Computational Management Science

, Volume 15, Issue 1, pp 55–86 | Cite as

A successive linear programming algorithm with non-linear time series for the reservoir management problem

Original Paper


This paper proposes a multi-stage stochastic programming formulation based on affine decision rules for the reservoir management problem. Our approach seeks to find a release schedule that balances flood control and power generation objectives while considering realistic operating conditions as well as variable water head. To deal with the non-convexity introduced by the variable water head, we implement a simple, yet effective, successive linear programming algorithm. We also introduce a novel non-linear inflow representation that captures serial correlation of arbitrary order. We test our method on a small real river system and discuss policy implications. Our results namely show that our method can decrease flood risk and increase production compared to the historical decisions, albeit at the cost of reduced final storages.


Mathematical programming Stochastic processes Forecasting Risk analysis 



The authors would like to thank Grégory Émiel, Louis Delorme, Pierre-Marc Rondeau, Sara Séguin, Jasson Pina and Pierre-Luc Carpentier. This research was supported by NSERC/Hydro-Québec through the Industrial Research Chair on the Stochastic Optimization of Electricity Generation and Grant 386416-2010.


  1. Bana e Costa CA, Vansnick JC (1997) A theoretical framework for measuring attractiveness by a categorical based evaluation technique (MACBETH). Springer, New YorkCrossRefGoogle Scholar
  2. Ben-Tal A, Goryashko E, Guslitzer A, Nemirovski A (2004) Adjustable robust solutions of uncertain linear programs. Math Program 99:351–378CrossRefGoogle Scholar
  3. Bezerra B, Veiga Á, Barroso LA, Pereira M (2017) Stochastic long-term hydrothermal scheduling with parameter uncertainty in autoregressive streamflow models. IEEE Trans Power Syst 32(2):999–1006Google Scholar
  4. Billingsley P (1995) Probability and measure, 3rd edn. Wiley, New YorkGoogle Scholar
  5. 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(3):1115–1124CrossRefGoogle Scholar
  6. Box GEP, Cox DR (1964) An analysis of transformations. J Roy Stat Soc 2:211–252Google Scholar
  7. Box GEP, Jenkins GM, Reinsel GC (2008) Time series analysis: forecasting and control, 4th edn. Wiley, HobokenCrossRefGoogle Scholar
  8. Braaten SV, Gjonnes O, Hjertvik K, Fleten SE (2016) Linear decision rules for seasonal hydropower planning: modelling considerations. Energy Procedia 87:28–35. 5th International Workshop on Hydro Scheduling in Competitive Electricity MarketsGoogle Scholar
  9. Brockwell PJ, Davis RA (1987) Time series: theory and methods. Springer, New YorkCrossRefGoogle Scholar
  10. Carpentier PL, Gendreau M, Bastin F (2013) Long-term management of a hydroelectric multireservoir system under uncertainty using the progressive hedging algorithm. Water Resour Res 49:2812–2827CrossRefGoogle Scholar
  11. Castelletti A, Pianosi F, Soncini-Sessa R (2008) Water reservoir control under economic, social and environmental constraints. Automatica 44(6):1595–1607CrossRefGoogle Scholar
  12. Castelletti A, Galetti S, Restelli M, Soncini-Sessa R (2010) Tree-based reinforcement learning for optimal water reservoir operation. Water Resour Res 46(W09):507Google Scholar
  13. Cerisola S, Latorre JM, Ramos A (2012) Stochastic dual dynamic programming applied to nonconvex hydrothermal models. Eur J Oper Res 218(3):687–697. CrossRefGoogle Scholar
  14. De Ladurantaye D, Gendreau M, Potvin JY (2007) Strategic bidding for price-taker hydroelectricity producers. IEEE Trans Power Syst 22(4):2187–2203CrossRefGoogle Scholar
  15. De Ladurantaye D, Gendreau M, Potvin JY (2009) Optimizing profits from hydroelectricity production. Comput Oper Res 36:499–529CrossRefGoogle Scholar
  16. Diniz AL, Maceira MEP (2000) A four-dimensional model of hydro generation for the shortterm hydrothermal dispatch problem considering head and spillage effects. IEEE Trans Power Syst 23(3):1298–1308CrossRefGoogle Scholar
  17. Diniz AL, Souza TM (2014) Short-term hydrothermal dispatch with river-level and routing constraints. IEEE Trans Power Syst 29(5):2427–2435CrossRefGoogle Scholar
  18. dos Santos TN, Diniz AL (2009) A new multiperiod stage definition for the multistage benders decomposition approach applied to hydrothermal scheduling. IEEE Trans Power Syst 24(3):1383–1392CrossRefGoogle Scholar
  19. Egging R, Fleten SE, Grønvik I, Hadziomerovic A, Ingvoldstad N (2017) Linear decision rules for hydropower scheduling under uncertainty. IEEE Trans Power Syst 32(1):103–113CrossRefGoogle Scholar
  20. Ehrgott M (2005) Multicriteria optimization, 2nd edn. Springer, New YorkGoogle Scholar
  21. Gauvin C, Delage E, Gendreau M (2017) Decision rule approximations for the risk averse reservoir management problem. Eur J Oper Res 261:317–336CrossRefGoogle Scholar
  22. Gauvin C, Delage E, Gendreau M (2016) A stochastic program with tractable time series and affine decision rules for the reservoir management problem. Technical report. G-2016-24, Les cahiers du GERADGoogle Scholar
  23. Gauvin C, Delage E, Gendreau M (2017) A successive linear programming algorithm with non-linear time series for the reservoir management problem. Technical reportGoogle Scholar
  24. Gjelsvik A, Mo B, Haugstad A (2010) Long- and medium-term operations planning and stochastic modelling in hydro-dominated power systems based on stochastic dual dynamic programming. Springer Berlin Heidelberg, Berlin, pp 33–55. Google Scholar
  25. Hamann A, Hug G, Rosinski S (2017) Real-time optimization of the mid-columbia hydropower system. IEEE Trans Power Syst 32(1):157–165CrossRefGoogle Scholar
  26. Klöckl B, Papaefthymiou G (2010) Multivariate time series models for studies on stochastic generators in power systems. Electr Power Syst Res 80:265–276Google Scholar
  27. Kuhn D, Wiesemann W, Georghiou A (2011) Primal and dual linear decision rules in stochastic and robust optimization. Math Program Ser A 130:177–209CrossRefGoogle Scholar
  28. Labadie J (2004) Optimal operation of multireservoir systems: state-of-the-art review. J Water Resour Plan Manag 130:93–111CrossRefGoogle Scholar
  29. Li X, Guo S, Liu P, Chen G (2010) Dynamic control of flood limited water level for reservoir operation by considering inflow uncertainty. J Hydrol 391(1):124–132CrossRefGoogle Scholar
  30. Lohmann T, Hering AS, Rebennack S (2016) Spatio-temporal hydro forecasting of multireservoir inflows for hydro-thermal scheduling. Eur J Oper Res 255(1):243–258. CrossRefGoogle Scholar
  31. Lorca Á, Sun XA, Litvinov E, Zheng T (2016) Multistage adaptive robust optimization for the unit commitment problem. Oper Res 64(1):32–51. CrossRefGoogle Scholar
  32. Lorca A, Sun XA (2015) Adaptive robust optimization with dynamic uncertainty sets for multi-period economic dispatch under significant wind. IEEE Trans Power Syst 30:1702–1713CrossRefGoogle Scholar
  33. Maceira M, Damázio J (2004) 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, pp 397–402Google Scholar
  34. Maceira M, Duarte V, Penna D, Moraes L, Melo A (2008) Ten years of application of stochastic dual dynamic programming in official and agent studies in brazil—description of the newave program. In: 16th PSCC, Glasgow, ScotlandGoogle Scholar
  35. Needham JT, Watkins DW, Lund JR, Nanda SK (2000) Linear programming for flood control in the iowa and des moines rivers. J Water Resour Plan Manag 126(3):118–127CrossRefGoogle Scholar
  36. Nocedal J, Wright SJ (2006) Numerical optimization. Springer Series in Operations Research, 2nd edn. Springer, New YorkGoogle Scholar
  37. Pan L, Housh M, Liu P, Cai X, Chen X (2015) Robust stochastic optimization for reservoir operation. Water Resour Res 51(1):409–429. CrossRefGoogle Scholar
  38. Phillpott A, Wahid F, Bonnans F (2016) Midas: a mixed integer dynamic approximation scheme. Technical report.
  39. Pianosi F, Soncini-Sessa R (2009) Real-time management of a multipurpose water reservoir with a heteroscedastic inflow model. Water Resour Res 45(10):1–12. CrossRefGoogle Scholar
  40. Poorsepahy-Samian H, Espanmanesh V, Zahraie B (2016) Improved inflow modeling in stochastic dual dynamic programming. J Water Resour Plan Manag 142(12):04016065CrossRefGoogle Scholar
  41. ReVelle CS, Kirby W (1970) Linear decision rule in reservoir management and design, 2, performance optimization. Water Resour Res 6:1033–1044CrossRefGoogle Scholar
  42. Séguin S, Côté P, Audet C (2016) Self-scheduling short-term unit commitment and loading problem. IEEE Trans Power Syst 31(1):133–142CrossRefGoogle Scholar
  43. Séguin S, Côté P, Audet C (2014) Short-term unit commitment and loading problem. Tech. rep, Les cahiers du GERADGoogle Scholar
  44. Shapiro A (2011) Analysis of stochastic dual dynamic programming method. Eur J Oper Res 209:63–72CrossRefGoogle Scholar
  45. Shapiro A, Tekaya W, da Costa JP, Soares MP (2013) Risk neutral and risk averse stochastic dual dynamic programming method. Eur J Oper Res 224(2):375–391. CrossRefGoogle Scholar
  46. Shapiro A, Tekaya W, da Costa JP, Soares MP (2012) Final report for technical cooperation between georgia institute of technology and ons - operador nacional do sistema eletrico. Tech. rep., Georgia Institute of Technology and Operador Nacional do Sistema EletricoGoogle Scholar
  47. Stedinger JR, Faber BA (2001) Reservoir optimization using sampling sdp with ensemble streamflow prediction (esp) forecast. J Hydrol 249:113–133CrossRefGoogle Scholar
  48. Steeger G, Rebennack S (2017) Dynamic convexification within nested benders decomposition using lagrangian relaxation: an application to the strategic bidding problem. Eur J Oper Res 257(2):669–686CrossRefGoogle Scholar
  49. Thome F, Pereira M, Granville S, Fampa M (2013) Non-convexities representation on hydrothermal operation planning using sddp (unpublished technical report)Google Scholar
  50. Tilmant A, Kelman R (2007) A stochastic approach to analyze trade-offs and risk associated with large-scale water resources systems. Water Resour Res 43(W06):425Google Scholar
  51. Tsay RS (2005) Analysis of financial time series, 2nd edn. Wiley, HobokenCrossRefGoogle Scholar
  52. Turgeon A (2005) Solving a stochastic reservoir management problem with multilag autocorrelated inflows. Water Resour Res 41(W12):414Google Scholar
  53. Turgeon A, Charbonneau R (1998) An aggregation-disaggregation approach to long-term reservoir management. Water Resour Res 34:3585–3594CrossRefGoogle Scholar
  54. Wei CC, Hsu NS (2009) Optimal tree-based release rules for real-time flood control operations on a multipurpose multireservoir system. J Hydrol 365(3):213–224CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Polytechnique MontréalMontrealCanada
  2. 2.HEC MontréalMontrealCanada

Personalised recommendations