Controlled approximation of the value function in stochastic dynamic programming for multi-reservoir systems

Abstract

We present a new approach for adaptive approximation of the value function in stochastic dynamic programming. Under convexity assumptions, our method is based on a simplicial partition of the state space. Bounds on the value function provide guidance as to where refinement should be done, if at all. Thus, the method allows for a trade-off between solution time and accuracy. The proposed scheme is experimented in the particular context of hydroelectric production across multiple reservoirs.

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References

  1. Barros MT, Tsai FT, Sl Yang, Lopes JE, Yeh WW (2003) Optimization of large-scale hydropower system operations. J Water Res Plan Manag 129(3):178–188

    Article  Google Scholar 

  2. Castelletti A, de Rigo D, Rizzoli AE, Soncini-Sessa R, Weber E (2007) Neuro-dynamic programming for designing water reservoir network management policies. Control Eng Pract 15(8):1031–1038

    Article  Google Scholar 

  3. Chandramouli V, Raman H (2001) Multireservoir modeling with dynamic programming and neural networks. J Water Res Plan Manag 127(2):89–98

    Article  Google Scholar 

  4. Dickinson PJ (2013) On the exhaustivity of simplicial partitioning. J Glob Optim 58.1:1–15

  5. Foufoula-Georgiou E, Kitanidis PK (1988) Gradient dynamic programming for stochastic optimal control of multidimensional water resources systems. Water Resour Res 24(8):1345–1359

    Article  ADS  Google Scholar 

  6. Gal S (1989) The parameter iteration method in dynamic programming. Manag Sci 35(6):675–684

    MATH  Article  Google Scholar 

  7. Hiriart-Urruty JB, Lemaréchal C (2001) Fundamentals of convex analysis. Springer, Berlin

  8. Kim YO, Eum HI, Lee EG, Ko IH (2007) Optimizing operational policies of a korean multireservoir system using sampling stochastic dynamic programming with ensemble streamflow prediction. J Water Resour Plan Manag 133(1):4–14

    Article  Google Scholar 

  9. Krau S, Émiel G, Merleau J (2014) Une méthode d’échantillonnage adaptatif de l’espace des états pour la programmation dynamique appliquée à la gestion de systèmes hydriques. Seminar presentation, CIRRELT, Montréal, January

  10. Lang P, Zéphyr L, Lamond B (2014) Computing the expected value function for a multi-reservoir system with highly correlated inflows. In: Guan Y, Liao H (eds) Proceedings of the industrial and systems engineering research conference (ISERC 2014), Montreal, Canada, May 31–June 3. Online: http://www.xcdsystem.com/iie2014/abstract/finalpapers/I111.pdf

  11. Moore DW (1992) Simplical mesh generation with applications. PhD thesis

  12. Munos R, Moore A (2002) Variable resolution discretization in optimal control. Mach Learn 49(2–3):291–323

    MATH  Article  Google Scholar 

  13. Paulavičius R, Žilinskas J (2013) Simplicial global optimization. Springer, Berlin

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

    MATH  MathSciNet  Article  Google Scholar 

  15. Powell WB (2009) What you should know about approximate dynamic programming. Naval Res Logist 56(3):239–249

    MATH  MathSciNet  Article  Google Scholar 

  16. Rani D, Moreira MM (2010) Simulation-optimization modeling: a survey and potential application in reservoir systems operation. Water Resour Manag 24(6):1107–1138

    Article  Google Scholar 

  17. Shawwash ZK, Siu TK, Russell SD (2000) The BC hydro short term hydro scheduling optimization model. IEEE Trans Power Syst 15(3):1125–1131

    Article  Google Scholar 

  18. Tilmant A, Pinte D, Goor Q (2008) Assessing marginal water values in multipurpose multireservoir systems via stochastic programming. Water Resour Res 44(12):W12,431

  19. Yurtal R, Seckin G, Ardiclioglu G (2005) Hydropower optimization for the lower seyhan system in Turkey using dynamic programming. Water Int 30(4):522–529

    Article  Google Scholar 

  20. Zéphyr L (2015) Optimisation stochastique des systémes multi-réservoirs par l’agrégation de scénarios et la programmation dynamique approximative. PhD thesis

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Correspondence to Luckny Zéphyr.

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Zéphyr, L., Lang, P. & Lamond, B.F. Controlled approximation of the value function in stochastic dynamic programming for multi-reservoir systems. Comput Manag Sci 12, 539–557 (2015). https://doi.org/10.1007/s10287-015-0242-1

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Keywords

  • Value function approximation
  • Stochastic dynamic programming
  • Simplicial decomposition
  • Regular grids
  • Separable grids
  • Reservoir networks