Skip to main content

Advertisement

SpringerLink
Log in

Stochastic competitive analysis of hydropower and water supplies within an energy–water nexus

  • Original Paper
  • Published:
Stochastic Environmental Research and Risk Assessment Aims and scope Submit manuscript

Abstract

Energy and water are scarce resources and understanding the complicated energy–water nexus is an important issue for effective resource management. The purpose of this research was to analyze the competitive and cooperative relationships involving energy and water production and use. Specifically, tradeoff and integrated management of hydropower generation and water supplies are analyzed for energy–water systems. A Nash–Cournot model was established to analyze strategic behaviors among participants in energy–water systems. In the model, tradeoff analysis and integrated management of hydropower and water supplies were simulated for a reservoir system. In addition, hydropower and thermal power generation in competitive energy markets was examined. A case study of Dajia River reservoirs in the Tai-Chung and Chang-Hwa energy–water systems is presented. Dajia River is the second longest river in central Taiwan; the reservoirs system of Dajia River generates hydropower with installed capacity of 1150 MW. Strategic competitive and cooperative behaviors regarding energy–water linkage were quantified in the results. The results show that integrated management of hydropower and water supplies can increase renewable energy production, lower electricity equilibrium price, and decrease carbon dioxide emission.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Abbreviations

\( {\text{f}},{\text{g}} \) :

Firm; \( {\text{f}},{\text{g}} = 1, 2, \ldots , {\text{F}} \)

\( {\text{h}} \) :

Generator; \( {\text{h}} = 1, 2, \ldots ,{\text{H}} \)

\( {\text{i}} \) :

Node of transmission network; \( {\text{i}} = 1, 2, \ldots ,{\text{I}} \)

\( {\text{s}} \) :

Stochastic scenarios; \( {\text{s}} = 1, 2, \ldots ,{\text{S}} \)

\( {\text{t}} \) :

Time; \( {\text{t}} = 1, 2, \ldots ,{\text{T}} \)

\( {\text{B}}_{\text{t}} \) :

Number of hours at time \( {\text{t}} \) (hr)

\( {\text{CP}}_{{{\text{f}},{\text{i}},{\text{h}}}} \) :

Generation capacity of generator h for firm f in node i (MW)

\( {\text{CT}}1 \) :

Hydropower generation cost (USD/MWh)

\( {\text{CT}}3_{{{\text{f}},{\text{i}},{\text{h}}}} \) :

Power generation cost of generator h for firm f in node i (USD/MWh)

\( {\text{CT}}4_{{{\text{f}},{\text{i}},{\text{h}}}} \) :

Water cost of generator h for firm f in node i (USD/m3)

\( {\text{DM}}1_{{{\text{t}},{\text{s}}}} \) :

Water demand of the upstream dam of scenario s at t (m3/h)

\( {\text{DM}}2_{{{\text{t}},{\text{s}}}} \) :

Water demand of the downstream dam of scenario s at t (m3/h)

\( {\text{HP}} \) :

Hydropower production rate (MWh/m3)

\( {\text{I}}1_{{{\text{t}},{\text{s}}}} \) :

Water inflow of the upstream dam of scenario s at t (m3/h)

\( {\text{I}}2_{{{\text{t}},{\text{s}}}} \) :

Water inflow of the downstream dam of scenario s at t (m3/h)

\( {\text{P}}_{\text{s}} \) :

Probability of stochastic scenario s (dimensionless)

\( {\text{PI}}_{{{\text{i}},{\text{t}},{\text{s}}}} \) :

Price intercept of the linear power demand curve in i of s at t (USD/MWh)

\( {\text{QI}}_{{{\text{i}},{\text{t}},{\text{s}}}} \) :

Quantity intercept of the linear power demand curve in i of s at t (MW)

\( {\text{ST}}1 \) :

Storage of the upstream dam (m3)

\( {\text{ST}}2 \) :

Storage of the downstream dam (m3)

\( {\text{WT}}_{{{\text{f}},{\text{i}},{\text{h}}}} \) :

Water demand of generator h for firm f in node i (m3/MWh)

\( {\text{d}}1_{{{\text{t}},{\text{s}}}} \) :

Water storage of the upstream dam of scenario s at t (m3)

\( {\text{d}}2_{{{\text{t}},{\text{s}}}} \) :

Water storage of the downstream dam of scenario s at t (m3)

\( {\text{f}}1_{{{\text{t}},{\text{s}}}} \) :

Water release from an upstream dam to a downstream dam of s at t (m3/h)

\( {\text{o}}1_{{{\text{t}},{\text{s}}}} \) :

Water supply of the upstream dam of scenario s at t (m3/h)

\( {\text{o}}2_{{{\text{t}},{\text{s}}}} \) :

Water supply of the downstream dam of scenario s at t (m3/h)

\( {\text{s}}1_{{{\text{i}},{\text{t}},{\text{s}}}} \) :

Hydropower sale in node i of scenario s at t (MW)

\( {\text{s}}3_{{{\text{f}},{\text{i}},{\text{t}},{\text{s}}}}^{{}} \) :

Power sale of firm f in node i at time t for scenario s (MW)

\( {\text{x}}1_{{{\text{t}},{\text{s}}}} \) :

Hydropower generation of the upstream dam of scenario s at t (MW)

\( {\text{x}}2_{{{\text{t}},{\text{s}}}} \) :

Hydropower generation of the downstream dam of scenario s at t (MW)

\( {\text{x}}3_{{{\text{f}},{\text{i}},{\text{h}},{\text{t}},{\text{s}}}} \) :

Power generation of generator h for firm f in node i of s at t (MW)

\( {\text{y}}3_{{{\text{f}},{\text{i}},{\text{h}},{\text{t}},{\text{s}}}} \) :

Water demand of power generator h for firm f in node i of s at t (m3/h)

\( \upalpha1_{{{\text{t}},{\text{s}}}} \) :

Dual variable of storage capacity constraint for upstream dam at t (USD/m3)

\( \upalpha2_{{{\text{t}},{\text{s}}}} \) :

Dual variable of storage capacity constraint for downstream dam at t (USD/m3)

\( \upbeta1_{{{\text{t}},{\text{s}}}} \) :

Dual variable of water supply constraint for upstream dam at t (USD/m3/h)

\( \upbeta2_{{{\text{t}},{\text{s}}}} \) :

Dual variable of water supply constraint for downstream dam at t (USD/m3/h)

\( \upgamma1_{{{\text{t}},{\text{s}}}} \) :

Dual variable of water balance constraint for upstream dam at t (USD/m3)

\( \upgamma2_{{{\text{t}},{\text{s}}}} \) :

Dual variable of water balance constraint for downstream dam at t (USD/m3)

\( \updelta1_{{{\text{t}},{\text{s}}}} \) :

Dual variable of hydropower constraint for upstream dam at t (USD/MW)

\( \updelta2_{{{\text{t}},{\text{s}}}} \) :

Dual variable of hydropower constraint for downstream dam at t (USD/MW)

\( \upvarepsilon_{{{\text{t}},{\text{s}}}} \) :

Dual variable of hydropower sale constraint at t (USD/MW)

\( \uptheta_{{{\text{f}},{\text{i}},{\text{h}},{\text{t}},{\text{s}}}} \) :

Dual variable of capacity constraint of h for firm f in node i at t (USD/MW)

\( \uplambda_{{{\text{f}},{\text{t}},{\text{s}}}} \) :

Dual variable of power sale constraint for firm f at t (USD/MW)

\( \upmu_{{{\text{f}},{\text{i}},{\text{h}},{\text{t}},{\text{s}}}} \) :

Dual variable of water demand constraint of h for firm f in i at t (USD/m3/h)

References

  • Ackerman F, Fisher J (2013) Is there a water–energy nexus in electricity generation? Long-term scenarios for the western United States. Energy Policy 59:235–241

    Article  Google Scholar 

  • Cottle RW, Pang JS, Stone RE (1992) The linear complementarity problem. Academic Press, San Diego

    Google Scholar 

  • Crampes C, Moreaux M (2001) Water resource and power generation. Int J Ind Organ 19:975–997

    Article  Google Scholar 

  • DeNooyer TA, Peschel JM, Zhang Z, Stillwell AS (2016) Integrating water resources and power generation: the energy–water nexus in Illinois. Appl Energy 162:363–371

    Article  Google Scholar 

  • Eleftheriadou E, Mylopoulos Y (2008) Game theoretical approach to conflict resolution in transboundary water resources management. J Water Res Plan Manag 134:466–473

    Article  Google Scholar 

  • Ferris MC, Munson TS (2000) GAMS/PATH user guide version 4.3. http://www.gams.com/docs/pdf/path.pdf. Accessed Aug 2016

  • Gabriel SA, Zhuang J, Kiet S (2005) A large-scale complementarity model of the North American natural gas market. Energy Econ 27:639–665

    Article  Google Scholar 

  • Ghimire BNS, Reddy MJ (2014) Optimization and uncertainty analysis of operational policies for multipurpose reservoir system. Stoch Environ Res Risk Assess 28:1815–1833

    Article  Google Scholar 

  • Hobbs BF (2001) Linear complementarity models of Nash–Cournot competition in bilaterial and POOLCO power markets. IEEE Trans Power Syst 16(2):194–202

    Article  Google Scholar 

  • Jafarzadegan K, Abed-Elmdoust A, Kerachian R (2014) A stochastic model for optimal operation of inter-basin water allocation systems: a case study. Stoch Environ Res Risk Assess 28:1343–1358

    Article  Google Scholar 

  • Kahrl F, Roland-Holst D (2008) China’s water–energy nexus. Water Policy 10(S1):51–65

    Article  Google Scholar 

  • Liu P, Lin K, Wei X (2015) A two-stage method of quantitative flood risk analysis for reservoir real-time operation using ensemble-based hydrologic forecasts. Stoch Environ Res Risk Assess 29:803–813

    Article  Google Scholar 

  • Liu XM, Huang GH, Wang S, Fan YR (2016) Water resources management under uncertainty: factorial multi-stage stochastic program with chance constraints. Stoch Environ Res Risk Assess 30:945–957

    Article  Google Scholar 

  • Luo B, Zhou DC (2009) Planning hydroelectric resources with recourse-based multistage interval-stochastic programming. Stoch Environ Res Risk Assess 23:65–73

    Article  Google Scholar 

  • Mas-Colell A, Whinston MD, Green JR (1995) Microeconomic theory. Oxford University Press, New York

    Google Scholar 

  • Mushtaq S, Maraseni TN, Maroulis J, Hafeez M (2009) Energy and water tradeoffs in enhancing food security: a selective international assessment. Energy Policy 37(9):3635–3644

    Article  Google Scholar 

  • Ozturk I (2015) Sustainability in the food–energy–water nexus: evidence from BRICS (Brazil, the Russian Federation, India, China, and South Africa) countries. Energy 93:999–1010

    Article  Google Scholar 

  • Qin Y, Curmi E, Kopec GM, Allwood JM, Richards KS (2015) China’s energy–water nexus—assessment of the energy sector’s compliance with the “3 Red Lines” industrial water policy. Energy Policy 82:131–143

    Article  Google Scholar 

  • Rosenthal RE (2008) GAMS: a user’s guide. http://www.gams.com/dd/docs/bigdocs/GAMSUsersGuide.pdf. Accessed Aug 2016

  • Santhosh A, Farid AM, Youcef-Toumi K (2014) Real-time economic dispatch for the supply side of the energy–water nexus. Appl Energy 122:42–52

    Article  Google Scholar 

  • Scott CA, Pierce SA, Pasqualetti MJ, Jones AL, Montz BE, Hoover JH (2011) Policy and institutional dimensions of the water–energy nexus. Energy Policy 39(10):6622–6630

    Article  Google Scholar 

  • Segurado R, Costa M, Duic N, Carvalho MG (2015) Integrated analysis of energy and water supply in islands. Case study of S. Vicente, Cape Verde. Energy 92:639–648

    Article  Google Scholar 

  • Siddiqi A, Anadon LD (2011) The water–energy nexus in Middle East and North Africa. Energy Policy 39(8):4529–4540

    Article  Google Scholar 

  • Sovacool BK, Sovacool KE (2009) Identifying future electricity–water tradeoffs in the United States. Energy Policy 37(7):2763–2773

    Article  Google Scholar 

  • Venkatesh G, Chan A, Brattebø H (2014) Understanding the water–energy–carbon nexus in urban water utilities: comparison of four city case studies and the relevant influencing factors. Energy 75:153–166

    Article  Google Scholar 

  • Vilanova MRN, Balestieri JAP (2015) Exploring the water-energy nexus in Brazil: the electricity use for water supply. Energy 85:415–432

    Article  Google Scholar 

  • Wang S, Chen B (2016) Energy–water nexus of urban agglomeration based on multiregional input–output tables and ecological network analysis: a case study of the Beijing–Tianjin–Hebei region. Appl Energy 178:773–783

    Article  Google Scholar 

Download references

Acknowledgements

The authors thank the editors and anonymous referees for their thoughtful comments and suggestions. The authors are responsible for the opinions and errors. This research was funded by the Ministry of Science and Technology, Taiwan (TW) under Grant MOST-105-2627-M-002-037.

Author information

Authors and Affiliations

  1. Department of Bioenvironmental Systems Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei, 10617, Taiwan

    Ming-Che Hu, Hwa-Lung Yu & Ching-Pin Tung

  2. Department of Urban Planning, National Cheng Kung University, No. 1, University Road, Tainan, 701, Taiwan

    Tailin Huang

Authors
  1. Ming-Che Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tailin Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hwa-Lung Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ching-Pin Tung
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tailin Huang.

Appendix

Appendix

The KKT optimality conditions of the profit-maximizing model of reservoirs system in Eqs. (1)–(11) is derived. The KKT conditions contain primal feasibility, dual feasibility, and complementarity slackness. First, dual feasibility equations and associated complementarity conditions are derived for eight primal variables in Eqs. (17)–(24). Then, Eqs. (25)–(33) establish primal equations and associated complementarity conditions for nine dual variables.

Similarly, KKT conditions of thermal power plants in Eqs. (12)–(16) are derived. The KKT conditions in Eqs. (34)–(36) show dual feasibility and the associated complementarity conditions for three primal variables: \( {\text{s}}3_{{{\text{f}},{\text{i}},{\text{t}},{\text{s}}}} \), \( {\text{x}}3_{{{\text{f}},{\text{i}},{\text{h}},{\text{t}},{\text{s}}}} \), and \( {\text{y}}3_{{{\text{f}},{\text{i}},{\text{h}},{\text{t}},{\text{s}}}} \). Meanwhile, conditions (37)–(39) show primal feasibility and the associated complementarity conditions for three dual variables: \( \uptheta_{{{\text{f}},{\text{i}},{\text{h}},{\text{t}},{\text{s}}}} \), \( \uplambda_{{{\text{f}},{\text{t}},{\text{s}}}} \), and \( \upmu_{{{\text{f}},{\text{i}},{\text{h}},{\text{t}},{\text{s}}}} \).

$$ \begin{array}{*{20}l} {{\text{For}}\;{\text{d}}1_{{{\text{t}},{\text{s}}}} \ge 0,} \hfill & {} \hfill \\ {\left[ {\upalpha1_{{{\text{t}},{\text{s}}}} -\upgamma1_{{{\text{t}},{\text{s}}}} +\upgamma1_{{{\text{t}} - 1,{\text{s}}}} } \right] \ge 0} \hfill & {\forall\,{\text{t,s}}} \hfill \\ {\left( {{\text{d}}1_{{{\text{t}},{\text{s}}}} } \right) \times \left[ {\upalpha1_{{{\text{t}},{\text{s}}}} -\upgamma1_{{{\text{t}},{\text{s}}}} +\upgamma1_{{{\text{t}} - 1,{\text{s}}}} } \right] = 0} \hfill & {\forall\,{\text{t,s}}} \hfill \\ \end{array} $$
(17)
$$ \begin{array}{*{20}l} {{\text{For}}\;{\text{d}}2_{{{\text{t}},{\text{s}}}} \ge 0,} \hfill & {} \hfill \\ {\left[ {\upalpha2_{{{\text{t}},{\text{s}}}} -\upgamma2_{{{\text{t}},{\text{s}}}} +\upgamma2_{{{\text{t}} - 1,{\text{s}}}} } \right] \ge 0} \hfill & {\forall\,{\text{t,s}}} \hfill \\ {\left( {{\text{d}}2_{{{\text{t}},{\text{s}}}} } \right) \times \left[ {\upalpha2_{{{\text{t}},{\text{s}}}} -\upgamma2_{{{\text{t}},{\text{s}}}} +\upgamma2_{{{\text{t}} - 1,{\text{s}}}} } \right] = 0} \hfill & {\forall\,{\text{t,s}}} \hfill \\ \end{array} $$
(18)
$$ \begin{array}{*{20}l} {{\text{For}}\;{\text{f}}1_{{{\text{t}},{\text{s}}}} \ge 0,} \hfill & {} \hfill \\ {\left[ {{\text{B}}_{t} \times\upgamma1_{{{\text{t}},{\text{s}}}} - {\text{B}}_{{{\text{t}},{\text{s}}}} \times\upgamma2_{{{\text{t}},{\text{s}}}} - {\text{HP}} \times\updelta1_{{{\text{t}},{\text{s}}}} } \right] \ge 0} \hfill & {\forall\,{\text{t,s}}} \hfill \\ {\left( {{\text{f}}1_{{{\text{t}},{\text{s}}}} } \right) \times \left[ {{\text{B}}_{\text{t}} \times\upgamma1_{{{\text{t}},{\text{s}}}} - {\text{B}}_{\text{t}} \times\upgamma2_{{{\text{t}},{\text{s}}}} - {\text{HP}} \times\updelta1_{{{\text{t}},{\text{s}}}} } \right] = 0} \hfill & {\forall\,{\text{t,s}}} \hfill \\ \end{array} $$
(19)
$$ \begin{array}{*{20}l} {{\text{For}}\;{\text{o}}1_{{{\text{t}},{\text{s}}}} \ge 0} \hfill & {} \hfill \\ {\left[ { -\upbeta1_{{{\text{t}},{\text{s}}}} + {\text{B}}_{t} \times\upgamma1_{{{\text{t}},{\text{s}}}} - {\text{HP}} \times\updelta1_{{{\text{t}},{\text{s}}}} } \right] \ge 0} \hfill & {\forall\,{\text{t,s}}} \hfill \\ {\left( {{\text{o}}1_{{{\text{t}},{\text{s}}}} } \right) \times \left[ { -\upbeta1_{{{\text{t}},{\text{s}}}} + {\text{B}}_{t} \times\upgamma1_{{{\text{t}},{\text{s}}}} - {\text{HP}} \times\updelta1_{{{\text{t}},{\text{s}}}} } \right] = 0} \hfill & {\forall\,{\text{t,s}}} \hfill \\ \end{array} $$
(20)
$$ \begin{array}{*{20}l} {{\text{For}}\;{\text{o}}2_{{{\text{t}},{\text{s}}}} \ge 0,} \hfill & {} \hfill \\ {\left[ { -\upbeta2_{{{\text{t}},{\text{s}}}} + {\text{B}}_{\text{t}} \times\upgamma2_{{{\text{t}},{\text{s}}}} - {\text{HP}} \times\updelta2_{{{\text{t}},{\text{s}}}} } \right] \ge 0} \hfill & {\forall\,{\text{t,s}}} \hfill \\ {\left( {{\text{o}}2_{{{\text{t}},{\text{s}}}} } \right) \times \left[ { -\upbeta2_{{{\text{t}},{\text{s}}}} + {\text{B}}_{t} \times\upgamma2_{{{\text{t}},{\text{s}}}} - {\text{HP}} \times\updelta2_{{{\text{t}},{\text{s}}}} } \right] = 0} \hfill & {\forall\,{\text{t,s}}} \hfill \\ \end{array} $$
(21)
$$ \begin{array}{*{20}l} {{\text{For}}\;{\text{s}}1_{{{\text{i}},{\text{t}},{\text{s}}}} \ge 0,} \hfill & {} \hfill \\ {\left[ {\upvarepsilon_{{{\text{t}},{\text{s}}}} - {\text{P}}_{\text{s}} \times {\text{B}}_{\text{t}} \times \left( {{\text{PI}}_{{{\text{i}},{\text{t}},{\text{s}}}} - \left( {{\text{PI}}_{{{\text{i}},{\text{t}},{\text{s}}}} /{\text{QI}}_{{{\text{i}},{\text{t}},{\text{s}}}} } \right) \times \left( {2 \times {\text{s}}1_{{{\text{i}},{\text{t}},{\text{s}}}} + \sum\nolimits_{\text{f}} {{\text{s}}3_{{{\text{f}},{\text{i}},{\text{t}},{\text{s}}}} } } \right)} \right)} \right] \ge 0} \hfill & {\forall\,{\text{i}},{\text{t,s}}} \hfill \\ {\left( {{\text{s}}1_{{{\text{i}},{\text{t}},{\text{s}}}} } \right) \times \, \left[ {\upvarepsilon_{{{\text{t}},{\text{s}}}} - {\text{P}}_{\text{s}} \times {\text{B}}_{\text{t}} \times \left( {{\text{PI}}_{{{\text{i}},{\text{t}},{\text{s}}}} - \left( {{\text{PI}}_{{{\text{i}},{\text{t}},{\text{s}}}} /{\text{QI}}_{{{\text{i}},{\text{t}},{\text{s}}}} } \right) \times \left( {2 \times {\text{s}}1_{{{\text{i}},{\text{t}},{\text{s}}}} + \sum\nolimits_{\text{f}} {{\text{s}}3_{{{\text{f}},{\text{i}},{\text{t}},{\text{s}}}} } } \right)} \right)} \right] = 0} \hfill & {\forall\,{\text{i}},{\text{t,s}}} \hfill \\ \end{array} $$
(22)
$$ \begin{array}{*{20}l} {{\text{For}}\;{\text{x}}1_{{{\text{t}},{\text{s}}}} \ge 0,} \hfill & {} \hfill \\ {\left[ {\updelta1_{\text{t,s}} -\upvarepsilon_{\text{t,s}} + {\text{P}}_{s} \times {\text{Bt}}_{t} \times {\text{CT}}1} \right] \ge 0} \hfill & {\forall\,{\text{t,s}}} \hfill \\ {\left( {{\text{x}}1_{{{\text{t}},{\text{s}}}} } \right) \times \left[ {\updelta1_{{{\text{t}},{\text{s}}}} -\upvarepsilon_{{{\text{t}},{\text{s}}}} + {\text{P}}_{\text{s}} \times {\text{B}}_{\text{t}} \times {\text{CT}}1} \right] = 0} \hfill & {\forall\,{\text{t,s}}} \hfill \\ \end{array} $$
(23)
$$ \begin{array}{*{20}l} {{\text{For}}\;{\text{x}}2_{{{\text{t}},{\text{s}}}} \ge 0,} \hfill & {} \hfill \\ {\left[ {\updelta2_{{{\text{t}},{\text{s}}}} -\upvarepsilon_{{{\text{t}},{\text{s}}}} + {\text{P}}_{\text{s}} \times {\text{B}}_{\text{t}} \times {\text{CT}}1} \right] \ge 0} \hfill & {\forall\,{\text{t,s}}} \hfill \\ {\left( {{\text{x}}2_{{{\text{t}},{\text{s}}}} } \right) \times \left[ {\updelta2_{{{\text{t}},{\text{s}}}} -\upvarepsilon_{{{\text{t}},{\text{s}}}} + {\text{P}}_{\text{s}} \times {\text{t}}_{\text{t}} \times {\text{CT}}1} \right] = 0} \hfill & {\forall\,{\text{t,s}}} \hfill \\ \end{array} $$
(24)
$$ \begin{array}{*{20}l} {{\text{For}}\;\upalpha1_{{{\text{t}},{\text{s}}}} \ge 0,} \hfill & {} \hfill \\ {\left[ {{\text{ST}}1 - {\text{d}}1_{{{\text{t}},{\text{s}}}} } \right] \ge 0} \hfill & {\forall\,{\text{t,s}}} \hfill \\ {\left( {\upalpha1_{{{\text{t}},{\text{s}}}} } \right) \times \left[ {{\text{ST}}1 - {\text{d}}1_{{{\text{t}},{\text{s}}}} } \right] = 0} \hfill & {\forall\,{\text{t,s}}} \hfill \\ \end{array} $$
(25)
$$ \begin{array}{*{20}l} {{\text{For}}\;\upalpha2_{{{\text{t}},{\text{s}}}} \ge 0,} \hfill & {} \hfill \\ {\left[ {{\text{ST}}2 - {\text{d}}2_{{{\text{t}},{\text{s}}}} } \right] \ge 0} \hfill & {\forall\,{\text{t,s}}} \hfill \\ {\left( {\upalpha2_{{{\text{t}},{\text{s}}}} } \right) \times \left[ {{\text{ST}}2 - {\text{d}}2_{{{\text{t}},{\text{s}}}} } \right] = 0} \hfill & {\forall\,{\text{t,s}}} \hfill \\ \end{array} $$
(26)
$$ \begin{array}{*{20}l} {{\text{For}}\;\upbeta1_{{{\text{t}},{\text{s}}}} \ge 0,} \hfill & {} \hfill \\ {\left[ {{\text{o}}1_{{{\text{t}},{\text{s}}}} - {\text{DM}}1_{{{\text{t}},{\text{s}}}} } \right] \ge 0} \hfill & {\forall\,{\text{t,s}}} \hfill \\ {\left( {\upbeta1_{{{\text{t}},{\text{s}}}} } \right) \times \left[ {{\text{o}}1_{{{\text{t}},{\text{s}}}} - {\text{DM}}1_{{{\text{t}},{\text{s}}}} } \right] = 0} \hfill & {\forall\,{\text{t,s}}} \hfill \\ \end{array} $$
(27)
$$ \begin{array}{*{20}l} {{\text{For}}\;\upbeta2_{{{\text{t}},{\text{s}}}} \ge 0,} \hfill & {} \hfill \\ {\left[ {{\text{o}}2_{{{\text{t}},{\text{s}}}} - {\text{DM}}2_{{{\text{t}},{\text{s}}}} } \right] \ge 0} \hfill & {\forall\,{\text{t,s}}} \hfill \\ {\left( {\upbeta2_{{{\text{t}},{\text{s}}}} } \right) \times \left[ {{\text{o}}2_{{{\text{t}},{\text{s}}}} - {\text{DM}}2_{{{\text{t}},{\text{s}}}} } \right] = 0} \hfill & {\forall\,{\text{t,s}}} \hfill \\ \end{array} $$
(28)
$$ \begin{array}{*{20}l} {{\text{For}}\;\upgamma1_{{{\text{t}},{\text{s}}}} \;{\text{is}}\;{\text{unrestricted}},} \hfill & {} \hfill \\ {{\text{d}}1_{{{\text{t}} + 1,{\text{s}}}} = {\text{d}}1_{{{\text{t}},{\text{s}}}} + {\text{B}}_{t} \times \left( {{\text{I}}1_{{{\text{t}},{\text{s}}}} - {\text{o}}1_{{{\text{t}},{\text{s}}}} - {\text{f}}1_{{{\text{t}},{\text{s}}}} } \right)} \hfill & {\forall\,{\text{t,s}}} \hfill \\ \end{array} $$
(29)
$$ \begin{array}{*{20}l} {{\text{For}}\;\upgamma2_{{{\text{t}},{\text{s}}}} \;{\text{is}}\;{\text{unrestricted}},} \hfill & {} \hfill \\ {{\text{d}}2_{{{\text{t}} + 1,{\text{s}}}} = {\text{d}}2_{{{\text{t}},{\text{s}}}} + {\text{B}}_{\text{t}} \left( {{\text{I}}2_{{{\text{t}},{\text{s}}}} - {\text{o}}2_{{{\text{t}},{\text{s}}}} + {\text{f}}1_{{{\text{t}},{\text{s}}}} } \right)} \hfill & {\forall\,{\text{t,s}}} \hfill \\ \end{array} $$
(30)
$$ \begin{array}{*{20}l} {{\text{For}}\;\updelta1_{{{\text{t}},{\text{s}}}} \ge 0,} \hfill & {} \hfill \\ {\left[ {{\text{HP}} \times \left( {{\text{o}}1_{{{\text{t}},{\text{s}}}} + {\text{f}}1_{{{\text{t}},{\text{s}}}} } \right) - {\text{x}}1_{{{\text{t}},{\text{s}}}} } \right] \ge 0} \hfill & {\forall\,{\text{t,s}}} \hfill \\ {\left( {\updelta1_{{{\text{t}},{\text{s}}}} } \right) \times \left[ {{\text{HP}} \times \left( {{\text{o}}1_{{{\text{t}},{\text{s}}}} + {\text{f}}1_{{{\text{t}},{\text{s}}}} } \right) - {\text{x}}1_{{{\text{t}},{\text{s}}}} } \right] = 0} \hfill & {\forall\,{\text{t,s}}} \hfill \\ \end{array} $$
(31)
$$ \begin{array}{*{20}l} {{\text{For}}\;\updelta2_{{{\text{t}},{\text{s}}}} \ge 0,} \hfill & {} \hfill \\ {\left[ {{\text{HP}} \times {\text{o}}2_{{{\text{t}},{\text{s}}}} - {\text{x}}2_{{{\text{t}},{\text{s}}}} } \right] \ge 0} \hfill & {\forall\,{\text{t,s}}} \hfill \\ {\left( {\updelta2_{{{\text{t}},{\text{s}}}} } \right) \times \left[ {{\text{HP}} \times {\text{o}}2_{{{\text{t}},{\text{s}}}} - {\text{x}}2_{{{\text{t}},{\text{s}}}} } \right] = 0} \hfill & {\forall\,{\text{t,s}}} \hfill \\ \end{array} $$
(32)
$$ \begin{array}{*{20}l} {{\text{For}}\;\upvarepsilon_{{{\text{t}},{\text{s}}}} \;{\text{is}}\;{\text{unrestricted}},} \hfill & {} \hfill \\ {\sum\nolimits_{i} {{\text{s}}1_{{{\text{i}},{\text{t}},{\text{s}}}} } = {\text{x}}1_{{{\text{t}},{\text{s}}}} + {\text{x}}2_{{{\text{t}},{\text{s}}}} } \hfill & {\forall\,{\text{t,s}}} \hfill \\ \end{array} $$
(33)
$$ \begin{array}{*{20}l} {{\text{For}}\;{\text{s}}3_{{{\text{f}},{\text{i}},{\text{t}},{\text{s}}}} \ge 0,} \hfill & {} \hfill \\ {\left[ {\uplambda_{{{\text{f}},{\text{t}},{\text{s}}}} - {\text{P}}_{\text{s}} \times {\text{B}}_{\text{t}} \times \left( {{\text{PI}}_{{{\text{i}},{\text{t}},{\text{s}}}} - \left( {{\text{PI}}_{{{\text{i}},{\text{t}},{\text{s}}}} /{\text{QI}}_{{{\text{i}},{\text{t}},{\text{s}}}} } \right) \times \left( {\sum\nolimits_{\text{f}} {{\text{s}}3_{{{\text{f}},{\text{i}},{\text{t}},{\text{s}}}} } + {\text{s}}1_{{{\text{i}},{\text{t}},{\text{s}}}} + {\text{s}}3_{{{\text{f}},{\text{i}},{\text{t}},{\text{s}}}} } \right)} \right)} \right] \ge 0} \hfill & {\forall\,{\text{f}},{\text{i}},{\text{t,s}}} \hfill \\ {\left( {{\text{s}}3_{{{\text{f}},{\text{i}},{\text{t}},{\text{s}}}} } \right) \times \left[ {\uplambda_{{{\text{f}},{\text{t}},{\text{s}}}} - {\text{P}}_{\text{s}} \times {\text{B}}_{\text{t}} \times \left( {{\text{PI}}_{{{\text{i}},{\text{t}},{\text{s}}}} - \left( {{\text{PI}}_{{{\text{i}},{\text{t}},{\text{s}}}} /{\text{QI}}_{{{\text{i}},{\text{t}},{\text{s}}}} } \right) \times \left( {\sum\nolimits_{\text{f}} {{\text{s}}3_{{{\text{f}},{\text{i}},{\text{t}},{\text{s}}}} } + {\text{s}}1_{{{\text{i}},{\text{t}},{\text{s}}}} + {\text{s}}3_{{{\text{f}},{\text{i}},{\text{t}},{\text{s}}}} } \right)} \right]} \right] = 0} \hfill & {\forall\,{\text{f}},{\text{i}},{\text{t,s}}} \hfill \\ \end{array} $$
(34)
$$ \begin{array}{*{20}l} {{\text{For}}\;{\text{x}}3_{{{\text{f}},{\text{i}},{\text{h}},{\text{t}},{\text{s}}}} \ge 0,} \hfill & {} \hfill \\ {\left[ {\uptheta_{{{\text{f}},{\text{i}},{\text{h}},{\text{t}},{\text{s}}}} -\uplambda_{{{\text{f}},{\text{t}},{\text{s}}}} + {\text{WT}}_{{{\text{f}},{\text{i}},{\text{h}}}} \times\upmu_{{{\text{f}},{\text{i}},{\text{h}},{\text{t}},{\text{s}}}} + {\text{P}}_{\text{s}} \times {\text{B}}_{\text{t}} \times {\text{CT}}3_{{{\text{f}},{\text{i}},{\text{h}}}} } \right] \ge 0} \hfill & {\forall\,{\text{f}},{\text{i}},{\text{h}},{\text{t,s}}} \hfill \\ {\left( {{\text{x}}3_{{{\text{f}},{\text{i}},{\text{h}},{\text{t}},{\text{s}}}} } \right) \left[ {\uptheta_{{{\text{f}},{\text{i}},{\text{h}},{\text{t}},{\text{s}}}} -\uplambda_{{{\text{f}},{\text{t}},{\text{s}}}} + {\text{WT}}_{{{\text{f}},{\text{i}},{\text{h}}}} \times\upmu_{{{\text{f}},{\text{i}},{\text{h}},{\text{t}},{\text{s}}}} + {\text{P}}_{\text{s}} \times {\text{B}}_{\text{t}} \times {\text{CT}}3_{{{\text{f}},{\text{i}},{\text{h}}}} } \right] = 0} \hfill & {\forall\,{\text{f}},{\text{i}},{\text{h}},{\text{t,s}}} \hfill \\ \end{array} $$
(35)
$$ \begin{array}{*{20}l} {{\text{For}}\;\text{y}3_{{{\text{f}},{\text{i}},{\text{h}},{\text{t}},{\text{s}}}} \ge 0,} \hfill & {} \hfill \\ {\left[ { -\upmu_{{{\text{f}},{\text{i}},{\text{h}},{\text{t}},{\text{s}}}} + {\text{P}}_{\text{s}} \times {\text{B}}_{\text{t}} \times {\text{CT}}4_{{{\text{f}},{\text{i}},{\text{h}}}} } \right] \ge 0} \hfill & {\forall\,{\text{f}},{\text{i}},{\text{h}},{\text{t,s}}} \hfill \\ {\left( {{\text{y}}3_{{{\text{f}},{\text{i}},{\text{h}},{\text{t}},{\text{s}}}} } \right) \times \left[ { -\upmu_{{{\text{f}},{\text{i}},{\text{h}},{\text{t}},{\text{s}}}} + {\text{P}}_{\text{s}} \times {\text{B}}_{\text{t}} \times {\text{CT}}4_{{{\text{f}},{\text{i}},{\text{h}}}} } \right] = 0} \hfill & {\forall\,{\text{f}},{\text{i}},{\text{h}},{\text{t,s}}} \hfill \\ \end{array} $$
(36)
$$ \begin{array}{*{20}l} {{\text{For}}\;\uptheta_{{{\text{f}},{\text{i}},{\text{h}},{\text{t}},{\text{s}}}} \ge 0,} \hfill & {} \hfill \\ {\left[ {{\text{CP}}_{{{\text{f}},{\text{i}},{\text{h}}}} - {\text{x}}3_{{{\text{f}},{\text{i}},{\text{h}},{\text{t}},{\text{s}}}} } \right] \ge 0} \hfill & {\forall\,{\text{f}},{\text{i}},{\text{h}},{\text{t,s}}} \hfill \\ {\left( {\uptheta_{{{\text{f}},{\text{i}},{\text{h}},{\text{t}},{\text{s}}}} } \right) \times \left[ {{\text{CP}}_{{{\text{f}},{\text{i}},{\text{h}}}} - {\text{x}}3_{{{\text{f}},{\text{i}},{\text{h}},{\text{t}},{\text{s}}}} } \right] = 0} \hfill & {\forall\,{\text{f}},{\text{i}},{\text{h}},{\text{t,s}}} \hfill \\ \end{array} $$
(37)
$$ \begin{array}{*{20}l} {{\text{For}}\;\uplambda_{{{\text{f}},{\text{t,s}}}} \;{\text{is}}\;{\text{unrestricted}},} \hfill & {} \hfill \\ {\sum\nolimits_{\text{i}} {{\text{s}}3_{{{\text{f}},{\text{i}},{\text{t}},{\text{s}}}} } - \sum\nolimits_{{{\text{i}},{\text{h}}}} {{\text{x}}3_{{{\text{f}},{\text{i}},{\text{h}},{\text{t}},{\text{s}}}} } = 0} \hfill & {\forall\,{\text{f}},{\text{t,s}}} \hfill \\ \end{array} $$
(38)
$$ \begin{array}{*{20}l} {{\text{For}}\;\upmu_{{{\text{f}},{\text{i}},{\text{h}},{\text{t,s}}}} \;{\text{is}}\;{\text{unrestricted}},} \hfill & {} \hfill \\ {{\text{WT}}_{{{\text{f}},{\text{i}},{\text{h}}}} \times {\text{x}}3_{{{\text{f}},{\text{i}},{\text{h}},{\text{t}},{\text{s}}}} - {\text{y}}3_{{{\text{f}},{\text{i}},{\text{h}},{\text{t}},{\text{s}}}} = 0} \hfill & {\forall\,{\text{f}},{\text{i}},{\text{h}},{\text{t,s}}} \hfill \\ \end{array} $$
(39)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hu, MC., Huang, T., Yu, HL. et al. Stochastic competitive analysis of hydropower and water supplies within an energy–water nexus. Stoch Environ Res Risk Assess 32, 2761–2769 (2018). https://doi.org/10.1007/s00477-017-1500-2

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00477-017-1500-2

Keywords

Search

Navigation