Skip to main content
SpringerLink
Log in

Combining sampling-based and scenario-based nested Benders decomposition methods: application to stochastic dual dynamic programming

  • Full Length Paper
  • Series A
  • Published:
Mathematical Programming Submit manuscript

Abstract

Nested Benders decomposition is a widely used and accepted solution methodology for multi-stage stochastic linear programming problems. Motivated by large-scale applications in the context of hydro-thermal scheduling, in 1991, Pereira and Pinto introduced a sampling-based variant of the Benders decomposition method, known as stochastic dual dynamic programming (SDDP). In this paper, we embed the SDDP algorithm into the scenario tree framework, essentially combining the nested Benders decomposition method on trees with the sampling procedure of SDDP. This allows for the incorporation of different types of uncertainties in multi-stage stochastic optimization while still maintaining an efficient solution algorithm. We provide an illustration of the applicability of our method towards a least-cost hydro-thermal scheduling problem by examining an illustrative example combining both fuel cost with inflow uncertainty and by studying the Panama power system incorporating both electricity demand and inflow uncertainties.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Batlle, C., Barquín, J.: Fuel prices scenario generation based on a multivariate GARCH model for risk analysis in a wholesale electricity market. Int. J. Electr. Power. Energy Syst. 26(4), 273–280 (2004)

    Article  Google Scholar 

  2. Benders, J.F.: Partitioning procedures for solving mixed variables programming problems. Numer. Math. 4, 238–252 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  3. Benders, J.F.: Partitioning procedures for solving mixed-variables programming problems. CMS 2, 3–19 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bezerra, B., Kelman, R., Barroso, L.A., Flach, B., Latorre, M.L., Campodonico, N., Pereira, M.V.F.: Integrated electricity–gas operations planning in hydrothermal systems. In Proc. Symp. Specialists in Electric Operational and Expansion Planning (SEPOPE), Brazil (2006)

  5. Birge, J.R., Louveaux, F.: Introduction to Stochastic Programming. Operations Research and Financial Engineering, 2nd edn. Springer, New York (2011)

    Book  Google Scholar 

  6. Casey, M.S., Sen, S.: The scenario generation algorithm for multistage stochastic linear programming. Math. Oper. Res. 30(3), 615–631 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chabar, R.M., Pereira, M.V.F., Granville, S., Barroso, L.A., Iliadis, N.A.: Optimization of fuel contracts management and maintenance scheduling for thermal plants under price uncertainty. In: IEEE Power Systems Conference and Exposition, pp. 923–930 (2006)

  8. Chabar, R.M., Granville, S., Pereira, M.V.F., Iliadis, N.A.: Energy, natural resources and environmental economics, chapter optimization of fuel contract management and maintenance scheduling for thermal plants in hydro-based power systems, pp. 201–219. Springer (2009)

  9. Chen, Z.-L., Powell, W.B.: A convergent cutting-plane and partial-sampling algorithm for multistage stochastic linear programs with recourse. J. Optim. Theory Appl. 103, 497–524 (1999)

    Article  MathSciNet  Google Scholar 

  10. Costa, L.C.: Considering reliability constraints in the optimal power systems expansion planning problem. Master’s thesis, COPPE/UFRJ, May (2008)

  11. de Matos, V.L., Finardi, E.C.: A computational study of a stochastic optimization model for long term hydrothermal scheduling. Int. J. Electr. Power Energy Syst. 43(1), 1443–1452 (2012)

    Article  Google Scholar 

  12. de Queiroza, A.R., Morton, D.P.: Sharing cuts under aggregated forecasts when decomposing multi-stage stochastic programs. Oper. Res. Lett. 41(3), 311–316 (2013)

    Article  MathSciNet  Google Scholar 

  13. Diniz, A.L., dos Santos, T.N.: Multi-period stage definition for the multi stage Benders decomposition approach applied to hydrothermal scheduling. In: EngOpt 2008—International Conference on Engineering Optimization, Rio de Janeiro, Brazil (2008)

  14. Donohue, C.J.: Stochastic network programming and the dynamic vehicle allocation problem. PhD thesis, University of Michigan (1996)

  15. Donohue, C.J., Birge, J.R.: The abridged nested decomposition method for multistage stochastic linear programs with relatively complete recourse. Algorithm. Oper. Res. 1(1), 20–30 (2006)

    MathSciNet  MATH  Google Scholar 

  16. dos Santos, T.N., Diniz, A.L.: A new multiperiod stage definition for the multistage Benders decomposition approach applied to hydrothermal scheduling. IEEE Trans. Power Syst. 24(3), 1383–1392 (2009)

    Article  Google Scholar 

  17. Dupačová, J., Gröwe-Kuska, N., Römisch, W.: Scenario reduction in stochastic programming: an approach using probability metrics. Math. Program. 95, 493–511 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gassmann, H.I.: MSLiP: a computer code for the multistage stochastic linear programming problem. Math. Program. 47, 407–423 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gjelsvik, A., Wallace, S.W.: Methods for stochastic medium-term scheduling in hydro-dominated power systems. Technical report, Norwegian Electric Power Research Institute, Trondheim. EFI TR A4438 (1996)

  20. Gjelsvik, A., Belsnes, M.M., Håland, M.: A case of hydro scheduling with a stochastic price model. In: Broch, E., Lysne, D.K., Flatabø, N., Helland-Hansen, E. (eds.) Procedings of the 3rd International Conference on Hydropower, pp. 211–218. Trondheim/Norway/30 June 2 July 1997. A.A. Balkema, Rotterdam (1997)

  21. Gjelsvik, A., Mo, B., Haugstad, A.: Long- and medium-term operations planning and stochastic modelling in hydro-dominated power systems based on stochastic dual dynamic programming. In: Rebennack, S., Pardalos, P.M., Pereira, M.V.F., Iliadis, N.A. (eds.) Handbook of Power Systems. Energy Systems. Springer, Berlin (2010)

    Google Scholar 

  22. Gorenstin, B., Costa, J.P., Pereira, M.V.F., Campodónico, N.M.: Power system expansion planning under uncertainty. IEEE Trans. Power Syst. 8(1), 129–136 (1993)

    Article  Google Scholar 

  23. Granville, S., Oliveira, G.C., Thome, L.M., Campodonico, N., Latorre, M.L., Pereira, M.V.F., Barroso, L.A.: Stochastic optimization of transmission constrained and large scale hydrothermal systems in a competitive framework. In: IEEE Power Engineering Society General Meeting, vol. 2. Toronto (2003)

  24. Gröwe-Kuska, N., Heitsch, H., Römisch, W.: Scenario reduction and scenario tree construction for power management problems. In: IEEE Power Tech Conference. Bologna, Italy (2003)

  25. Heitsch, H., Römisch, W.: Scenario reduction algorithms in stochastic programming. Comput. Optim. Appl. 24(2–3), 187–206 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  26. Heitsch, H., Römisch, W.: Scenario tree modeling for multistage stochastic programs. Math. Program. 118, 371–406 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Heitsch, H., Römisch, W., Strugarek, C.: Stability of multistage stochastic programs. SIAM J. Optim. 17, 511–525 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  28. Homem-de-Mello, T., de Matos, V.L., Finardi, E.C.: Sampling strategies and stopping criteria for stochastic dual dynamic programming: a case study in long-term hydrothermal scheduling. Energy Syst. 2(1), 1–31 (2011)

    Article  Google Scholar 

  29. Høyland, K., Wallace, S.W.: Generating scenario trees for multistage decision problems. Manag. Sci. 47(2), 295–307 (2001)

    Article  Google Scholar 

  30. Iliadis, N.A.: Financial risk modelling in electricity portfolio optimisation. PhD thesis, Doctoral School of EPFL, August (2006)

  31. Iliadis, N.A., Perira, M.V.F., Granville, S., Finger, M., Haldi, P.-A., Barroso, L.-A.: Bechmarking of hydroelectric stochastic risk management models using financial indicators. In: Power Engineering Society General Meeting, pp. 1–8 (2006)

  32. Infanger, G., Morton, D.P.: Cut sharing for multistage stochastic linear programs with interstage dependency. Math. Program. 75, 241–256 (1996)

    MathSciNet  MATH  Google Scholar 

  33. Kuhn, D.: Aggregation and discretization in multistage stochastic programming. Math. Program. 113, 61–94 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  34. Maceira, M.E.P., Damázio, J.M.: The use of PAR(p) model in the stochastic dual dynamic programming optimization scheme used in the operation planning of the Brazilian hydropower system. In: 8th International Conference on Probabilistic Methods Applied to Power Systems, Iowa State University, Ames, Iowa, Sept 12–16 (2004)

  35. Maceira, M.E.P., Duarte, V.S., Penna, D.D.J., Moraes, L.A.M., Melo, A.C.G.: Ten years of application of stochastic dual dynamic programming in official and agent studies in Brazil—description of the NEWAVE program. In: 16th Power Systems Computation Conference—PSCC, Glasgow, SCO, July (2008)

  36. Mirkov, R., Pflug, GCh.: Tree approximations of dynamic stochastic programs. SIAM J. Optim. 18(3), 1082–1105 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  37. Mo, B., Gjelsvik, A., Grundt, A.: Integrated risk management of hydro power scheduling and contract management. IEEE Trans. Power Syst. 16(2), 216–221 (2001)

    Article  Google Scholar 

  38. Morton, D.P.: An enhanced decomposition algorithm for multistage stochastic hydroelectric scheduling. Ann. Oper. Res. 64, 211–235 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  39. Nowak, M.P., Römisch, W.: Stochastic Lagrangian relaxation applied to power scheduling in a hydro-thermal system under uncertainty. Ann. Oper. Res. 100(1–4), 251–272 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  40. Olsen, P.: Discretization of multistage stochastic programming problems. Math. Program. Stud. 6, 111–124 (1976)

    Article  MathSciNet  Google Scholar 

  41. Pennanen, T.: Epi-convergent discretization of multistage stochastic programs. Math. Oper. Res. 30(1), 245–256 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  42. Pennanen, T.: Epi-convergent discretizations of multistage stochastic programs via integration quadratures. Math. Program. 116, 461–479 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  43. Pereira, M.V.F., Pinto, L.M.V.G.: Stochastic optimization of a multireservoir hydroelectric system: a decomposition approach. Water Resour. Res. 21(6), 779–792 (1985)

    Article  Google Scholar 

  44. Pereira, M.V.F., Pinto, L.M.V.G.: Multi-stage stochastic optimization applied to energy planning. Math. Program. 52, 359–375 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  45. Pereira, M.V.F., Campodnico, N., Kelman, R.: Application of stochastic dual DP and extensions to hydrothermal scheduling. Technical report 2.0, PSRI, April 1999. PSRI Technical Report 012/99

  46. Philpott, A.B., de Matos, V.L.: Dynamic sampling algorithms for multi-stage stochastic programs with risk aversion. Eur. J. Oper. Res. 218(2), 470–483 (2012)

    Article  MATH  Google Scholar 

  47. Philpott, A.B., Guan, Z.: On the convergence of stochastic dual dynamic programming and related methods. Oper. Res. Lett. 36(4), 450–455 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  48. Powell, W.B.: Approximate Dynamic Programming: Solving the Curses of Dimensionality, 2nd edn. Wiley, New York (2011)

    Book  Google Scholar 

  49. Read, E.G.: A dual approach to stochastic dynamic programming for reservoir release scheduling. In: Esogbue, A.O. (ed.) Dynamic Programming for Optimal Water Resources System Management, pp. 361–372. Prentice Hall, NY (1989)

    Google Scholar 

  50. Read, E.G., Hindsberger, M.: Constructive dual DP for reservoir optimization. In: Rebennack, S., Pardalos, P.M., Pereira, M.V.F., Iliadis, N.A. (eds.) Handbook of Power Systems. Energy Systems. Springer, Berlin (2010)

    Google Scholar 

  51. Read, E.G., Culy, J.G., Halliburton, T.S., Winter, N.L.: A simulation model for long-term planning of the New Zealand power system. In: Rand, G.K. (ed.) Operational Research, pp. 493–507. North Holland, New York (1987)

    Google Scholar 

  52. Rebennack, S., Flach, B., Pereira, M.V.F., Pardalos, P.M.: Stochastic hydro-thermal scheduling under CO\(_2\) emission constraints. IEEE Trans. Power Syst. 27(1), 58–68 (2012)

    Article  Google Scholar 

  53. Rockafellar, R.T., Uryasev, S.: Optimization of conditional value-at-risk. J. Risk 2(3), 21–42 (2000)

    Google Scholar 

  54. Shapiro, A.: Analysis of stochastic dual dynamic programming method. Eur. J. Oper. Res. 209, 63–72 (2011)

    Article  MATH  Google Scholar 

  55. Shapiro, A., Tekaya, W., da Costa, J.P., Soares, M.P.: Risk neutral and risk averse stochastic dual dynamic programming method. Eur. J. Oper. Res. 224, 375–391 (2013)

    Article  MATH  Google Scholar 

  56. Shrestha, G.B., Pokharel, B.K., Lie, T.T., Fleten, S.-E.: Medium term power planning with bilateral contracts. IEEE Trans. Power Syst. 20(5), 627–633 (2005)

    Article  Google Scholar 

  57. Velásquez, J.: GDDP: generalized dual dynamic programming theory. Ann. Oper. Res. 117, 21–31 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  58. Wallace, S.W., Fleten, S.-E.: Stochastic programming, volume 10 of Handbooks in Operations Research and Management Science, chapter Stochastic programming models in energy, pp. 637–677. North-Holland (2003)

  59. Wets, R.J.-B.: Stochastic programs with fixed recourse: the equivalent deterministic program. SIAM Rev. 16(3), 309–339 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  60. Yakowitz, S.: Dynamic programming applications in water resources. Water Resour. Res. 18(4), 673–696 (1982)

    Article  Google Scholar 

  61. Zhou, Q., Tesfatsion, L., Liu, C.-C.: Scenario generation for price forecasting in restructured wholesale power markets. In: Power Systems Conference and Exposition (2009)

  62. Zimmermann, H.-J.: An application-oriented view of modeling uncertainty. Eur. J. Oper. Res. 122(2), 190–198 (2000)

    Article  MATH  Google Scholar 

Download references

Acknowledgments

The author thanks Mario Pereira (PSR) for his discussions on this research. He also thanks Panos M. Pardalos (University of Florida), David P. Morton (The University of Texas at Austin) and Bruno Flach (IBM) for their comments; Steven Frank, Timo Lohmann, Gregory Steeger (all Colorado School of Mines) and Josef Kallrath (BASF) for proofreading of the paper. The author also thanks the editor and the two reviewers for their thoughtful comments and suggestions.

Author information

Authors and Affiliations

  1. Division of Economics and Business, Colorado School of Mines, Golden, CO, USA

    Steffen Rebennack

Authors
  1. Steffen Rebennack
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Steffen Rebennack.

Appendix: Nomenclature

Appendix: Nomenclature

The nomenclature throughout this article is summarized in Tables 7, 8 and 9.

Table 7 Indices, sets and random variables
Full size table
Table 8 Decision variables, functions and values obtained through optimization
Full size table
Table 9 Data (no entry in the column “Dimension” means one-dimensional data)
Full size table

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rebennack, S. Combining sampling-based and scenario-based nested Benders decomposition methods: application to stochastic dual dynamic programming. Math. Program. 156, 343–389 (2016). https://doi.org/10.1007/s10107-015-0884-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-015-0884-3

Keywords

Mathematics Subject Classification

Search

Navigation