. IEEE Trans. Power Syst. 19(3), 1589–1598 (2004)CrossRefGoogle Scholar
  45. 45.
    Padhy, N.P.: Unit commitment-a bibliographical survey. IEEE Trans. Power Syst. 19(2), 1196–1205 (2004)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Rajan, D., Takriti, S.: Minimum up/down polytopes of the unit commitment problem with start-up costs. Technical report, IBM Research Report (2005)Google Scholar
  47. 47.
    Ruiz, P.A., Philbrick, C.R., Zak, E., Cheung, K.W., Sauer, P.W.: Uncertainty management in the unit commitment problem. IEEE Trans. Power Syst. 24(2), 642–651 (2009)CrossRefGoogle Scholar
  48. 48.
    Shahidehpour, M., Yamin, H., Li, Z.: Market operations in electric power systems: forecasting, scheduling, and risk management. Wiley, New York (2003)Google Scholar
  49. 49.
    Sherali, H.D., Smith, J.C.: Improving zero-one model representations via symmetry considerations. Manag. Sci. 47(10), 1396–1407 (2001)CrossRefMATHGoogle Scholar
  50. 50.
    Sherali, H.D., Tuncbilek, C.H.: A global optimization algorithm for polynomial programming problems using a reformulation-linearization technique. J. Global Optim. 2, 101–112 (1992)MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Simoglou, C.K., Biskas, P.N., Bakirtzis, A.G.: Optimal self-scheduling of a thermal producer in short-term electricity markets by MILP. IEEE Trans. Power Syst. 25(4), 1965–1977 (2010)CrossRefGoogle Scholar
  52. 52.
    Sioshansi, R., O’Neill, R., Oren, S.S.: Economic consequences of alternative solution methods for centralized unit commitment in day-ahead electricity markets. IEEE Trans. Power Syst. 23(2), 344–352 (2008)CrossRefGoogle Scholar
  53. 53.
    Taylor, J.A., Hover, F.S.: Linear relaxations for transmission system planning. IEEE Trans. Power Syst. 26(4), 2533–2538 (2011)CrossRefGoogle Scholar
  54. 54.
    Troy, N., Denny, E., O’Malley, M.: Base-load cycling on a system with significant wind penetration. IEEE Trans. Power Syst. 25(2), 1088–1097 (2010)CrossRefGoogle Scholar
  55. 55.
    Tuohy, A., Meibom, P., Denny, E., O’Malley, M.: Unit commitment for systems with significant wind penetration. IEEE Trans. Power Syst. 24(2), 592–601 (2009)CrossRefGoogle Scholar
  56. 56.
    Wang, Q., Guan, Y., Wang, J.: A chance-constrained two-stage stochastic program for unit commitment with uncertain wind power output. IEEE Trans. Power Syst. 27(1), 206–215 (2012)CrossRefGoogle Scholar
  57. 57.
    Wiebking, R.: Stochastische modelle zur optimalen lastverteilung in einem kraftwerksverbund. Zeitschrift für Oper. Res. 21(6), B197–B217 (1977)MATHGoogle Scholar
  58. 58.
    Wu, L.: A tighter piecewise linear approximation of quadratic cost curves for unit commitment problems. IEEE Trans. Power Syst. 26(4), 2581–2583 (2011)CrossRefGoogle Scholar
  59. 59.
    Wu, L., Shahidehpour, M., Tao, L.: Stochastic security-constrained unit commitment. IEEE Trans. Power Syst. 22(2), 800–811 (2007)CrossRefGoogle Scholar
  60. 60.
    Wu, L., Shahidehpour, M., Li, T.: Cost of reliability analysis based on stochastic unit commitment. IEEE Trans. Power Syst. 23(3), 1364–1374 (2008)CrossRefGoogle Scholar
  61. 61.
    Yamin, H.Y.: Review on methods of generation scheduling in electric power systems. Electric Power Syst. Res. 69(2–3), 227–248 (2004)CrossRefGoogle Scholar
  62. 62.
    Zhao, C., Wang, J., Watson, J.-P., Guan, Y.: Multi-stage robust unit commitment considering wind and demand response uncertainties. IEEE Trans. Power Syst. 28(3), 2708–2717 (2013)CrossRefGoogle Scholar
  63. 63.
    Zhao, L., Zeng, B.: Robust unit commitment problem with demand response and wind energy. In: Power and Energy Society General Meeting, 2012 IEEE, pp. 1–8 (2012)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Canada Research Chair in Discrete Nonlinear Optimization in EngineeringGERAD and École Polytechnique de MontréalMontrealCanada

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