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


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.

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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.


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).

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Correspondence to Alexandra Newman.

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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.


\(t \in \mathcal {T}\) 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


\(f_{jr^{\prime }rt}\) 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
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
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
p e demand exceedance penalty ($/AF)
M “sufficiently large” value


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


Capacities (see “Capacities”)

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

Penalties (see “Penalties”)

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

Flow Balance (see “Flow Balance”)

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

Inventory Flow (see “Inventory Flow”)

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

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

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

Construction of Reservoirs (see “Construction of Reservoirs”)

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

Construction of Pipelines (see “Construction of Pipelines”)

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

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

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

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

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

South Platte Compact (see “South Platte Compact”)

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

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

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


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.


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 ([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.

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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).

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