# Life Cycle Network Modeling Framework and Solution Algorithms for Systems Analysis and Optimization of the Water-Energy Nexus

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Approach

#### 2.1. Data Collection and Life Cycle Optimization Approach

#### 2.2. Model Formulation

#### 2.2.1. Sets and Notation

#### 2.2.2. Objective Functions

#### 2.2.3. Economic Constraints

_{j}is the fixed cost factor for technology j, cchf is the capital charge factor required to annualize the capital cost for appropriate calculation of the NPV, CC

_{j}is the capital cost of technology j, X

_{j}is the capacity of technology j, refc

_{j}is the reference capacity for technology j, refoc

_{j}is the reference operating cost for technology j, P

_{i}is the quantity purchased of material/compound i, fp

_{i}is the feedstock price of compound i, vtc

_{i}is the variable transportation cost of feedstock i, ftc

_{i}is the fixed transportation cost of feedstock i, ec is the cost of electricity, ue

_{j}is the unit energy consumption of technology j, S

_{i}is the quantity sold of final product i, and sp

_{i}is the selling price of final product i. We note that the fixed transportation cost term ftc

_{i}is in distance-fixed units of $/kg, so it must be multiplied by the quantity purchased P

_{i}. Similarly, the variable transportation costs are variable costs that change based on the distance traveled. The transportation distance has been assumed to be pre-determined by the user and is already incorporated into each parameter ftc

_{i}and vtc

_{i}. The capital charge factor cchf

_{i}can be calculated from an interest rate r and anticipated processing pathway lifetime n:

_{j}is the reference capital cost of technology j corresponding to refc

_{j}, sf

_{j}is the scaling factor for capital cost with capacity, and ccf is the capital cost factor that takes into account the chemical engineering plant cost index (CEPCI) from the year the technology was reported to the current year.

_{j}is the lower bound on capacity for technology j, BD

_{j}is the binary decision variable that determines whether technology j is chosen to be in the pathway or not, and ucap

_{j}is the upper bound on capacity for technology j.

#### 2.2.4. Mass Balance Constraints

_{ij}is the productive yield of compound/material i in technology j, and dy

_{ij}is the destructive yield of compound/material i in technology j. In other words, if compound i is produced from technology j, then the corresponding py

_{ij}value will be positive, with no value for the corresponding dy

_{ij}. If compound i is consumed in technology j, then the corresponding value for dy

_{ij}will be negative with no value for the corresponding py

_{ij}.

_{i}is the demand for final product i.

_{i}is the minimum availability of input i and mxa

_{i}is the maximum availability of input i.

_{i}is the product energy content of final product i. For liquid energy products, for example, this could be the higher heating value (HHV) of each product.

#### 2.2.5. Water Constraint

_{i}is the water used during production of input i and wp

_{j}is the water consumption rate of technology j in the processing pathway. The water consumption rate for each technology must take into account direct water consumption (e.g., water used in reactions), and indirect water consumption (e.g., makeup water required for heating and/or cooling).

_{j}that determine if technology j is chosen to be in the final processing pathway or not. Nonlinear terms arise in Equation (6) that represent the nonlinear scaling of capital costs (CC

_{j}) with technology capacity (X

_{j}). The scaling factor sf

_{j}in Equation (6) usually lies in the range of 0.5–0.8, with 0.6 often chosen as a representative value following the “six-tenths rule.” [40] Scaling factors in this range result in terms that are not only nonlinear but also nonconvex, increasing the difficulty of solving the optimization problem. A solution method that aims to decrease computational difficulties arising from the structure of these models is presented in the following subsection.

#### 2.3. Solution Method

_{FE}is the ε-constraint parameter for the energy efficiency of water objective, and ε

_{NPV}is the ε-constraint parameter for the economic efficiency of water objective. We note that Equations (14) and (15) are still required to calculate the upper and lower bounds for each ε.

_{j,n}is a weighted variable to determine where in the piecewise interval the solution lies, u

_{j,n}is a parameter that acts as a stand-in for the capacity X

_{j}, val

_{j,n}calculates the capital cost at each point u

_{j,n}, and EX

_{j,n}are SOS1 variables that ensures only one solution is present across all intervals in the piecewise linear approximation.

Algorithm. Parametric algorithm outer loop with branch and refine (B&R) + piecewise linear approximation algorithm with NLP subproblems in the inner loop | |||||||||||

1: | Initialization. | ||||||||||

2: | Set lowerbound:= −Inf, upperbound:= +Inf | ||||||||||

3: | Set initial piecewise linear approximations at lcap and ucap | ||||||||||

4: | Set initial parametric parameter value: = 0 | ||||||||||

5: | Set initial parametric gap: = +Inf | ||||||||||

6: | while the parametric gap is larger than the parametric tolerance | ||||||||||

7: | while the B&R gap is larger than the B&R tolerance | ||||||||||

8: | Solve MILP with SOS1 variable approximation for capital cost and parametric form for economic efficiency of energy | ||||||||||

9: | if NPV < upperbound | ||||||||||

10: | Set upperbound: = NPV | ||||||||||

11: | end if | ||||||||||

12: | Fix binary technology decision variables | ||||||||||

13: | Solve NLP problem with solution from step 8 | ||||||||||

14: | if feasible solution from step 13 is found & solution NPV > lowerbound | ||||||||||

14: | Set lowerbound: = NPV from step 13 | ||||||||||

15: | end if | ||||||||||

16: | Calculate gap between upper and lower bounds | ||||||||||

17: | if B&R gap < B&R tolerance | ||||||||||

18: | Terminate with upperbound as solution for the NPV | ||||||||||

19: | else B&R gap > B&R tolerance | ||||||||||

20: | Determine in which interval the solution from step 8 is located | ||||||||||

21: | Place a new node for piecewise linear approximation at that solution | ||||||||||

22: | Return to step 8. | ||||||||||

23: | end if | ||||||||||

24: | Calculate the parametric gap with solution from step 13 | ||||||||||

25: | if B&R gap < B&R tolerance | ||||||||||

26: | if parametric gap > parametric tolerance | ||||||||||

27: | Update parametric parameter value. | ||||||||||

28: | end if | ||||||||||

29: | if parametric gap < parametric tolerance | ||||||||||

30: | Calculate economic efficiency of energy from solution from step 13 as optimal solution | ||||||||||

31: | end if | ||||||||||

32: | end while | ||||||||||

33: | Reset upperbound and lowerbound values to Initialization values (step 2). | ||||||||||

34: | end while |

_{j}). If the values of these variables are fixed after finding a solution to the aforementioned branch and refine algorithm, the original MINLFP problem can be reformulated as an NLP. Any feasible solution to this NLP can function as a valid global lower bound to the original MINLFP. Conceptually, this method aims to find any feasible configuration of the technologies chosen from the solution of the branch-and-refine algorithm. If such a configuration is found, then it can be used to calculate the valid lower bound for the NPV. The upper and lower bounds for the NPV can then be used to calculate the valid global upper and lower bounds.

## 3. Case Study with a Bioconversion Network, Results, and Discussion

^{−2}.

#### 3.1. Description of Case Study

**Figure 1.**Sample subset of technologies in the model and their possible connections within a processing pathway.

**Figure 2.**Abstract representation of the bioconversion network used in this study. Yellow nodes correspond to biofuel products, orange nodes correspond to byproducts/intermediates, teal nodes correspond to biomass feedstocks, purple nodes correspond to initial/pretreatment technologies, and blue nodes correspond to final upgrading technologies.

Biomass Feedstock | Water Consumption Rate for Cultivation (L/kg Biomass) |
---|---|

Soybean | 2145 |

Corn | 1222 |

Sugarcane | 210 |

Corn Stover | 1222 |

Hardwood | 0.357 |

Softwood | 0.268 |

Switchgrass | 0 |

_{j}is taken as 5% of the capital cost of technology j [40]. Technology selection, sizing, feedstock selection and quantities purchased, levels of the three fuels produced, capital costs, operating costs, and the water footprint are all decision variables. The HHV for each fuel was used to determine the energy efficiency of water for each processing pathway. The case study has a system boundary of “cradle to grave”, and it is assumed that all biomass that is purchased is processed, and all biofuels produced are sold and consumed. Water consumption is tallied from biomass cultivation to biomass processing and fuel production. Current market prices for feedstocks and fuels are used [53,54], and these prices are compiled in Table 2.

**Table 2.**Market prices of all raw materials (biomass feedstocks) and products (biofuels) available in this work.

Biomass Feedstock/Biofuel | Market Price ($/kg) |
---|---|

Soybean | 0.1085 |

Corn | 0.0317 |

Sugarcane | 0.0925 |

Corn Stover | 0.0881 |

Hardwood | 0.0728 |

Softwood | 0.0728 |

Switchgrass | 0.0878 |

Ethanol | 0.61 |

Gasoline | 0.83 |

Diesel and Biodiesel | 0.92 |

**Figure 3.**Pareto-optimal surface with key features and extreme points highlighted. The minimum water consumption solution is denoted with a blue circle (

**far left**), the maximum energy efficiency of water solution is denoted with a green circle (

**middle**), and the maximum economic efficiency of water solution is denoted with a yellow circle (

**right**).

#### 3.2. Minimum Water Footprint Solution

**Figure 4.**Optimal processing pathway, water footprint, economic efficiency of water, overall capital cost, and energy efficiency of water of the minimum water footprint solution. A water drop over a technology in the pathway denotes process water recycling is incorporated in the input process data.

#### 3.3. Maximum Economic Efficiency of Water Solution

**Figure 5.**Optimal processing pathway, water footprint, economic efficiency of water, overall capital cost, and energy efficiency of water of the maximum economic efficiency of water solution.

#### 3.4. Maximimum Energy Efficiency of Water Solution

**Figure 6.**Optimal processing pathway, water footprint, economic efficiency of water, overall capital cost, and energy efficiency of water of the maximum energy efficiency of water solution.

Metric | Minimum Water Footprint | Maximum Economic Efficiency of Water | Maximum Energy Efficiency of Water |
---|---|---|---|

Energy Produced (in Biofuel Form) (MJ/year) | 1.34 × 10^{9} | 1.56 × 10^{9} | 1.66 × 10^{9} |

Water Footprint (ML/year) | 55.1 | 87.6 | 59.3 |

Energy Efficiency of Water (MJ/L) | 24.62 | 15.32 | 27.98 |

Capital Cost ($M) | 250 | 250 | 250 |

Economic Efficiency of Water ($/L) | −1.31 | 0.76 | −1.19 |

## 4. Conclusions

## Supplementary Files

Supplementary File 1## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix

#### A.1. Notation

Sets | |

I | Set of materials/compounds |

J | Set of technologies |

N | Set of points in the piecewise linear approximations |

Subsets | |

B | Subset of biomass feedstocks |

F | Subset of biofuel products |

Continuous Variables | |

CCj | Capital cost of technology j |

FE | Energy available in all final fuel products |

NPV | Annualized net present value of the processing pathway |

OBJ_{EcEW} | Objective function for the economic efficiency of water |

OBJ_{EEW} | Objective function for the energy efficiency of water |

OBJ_{water} | Objective function for the water footprint |

P_{i} | Amount of biomass feedstock i purchased |

S_{i} | Amount of biofuel product i produced and sold |

W_{j,n} | Weighted variable to determine where along the piecewise linear approximations in interval n the solution lies for technology j |

WF | The water footprint of the processing pathway |

X_{j} | The capacity of technology j |

Discrete Variables | |

BD_{j} | A decision variable that determines if technology j is included in the final processing pathway |

SOS1 Variables | |

EX_{j,n} | SOS1 variable that ensures only one solution is present for technology j along the piecewise linear approximations for technology j over intervals n |

Parameters | |

ccb | The capital cost budget of the processing pathway |

ccf | Capital cost factor |

cchf | Capital charge factor |

dem_{i} | The demand to be satisfied for fuel i |

dy_{ij} | The destructive yield of compound/material i in technology j |

ec | Cost of electricity |

fcf_{j} | The fixed cost factor for technology j |

fp_{i} | Price of feedstock i |

ftc_{i} | Distance-fixed transportation cost for feedstock i |

mna_{i} | Minimum availability for feedstock i |

mxa_{i} | Minimum availability for feedstock i |

n | Expected lifetime of the processing pathway |

pec_{i} | Product energy content of fuel product i |

py_{ij} | Productive yield of compound/material i in technology j |

qe | Parametric parameter for the economic efficiency of water objective |

qn | Parametric parameter for the energy efficiency of water objective |

r | Interest rate |

refc_{j} | Reference capacity of technology j |

refcc_{j} | Reference capital cost of technology j |

refoc_{j} | Reference operating cost of technology j |

sf_{j} | Capital cost scaling factor for technology j |

sp_{i} | The selling price of compound/material i |

u_{j,n} | Parameter used to represent the capacity of technology j at point n of the piecewise linear approximation |

ue_{j} | Unit electricity requirement of technology j |

val_{j,n} | Parameter used, along with the variable W_{j,n}, to represent the capital cost of technology j at point n of the piecewise linear approximation |

vtc_{i} | Variable transportation cost of feedstock i |

wc_{i} | Unit rate of water consumption for cultivation of feedstock i |

wp_{j} | Unit rate of water consumption of technology j |

ε_{FE} | ε-constraint parameter for the energy efficiency of energy |

ε_{NPV} | ε-constraint parameter for the economic efficiency of water |

#### A.2. Computational Performance Results

**Table A.1.**Comparative computational results for calculating the maximum energy efficiency of water solution point.

Model Property | Original MINLFP Problem | MILP with NLP Subproblems |
---|---|---|

Objective Value (MJ/L) | 27.98 | 27.98 |

Constraints | 1,512 | 4112 (MILP); 1512 (NLP subproblems) |

Continuous Variables | 891 | 4291 (MILP); 891 (NLP subproblem) |

Discrete Variables | 200 | 400 (MILP); 0 (NLP subproblem) |

Solver | BARON 14.4.0 | CPLEX 12.6.1/CONOPT3 |

Solution Time (CPUs) | 56.5 | 7.6 |

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## Share and Cite

**MDPI and ACS Style**

Garcia, D.J.; You, F.
Life Cycle Network Modeling Framework and Solution Algorithms for Systems Analysis and Optimization of the Water-Energy Nexus. *Processes* **2015**, *3*, 514-539.
https://doi.org/10.3390/pr3030514

**AMA Style**

Garcia DJ, You F.
Life Cycle Network Modeling Framework and Solution Algorithms for Systems Analysis and Optimization of the Water-Energy Nexus. *Processes*. 2015; 3(3):514-539.
https://doi.org/10.3390/pr3030514

**Chicago/Turabian Style**

Garcia, Daniel J., and Fengqi You.
2015. "Life Cycle Network Modeling Framework and Solution Algorithms for Systems Analysis and Optimization of the Water-Energy Nexus" *Processes* 3, no. 3: 514-539.
https://doi.org/10.3390/pr3030514