Reservoir Design and Operation for the Food-Energy-Water Nexus

Abstract

As populations grow concurrently with changing climates, expanding economies and urbanization, competition for food, energy, and water resources increases. The intersection of these areas, sometimes referred to as the food-energy-water nexus, poses significant challenges. Using mixed-integer linear programming, this paper considers the impact of nexus decisions related to agricultural irrigation, water storage, and power generation on a river basin in northeastern Colorado. The model minimizes the cost of mitigating agricultural water shortages by designing additional storage for, and assigning flow of, excess water while identifying the location of the highest, most consistent volume to facilitate thermal power generation, while adhering to physical and topographical constraints that govern the movement of the river. We find that the optimal solution is a series of small reservoirs (cumulative storage volume of 31,023 acre-feet) to mitigate unmet agricultural demands, and the lowermost portion of the river has the highest, most consistent flow to facilitate thermal power generation. However, there exists enough water in the river during the time horizon of the study to support energy generation at any point along the river. Our optimization model can be used by long-range planners to make strategic food, energy, and water infrastructure decisions.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

References

  1. 1.

    Abbott M, Bazilian M, Egel D, Willis HH. Examining the food–energy–water and conflict nexus. Current Opinion Chem Eng 2017;18:55–60.

    Article  Google Scholar 

  2. 2.

    Hague W. 2010. The diplomacy of climate change. Speech to the Council on Foreign Relations.

  3. 3.

    United Nations. 2018. U.N. Water: Coordinating the UN’s work on water and sanitation. http://www.unwater.org/water-facts/water-food-and-energy/, accessed: 2018-08-17.

  4. 4.

    Waskom R, Akhbari M, Grigg N. 2014. U.S. Perspective on the Water-Energy-Food Nexus. Tech. rep., Colorado Water Institute.

  5. 5.

    Howells M, Hermann S, Welsch M, Bazilian M, Segerström R, Alfstad T, Gielen D, Rogner H, Fischer G, Van Velthuizen H, et al. Integrated analysis of climate change, land-use, energy and water strategies. Nature Climate Change 2013;3(7):621.

    Article  Google Scholar 

  6. 6.

    Kaddoura S, El Khatib S. Review of water-energy-food nexus tools to improve the nexus modeling approach for integrated policy making. Environ Sci Policy 2017;77:114–121.

    Article  Google Scholar 

  7. 7.

    Inas E, Grigg N, Waskom R. Water-food-energy: nexus and non-nexus approaches for optimal cropping pattern. Water Res Manag 2017;31(15):4971–4980.

    Article  Google Scholar 

  8. 8.

    Office of Energy Policy and Systems Analysis. 2017. The Quadrennial Energy Review: Environmental Baseline Volume 4 Energy-Water Nexus. Tech. rep., U.S. Department of Energy.

  9. 9.

    Daher B, Hanibal B, Portney KE, Mohtar RH. 2018. Toward creating an environment of cooperation between water, energy, and food stakeholders in San Antonio. Science of The Total Environment.

  10. 10.

    Hellegers P, Zilberman D, Steduto P, McCornick P. Interactions between water, energy, food and environment: evolving perspectives and policy issues. Water Policy 2008;10(S1):1–10.

    Article  Google Scholar 

  11. 11.

    Sandia National Laboratories. 2015. Energy: secure and sustainable energy future. https://energy.sandia.gov/climate-earth-systems/energy-water-nexus/, accessed: 2018-08-14.

  12. 12.

    Nonhebel S. Renewable energy and food supply: will there be enough land?. Renew Sustainable Energy Rev 2005;9(2):191–201.

    Article  Google Scholar 

  13. 13.

    Khan S, Hanjra MA. Footprints of water and energy inputs in food production–Global perspectives. Food Policy 2009;34(2):130–140.

    Article  Google Scholar 

  14. 14.

    Dieter C, Maupin M, Caldwell R, Harris M, Ivahnenko T, Lovelace J, Barber N, Linsey K. 2018. Estimated use of water in the United States in 2015. Tech. rep., US Geological Survey.

  15. 15.

    US Environmental Protection Agency. 2013. Drinking Water Infrastructure Needs Survey and Assessment Fifth Report to Congress 2013. Tech. rep., U.S. Environmental Protection Agency.

  16. 16.

    Welsch M, Hermann S, Howells M, Rogner HH, Young C, Ramma I, Bazilian M, Fischer G, Alfstad T, Gielen D, et al. Adding value with CLEWS–Modeling the energy system and its interdependencies for Mauritius. Appl energy 2014;113:1434–1445.

    Article  Google Scholar 

  17. 17.

    Sachs I, Silk D, et al. Food and energy: strategies for sustainable development. Tokyo: United Nations University Press; 1990.

    Google Scholar 

  18. 18.

    Endo A, Tsurita I, Burnett K, Orencio PM. A review of the current state of research on the water, energy, and food nexus. J Hydrology: Regional Stud 2017;11:20–30.

    Google Scholar 

  19. 19.

    Hoff H. 2011. Understanding the nexus: background paper for the Bonn 2011Nexus Conference.

  20. 20.

    Albrecht TR, Crootof A, Scott CA. The Water-Energy-Food Nexus: a systematic review of methods for nexus assessment. Environ Res Lett 2018;13(4):043002.

    Article  Google Scholar 

  21. 21.

    Bonsch M, Humpenöder F, Popp A, Bodirsky B, Dietrich JP, Rolinski S, Biewald A, Lotze-Campen H, Weindl I, Gerten D, et al. Trade-offs between land and water requirements for large-scale bioenergy production. GCB Bioenergy 2016;8(1):11–24.

    Article  Google Scholar 

  22. 22.

    Karlberg L, Hoff H, Amsalu T, Andersson K, Binnington T, Flores-López F, de Bruin A, Gebrehiwot SG, Gedif B, Johnson O, et al. Tackling complexity: understanding the food-energy-environment nexus in Ethiopia’s Lake Tana sub-basin. Water Alternatives 2015;8:1.

    Google Scholar 

  23. 23.

    Walsh MJ, Van Doren LG, Sills DL, Archibald I, Beal CM, Lei XG, Huntley ME, Johnson Z, Greene CH. Algal food and fuel coproduction can mitigate greenhouse gas emissions while improving land and water-use efficiency. Environ Res Lett 2016;11(11):114006.

    Article  Google Scholar 

  24. 24.

    Bekchanov M, Lamers J. The effect of energy constraints on water allocation decisions: The elaboration and application of a system-wide economic-water-energy model (SEWEM). Water 2016;8(6):253.

    Article  Google Scholar 

  25. 25.

    Yang YE, Ringler C, Brown C, Mondal MAH. Modeling the Agricultural Water–Energy–Food Nexus in the Indus River Basin, Pakistan. J Water Res Plan Manag 2016;142(12):04016062.

    Article  Google Scholar 

  26. 26.

    Endo A, Burnett K, Orencio PM, Kumazawa T, Wada CA, Ishii A, Tsurita I, Taniguchi M. Methods of the water-energy-food nexus. Water 2015;7(10):5806–5830.

    Article  Google Scholar 

  27. 27.

    Bazilian M, Rogner H, Howells M, Hermann S, Arent D, Gielen D, Steduto P, Mueller A, Komor P, Tol RS, et al. Considering the energy, water and food nexus: towards an integrated modeling approach. Energy Policy 2011;39(12):7896–7906.

    Article  Google Scholar 

  28. 28.

    International Atomic Energy Agency. 2009. Annex VI: Seeking sustainable climate, land, energy and water (CLEW) strategies. Tech. rep., International Atomic Energy Agency.

  29. 29.

    Colorado Water Conservation Board. 2018. Colorado’s Water Plan: Basins. https://www.colorado.gov/pacific/cowaterplan/basins/, accessed: 2018-09-10.

  30. 30.

    The World Bank. 2018. Thirsty Energy: Water-Smart Energy Planning in South Africa. http://www.worldbank.org/en/news/feature/2017/06/15/thirsty-energy-water-smart-energy-planning-in-south-africa/ http://www.worldbank.org/en/news/feature/2017/06/15/thirsty-energy-water-smart-energy-planning-in-south-africa/, accessed: 2018-10-01.

  31. 31.

    Burrow A, Newman A. 2018. Optimal Design and Operation of River Basin Storage. Omega, under revision.

  32. 32.

    Meyer B. 2017. Colorado Agricultural Statistics 2017. Tech. rep., United States Department of Agriculture.

  33. 33.

    Schantz B. 2018. Personal communication on 2018-09-05 about South Platte diversions below the Kersey Gauge.

  34. 34.

    Johnson G. 2018. Scenario Planning and Gap Analysis Methodology. Tech. rep., Colorado Water Conservation Board.

  35. 35.

    Water Encyclopedia. 2018. Prior Appropriation. http://www.waterencyclopedia.com/Po-Re/Prior-Appropriation.html/ http://www.waterencyclopedia.com/Po-Re/Prior-Appropriation.html/, accessed: 2018-10-02.

  36. 36.

    Allen T. 2015. Colorado’s Water Plan. Tech. rep., Colorado Water Conservation Board.

  37. 37.

    Colorado House of Representatives. 2016. South Platte Water Storage Study.

  38. 38.

    Cook M. 2015. South Platte Basin Implementation Plan. Tech. rep. West Sage Water Consultants.

  39. 39.

    Burrow A, Newman A, Hering M, Morton D. 2018. Optimal Design and Operation of River Basin Storage Under Stochastic Conditions. Journal of Water Resources Planning and Management, working paper.

  40. 40.

    Macknick J, Newmark R, Heath G, Hallett KC. Operational water consumption and withdrawal factors for electricity generating technologies: a review of existing literature. Environ Res Lett 2012;7(4):045802.

    Article  Google Scholar 

  41. 41.

    Martin A. 2015. Water for thermal power plants: understanding a piece of the water energy nexus. http://www.globalwaterforum.org/2015/06/22/water-for-thermal-power-plants-understanding-a-piece-of-the-water-energy-nexus/ http://www.globalwaterforum.org/2015/06/22/water-for-thermal-power-plants-understanding-a-piece-of-the-water-energy-nexus/ http://www.globalwaterforum.org/2015/06/22/water-for-thermal-power-plants-understanding-a-piece-of-the-water-energy-nexus/, accessed: 2018-09-07.

  42. 42.

    Cohen S. 2018. Personal communication on 2018-08-30 about thermal power plants in Colorado.

  43. 43.

    The USGS Water Science School. 2018. Hydroelectric power water use. https://water.usgs.gov/edu/wuhy.html/, accessed: 2018-10-01.

  44. 44.

    Kenny JF, Barber NL, Hutson SS, Linsey KS, Lovelace JK, Maupin MA. 2009. Estimated use of water in the United States in 2005. Tech. rep., US Geological Survey.

  45. 45.

    Xcel Energy. 2018. Pawnee Generating Station. https://www.xcelenergy.com/energy_portfolio/electricity/power_plants/pawnee/, accessed: 2018-09-10.

  46. 46.

    Issakhov A. Mathematical modeling of the discharged heat water effect on the aquatic environment from thermal power plant under various operational capacities. Appl Math Model 2016;40(2):1082–1096.

    MathSciNet  Article  Google Scholar 

  47. 47.

    Lyubimova T, Lepikhin A, Parshakova Y, Lyakhin Y, Tiunov A. The modeling of the formation of technogenic thermal pollution zones in large reservoirs. Int J Heat Mass Trans 2018;126:342–352.

    Article  Google Scholar 

  48. 48.

    Raptis CE, van Vliet MT, Pfister S. Global thermal pollution of rivers from thermoelectric power plants. Environ Res Lett 2016;11(10):104011.

    Article  Google Scholar 

  49. 49.

    Raptis CE, Boucher JM, Pfister S. Assessing the environmental impacts of freshwater thermal pollution from global power generation in LCA. Sci Total Environ 2017;580:1014– 1026.

    Article  Google Scholar 

  50. 50.

    Colorado Water Conservation Board. 2016. State of Colorado’s Water Supply Model (StateMod) Version 15.

  51. 51.

    Carrión M, Arroyo JM. A computationally efficient mixed-integer linear formulation for the thermal unit commitment problem. IEEE Trans Power Syst 2006;21(3):1371–1378.

    Article  Google Scholar 

  52. 52.

    Lara CL, Mallapragada DS, Papageorgiou DJ, Venkatesh A, Grossmann IE. Deterministic electric power infrastructure planning: mixed-integer programming model and nested decomposition algorithm. European J Oper Res 2018;271(3):1037–1054.

    MathSciNet  Article  MATH  Google Scholar 

  53. 53.

    Samsatli S, Samsatli NJ. A general mixed integer linear programming model for the design and operation of integrated urban energy systems. J Cleaner Prod 2018;191:458–479.

    Article  Google Scholar 

  54. 54.

    US Energy Information Administration. 2018. Today in Energy. https://www.eia.gov/todayinenergy/detail.php/id=25652/, accessed: 2018-09-21.

  55. 55.

    Nosratabadi SM, Hooshmand RA, Gholipour E. A comprehensive review on microgrid and virtual power plant concepts employed for distributed energy resources scheduling in power systems. Renew Sustainable Energy Rev 2017;67:341–363.

    Article  Google Scholar 

  56. 56.

    Ondeck AD, Edgar TF, Baldea M. Optimal operation of a residential district-level combined photovoltaic/natural gas power and cooling system. Applied Energy 2015;156:593–606.

    Article  Google Scholar 

  57. 57.

    Tajeddini MA, Rahimi-Kian A, Soroudi A. Risk averse optimal operation of a virtual power plant using two stage stochastic programming. Energy 2014;73:958–967.

    Article  Google Scholar 

  58. 58.

    Wood AJ, Wollenberg BF. Power generation, operation, and control. Hoboken: Wiley; 2012.

    Google Scholar 

  59. 59.

    US Energy Information Administration. 2018. Electricity. https://www.eia.gov/electricity/wholesale//, accessed: 2018-09-21.

  60. 60.

    Biliyok C, Yeung H. Evaluation of natural gas combined cycle power plant for post-combustion CO2 capture integration. Int J Greenhouse Gas Control 2013;19:396–405.

    Article  Google Scholar 

  61. 61.

    Etxegarai A, Eguia P, Torres E, Iturregi A, Valverde V. Review of grid connection requirements for generation assets in weak power grids. Renew Sustainable Energy Rev 2015;41:1501– 1514.

    Article  Google Scholar 

  62. 62.

    Sipöcz N, Tobiesen FA. Natural gas combined cycle power plants with CO2 capture–opportunities to reduce cost. Int J Greenhouse Gas Control 2012;7:98–106.

    Article  Google Scholar 

  63. 63.

    Spath PL, Mann MK. 2000. Life cycle assessment of a natural gas combined-cycle power generation system. National Renewable Energy Laboratory Golden, CO.

  64. 64.

    Conejo AJ, Contreras J, Espinola R, Plazas MA. Forecasting electricity prices for a day-ahead pool-based electric energy market. Int J Forecasting 2005;21(3):435–462.

    Article  Google Scholar 

  65. 65.

    Dowling AW, Kumar R, Zavala VM. A multi-scale optimization framework for electricity market participation. Appl Energy 2017;190:147–164.

    Article  Google Scholar 

  66. 66.

    Palensky P, Dietrich D. Demand side management: demand response, intelligent energy systems, and smart loads. IEEE Trans Industrial Inform 2011;7(3):381–388.

    Article  Google Scholar 

  67. 67.

    Vahid-Pakdel M, Nojavan S, Mohammadi-Ivatloo B, Zare K. Stochastic optimization of energy hub operation with consideration of thermal energy market and demand response. Energy Conversion Manag 2017; 145:117–128.

    Article  Google Scholar 

  68. 68.

    US Energy Information Administration. 2018. What is the efficiency of different types of power plants? https://www.eia.gov/tools/faqs/faq.php?id=107&t=3/, accessed: 2018-09-21.

  69. 69.

    Paulson C. 2017. HB16-1256 South Platte Storage Study. Tech. rep., Stantec.

  70. 70.

    AMPL Optimization LLC. 2018. AMPL Version 20170207.

  71. 71.

    Fourer R, Gay D, Kernighan B. 2003. AMPL - A Modeling Language for Mathematical Programming. Thomson Brooks/Cole.

  72. 72.

    International Business Machines. 2018. CPLEX Version 12.7.0.0.

  73. 73.

    IBMILOG AMPL. 2010. Version 12.2 User’s Guide: Standard (Command-line) Version Including CPLEX Directives.

  74. 74.

    Pochet Y, Wolsey LA. 2006. Production planning by mixed integer programming. Springer Science & Business Media.

  75. 75.

    Cullenbine C, Wood RK, Newman A. A sliding time window heuristic for open pit mine block sequencing. Optim Lett 2011;5(3):365–377.

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgements

We thank the following individuals for their direction and insights: (1) Stuart Cohen, National Renewable Energy Laboratory, (2) Kelly Eurek, National Renewable Energy Laboratory, (3) Andres Guerra, Department of Civil and Environmental Engineering, Colorado School of Mines, (4) William Hamilton, Department of Mechanical Engineering, Colorado School of Mines, (5) Tissa Illangasekare, Department of Civil and Environmental Engineering, Colorado School of Mines, (6) Wil Kircher (Petroleum engineer), (7) Dinesh Mehta, Department of Computer Science, Colorado School of Mines, (8) Brent Schantz, Colorado Department of Natural Resources, and (9) Regan Waskom, Colorado Water Institute, Colorado State University - Fort Collins.

Funding

The authors gratefully acknowledge the financial sponsorship of the Veteran’s Administration via the G.I. Bill, as well as a grant from the Colorado Water Conservation Board and the South Platte Basin Roundtable (POGGI-2017-906).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Alexandra Newman.

Ethics declarations

Conflict of interests

The authors declare that they have no conflict of interest.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article is part of the Topical Collection on Energy Markets

Appendices

Appendix A: Model

A MILP formulation, (\(\mathcal {P}\)), combines costs, penalties, excess supply, unmet demands, topography, and capacities to determine the location of new storage and corresponding flow needed to meet agricultural demand and support thermal power plant operation while exploiting revenue gained from potential electricity generation. The objective function minimizes: (1) construction as well as operation and maintenance costs of new infrastructure, (2) unmet demand penalties, and (3) penalties for exceeding demand while maximizing revenue generation.

Sets

\(t \in \mathcal {T}\) set of all monthly time periods
\(t \in \mathcal {T}^{\prime }\) set of all time periods in which the South
  Platte Compact applies
\(d \in \mathcal {D}_t\) set of all demand sites at time t
\(n \in N_{t} \subset \mathcal {D}_{t}\) set of all demand nodes at time t and within
  the lower half of the river (District 64) that
  hold a water right junior to that of the South
  Platte Compact
\(r \in \mathcal {R}\) set of all reservoirs which includes existing
  and potential, where r is a downstream reser
  voir from r
\(\mathcal {E}\subset \mathcal {R}\) set of existing reservoirs (surface only)
\(\mathcal {P}^s\subset \mathcal {R}\) set of potential reservoirs (surface only)
\(\mathcal {P}^u\subset \mathcal {R}\) set of potential reservoirs (underground only)
\(s \in \mathcal {S}_t\) set of all supply sites at time t
\(j \in \mathcal {J}\) set of all pipeline flow capacities

Parameters

\(f_{jr^{\prime }rt}\) fixed cost to construct a pipeline with flow level j
  from reservoir r to upstream reservoir r in time
  t ($)
\(\hat {f}_{jr^{\prime }rt}\) fixed monthly cost to maintain a pipeline with flow
  level j from reservoir r to upstream reservoir r in
  time t ($)
\(\mathring {f}_{jr^{\prime } rt}\) fixed monthly cost to operate a pipeline with flow
  level j from reservoir r to upstream reservoir r in
  time t ($)
\(\breve {u}_{rt}\) per unit cost to store water in reservoir r in time t
  ($/AF)
ů r t per unit cost to operate and maintain reservoir site
  r in time t ($/AF)
\(\hat {c}_{st}\) capacity of supply site s in time t (AF)
\(\bar {c}_{rt}\) maximum capacity of reservoir r in time t (AF)
\(\underline {c}_{rt}\) minimum capacity of reservoir r in time t (AF)
\(c_{r}^{s}\) capacity of the reservoir discharge ditch r (AF)
c r capacity of the South Platte River (AF)
\(c^{p}_{j}\) capacity of a water pipeline with flow level j (AF)
\(c^{d}_{r}\) capacity of the intake diversion ditch at reservoir r
  (AF)
d d t demand at site d in time period t (AF)
\(v^{+}_{dt}\) volume of river water entering demand site d in
  time period t (AF)
\(v^{-}_{st}\) volume of river water leaving supply site s in time
  period t (AF)
\(v^{d}_{rt}\) volume of river water already diverted into
  diversion ditch at reservoir r in time period t (AF)
\(v^{r}_{rt}\) volume of river water already released by reservoir
  r in time period t (AF)
p d t unmet demand penalty at site d in time period t
  ($/AF)
p e demand exceedance penalty ($/AF)
M “sufficiently large” value

Variables

\(\bar {X}_{sdt}=\) amount of water diverted from supply site s to
  demand site d in time t (AF)
\(\tilde {X}_{gsrt}=\) within reservoir group g, amount of water
  diverted from supply site s to reservoir r in
  time t (AF)
\(\hat {X}_{r^{\prime } rt}=\) amount of water pumped from reservoir r to
  upstream reservoir r in time t (AF)
\(\breve {X}_{gr}=\) within reservoir group g, maximum flow from
  reservoir r across all time periods (AF)
Ẍgrt = within reservoir group g, maximum flow from
  reservoir r by time t (AF)
\(Z^{+}_{dt}=\) amount in excess of demand at site d in time t
  (AF)
\(Z^{-}_{dt}=\) amount of the unmet demand at site d in time t
  (AF)
Irt = inventory amount for reservoir r in time t (AF)
\(\breve {W}_{grt}=\) within reservoir group g, maximum size of
  reservoir r in time t (AF)
$$ \mathring{Y}_{grt}=\!\hspace{5mm}\left\{ \begin{array}{l} 1\text{ if within group } g \text{ operation and maintenance}\\ \hspace{3mm} \text{costs are incurred at reservoir } r \text{ by time }\\ \hspace{3mm} \text{period }t\\ \hspace{0cm}0 \text{ otherwise} \end{array} \right. $$
$$ \tilde{Y}_{jr^{\prime} rt}=\hspace{3mm}\left\{ \begin{array}{l} 1\text{ if a pipeline with flow capacity } j\hspace{1mm} \text{ is }\\ \hspace{3mm} \text{constructed between reservoir } r^{\prime} \text{ and}\\ \hspace{3mm}\text{upstream reservoir } r \textit{ in} \text{ time period } \textit{t}\\ \hspace{0cm}0 \text{ otherwise} \end{array} \right. $$
$$ \tilde{Y}^{\prime}_{jr^{\prime} rt}=\hspace{3.3mm}\left\{ \begin{array}{l} 1\text{ if a pipeline with flow capacity } j\hspace{1mm} \text{ is}\\ \hspace{3mm} \text{constructed between reservoir } r^{\prime} \text{ and }\\ \hspace{3mm}\text{upstream reservoir } r \textit{ by } \text{ time period } t\\ \hspace{0cm}0 \text{ otherwise} \end{array} \right. $$
$$ \tilde{Y}^{\prime \prime}_{jr^{\prime} rt}=\!\hspace{3.5mm}\left\{ \begin{array}{l} 1\text{ if a pipeline with flow capacity } j\hspace{1mm} \text{ is used}\\ \hspace{3mm} \text{between reservoir }r^{\prime} \text{ and upstream reservoir }\\ \hspace{3mm}r \text{ in time period } t\\ \hspace{0cm}0 \text{ otherwise} \end{array} \right. $$
$$ \alpha_{rt}=\hspace{7mm}\left\{ \begin{array}{l} 1 \text{ if the amount of water stored in reservoir } r \\ \hspace{2mm}\text{ in time period } t \text{ is greater than the}\\ \hspace{2mm}\text{ minimum capacity of } r\\ \hspace{0cm}0 \text{ otherwise } \end{array} \right. $$
$$ \beta_{dt}=\hspace{8mm}\left\{ \begin{array}{l} 1 \text{ if the total amount of water allocated to }\\ \hspace{1.9mm} \text{ demand site } d \text{ in time period } t \text{ exceeds the}\\ \hspace{1.9mm}\text{ requirement of the South}\\ \hspace{2.5mm}\text{ Platte Compact}\\ \hspace{0cm}0 \text{ otherwise } \end{array} \right. $$
$$ W_{grt}=\hspace{6mm}\left\{ \begin{array}{l} 1\text{ if within group } g \text{ reservoir } r \text{ is first used at}\\ \hspace{2mm}\text{ time period } t\\ \hspace{0cm}0 \text{ otherwise} \end{array} \right. $$

Constraints

Capacities (see “Capacities”)

$$ \displaystyle \sum\limits_{s\leq r}\tilde{X}_{gsrt} \leq ({c^{d}_{r}}-v_{rt}^{d})Y_{grt}\hspace{7mm} \forall\hspace{3mm} g \in \mathcal{G}, r \in \mathcal{R}\text{ and } t \in \mathcal{T} $$
(A.1a)
$$ \hat{X}_{r^{\prime} rt}\leq \sum\limits_{j \in\mathcal{J}}{c^{p}_{j}}\tilde{Y}^{\prime \prime}_{jr^{\prime} rt}\hspace{3mm} \forall\hspace{3mm} r^{\prime}, r \in \mathcal{R}\text{ and } t \in \mathcal{T} \\ $$
(A.1b)
$$ \displaystyle \sum\limits_{s\leq d}\bar{X}_{sdt}+\sum\limits_{g \in\mathcal{G}}{\sum}_{r\leq d}X_{grdt}\!\leq\! c^{r}-v^{+}_{dt}\hspace{7mm} \forall\hspace{3mm} d \!\in\! \mathcal{D}_{t}\text{ and } t \!\in\! \mathcal{T} \\ $$
(A.1c)
$$ \displaystyle \sum\limits_{d\geq s}\bar{X}_{sdt}+\sum\limits_{g \in\mathcal{G}}\sum\limits_{r\geq s}\tilde{X}_{gsrt}\leq c^{r}-v^{-}_{st}\hspace{7mm} \forall\hspace{3mm} s\!\in\! \mathcal{S}_{t}\text{ and } t \!\in\! \mathcal{T} \\ $$
(A.1d)

Penalties (see “Penalties”)

$$ Z^{+}_{dt}\geq {\sum}_{g \in\mathcal{G}}{\sum}_{r\leq d}X_{grdt}+\sum\limits_{s\leq d}\bar{X}_{sdt}-d_{dt} \hspace{3mm} \forall\hspace{3mm}d \!\in\!\mathcal{D}_{t}\text{ and } t \in \mathcal{T} \\ $$
(A.2a)
$$ \displaystyle Z^{-}_{dt} \geq d_{dt}-\sum\limits_{g \in\mathcal{G}}{\sum}_{r\leq d}X_{grdt}-\sum\limits_{s\leq d}\bar{X}_{sdt}{}\hspace{3mm}\forall\hspace{3mm} d \!\in\! \mathcal{D}_{t}\text{ and } t \!\in\! \mathcal{T} \\ $$
(A.2b)

Flow Balance (see “Flow Balance”)

$$ \displaystyle \hat{c}_{st} = \sum\limits_{d\geq s}\bar{X}_{sdt}+\sum\limits_{g \in\mathcal{G}}\sum\limits_{r\geq s}\tilde{X}_{gsrt}\hspace{3mm}\forall\hspace{3mm} s \in \mathcal{S}_{t}\text{ and } t \in \mathcal{T} \\ $$
(A.3a)
$$ {\sum}_{g \in\mathcal{G}}\sum\limits_{d\geq r}X_{grdt}=0 \hspace{3mm}\forall\hspace{3mm} r \in \mathcal{R}\text{ and } t \in \mathcal{T}: t=1 $$
(A.3b)
$$ {\sum}_{r< r^{\prime}}\hat{X}_{r^{\prime} rt}=0 \hspace{3mm}\forall\hspace{3mm} r^{\prime} \in \mathcal{R}\text{ and } t \in \mathcal{T}: t=1 $$
(A.3c)

Inventory Flow (see “Inventory Flow”)

$$ I_{rt}\leq M\sum\limits_{g \in\mathcal{G}}Y_{grt} \hspace{3mm} \forall\hspace{3mm} r \in \mathcal{R}\text{ and } t \in \mathcal{T} $$
(A.4a)
$$ I_{rt}\geq \underline{c}_{rt}-M(1-\alpha_{rt}) \hspace{3mm} \forall\hspace{3mm} r \in \mathcal{R}\text{ and } t \in \mathcal{T} $$
(A.4b)
$$ \sum\limits_{d\geq r}X_{grdt} \leq M\alpha_{rt} \hspace{3mm} \forall\hspace{3mm} g \in \mathcal{G},r \in \mathcal{R}\text{ and } t \in \mathcal{T} $$
(A.4c)

Total Quantity of New Infrastructure (see “Total Quantity of New Infrastructure”)

$$ \displaystyle {\sum}_{j \in\mathcal{J}}\sum\limits_{t\in\mathcal{T}}\tilde{Y}_{jr^{\prime} rt} \leq 1 \hspace{3mm} \forall\hspace{3mm} r^{\prime} ,r \in\mathcal{R} $$
(A.5a)
$$ \displaystyle \sum\limits_{r\in\mathcal{P}^{s}}Y_{1rt} \leq 1\hspace{3mm} \forall\hspace{3mm}t \in\mathcal{T} $$
(A.5b)
$$ \displaystyle \sum\limits_{r\in\mathcal{E}}Y_{1rt} \leq 1\hspace{3mm} \forall\hspace{3mm}t \in\mathcal{T} $$
(A.5c)
$$ \displaystyle \sum\limits_{r\in\mathcal{P}^{u}}Y_{1rt} \leq 1\hspace{3mm} \forall\hspace{3mm}t \in\mathcal{T} $$
(A.5d)
$$ \displaystyle Y_{grt}\leq Y_{gr,t+1} \hspace{3mm} \forall\hspace{3mm}g \in\mathcal{G},r \in\mathcal{R}\text{ and } t \in \mathcal{T}:t\leq|\mathcal{T}|-1 $$
(A.5e)
$$ \displaystyle \sum\limits_{r\in\mathcal{R}}Y_{2rt} \leq 1\hspace{3mm} \forall\hspace{3mm}t \in\mathcal{T} $$
(A.5f)
$$ \displaystyle {\sum}_{g \in\mathcal{G}}Y_{grt} \leq 1\hspace{3mm} \forall\hspace{3mm}r \in\mathcal{R},t \in\mathcal{T} $$
(A.5g)

Construction of Reservoirs (see “Construction of Reservoirs”)

$$ \displaystyle \breve{W}_{grt}\geq\breve{X}_{gr}-M(1-W_{grt})\hspace{3mm} \forall\hspace{3mm}g \in\mathcal{G},r \in\mathcal{R}\text{ and } t \in \mathcal{T} $$
(A.6a)
$$ \displaystyle Y_{grt}-Y_{gr,t-1}=W_{grt}\hspace{3mm} \forall\hspace{3mm}g \in\mathcal{G},r \in\mathcal{R}\text{ and } t \in \mathcal{T} $$
(A.6b)

Construction of Pipelines (see “Construction of Pipelines”)

$$ \displaystyle \tilde{Y}^{\prime \prime}_{jr^{\prime} rt} \leq \sum\limits^{t-1}_{t^{\prime} = 1}\tilde{Y}_{jr^{\prime} rt^{\prime}} \hspace{3mm} \forall\hspace{3mm}j \in\mathcal{J}, r^{\prime} ,r \in\mathcal{R}\text{ and } t \in \mathcal{T}: t\geq2 $$
(A.7)

Operation and Maintenance of Reservoirs (see “Operation and Maintenance of Pipelines”)

$$ \displaystyle \sum\limits_{g\in\mathcal{G}}\breve{X}_{gr}\geq I_{rt} \hspace{3mm} \forall\hspace{3mm}r \in\mathcal{R}\text{ and } t \in \mathcal{T} $$
(A.8a)
$$ \displaystyle \breve{X}_{gr}\leq \mathring{X}_{grt}+M(1-\mathring{Y}_{grt}) \hspace{3mm} \forall\hspace{3mm}g \in\mathcal{G},r \in\mathcal{R}\text{ and } t \in \mathcal{T} $$
(A.8b)
$$ \displaystyle Y_{grt}\leq \mathring{Y}_{gr,t+1} \hspace{3mm} \forall\hspace{3mm}g \in\mathcal{G},r \in\mathcal{R}\text{ and } t \in \mathcal{T}:t\leq|\mathcal{T}|-1 $$
(A.8c)

Operation and Maintenance of Pipelines (see “Operation and Maintenance of Pipelines”)

$$ \displaystyle \tilde{Y}^{\prime \prime}_{jr^{\prime} rt} \leq \tilde{Y}^{\prime}_{jr^{\prime} rt} \hspace{3mm} \forall\hspace{3mm}j \in\mathcal{J}, r^{\prime} ,r \in\mathcal{R}\text{ and } t \in \mathcal{T} $$
(A.9a)
$$ \displaystyle \tilde{Y}^{\prime}_{jr^{\prime} rt} {\leq} \tilde{Y}^{\prime}_{jr^{\prime} r,t+1} \hspace{3mm} \forall\hspace{3mm}j \in\mathcal{J}, r^{\prime} ,r \in\mathcal{R}\text{ and } t \in \mathcal{T}, t\leq|\mathcal{T}|-1 $$
(A.9b)

South Platte Compact (see “South Platte Compact”)

$$ \begin{array}{@{}rcl@{}} &&{}\displaystyle \sum\limits_{g \in\mathcal{G}}\sum\limits_{r \in\mathcal{R}}X_{grdt}{+}\!{\sum}_{s \in\mathcal{S}_{t}}\bar{X}_{sdt}{+}\sum\limits_{s \in\mathcal{S}_{t}}\hat{c}_{st}\\ &&~~~~~~~~~~~~~~~~~~~~~ {\geq} d_{dt}{-}M^{\prime}(1{-}\beta_{dt})\hspace{2mm} \forall\hspace{2mm}d\in\mathcal{D}_{t}\text{ and } t \in \mathcal{T}^{\prime}\\ \end{array} $$
(A.10a)
$$ \begin{array}{@{}rcl@{}} &&{}{\sum}_{g \in \mathcal{G}}{\sum}_{r \geq n}{\sum}_{n\in\mathcal{N}_{t}}X_{grnt}{+}{\sum}_{s\geq n}\!{\sum}_{n\in\mathcal{N}_{t}}\!\bar{X}_{snt}\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~ {\leq} 0{+}M^{\prime}\beta_{dt} \hspace{1.5mm} \forall\hspace{1.25mm} d \!\in \! \mathcal{D}_{t}\text{ and } t \!{\kern-.5pt}\in{\kern-.5pt} \! \mathcal{T}^{\prime}\\ \end{array} $$
(A.10b)

Non-Negativity and Binary Restrictions (see “Non-Negativity and Binary Restrictions”)

$$ \begin{array}{@{}rcl@{}} &&X_{grdt},\hat{X}_{r^{\prime} rt},\bar{X}_{sdt},\tilde{X}_{gsrt}, \mathring{X}_{grt}, \breve{X}_{gr}, I_{rt}, Z^{+}_{dt},Z^{-}_{dt},\\ &&\breve{W}_{grt} \geq 0 \hspace{2mm} \forall\hspace{2mm} g \in \mathcal{G}\text{, } r^{\prime} ,r \in \mathcal{R}\text{, } d \in \mathcal{D}_{t}\text{, } s \in\mathcal{S}_{t}\text{, }{}\text{ and }{} t \in \mathcal{T}\\ \end{array} $$
(A.11a)
$$ \begin{array}{@{}rcl@{}} &&Y_{grt},\tilde{Y}_{jr^{\prime} rt},\tilde{Y}^{\prime}_{jr^{\prime} rt},\tilde{Y}^{\prime \prime}_{jr^{\prime} rt},\mathring{Y}_{grt},\alpha_{rt},\beta_{dt}, W_{grt}\hspace{1.5mm}{\text{binary}}\\ && \forall\hspace{2mm} g \in \mathcal{G}\text{, } r^{\prime} ,r \in \mathcal{R}\text{ , } j \in \mathcal{J}\text{, } d \in \mathcal{D}_{t}\text{, }s \in\mathcal{S}_{t}\text{ and } t \in \mathcal{T}\\ \end{array} $$
(A.11b)

Capacities

Constraints (A.1a) safeguard the total volume released from all upstream supply nodes to each reservoir such that the available capacity of each reservoir intake ditch in each time period is not surpassed. Constraints (A.1b) guarantee that the pumped water volume is less than pipe capacity. Constraints (A.1c) ensure that the cumulative water released to meet demand does not violate the available river capacity. Constraints (A.1d) make certain that the cumulative volume released from each supply node is within the available river capacity.

Penalties

Constraints (A.2a) apply a penalty if demand is exceeded. If demand goes unmet, constraints (A.2b) apply a penalty, increasing with the seniority of the water right.

Flow Balance

Constraints (A.3a) empty each supply node into storage and/or to meet demand. Constraints (A.3b)-(A.3c) preclude the release of or pumping water from any reservoir in time period 1.

Inventory Flow

Constraints (A.4a) ensure that inventory is only held in constructed reservoirs. Constraints (A.4b)-(A.4c) maintain the minimum required reservoir volume.

Total Quantity of New Infrastructure

Constraints (A.5a) construct pipelines between nodes only once. Constraints (A.5b)-(A.5d) and (A.5f)-(A.5g) enforce the specified number and type of reservoirs to be considered. Constraints (A.5e) ensure a constructed reservoir incurs operational costs in all future time periods.

Construction of Reservoirs

In order to calculate the reservoir construction costs, the maximum reservoir size is needed. Constraints (A.6a)-(A.6b) apply the corresponding cost during the initial time period of use.

Construction of Pipelines

Constraints (A.7) ensure pipeline construction costs are incurred in the time period preceding first use.

Operation and Maintenance of Reservoirs

Constraints (A.8a)-(A.8b) apply the reservoir operation and maintenance costs using the maximum size of the new reservoir at the time of construction. By constraints (A.8c), the operation and maintenance costs are incurred in every year following construction.

Operation and Maintenance of Pipelines

Constraints (A.9a)-(A.9b) apply the operation and maintenance costs for pipeline use during the initial time period of use and in every time period thereafter.

South Platte Compact

The South Platte Compact with Nebraska is enforced annually from April through October in District 64 (lower portion) of the river. Constraints (A.10a)-(A.10b) enforce the South Platte Compact with Nebraska and only apply at node 50 (Nebraska border) in the amount of 7,244 AF/month.

Non-Negativity and Binary Restrictions

Constraints (A.11a) dictate the non-negativity of continuous variables. Constraints (A.11b) guarantee that the appropriate variables are binary.

Appendix B: Solution

The model (\(\mathcal {P}^T\)) takes as input: (1) simulated river flow data (both shortages and excesses), (2) regional topography, (3) capacities, (4) costs, and (5) penalties in order to minimize the cost of shortage mitigation (via new storage infrastructure) while exploiting revenue gained from potential electricity generation from a thermal plant. The model runs on a SuperServer 1028GR-TR with a 1TB hard drive, 3 Intel Xeon-PHI coprocessors and 164GB of RAM operating under a Linux environment; we employ eight threads. We code model instances in AMPL ([70], [71]) and solve them with CPLEX version 12.8.0.0 ([72], [73]); these instances contain more than 1,500,000 variables, of which at least 183,000 are binary, and more than 890,000 constraints. AMPL and CPLEX parameter settings are used to reduce solve time and improve solution quality by invoking presolve as well as Implied Bound, Flow, and Mixed-Integer Rounding cuts. Using an initial feasible solution obtained via a sliding time window heuristic (e.g., [74], [75]), and the settings above, all instances of (\(\mathcal {P}^T\)) solve to within 0.5% of optimality in fewer than 45 seconds. See [31] for more details.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Burrow, A., Newman, A. & Bazilian, M. Reservoir Design and Operation for the Food-Energy-Water Nexus. Curr Sustainable Renewable Energy Rep 6, 71–89 (2019). https://doi.org/10.1007/s40518-019-00126-3

Download citation

Keywords

  • Mixed-integer programming
  • Reservoir design
  • River basin management
  • Integrated water resource planning
  • Climate-Land-Energy-Water