Large-Scale Stochastic Mixed-Integer Programming Algorithms for Power Generation Scheduling



This chapter presents a stochastic unit commitment model for power systems and revisits parallel decomposition algorithms for these types of models. The model is a two-stage stochastic programming problem with first-stage binary variables and second-stage mixed-binary variables. The here-and-now decision is to find day-ahead schedules for slow thermal power generators. The wait-and-see decision consists of dispatching power and scheduling fast-start generators. We discuss advantages and limitations of different decomposition methods and provide an overview of available software packages. A large-scale numerical example is presented using a modified IEEE 118-bus system with uncertain wind power generation.


Master Problem Unit Commitment Bundle Method Subgradient Method Wind Power Generation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This material is based upon work supported by the U.S. Department of Energy, Office of Science, under contract number DE-AC02-06CH11357. We gratefully acknowledge the computing resources provided on Blues, a high-performance computing cluster operated by the Laboratory Computing Resource Center at Argonne National Laboratory. We thank Julie Bessac for providing wind speed prediction data.


  1. Achterberg, T.: Scip: solving constraint integer programs. Mathematical Programming Computation 1(1), 1–41 (2009)Google Scholar
  2. Ahmed, S., Tawarmalani, M., Sahinidis, N.V.: A finite branch-and-bound algorithm for two-stage stochastic integer programs. Mathematical Programming 100(2), 355–377 (2004)Google Scholar
  3. Bertsekas, D.P., Scientific, A.: Convex Optimization Algorithms. Athena Scientific (2015)Google Scholar
  4. Birge, J.R., Dempster, M.A., Gassmann, H.I., Gunn, E.A., King, A.J., Wallace, S.W.: A standard input format for multiperiod stochastic linear programs. IIASA Laxenburg Austria (1987)Google Scholar
  5. CarøE, C.C., Schultz, R.: Dual decomposition in stochastic integer programming. Operations Research Letters 24(1), 37–45 (1999)Google Scholar
  6. Fisher, M.L.: An applications oriented guide to Lagrangian relaxation. Interfaces 15(2), 10–21 (1985)Google Scholar
  7. Forrest, J.: Cbc. URL
  8. Gade, D., Küçükyavuz, S., Sen, S.: Decomposition algorithms with parametric gomory cuts for two-stage stochastic integer programs. Mathematical Programming 144(1–2), 39–64 (2014)Google Scholar
  9. Geoffrion, A.M.: Lagrangean relaxation for integer programming. Springer (1974)Google Scholar
  10. Gondzio, J., Gonzalez-Brevis, P., Munari, P.: New developments in the primal–dual column generation technique. European Journal of Operational Research 224(1), 41–51 (2013)Google Scholar
  11. Huchette, J., Lubin, M., Petra, C.: Parallel algebraic modeling for stochastic optimization. In: Proceedings of the 1st First Workshop for High Performance Technical Computing in Dynamic Languages, pp. 29–35. IEEE Press (2014)Google Scholar
  12. Kim, K., Mehrotra, S.: A two-stage stochastic integer programming approach to integrated staffing and scheduling with application to nurse management. Operations Research 63(6), 1431–1451 (2015)Google Scholar
  13. Kim, K., Zavala, V.M.: Algorithmic innovations and software for the dual decomposition method applied to stochastic mixed-integer programs. Optimization Online (2015)Google Scholar
  14. Lemaréchal, C.: Lagrangian relaxation. In: Computational combinatorial optimization, pp. 112–156. Springer (2001)Google Scholar
  15. Lubin, M., Martin, K., Petra, C.G., Sandkç, B.: On parallelizing dual decomposition in stochastic integer programming. Operations Research Letters 41(3), 252–258 (2013)Google Scholar
  16. Märkert, A., Gollmer, R.: Users Guide to ddsip–A C package for the dual decomposition of two-stage stochastic programs with mixed-integer recourse (2014)Google Scholar
  17. Pyro: Python remote objects (2015). URL
  18. Rockafellar, R.T., Wets, R.J.B.: Scenarios and policy aggregation in optimization under uncertainty. Mathematics of operations research 16(1), 119–147 (1991)Google Scholar
  19. Sen, S., Higle, J.L.: The C 3 theorem and a D 2 algorithm for large scale stochastic mixed-integer programming: set convexification. Mathematical Programming 104(1), 1–20 (2005)Google Scholar
  20. Sherali, H.D., Fraticelli, B.M.: A modification of benders’ decomposition algorithm for discrete subproblems: An approach for stochastic programs with integer recourse. Journal of Global Optimization 22(1–4), 319–342 (2002)Google Scholar
  21. U.S. Energy Information Administration: Monthly energy review. Tech. rep., U.S. Energy Information Administration (2015)Google Scholar
  22. Watson, J.P., Woodruff, D.L.: Progressive hedging innovations for a class of stochastic mixed-integer recsource allocation problems. Computational Management Science 8(4), 355–370 (2011)Google Scholar
  23. Watson, J.P., Woodruff, D.L., Hart, W.E.: PySP: modeling and solving stochastic programs in Python. Mathematical Programming Computation 4(2), 109–149 (2012)Google Scholar
  24. Zhang, M., Küçükyavuz, S.: Finitely convergent decomposition algorithms for two-stage stochastic pure integer programs. SIAM Journal on Optimization 24(4), 1933–1951 (2014)Google Scholar

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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Mathematics and Computer Science DivisionArgonne National LaboratoryLemontUSA
  2. 2.Department of Chemical and Biological EngineeringUniversity of Wisconsin-MadisonMadisonUSA

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