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

- 911 Downloads
- 16 Citations

## 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.

## Keywords

Stochastic dual dynamic programming Hydro-thermal power system Nested Benders decomposition Sampling Scenario tree Electricity demand and inflow uncertainty## Mathematics Subject Classification

90C15 90C05 90C39 90C90## Notes

### 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.

## 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)CrossRefGoogle Scholar - 2.Benders, J.F.: Partitioning procedures for solving mixed variables programming problems. Numer. Math.
**4**, 238–252 (1962)CrossRefMathSciNetzbMATHGoogle Scholar - 3.Benders, J.F.: Partitioning procedures for solving mixed-variables programming problems. CMS
**2**, 3–19 (2005)CrossRefMathSciNetzbMATHGoogle 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)Google Scholar
- 5.Birge, J.R., Louveaux, F.: Introduction to Stochastic Programming. Operations Research and Financial Engineering, 2nd edn. Springer, New York (2011)CrossRefGoogle Scholar
- 6.Casey, M.S., Sen, S.: The scenario generation algorithm for multistage stochastic linear programming. Math. Oper. Res.
**30**(3), 615–631 (2005)CrossRefMathSciNetzbMATHGoogle 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)Google Scholar
- 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)Google Scholar
- 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)CrossRefMathSciNetGoogle Scholar - 10.Costa, L.C.: Considering reliability constraints in the optimal power systems expansion planning problem. Master’s thesis, COPPE/UFRJ, May (2008)Google Scholar
- 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)CrossRefGoogle 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)CrossRefMathSciNetGoogle 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)Google Scholar
- 14.Donohue, C.J.: Stochastic network programming and the dynamic vehicle allocation problem. PhD thesis, University of Michigan (1996)Google Scholar
- 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)MathSciNetzbMATHGoogle 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)CrossRefGoogle 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)CrossRefMathSciNetzbMATHGoogle Scholar - 18.Gassmann, H.I.: MSLiP: a computer code for the multistage stochastic linear programming problem. Math. Program.
**47**, 407–423 (1990)CrossRefMathSciNetzbMATHGoogle 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)Google Scholar
- 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)Google Scholar
- 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)CrossRefGoogle 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)Google Scholar
- 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)Google Scholar
- 25.Heitsch, H., Römisch, W.: Scenario reduction algorithms in stochastic programming. Comput. Optim. Appl.
**24**(2–3), 187–206 (2003)CrossRefMathSciNetzbMATHGoogle Scholar - 26.Heitsch, H., Römisch, W.: Scenario tree modeling for multistage stochastic programs. Math. Program.
**118**, 371–406 (2009)CrossRefMathSciNetzbMATHGoogle Scholar - 27.Heitsch, H., Römisch, W., Strugarek, C.: Stability of multistage stochastic programs. SIAM J. Optim.
**17**, 511–525 (2006)CrossRefMathSciNetzbMATHGoogle 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)CrossRefGoogle Scholar - 29.Høyland, K., Wallace, S.W.: Generating scenario trees for multistage decision problems. Manag. Sci.
**47**(2), 295–307 (2001)CrossRefGoogle Scholar - 30.Iliadis, N.A.: Financial risk modelling in electricity portfolio optimisation. PhD thesis, Doctoral School of EPFL, August (2006)Google Scholar
- 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)Google Scholar
- 32.Infanger, G., Morton, D.P.: Cut sharing for multistage stochastic linear programs with interstage dependency. Math. Program.
**75**, 241–256 (1996)MathSciNetzbMATHGoogle Scholar - 33.Kuhn, D.: Aggregation and discretization in multistage stochastic programming. Math. Program.
**113**, 61–94 (2008)CrossRefMathSciNetzbMATHGoogle 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)Google Scholar
- 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)Google Scholar
- 36.Mirkov, R., Pflug, GCh.: Tree approximations of dynamic stochastic programs. SIAM J. Optim.
**18**(3), 1082–1105 (2007)CrossRefMathSciNetzbMATHGoogle 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)CrossRefGoogle Scholar - 38.Morton, D.P.: An enhanced decomposition algorithm for multistage stochastic hydroelectric scheduling. Ann. Oper. Res.
**64**, 211–235 (1996)CrossRefMathSciNetzbMATHGoogle 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)CrossRefMathSciNetzbMATHGoogle Scholar - 40.Olsen, P.: Discretization of multistage stochastic programming problems. Math. Program. Stud.
**6**, 111–124 (1976)CrossRefMathSciNetGoogle Scholar - 41.Pennanen, T.: Epi-convergent discretization of multistage stochastic programs. Math. Oper. Res.
**30**(1), 245–256 (2005)CrossRefMathSciNetzbMATHGoogle Scholar - 42.Pennanen, T.: Epi-convergent discretizations of multistage stochastic programs via integration quadratures. Math. Program.
**116**, 461–479 (2009)CrossRefMathSciNetzbMATHGoogle 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)CrossRefGoogle 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)CrossRefMathSciNetzbMATHGoogle 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 012/99Google Scholar
- 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)CrossRefzbMATHGoogle 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)CrossRefMathSciNetzbMATHGoogle Scholar - 48.Powell, W.B.: Approximate Dynamic Programming: Solving the Curses of Dimensionality, 2nd edn. Wiley, New York (2011)CrossRefGoogle 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)CrossRefGoogle 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)CrossRefzbMATHGoogle 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)CrossRefzbMATHGoogle 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)CrossRefGoogle Scholar - 57.Velásquez, J.: GDDP: generalized dual dynamic programming theory. Ann. Oper. Res.
**117**, 21–31 (2002)CrossRefMathSciNetzbMATHGoogle 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)Google Scholar
- 59.Wets, R.J.-B.: Stochastic programs with fixed recourse: the equivalent deterministic program. SIAM Rev.
**16**(3), 309–339 (1974)CrossRefMathSciNetzbMATHGoogle Scholar - 60.Yakowitz, S.: Dynamic programming applications in water resources. Water Resour. Res.
**18**(4), 673–696 (1982)CrossRefGoogle 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)Google Scholar
- 62.Zimmermann, H.-J.: An application-oriented view of modeling uncertainty. Eur. J. Oper. Res.
**122**(2), 190–198 (2000)CrossRefzbMATHGoogle Scholar