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Bayesian Stochastic Dynamic Programming for Hydropower Generation Operation Based on Copula Functions

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Abstract

Bayesian stochastic dynamic programming (BSDP) has been widely used in hydropower generation operation, as natural inflow and forecast uncertainties can be easily determined by transition probabilities. In this study, we propose a theoretical estimation method (TEM) based on copula functions to calculate the transition probability under conditions of limited historical inflow samples. The explicit expression of the conditional probability is derived using copula functions and then used to calculate prior and likelihood probabilities, and the prior probability can be revised to the posterior probability once new forecast information is available by Bayesian formulation. The performance of BSDP models in seven forecast scenarios and two extreme conditions considering no or perfect forecast information is evaluated and compared. The case study in the Ertan hydropower station in China shows that (1) TEM can avoid the shortcomings of empirical estimation method (EMM) in calculating the transition probability, so that the prior and likelihood probability matrices can be distributed more uniformly with less zeros, and the problem that the posterior probability cannot be calculated can be avoided; (2) there is a positive correlation between operating benefit and forecast accuracy; and (3) the operating policy considering reliable forecast information can improve hydropower generation. However, an incorrect decision may be made in the case of low forecast accuracy.

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Notes

  1. In China, a month can be divided into three periods of approximately 10 days each: the first 10 days (early), the middle 10 days (mid) and the rest days (late) of the month.

References

  • Ahmadianfar I, Samadi-Koucheksaraee A, Bozorg-Haddad O (2017) Extracting optimal policies of hydropower multi-reservoir systems utilizing enhanced differential evolution algorithm. Water Resour Manag 31:1–23

    Article  Google Scholar 

  • Birhanu K, Alamirew T, Dinka MO, Ayalew S, Aklog D (2014) Optimizing reservoir operation policy using chance constraint nonlinear programming for Koga Irrigation Dam, Ethiopia. Water Resour Manag 28:4957–4970

    Article  Google Scholar 

  • Celeste AB, Billib M (2009) Evaluation of stochastic reservoir operation optimization models. Adv Water Resour 32:1429–1443

    Article  Google Scholar 

  • Chandramouli V, Deka P (2005) Neural network based decision support model for optimal reservoir operation. Water Resour Manag 19:447–464

    Article  Google Scholar 

  • Genest C, Rémillard B, Beaudoin D (2009) Goodness-of-fit tests for copulas: a review and a power study. Insurance Math Econ 44:199–213

    Article  Google Scholar 

  • Harboe R (1993) Explicit stochastic optimization. Springer, Dordrecht

    Book  Google Scholar 

  • Karamouz M, Vasiliadis HV (1992) Bayesian stochastic optimization of reservoir operation using uncertain forecasts. Water Resour Res 28:1221–1232

    Article  Google Scholar 

  • Karmakar S, Simonovic SP (2010) Bivariate flood frequency analysis. Part 2: A copula-based approach with mixed marginal distributions. J Flood Risk Manag 2:32–44

    Article  Google Scholar 

  • Kim YO, Palmer RN (1997) Value of seasonal flow forecasts in Bayesian Stochastic Programming. J Water Resour Plan Manag 123:327–335

    Article  Google Scholar 

  • Lei XH, Tan QF, Wang X, Wang H, Wen X, Wang C, Zhang JW (2017) Stochastic optimal operation of reservoirs based on copula functions. J Hydrol:557, 265–275

    Article  Google Scholar 

  • Li L, Liu P, Rheinheimer DE, Deng C, Zhou Y (2014) Identifying explicit formulation of operating rules for multi-reservoir systems using genetic programming. Water Resour Manag 28:1545–1565

    Article  Google Scholar 

  • Liao S, Liu B, Cheng C, Li Z, Wu X (2017) Long-term generation scheduling of hydropower system using multi-core parallelization of particle swarm optimization water resources management an international. Water Resour Manag 31(9):2791–2807

    Article  Google Scholar 

  • Massey FJ (1951) The Kolmogorov-Smirnov test for goodness of fit. Publ Am Stat Assoc 46:68–78

    Article  Google Scholar 

  • Mcyer PL (1970) Introductory probability and STATISTICAL application. Addison-Wesley Publishing Company, Hoboken

    Google Scholar 

  • Mehta R, Jain SK (2009) Optimal operation of a multi-purpose reservoir using Neuro-Fuzzy technique. Water Resour Manag 23:509–529

    Article  Google Scholar 

  • Mujumdar PP, Nirmala B (2007) A Bayesian Stochastic optimization model for a multi-reservoir hydropower system. Water Resour Manag 21:1465–1485

    Article  Google Scholar 

  • Opan M (2010) Irrigation-energy management using a DPSA-based optimization model in the Ceyhan Basin of Turkey, 385. J Hydrol:353–360

    Article  Google Scholar 

  • Reddy MJ (2012) Bivariate flood frequency analysis of upper godavari river flows using Archimedean Copulas. Water Resour Manag 26:3995–4018

    Article  Google Scholar 

  • Ren D (2016) Outlook for renewable energy development of 13th Five Year Plan Science & Technology Review

  • Shi Y, Yong P, Wei XU (2016) Optimal operation model of cascade reservoirs based on grey discrete differential dynamic programming. J Hydroelectr Eng 35(12):35–44

    Google Scholar 

  • Simonovic SP, Burn DH (1989) An improved methodology for short-term operation of a single multipurpose reservoir. Water Resour Res 25:1–8

    Article  Google Scholar 

  • Sklar M (1959) Fonctions de repartition a n dimensions et leurs marges. Publ. Inst. Statist. Univ., Paris, p 8

    Google Scholar 

  • Tan QF, Wang X, Liu P, Lei XH, Cai SY, Wang H, Ji Y (2017) The dynamic control bound of flood limited water level considering capacity compensation regulation and flood spatial pattern uncertainty. Water Resour Manag 31:143–158

    Article  Google Scholar 

  • Tan QF, Lei XH, Wang X, Wang H, Wen X, Ji Y, Kang AQ (2018) An adaptive middle and long-term runoff forecast model using EEMD-ANN hybrid approach. J Hydrol 567:767–780

    Article  Google Scholar 

  • Tan QF, Lei XH, Wen X, Fang GH, Wang X, Wang C, Ji Y, Huang XF (2019) Two-stage stochastic optimal operation model for hydropower station based on the approximate utility function of the carryover stage. Energy 183:670–682

    Article  Google Scholar 

  • Tang GL, Zhou HC, Li NN, Feng W, Wang YJ, Jian DP (2010) Value of medium-range precipitation forecasts in inflow prediction and hydropower optimization. Water Resour Manag 24:2721–2742

    Article  Google Scholar 

  • Willis R, Finney BA, Chu WS (1984) Monte Carlo optimization for reservoir operation. Water Resour Res 20:1177–1182

    Article  Google Scholar 

  • Xie M, Zhou J, Li C, Zhu S (2015) Long-term generation scheduling of Xiluodu and Xiangjiaba cascade hydro plants considering monthly streamflow forecasting error. Energy Convers Manag 105:368–376

    Article  Google Scholar 

  • Xu W, Zhang C, Peng Y, Fu G, Zhou H (2014) A two stage Bayesian stochastic optimization model for cascaded hydropower systems considering varying uncertainty of flow forecasts. Water Resour Res 50:9267–9286

    Article  Google Scholar 

  • Yan B, Guo S, Guo J, Chen L, Liu P, Chen H (2010) Regional design flood composition based on Copula function. J Hydroelectr Eng 29:60–65

    Google Scholar 

  • Yazdi J, Moridi A (2018) Multi-objective differential evolution for Design of Cascade Hydropower Reservoir Systems. Water Resour Manag 32(14):4779–4791. https://doi.org/10.1007/s11269-018-2083-5

    Article  Google Scholar 

  • Yun R, Singh VP (2008) Multiple duration limited water level and dynamic limited water level for flood control, with implications on water supply. J Hydrol 354:160–170

    Article  Google Scholar 

  • Zhao T, Cai X, Lei X, Wang H (2011) Improved dynamic programming for reservoir operation optimization with a concave objective function. J Water Resour Plan Manag 138:590–596

    Article  Google Scholar 

  • Zhao T, Yang D, Cai X, Zhao J, Wang H (2012) Identifying effective forecast horizon for real-time reservoir operation under a limited inflow forecast. Water Resour Res 48:1540

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (U51909063, U1765201), the China Postdoctoral Science Foundation (2019 M651681), the Fundamental Research Funds for the Central Universities of China (2019B11514), the National Key Research and Development Project of China (2018YFC1508200). The authors also thank the support from international Clean Energy Talent Program (iCET) from China Scholarship Council.

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Correspondence to Xin Wen.

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Tan, Qf., Fang, Gh., Wen, X. et al. Bayesian Stochastic Dynamic Programming for Hydropower Generation Operation Based on Copula Functions. Water Resour Manage 34, 1589–1607 (2020). https://doi.org/10.1007/s11269-019-02449-8

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