Handbook of Power Systems I pp 177-208

Part of the Energy Systems book series (ENERGY) | Cite as

Recent Progress in Two-stage Mixed-integer Stochastic Programming with Applications to Power Production Planning



We present recent developments in two-stage mixed-integer stochastic programming with regard to application in power production planning. In particular, we review structural properties, stability issues, scenario reduction, and decomposition algorithms for two-stage models. Furthermore, we describe an application to stochastic thermal unit commitment.


Decomposition algorithms Discrepancy Mixed-integer Two-stage Scenario reduction Stability Stochastic programming Unit commitment 


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  1. Ahmed S, Tawarmalani M, Sahinidis NV (2004) A finite branch and bound algorithm for two-stage stochastic integer programs. Math Program 100:355–377CrossRefMATHMathSciNetGoogle Scholar
  2. Alonso-Ayuso A, Escudero LF, Teresa Ortuńo M (2003) BFC, a branch-and-fix coordination algorithmic framework for solving some types of stochastic pure and mixed 0-1 programs. Eur J Oper Res 151:503–519CrossRefMATHGoogle Scholar
  3. Balas E (1998) Disjunctive programming: Properties of the convex hull of feasible points. Discrete Appl Math 89:3–44; originally MSRR#348, Carnegie Mellon University (1974)Google Scholar
  4. Bank B, Guddat J, Kummer B, Klatte D, Tammer K (1982) Non-linear parametric optimization. Akademie-Verlag, BerlinCrossRefGoogle Scholar
  5. Bertsekas DP (1982) Constrained optimization and Lagrange multiplier methods. Academic Press, NYMATHGoogle Scholar
  6. Birge JR (1985) Decomposition and partitioning methods for multistage stochastic programming. Oper Res 33(5):989–1007CrossRefMATHMathSciNetGoogle Scholar
  7. Blair CE, Jeroslow RG (1982) The value function of an integer program. Math Program 23:237–273CrossRefMATHMathSciNetGoogle Scholar
  8. Carøe CC, Schultz R (1999) Dual decomposition in stochastic integer programming. Oper Res Lett 24(1–2):37–45CrossRefMATHMathSciNetGoogle Scholar
  9. Carøe CC, Tind J (1997) A cutting-plane approach to mixed 0–1 stochastic integer programs. Eur J Oper Res 101(2):306–316CrossRefGoogle Scholar
  10. Carøe CC, Tind J (1998) L-shaped decomposition of two-stage stochastic programs with integer recourse. Math Program 83(3A):451–464Google Scholar
  11. Carpentier P, Cohen G, Culioli JC, Renaud (1996) A stochastic optimization of unit commitment: A new decomposition framework. IEEE Trans Power Syst 11:1067–1073Google Scholar
  12. Dentcheva D, Römisch W (2004) Duality gaps in nonconvex stochastic optimization. Math Program 101(3A):515–535Google Scholar
  13. Dupačová J, Gröwe-Kuska N, Römisch W (2003) Scenarios reduction in stochastic programming: An approach using probability metrics. Math Program 95:493–511CrossRefMATHMathSciNetGoogle Scholar
  14. Eichhorn A, Heitsch H, Römisch W (2010) Stochastic optimization of electricity portfolios: Scenario tree modeling and risk management. In: Rebennack S, Pardalos PM, Pereira MVF, Illiadis NA (eds.) Handbook of Power Systems, vol. II. Springer, Berlin, pp. 405–432CrossRefGoogle Scholar
  15. Eichhorn A, Römisch W (2005) Polyhedral risk measures in stochastic programming. SIAM J Optim 16:69–95CrossRefMATHMathSciNetGoogle Scholar
  16. Eichhorn A, Römisch W (2007) Stochastic integer programming: Limit theorems and confidence intervals. Math Oper Res 32:118–135CrossRefMATHMathSciNetGoogle Scholar
  17. Eichhorn A, Römisch W, Wegner I (2005) Mean-risk optimization of electricity portfolios using multiperiod polyhedral risk measures. In: IEEE St. Petersburg PowerTech ProceedingsGoogle Scholar
  18. Escudero LF, Garín A, Merino M, Pérez G (2007) A two-stage stochastic integer programming approach as a mixture of branch-and-fix coordination and Benders decomposition schemes. Ann Oper Res 152:395–420CrossRefMATHMathSciNetGoogle Scholar
  19. Gröwe-Kuska N, Kiwiel KC, Nowak MP, Römisch W, Wegner I (2002) Power management in a hydro-thermal system under uncertainty by Lagrangian relaxation. In: Greengard C, Ruszczyński A (eds.) Decision making under uncertainty: Energy and power, Springer, NY, pp. 39–70CrossRefGoogle Scholar
  20. Gröwe-Kuska N, Römisch W (2005) Stochastic unit commitment in hydro-thermal power production planning. In: Wallace SW, Ziemba WT (eds.) Applications of stochastic programming. MPS/SIAM series on optimization, SIAM, Philadelphia, pp. 633–653CrossRefGoogle Scholar
  21. Heitsch H, Römisch W (2007) A note on scenario reduction for two-stage stochastic programs. Oper Res Lett 35:731–738CrossRefMATHMathSciNetGoogle Scholar
  22. Henrion R, Küchler C, Römisch W (2008) Discrepancy distances and scenario reduction in two-stage stochastic mixed-integer programming. J Ind Manag Optim 4:363–384CrossRefMATHMathSciNetGoogle Scholar
  23. Henrion R, Küchler C, Römisch W (2009) Scenario reduction in stochastic programming with respect to discrepancy distances. Comput Optim Appl 43:67–93CrossRefMATHMathSciNetGoogle Scholar
  24. Henrion R, Küchler C, Römisch W (in preparation) A scenario reduction heuristic for two-stage stochastic integer programsGoogle Scholar
  25. Kiwiel KC (1990) Proximity control in bundle methods for convex nondifferentiable optimization. Math Program 46:105–122CrossRefMATHMathSciNetGoogle Scholar
  26. Klein Haneveld WK, Stougie L, van der Vlerk MH (2006) Simple integer recourse models: convexity and convex approximations. Math Program 108(2–3B):435–473Google Scholar
  27. Küchler C, Vigerske S (2007) Decomposition of multistage stochastic programs with recombining scenario trees. Stoch Program E-Print Series 9 www.speps.org
  28. Laporte G, Louveaux FV (1993) The integer L-shaped method for stochastic integer programs with complete recourse. Oper Res Lett 13(3):133–142CrossRefMATHMathSciNetGoogle Scholar
  29. Louveaux FV, Schultz R (2003) Stochastic integer programming. In: Ruszczyński A, Shapiro A (eds.) Stochastic programming, pp. 213–266; Handbooks in operations research and management science vol. 10 ElsevierGoogle Scholar
  30. Lulli G, Sen S (2004) A branch and price algorithm for multi-stage stochastic integer programs with applications to stochastic lot sizing problems. Manag Sci 50:786–796CrossRefMATHGoogle Scholar
  31. Niederreiter H (1992) Random number generation and Quasi-Monte Carlo methods. SIAM, PhiladelphiaCrossRefMATHGoogle Scholar
  32. Nowak MP, Römisch W (2000) Stochastic Lagrangian relaxation applied to power scheduling in a hydro-thermal system under uncertainty. Ann Oper Res 100:251–272CrossRefMATHMathSciNetGoogle Scholar
  33. Nowak MP, Schultz R, Westphalen M (2005) A stochastic integer programming model for incorporating day-ahead trading of electricity into hydro-thermal unit commitment. Optim Eng 6:163–176CrossRefMATHMathSciNetGoogle Scholar
  34. Ntaimo L, Sen S (2005) The million-variable “march” for stochastic combinatorial optimization. J Global Optim 32(3):385–400CrossRefMATHMathSciNetGoogle Scholar
  35. Ntaimo L, Sen S (2008) A comparative study of decomposition algorithms for stochastic combinatorial optimization. Comput Optim Appl 40(3):299–319CrossRefMATHMathSciNetGoogle Scholar
  36. Ntaimo L, Sen S (2008) A branch-and-cut algorithm for two-stage stochastic mixed-binary programs with continuous first-stage variables. Int J Comp Sci Eng 3:232–241Google Scholar
  37. Nürnberg R, Römisch W (2002) A two-stage planning model for power scheduling in a hydro-thermal system under uncertainty. Optim Eng 3:355–378CrossRefMATHMathSciNetGoogle Scholar
  38. Philpott AB, Craddock M, Waterer H (2000) Hydro-electric unit commitment subject to uncertain demand. Eur J Oper Res 125:410–424CrossRefMATHGoogle Scholar
  39. Rachev ST (1991) Probability Metrics and the Stability of Stochastic Models. Wiley, ChichesterMATHGoogle Scholar
  40. Römisch W, Schultz R (2001) Multistage stochastic integer programs: An introduction. In: Grötschel M, Krumke SO, Rambau J (eds.) Online optimization of large scale systems, Springer, Berlin, pp. 581–600CrossRefGoogle Scholar
  41. Römisch W, Vigerske S (2008) Quantitative stability of fully random mixed-integer two-stage stochastic programs. Optim Lett 2:377–388CrossRefMATHMathSciNetGoogle Scholar
  42. Schultz R (1996) Rates of convergence in stochastic programs with complete integer recourse. SIAM J Optim 6:1138–1152CrossRefMATHMathSciNetGoogle Scholar
  43. Schultz R (2003) Stochastic programming with integer variables. Math Program 97:285–309MATHMathSciNetGoogle Scholar
  44. Schultz R, Stougie L, van der Vlerk MH (1998) Solving stochastic programs with integer recourse by enumeration: a framework using Gröbner basis reductions. Math Program 83(2A):229–252Google Scholar
  45. Schultz R, Tiedemann S (2006) Conditional value-at-risk in stochastic programs with mixed-integer recourse. Math Program 105:365–386CrossRefMATHMathSciNetGoogle Scholar
  46. Sen S (2005) Algorithms for stochastic mixed-integer programming models. In: Aardal K, Nemhauser GL, Weismantel R (eds.) Handbook of discrete optimization, North-Holland Publishing Co. pp. 515–558Google Scholar
  47. Sen S, Higle JL (2005) The \(\mathrm{{C}^{3}}\) theorem and a \(\mathrm{{D}^{2}}\) algorithm for large scale stochastic mixed-integer programming: set convexification. Math Program 104(A):1–20Google Scholar
  48. Sen S, Sherali HD (2006) Decomposition with branch-and-cut approaches for two-stage stochastic mixed-integer programming. Math Progr 106(2A):203–223Google Scholar
  49. Sen S, Yu L, Genc T (2006) A stochastic programming approach to power portfolio optimization. Oper Res 54:55–72CrossRefMATHMathSciNetGoogle Scholar
  50. Takriti S, Birge JR, Long E (1996) A stochastic model for the unit commitment problem. IEEE Trans Power Syst 11:1497–1508CrossRefGoogle Scholar
  51. Takriti S, Krasenbrink B, Wu LSY (2000) Incorporating fuel constraints and electricity spot prices into the stochastic unit commitment problem. Oper Res 48:268–280CrossRefGoogle Scholar
  52. Tind J, Wolsey LA (1981) An elementary survey of general duality theory in mathematical programming. Math Program 21:241–261CrossRefMATHMathSciNetGoogle Scholar
  53. van den Bosch PPJ, Lootsma FA (1987) Scheduling of power generation via large-scale nonlinear optimization. J Optim Theor Appl 55:313–326CrossRefMATHGoogle Scholar
  54. van der Vlerk MH (2004) Convex approximations for complete integer recourse models. Math Program 99(2A):297–310Google Scholar
  55. van der Vlerk MH (1996-2007) Stochastic integer programming bibliography. http://mally.eco.rug.nl/spbib.html
  56. van der Vlerk MH (2005) Convex approximations for a class of mixed-integer recourse models; Ann Oper Res (to appear); Stoch Program E-Print SeriesGoogle Scholar
  57. Van Slyke RM, Wets R (1969) L-Shaped Linear Programs with Applications to Optimal Control and Stochastic Programming. SIAM J Appl Math 17(4):638–663CrossRefMATHMathSciNetGoogle Scholar
  58. Wolsey LA (1981) Integer programming duality: Price functions and sensitivity analysis. Math Program 20:173–195CrossRefMATHMathSciNetGoogle Scholar
  59. Zhuang G, Galiana FD (1988) Towards a more rigorous and practical unit commitment by Lagrangian relaxation. IEEE Trans Power Syst 3:763–773CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Humboldt UniversityBerlinGermany

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