Tighter MIP formulations for the discretised unit commitment problem with min-stop ramping constraints

  • Nicolas Dupin
Original Paper


This paper elaborates compact MIP formulations for a discrete unit commitment problem with minimum stop and ramping constraints. The variables can be defined in two different ways. Both MIP formulations are tightened with clique cuts and local constraints. The projection of constraints from one variable structure to the other allows to compare and tighten the MIP formulations. This leads to several equivalent formulations in terms of polyhedral descriptions and thus in LP relaxations. We analyse how MIP resolutions differ in the efficiency of the cuts, branching and primal heuristics. The resulting MIP implementation allows to tackle real size instances for an industrial application.


OR in energy Unit commitment problem Ramping constraints Mixed integer programming Polyhedron Constraint reformulation 

Mathematics Subject Classification

90C11 Mixed integer programming 90C90 Applications of mathematical programming 90B30 Production models 



This paper was written in the Ph.D. thesis (Dupin 2015) financed by the French Defence Procurement Agency of the French Ministry of Defence (DGA). Some results of this paper were presented in the ROADEF 2011 congress organised by the French association for operations research and decision making support. This work was awarded with a prize for junior researchers in the ROADEF 2011 congress. The author addresses special thanks to Pierre-Edouard Adenot and Pierre Bazot for their support, and to Catherine Cook for her help in writing the English version of this paper.


  1. Achterberg T, Koch T, Martin A (2005) Branching rules revisited. Oper Res Lett 33(1):42–54CrossRefGoogle Scholar
  2. Arroyo J, Carrion M (2006) A computationally efficient mixed-integer linear formulation for the thermal unit commitment problem. IEEE Trans Power Syst 21(3):1371–1378CrossRefGoogle Scholar
  3. Arroyo J, Conejo A (2004) Modeling of start-up and shut-down power trajectories of thermal units. IEEE Trans Power Syst 19(3):1562–1568CrossRefGoogle Scholar
  4. Atamtürk A, Nemhauser GL, Savelsbergh MW (2000) Conflict graphs in solving integer programming problems. Eur J Oper Res 121(1):40–55CrossRefGoogle Scholar
  5. Beale EML, Tomlin JA (1970) Special facilities in a general mathematical programming system for non-convex problems using ordered sets of variables. OR 69:447–454Google Scholar
  6. Bertacco L, Fischetti M, Lodi A (2007) A feasibility pump heuristic for general mixed-integer problems. Discrete Optim 4(1):63–76CrossRefGoogle Scholar
  7. Bertsimas D, Litvinov E et al (2013) Adaptive robust optimization for the security constrained unit commitment problem. IEEE Trans Power Syst 28(1):52–63CrossRefGoogle Scholar
  8. Borghetti A, D’Ambrosio C, Lodi A, Martello S (2008) An MILP approach for short-term hydro scheduling and unit commitment with head-dependent reservoir. IEEE Trans Power Syst 23(1):1115–1124CrossRefGoogle Scholar
  9. Cheng CP, Liu CW, Liu CC (2000) Unit commitment by Lagrangian relaxation and genetic algorithms. IEEE Trans Power Syst 15(2):707–714CrossRefGoogle Scholar
  10. Cong G, Meyers C, Rajan D, Parriani T (2015) Parallel strategies for solving large unit commitment problems in the California ISO planning model. In: IEEE parallel and distributed processing symposium (IPDPS), pp 710–719Google Scholar
  11. Damcı-Kurt P, Küçükyavuz S, Rajan, D, Atamtürk A (2013) A polyhedral study of production ramping. Mathematical Programming, pp 1–31Google Scholar
  12. Danna E, Rothberg E, Pape CL (2005) Exploring relaxation induced neighborhoods to improve MIP solutions. Math Program Ser A 102:71–90CrossRefGoogle Scholar
  13. Dubost L, Gonzalez R, Lemaréchal C (2005) A primal-proximal heuristic applied to the French Unit-commitment problem. Math Program 104(1):129–151CrossRefGoogle Scholar
  14. Dupin N (2015) Modélisation et résolution de grands problèmes stochastiques combinatoires: application à la gestion de production d’électricité. Ph.D. thesis Université Lille 1Google Scholar
  15. Fischetti M, Lodi A (2003) Local branching. Math Program 98(1–3):23–47CrossRefGoogle Scholar
  16. Frangioni A, Gentile C, Lacalandra F (2008) Solving unit commitment problems with general ramp constraints. J Electr Power Energy Syst 30(5):316–326CrossRefGoogle Scholar
  17. Frangioni A, Gentile C, Lacalandra F (2009) Tighter approximated MILP formulations for unit commitment problems. IEEE Trans Power Syst 24(1):105–113CrossRefGoogle Scholar
  18. Gu Z, Nemhauser GL, Savelsbergh MW (1998) Lifted cover inequalities for 0–1 integer programs: computation. INFORMS J Comput 10(4):427–437CrossRefGoogle Scholar
  19. Lee J, Leung J, Margot F (2004) Min-up/min-down polytopes. Discrete Optim 1:77–85CrossRefGoogle Scholar
  20. Lusby R, Muller L, Petersen B (2013) A solution approach based on Benders decomposition for the preventive maintenance scheduling problem of a stochastic large-scale energy system. J Sched 16(6):605–628CrossRefGoogle Scholar
  21. Morales-España G, Latorre JM, Ramos A (2013) Tight and compact MILP formulation for the thermal unit commitment problem. IEEE Trans Power Syst 28(4):4897–4908Google Scholar
  22. Morales-España G, Latorre JM, Ramos A (2013) Tight and compact MILP formulation formulation of start-up and shut-down ramping in unit commitment. IEEE Trans Power Syst 28(2):1288–1296Google Scholar
  23. Morales-España G, Gentile C, Ramos A (2015) Tight MIP formulations of the power-based unit commitment problem. OR Spectrum, pp 1–22Google Scholar
  24. Ostrowski J, Anjos M, Vannelli A (2012) Tight mixed integer linear programming formulations for the unit commitment problem. IEEE Trans Power Syst 27:39–46CrossRefGoogle Scholar
  25. Rajan D, Takriti S (2005) Min-up/down polytopes of the unit commitment problem with start-up costs. Tech. rep, IBM Research ReportGoogle Scholar
  26. Renaud A (1993) Daily generation management at Electricité de France: from planning towards real time. IEEE Trans Autom Control 38(7):1080–1093CrossRefGoogle Scholar
  27. Savelsbergh MW (1994) Preprocessing and probing techniques for mixed integer programming problems. ORSA J Comput 6(4):445–454CrossRefGoogle Scholar
  28. Takriti S, Krasenbrink B, Wu L (2000) Incorporating fuel constraints and electricity spot prices into the stochastic unit commitment problem. Oper Res 48:268–280CrossRefGoogle Scholar
  29. Vielma JP (2015) Mixed integer linear programming formulation techniques. SIAM Rev 57(1):3–57CrossRefGoogle Scholar
  30. Yıldız S, Vielma JP (2013) Incremental and encoding formulations for mixed integer programming. Oper Res Lett 41(6):654–658CrossRefGoogle Scholar

Copyright information

© EURO - The Association of European Operational Research Societies 2016

Authors and Affiliations

  1. 1.Direction Générale de l’Armement (DGA)Univ. Lille, UMR 9189, CRIStAL, Centre de Recherche en Informatique Signal et Automatique de LilleLilleFrance

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