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Inexact stochastic dynamic programming method and application to water resources management in Shandong China under uncertainty

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Abstract

In this study, an interval parameter multistage joint-probability programming (IMJP) approach has been developed to deal with water resources allocation under uncertainty. The IMJP can be used not only to deal with uncertainties in terms of joint-probability and intervals, but also to examine the risk of violating joint probabilistic constraints in the context of multistage. The proposed model can handle the economic expenditure caused by regional water shortage and flood control. The model can also reflect the related dynamic changes in the multi-stage cases and the system safety under uncertainty. The developed method is applied to a case study of water resources allocation in Shandong, China, under multistage, multi-reservoir and multi-industry. The violating reservoir constraints are addressed in terms of joint-probability. Different risk levels of constraint lead to different planning. The obtained results can help water resources managers to identify desired system designs under various economic, environment and system reliability scenarios.

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Abbreviations

\( f^{ \pm } \) :

Objective function, total benefits of the system

\( R_{{tk_{1} }}^{ \pm } \) :

Auxiliary variable, distributary volume from reservoir 1 (Tianzhuang reservoir) under the scenario of K1 in period t

\( R_{{tk_{1} k_{2} }}^{ \pm } \) :

Auxiliary variable, distributary volume from reservoir 2 (Bashan reservoir) in period t under the situation when scenario K1 and K2 are in joint probability form

\( S_{{tk_{1} }}^{ \pm } \) :

Auxiliary variable, storage water volume of reservoir 1 (Tianzhuang reservoir) under the scenario of K1 in period t

\( S_{{tk_{1} k_{2} }}^{ \pm } \) :

Auxiliary variable, storage water volume of reservoir 2 (Bashan reservoir) in period t under the situation when scenario K1 and K2 are in joint probability form

\( X_{t}^{ \pm } \) :

Decision variable, the promised water supply to county from the reservoir

\( X_{it}^{ \pm } \) :

Decision variable, the promised water supply to each industry from the reservoir

\( SH_{{tk_{1} k_{2} }}^{ \pm } \) :

Decision variable, the gap of water shortage between the actual water supply and the promised water supply to three industries from the reservoir in period t in the situation when scenario K1 and K2 are in joint probability form

\( SH_{{itk_{1} k_{2} }}^{ \pm } \) :

Decision variable, the gap of water shortage between the actual water supply and the promised water supply to each industry from the reservoir in period t in the situation when scenario K1 and K2 are in joint probability form

\( SU_{{tk_{1} k_{2} }}^{ \pm } \) :

Decision variable, distributary volume from reservoir in period t under the situation when scenario K1 and K2 are in joint probability form

\( C_{CL} \) :

Canal loss coefficient

\( C_{SL} \) :

Seepage loss coefficient

\( C_{EL} \) :

Evaporation loss coefficient

\( De_{t}^{\hbox{min} } \) :

Minimum water demand of three industries in period t

\( De_{t}^{\hbox{max} } \) :

Maximum water demand of three industries in period t

\( De_{it}^{\hbox{min} } \) :

Minimum water demand of each industry in period t

\( De_{it}^{\hbox{max} } \) :

Maximum water demand of each industry in period t

\( De_{et}^{\pm} \) :

Ecological water requirement of county in period t

\( De_{lt}^{\pm} \) :

Domestic water requirement of county in period t

\( DR \) :

Dead storage for reservoir

\( NB_{it} \) :

Net benefit per unit of water allocated to each industry in period t

\( PE_{t}^{ \pm } \) :

Penalty per unit of shortage water not delivered to three industries

\( PE_{it}^{ \pm } \) :

Penalty per unit of shortage water not delivered to each industry

\( P_{{tk_{1} }} \) :

Probability of the occurrence of scenario k 1 in period t

\( P_{{tk_{2} }} \) :

Probability of the occurrence of scenario k 2 in period t

\( Q \) :

Joint probability of exceeding constraints of the reservoir-storage capacities

\( q1 \) :

Probability of exceeding constraint of the storage capacity of reservoir 1

\( q2 \) :

Probability of exceeding constraint of the storage capacity of reservoir 2

\( \tilde{Q}_{{tk_{1} }}^{ \pm } \) :

The flow of stream 1 in period t under the scenario \( k_{1} \)

\( \tilde{Q}_{{tk_{2} }}^{ \pm } \) :

The flow of stream 1 in period t under the scenario \( k_{2} \)

\( TR^{\pm} \) :

Total reservoir capacity

\( UR \) :

Useful reservoir capacity

\( VC_{t}^{ \pm } \) :

The float charge of diversion in the period t

\( i \) :

Industry (primary industry, secondary industry, tertiary industry)

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Acknowledgments

This research was supported by the National Natural Science Foundation of China (No. 41271536, 71071154, 91125017), National High Technology Research and Development Program of China (863 Program) (No. 2011AA100502), the Governmental Public Research Funds for Projects of Ministry of Agriculture (No. 201203077) and Ministry of Water Resources (No. 200901083, 201001060, and 201001061). The authors would like to thank the anonymous reviewers for their insightful and helpful comments and suggestions that were very helpful for improving the manuscript.

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Authors and Affiliations

  1. Institute of Ecological Simulation and Urban Ecology, Beijing Normal University, Beijing, 100875, China

    J. J. Gu

  2. Centre for Agricultural Water Research in China, China Agricultural University, Beijing, 100083, China

    P. Guo

  3. Environmental Science and Engineering Program, University of Northern British Columbia, Prince George, BC, V2N 4Z9, Canada

    P. Guo

  4. Environmental Systems Engineering Program, University of Regina, Regina, Saskatchewan, S4S 0A2, Canada

    G. H. Huang

  5. Energy and Environmental Research Center, North China Electric Power University, Beijing, 102206, China

    G. H. Huang

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  1. J. J. Gu
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Correspondence to P. Guo.

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Gu, J.J., Guo, P. & Huang, G.H. Inexact stochastic dynamic programming method and application to water resources management in Shandong China under uncertainty. Stoch Environ Res Risk Assess 27, 1207–1219 (2013). https://doi.org/10.1007/s00477-012-0657-y

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