# Approximating Nonlinear Relationships for Optimal Operation of Natural Gas Transport Networks

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Common Natural Gas Transportation Model

#### 2.1. Network Topography

#### 2.2. Gas Flow Model

#### 2.2.1. Compressibility Factor

#### 2.2.2. Friction Factor

#### 2.3. Compressor Equations

#### 2.3.1. Isentropic Exponent

#### 2.3.2. Compressor Operating Envelop

#### 2.3.3. Compressor Efficiency

#### 2.4. Common FCMP Model

## 3. Optimal Piecewise-Linear Function Generation

- Decide what is an acceptable amount of maximum relative error between the piecewise-linear function being generated and the relationship being approximated, this is the stopping criteria ${E}^{tol}$. Typically, 1% is appropriate.
- Start with $m=M$ and $p=1$, where M is some relatively small number of data points equally dispersed over the approximation domain. Solve the MILP for the piecewise-linear function, ${\mathbb{F}}_{m}^{p}$.
- Calculate the maximum error between ${\mathbb{F}}_{m}^{p}$ and the entire dataset of the relationship being approximated, ${E}_{m}^{max,p}$.
- Increase m to $m+dM$, and solve the MILP for ${\mathbb{F}}_{m}^{p}$.
- Calculate ${E}_{m}^{max,p}$.
- Iterate Steps 4 and 5 until the change in ${E}_{m}^{max,p}$ is less than 1% for two consecutive iterations, then move to Step 7. If the MILP becomes intractable before this, move to Step 7.
- If the smallest ${E}_{m}^{max,p}$ is less than ${E}^{tol}$, or has changed less than 1% for two consecutive increases in p, terminate, the corresponding ${\mathbb{F}}_{m}^{p}$ is either an acceptable approximation or as good as the PLFG method is likely to yield. Else, increase p to $p+1$, reinitialize $m=M$, and return to Step 4. If the MILP becomes intractable before meeting the stopping criteria, terminate and consider breaking up the domain of approximation.

## 4. Relationship Approximation

#### 4.1. Compressibility Factor

#### 4.2. Friction Factor

#### 4.3. Isentropic Exponent

#### 4.4. Compressor Model

#### 4.4.1. Compressor Operating Envelop

#### 4.4.2. Compressor Efficiency

## 5. FCMP Model with Novel Relationship Approximations

#### 5.1. Compressibility Factor

#### 5.2. Friction Factor

#### 5.3. Isentropic Exponent

#### 5.4. Compressor Operating Envelop

#### 5.5. Compressor Efficiency

## 6. Case Study

#### 6.1. FCMP Models

- The pressure of the first node is set as the first node pressure from the optimal solution.
- If the next arc is a pipe section, then the pressure of the next node is calculated using the pipeline resistance equation, where Z and $\lambda $ are calculated using their rigorous relations. Else, move to Step 3.
- The next arc is a compressor station, so the outlet pressure is set as the outlet pressure from the optimal solution. The simulated compressor fuel cost is then calculated using the rigorous calculations for Z, $\kappa $, ${H}_{ad}$, and ${\eta}_{ad}$ so that a theoretically exact fuel cost is obtained.
- Return to Step 2 and iterate until all node pressures and compressor station fuel costs are calculated.

#### 6.2. Example 1

#### 6.3. Example 2

#### 6.4. Example 3

## 7. Conclusions and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

FCMP | Fuel Cost Minimization Problem |

NLP | Nonlinear Program |

MINLP | Mixed-Integer Nonlinear Program |

PLFG | Piecewise-Linear Function Generation |

MILP | Mixed-Integer Linear Program |

## Appendix A. Novel FCMP Model (FCMP_{N})

## Appendix B. Simplified FCMP Model (FCMP_{S})

## Appendix C. Rigorous FCMP Model (FCMP_{R})

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**Figure 1.**Comparison of the rigorously calculated gas compressibility factor with the American Gas Association and Papay empirical models for natural gas with a composition of: 85% methane, 14% ethane, and 1% nitrogen.

**Figure 2.**Comparison of the Colebrook–White and Nikuradse friction factor on a pipeline with a diameter of $0.9$ m, and roughness of $0.05$ mm.

**Figure 3.**Comparison of the rigorously calculated isentropic exponent with a common simplification for natural gas with a composition of: 85%methane, 14% ethane, and 1% nitrogen.

**Figure 4.**Typical operating envelop of a centrifugal compressor unit, with compressor speed and efficiency contours shown.

**Figure 5.**(

**a**) Example of potential approximation produced when only the ${f}_{i}\ge {c}^{k}{x}^{i}+{d}_{k}$ constraint is used in the PLFG MILP; (

**b**) Example of potential approximation produced when all of the constraints are used in the PLFG MILP (PLFG: Piecewise-Linear Function Generation; MILP: mixed-integer linear program).

**Figure 6.**(

**a**) Example of approximation produced when curve cross-over is allowed; (

**b**) Example of approximation when restricted to entirely above the curve; (

**c**) Example of approximation when restricted to entirely below the curve.

**Figure 7.**(

**a**) Three-dimensional compressibility factor relationship with temperature–pressure; and (

**b**) two-dimensional compressibility factor isothermal relationship with pressure for natural gas with a composition of: 85% methane, 14% ethane, and 1% nitrogen.

**Figure 8.**Piecewise-linear function generated for approximating 289.5 K compressibility factor isotherm, consisting of two line segments.

**Figure 9.**(

**a**) Relationship between $\zeta \left(q\right)$ and mass flow rate, and quadratic regression approximation; (

**b**) Comparison of error for $\zeta \left(q\right)$ from the quadratic regression and constant friction factor. All for a pipeline with a diameter of 0.9144 m and absolute roughness of 0.05 mm.

**Figure 10.**(

**a**) Rigorous three-dimensional relationship between m and temperature–pressure; and (

**b**) piecewise- linear approximation of m using four planes for natural gas with a composition of: 85% methane, 14% ethane, and 1% nitrogen.

**Figure 11.**Approximation of the compressor operating envelop shown in Figure 4.

**Figure 12.**Rigorous relationship of compressor efficiency with compressor head and volumetric flow rate.

**Figure 13.**(

**a**) Rigorous relationship of $\frac{{H}_{ad}}{{\eta}_{ad}}$ with compressor head and volumetric flow rate. (

**b**) Approximation of rigorous relationship of $\frac{{H}_{ad}}{{\eta}_{ad}}({H}_{ad},Q)$ with a maximum relative error of 1.12%.

**Figure 14.**(

**a**) Convex feasible region produced by bounding a parameter to be greater than a set of convex line segments. (

**b**) Concave feasible region produced by bounding a parameter to be less than a set of concave line segments.

**Figure 15.**Flow chart detailing the components of each FCMP model applied in the case study, and the rigorous simulation model.

**Figure 16.**Example 1 gas network consisting of three pipes and two compressor stations. ${P}_{i}^{L}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}$4.14 MPa, and ${P}_{i}^{H}=$ 5.52 MPa for all $i\in N$. ${D}_{ij}=$ 0.9144 m and ${L}_{ij}=$ 80.47 km for all $(i,j)\in {A}_{p}$.

**Figure 17.**Example 2 gas network consisting of six pipes and three compressor stations. ${P}_{1}^{L}={P}_{2}^{L}=$ 4.14 MPa, ${P}_{3}^{L}={P}_{5}^{L}={P}_{6}^{L}={P}_{7}^{L}={P}_{9}^{L}=$ 3.10 MPa, ${P}_{4}^{L}=$ 3.45 MPa, ${P}_{8}^{L}=$ 3.79 MPa, ${P}_{10}^{L}=$ 2.76 MPa. ${P}_{1}^{H}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}$ 4.83 MPa, ${P}_{i}^{H}=$ 5.52 MPa for all $i>1$. ${D}_{ij}=$ 0.9144 m and ${L}_{ij}=$ 80.47 km for all $(i,j)\in {A}_{p}$.

**Figure 18.**Example 3 gas network consisting of 12 pipes and five compressor stations. ${P}_{1}^{L}={P}_{2}^{L}=$ 4.14 MPa, ${P}_{3}^{L}={P}_{5}^{L}={P}_{6}^{L}={P}_{7}^{L}={P}_{9}^{L}=$ 3.10 MPa, ${P}_{4}^{L}=$ 3.45 MPa, ${P}_{8}^{L}=$ 3.79 MPa, ${P}_{10}^{L}=$ 2.76 MPa, ${P}_{11}^{L}={P}_{12}^{L}={P}_{13}^{L}={P}_{14}^{L}={P}_{15}^{L}={P}_{16}^{L}={P}_{17}^{L}={P}_{18}^{L}=$ 0.69 MPa. ${P}_{1}^{H}=$ 4.83 MPa, ${P}_{i}^{H}=$ 5.52 MPa for all $i>1$. ${D}_{ij}=$ 0.9144 m and ${L}_{ij}=$ 80.47 km for all $(i,j)\in {A}_{p}$.

Symbol | Description | Unit |
---|---|---|

q | Mass flow rate | kg s${}^{-1}$ |

P | Pressure | MPa |

T | Temperature | K |

L | Length | m |

D | Diameter | m |

A | Cross-sectional area | m${}^{2}$ |

$\lambda $ | Friction coefficient | - |

Z | Compressibility factor | - |

R | Gas constant | J mol${}^{-1}$ K${}^{-1}$ |

${M}_{w}$ | Molecular weight | kg mol${}^{-1}$ |

B | Second virial coefficient | m${}^{3}$ kmol${}^{-1}$ |

${\rho}_{m}$ | Molar density | kmol m${}^{-3}$ |

K | Size parameter | m kmol${}^{-1/3}$ |

${C}_{n}^{\ast}$ | Compressibility factor coefficient | - |

${b}_{n},{c}_{n},{k}_{n}$ | Compressibility factor constant | - |

${T}_{c}$ | Critical temperature | K |

${P}_{c}$ | Critical pressure | MPa |

$\u03f5$ | Absolute roughness | m |

$Re$ | Reynolds Number | - |

$\mu $ | Gas dynamic viscosity | Pa s |

${H}_{ad}$ | Specific change in adiabatic enthalpy | J kg${}^{-1}$ |

$\kappa $ | Isentropic exponent | - |

Q | Volumetric flow rate | m${}^{3}$s${}^{-1}$ |

S | Compressor speed | rpm |

${S}_{min},{S}_{max}$ | Compressor speed limit | rpm |

$surge,stonewall$ | Compressor throughput/speed limit | m${}^{3}$s${}^{-1}$rpm${}^{-1}$ |

${\eta}_{ad}$ | Adiabatic efficiency | - |

**Table 2.**Parameters and variables for implementing the approximations into the Fuel Cost Minimization Problem (FCMP) model.

Symbol | Description | Unit |
---|---|---|

${b}_{k}^{Z}$ | Compressibility factor binary variable | - |

${c}_{k}^{Z}$, ${d}_{k}^{Z}$ | Compressibility factor line coefficients | MPa${}^{-1}$, - |

${M}_{Big}^{Z}$ | Compressibility factor sufficiently large number | - |

$\alpha $, $\beta $ | Friction factor regression coefficients | -, kg s${}^{-1}$ |

${b}_{k}^{m}$ | m binary variable | - |

${c}_{1,k}^{m}$, ${c}_{2,k}^{\kappa}$, ${d}_{k}^{m}$ | m plane coefficients | K${}^{-1}$, MPa${}^{-1}$, - |

${c}_{k}^{stw}$, ${d}_{k}^{stw}$ | Stonewall limit line coefficients | kJ-s kg${}^{-1}$-m${}^{-3}$, kJ kg${}^{-1}$ |

${c}_{k}^{Smax}$, ${d}_{k}^{Smax}$ | Smax limit line coefficients | kJ-s kg${}^{-1}$-m${}^{-3}$, kJ kg${}^{-1}$ |

${b}_{k}^{Smin}$ | Smin limit binary variable | - |

${c}_{k}^{Smin}$, ${d}_{k}^{Smin}$ | Smin limit line coefficients | kJ-s kg${}^{-1}$-m${}^{-3}$, kJ kg${}^{-1}$ |

${M}_{Big}^{Smin}$ | Smin limit sufficiently large number | kJ kg${}^{-1}$ |

${L}_{ij}^{Smin}$ | Smin limit bound variable | kJ kg${}^{-1}$ |

${b}_{k}^{srg}$ | Surge limit binary variable | - |

${c}_{k}^{srg}$, ${d}_{k}^{srg}$ | Surge limit line coefficients | kJ-s kg${}^{-1}$-m${}^{-3}$, kJ kg${}^{-1}$ |

${M}_{Big}^{srg}$ | Surge limit sufficiently large number | kJ kg${}^{-1}$ |

${L}_{ij}^{srg}$ | Surge limit bound variable | kJ kg${}^{-1}$ |

${c}_{1,k}^{g}$, ${c}_{2,k}^{g}$, ${d}_{k}^{g}$ | ${g}_{c}$ plane coefficients | kJ-s kg${}^{-1}$-m${}^{-3}$, -, kJ kg${}^{-1}$ |

**Table 3.**Example 1 case study results. Both compressor stations only have one unit active in all three model solutions. Lines in bold represent outlet pressures from compressor stations.

${\mathit{FCMP}}_{\mathit{S}}$ | ${\mathit{FCMP}}_{\mathit{N}}$ | ${\mathit{FCMP}}_{\mathit{PR}}$ | ||||
---|---|---|---|---|---|---|

Variable | Optimization | Simulation | Optimization | Simulation | Optimization | Simulation |

${P}_{1}$ (MPa) | 4.99 | 4.99 | 5.01 | 5.01 | 5.01 | 5.01 |

${P}_{2}$ | 4.47 | 4.37 | 4.39 | 4.39 | 4.39 | 4.39 |

${\mathbf{P}}_{\mathbf{3}}$ | 4.87 | 4.87 ** | 4.92 | 4.92 | 4.92 | 4.92 |

${P}_{4}$ | 4.33 | 4.22 | 4.28 | 4.29 | 4.28 | 4.29 |

${\mathbf{P}}_{\mathbf{5}}$ | 4.70 | 4.70 ** | 4.80 | 4.80 | 4.80 | 4.80 |

${P}_{6}$ | 4.14 | 4.02 * | 4.14 | 4.14 | 4.14 | 4.14 |

Fuel Cost (kJ s${}^{-1}$) | 39.76 | 44.88 *** | 47.31 | 47.33 | 47.58 | 47.28 |

% Diff | 11.41% | 0.05% | 0.61% | |||

Solution Time (s) | 0.32 s | 1.26 s | 5.97 s | |||

# of Constraints | 31 | 123 | 86 | |||

# of Continuous Variables | 23 | 71 | 68 | |||

# of Discrete Variables | 2 | 34 | 20 |

**Table 4.**Example 2 case study results. All three compressor stations only have one unit active in all three model solutions. Lines in bold represent outlet pressures from compressor stations.

${\mathit{FCMP}}_{\mathit{S}}$ | ${\mathit{FCMP}}_{\mathit{N}}$ | ${\mathit{FCMP}}_{\mathit{PR}}$ | ||||
---|---|---|---|---|---|---|

Variable | Optimization | Simulation | Optimization | Simulation | Optimization | Simulation |

${P}_{1}$ (MPa) | 4.26 | 4.26 | 4.38 | 4.38 | 4.38 | 4.38 |

${\mathbf{P}}_{\mathbf{2}}$ | 4.53 | 4.53 | 4.69 | 4.69 | 4.69 | 4.69 |

${P}_{3}$ | 3.42 | 3.15 | 3.37 | 3.38 | 3.38 | 3.38 |

${\mathbf{P}}_{\mathbf{4}}$ | 3.79 | 3.79 ** | 3.79 | 3.79 | 3.79 | 3.79 |

${P}_{5}$ | 3.49 | 3.42 | 3.42 | 3.42 | 3.42 | 3.42 |

${P}_{6}$ | 3.45 | 3.36 | 3.36 | 3.36 | 3.36 | 3.36 |

${P}_{7}$ | 3.45 | 3.36 | 3.36 | 3.36 | 3.36 | 3.36 |

${\mathbf{P}}_{\mathbf{8}}$ | 3.79 | 3.79 ** | 3.79 | 3.79 | 3.79 | 3.79 |

${P}_{9}$ | 3.49 | 3.42 | 3.42 | 3.42 | 3.42 | 3.42 |

${P}_{10}$ | 3.31 | 3.18 | 3.18 | 3.18 | 3.18 | 3.18 |

Fuel Cost (kJ s${}^{-1}$) | 56.55 | 79.51 *** | 58.47 | 57.88 | 58.17 | 57.72 |

% Diff | 28.87% | 1.02% | 0.78% | |||

Solution Time (s) | 1.03 s | 1.88 s | 54.26 s | |||

# of Constraints | 50 | 197 | 146 | |||

# of Continuous Variables | 30 | 115 | 115 | |||

# of Discrete Variables | 3 | 54 | 33 |

**Table 5.**Example 3 case study results. Lines in bold represent outlet pressures from compressor stations.

${\mathit{FCMP}}_{\mathit{S}}$ | ${\mathit{FCMP}}_{\mathit{N}}$ | ${\mathit{FCMP}}_{\mathit{PR}}$ | ||||
---|---|---|---|---|---|---|

Variable | Optimization | Simulation | Optimization | Simulation | Optimization | Simulation |

${P}_{1}$ (MPa) | 4.26 | 4.26 | 4.46 | 4.46 | 4.45 | 4.45 |

${\mathbf{P}}_{\mathbf{2}}$ | 4.53 | 4.53 ** | 4.81 | 4.81 | 4.80 | 4.80 |

${P}_{3}$ | 3.42 | 3.15 | 3.56 | 3.56 | 3.54 | 3.54 |

${\mathbf{P}}_{\mathbf{4}}$ | 3.79 | 3.79 ** | 4.01 | 4.01 | 4.01 | 4.01 |

${P}_{5}$ | 3.49 | 3.42 | 3.66 | 3.66 | 3.66 | 3.66 |

${P}_{6}$ | 3.45 | 3.36 | 3.60 | 3.60 | 3.60 | 3.60 |

${P}_{7}$ | 3.45 | 3.36 | 3.60 | 3.60 | 3.60 | 3.60 |

${\mathbf{P}}_{\mathbf{8}}$ | 3.79 | 3.79 ** | 4.01 | 4.01 | 3.98 | 3.98 |

${P}_{9}$ | 3.49 | 3.42 | 3.66 | 3.66 | 3.63 | 3.63 |

${P}_{10}$ | 3.31 | 3.18 | 3.44 | 3.44 | 3.41 | 3.41 |

${\mathbf{P}}_{\mathbf{11}}$ | 3.66 | 3.66 ** | 3.84 | 3.84 | 3.84 | 3.84 |

${P}_{12}$ | 2.76 | 2.49 | 2.75 | 2.76 | 2.75 | 2.76 |

${P}_{13}$ | 2.32 | 1.85 | 2.19 | 2.19 | 2.19 | 2.19 |

${P}_{14}$ | 2.59 | 2.26 | 2.55 | 2.55 | 2.55 | 2.55 |

${\mathbf{P}}_{\mathbf{15}}$ | 2.76 | 2.76 ** | 2.77 | 2.77 | 2.77 | 2.77 |

${P}_{16}$ | 1.49 | 0.85 | 0.88 | 0.88 | 0.88 | 0.88 |

${P}_{17}$ | 1.49 | 0.85 | 0.88 | 0.88 | 0.88 | 0.88 |

${P}_{18}$ | 1.42 | 0.65 * | 0.69 | 0.69 | 0.69 | 0.69 |

${n}_{1,2}^{s}$ | 2 | 1 | 1 | |||

${n}_{3,4}^{s}$ | 1 | 1 | 1 | |||

${n}_{3,8}^{s}$ | 1 | 1 | 1 | |||

${n}_{7,11}^{s}$ | 2 | 1 | 1 | |||

${n}_{14,15}^{s}$ | 4 | 3 | 3 | |||

Fuel Cost (kJ s${}^{-1}$) | 98.21 | 187.08 *** | 123.02 | 122.26 | 123.74 | 122.77 |

% Diff | 47.50% | 0.62% | 0.79% | |||

Solution Time (s) | 3.99 s | 8.21 s | 994.98 s | |||

# of Constraints | 88 | 345 | 266 | |||

# of Continuous Variables | 68 | 205 | 211 | |||

# of Discrete Variables | 5 | 94 | 59 |

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Kazda, K.; Li, X.
Approximating Nonlinear Relationships for Optimal Operation of Natural Gas Transport Networks. *Processes* **2018**, *6*, 198.
https://doi.org/10.3390/pr6100198

**AMA Style**

Kazda K, Li X.
Approximating Nonlinear Relationships for Optimal Operation of Natural Gas Transport Networks. *Processes*. 2018; 6(10):198.
https://doi.org/10.3390/pr6100198

**Chicago/Turabian Style**

Kazda, Kody, and Xiang Li.
2018. "Approximating Nonlinear Relationships for Optimal Operation of Natural Gas Transport Networks" *Processes* 6, no. 10: 198.
https://doi.org/10.3390/pr6100198