# A Model of Optimal Gas Supply to a Set of Distributed Consumers

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{3}) and below 6 $/MMBtu (about 0.21 $/m

^{3}) for the low income markets. The transportation to the customer and the supply security are also important issues. Today, natural gas is delivered to Europe mainly by pipeline from the Russian Federation and Norway [7].

## 2. Model Description

#### 2.1. Model Assumptions

_{p}, and molar mass, $\overline{M}$. The biogas injected into the pipeline network is for the case of simplicity taken to be upgraded to the same quality as the natural gas. Therefore, the different gases can be interchanged freely in the supply chain.

- The mass flows in the system are balanced.
- Fuel in adequate quantity covers the customers’ demands.
- Technical and physical constraints are obeyed.
- Customers supplied by LNG truck must have adequate storing facilities.

#### 2.2. Constraints

#### 2.2.1. Pipe Transportation

#### 2.2.2. Truck Supply

#### 2.3. Costs and Objective Function

#### 2.4. Computational Solution

## 3. Case Study

#### 3.1. Parameters for the Local Gas Supply Problem

#### 3.2. Background of Case Study

^{3}, with a maximum regasification rate of 15 kg/s. The maximum biogas supply is 3 kg/s. Color coding (blue for local LNG, orange for distant LNG, yellow for CNG and green for biogas) will in the following be used to represent the fuel supplied in the figures representing the optimal solutions under different conditions. This formulation resulted in about 55,000 constraints and 38,000 variables (out of which were 14,000 integer variables). The optimization of each case took 5–30 min on a standard PC.

#### 3.3. Base Case Solution

## 4. Sensitivity Analysis

#### 4.1. Effect of Gas Price and Investment Costs

#### 4.1.1. Case 1

#### 4.1.2. Case 2

#### 4.1.3. Case 3

#### 4.1.4. Case 4

#### 4.2. Detailed Effect of Alternative Gas Price

#### 4.3. Effect of Gas Demand

#### 4.3.1. Low Demand

#### 4.3.2. High Demand

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

Binary and Integer Variables | |

b | integer controlling storages |

f | truck supply existence variable |

g | gasification existence variable |

s | number of tank lines |

w | CNG loading line binary variable |

y | variable for existing connections |

Continuous variables | |

L | truck supply, kg/s |

m | mass flow rate, kg/s |

N | number of trucks |

O | outflow of natural gas, kg/s |

p | pressure, bar |

S | supply of natural gas, kg/s |

T | temperature, K |

$\tilde{T}$ | temperature after ideal compression, K |

Parameters | |

c_{p} | specific heat capacity, kJ/(kg K) |

C | cost, € |

d | pipe diameter, m |

D | energy demand at node, MW |

H | heating value, MJ/kg |

K | life length of investment, a |

l | pipe length, m |

M | large positive constant (“big M”), - |

$\overline{M}$ | average molar mass of natural gas, kg/kmol |

n | number of compression steps |

O | energy outflow at node, MW |

R_{g} | universal gas constant, J/(mol K) |

t | duration of time period, h |

u | interest rate, - |

U | capacity, kg |

ν | unit cost, €, €/kWh, €/m or €/kg |

Sets | |

A | storage type a ∊ A |

I | nodes i ∊ I |

J | nodes j ∊ J |

R | pipe diameter type r ∊ R |

Greek | |

η | efficiency factor, - |

ζ | friction factor, - |

ρ | density, kg/m^{3} |

Superscripts | |

ALT | alternative fuel |

BIO | biogas |

CNG | compressed natural gas |

dist | distance travelled |

k | fuels by truck: LNG, CNG, ALT |

load | LNG load line |

LNG | liquefied natural gas |

max | maximum amount |

NG | natural gas |

pipe | pipe |

pow | power |

stor | storage |

tank | tanking |

time | travelling time |

truck | truck transportation |

Subscripts | |

a | storage type |

ALT | alternative fuel |

amb | ambient |

BIO | biogas |

CNG | compressed natural gas |

comp | compressor |

gasif | gasification cost |

i | node |

invest | investment cost |

j | node |

k | fuels by truck: LNG, CNG, ALT |

LNG | liquefied natural gas |

load | LNG loading line |

mult | multi-day |

NG | natural gas |

oper | operational cost |

pipe | pipe investment |

r | pipe type |

tank | tanking |

truck | truck transportation |

stor | storage |

sup | supply node |

year | yearly operation |

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**Figure 1.**Gas suppliers and consumers. Left: Location of the region studied (Vasa) and the distant LNG terminal (Pori). Right: Consumers in the region with demands reported in boxes (Background map source: © OpenStreetMap contributors).

**Figure 2.**Network scheme with the location of the potential pipe connections (lines), customers (black dots) and suppliers (color dots). Node numbers are reported in red (Background map source: © OpenStreetMap contributors).

**Figure 3.**Optimal supply chain for the Base Case. The fuel supplied to the nodes is indicated in the colored boxes (Background map source: © OpenStreetMap contributors).

**Figure 4.**Optimal supply network in Case 1 with storages (triangles) and pipelines (blue lines). The type of fuel used is denoted by color in the rectangles, which also reports the fuel demand. The node number is given by the bold number in the rectangle. The node with the storage supplying regasified LNG into the pipeline is denoted by the red framed rectangle (Background map source: © OpenStreetMap contributors).

**Figure 7.**Optimal supply network in Case 4. For a definition of the symbols, see caption of Figure 4 (Background map source: © OpenStreetMap contributors).

**Figure 8.**Effect of alternative gas price on the (

**a**) total cost and the constructed pipeline length, (

**b**) distribution between different gas sources.

**Figure 9.**Optimal gas supply chains for an alternative fuel price of (

**a**) 75.0%, (

**b**) 81.2%, (

**c**) 85.4%, and (

**d**) 91.7% of the local fuel price (Background map source: © OpenStreetMap contributors).

**Figure 10.**Optimal supply network for the scenario where the demand is half of the nominal one. For a definition of the symbols, see caption of Figure 4 (Background map source: © OpenStreetMap contributors).

**Figure 11.**Optimal supply network for the scenario where the demand is double the nominal demand. For a definition of the symbols, see caption of Figure 4 (Background map source: © OpenStreetMap contributors).

Component | Specification (Symbol) | Unit Cost | K (a) |
---|---|---|---|

Fuel | LNG (${v}^{\mathrm{LNG}}$) | 86.4 €/MWh | - |

CNG (${v}^{\mathrm{CNG}}$) | 86.4 €/MWh | - | |

BIO (${v}^{\mathrm{BIO}}$) | 86.4 €/MWh | - | |

ALT (${v}^{\mathrm{ALT}}$) | 86.4 €/MWh | - | |

Pipe | 0.15 m (${v}_{1}^{\mathrm{pipe}}$) | 328 €/m | 30 |

0.25 m (${v}_{2}^{\mathrm{pipe}}$) | 386 €/m | 30 | |

0.40 m (${v}_{3}^{\mathrm{pipe}}$) | 491 €/m | 30 | |

0.50 m (${v}_{4}^{\mathrm{pipe}}$) | 578 €/m | 30 | |

LNG infrastructure | S1 (${v}_{\mathrm{S}1}^{\mathrm{stor}}$) | 1800 k€ | 30 |

S2 (${v}_{\mathrm{S}2}^{\mathrm{stor}}$) | 7000 k€ | 30 | |

S3 (${v}_{\mathrm{S}3}^{\mathrm{stor}}$) | 13,000 k€ | 30 | |

LNG loading (${v}^{\mathrm{load}}$) | 450 k€ | 20 | |

LNG gasification (${v}^{\mathrm{gasif}}$) | 2000 k€ | 20 | |

CNG infrastructure | CNG container (${v}^{\mathrm{cont}}$) | 90 k€ | 15 |

CNG tanking (${v}^{\mathrm{tank}}$) | 600 k€ | 20 | |

CNG filling (${v}^{\mathrm{fill}}$) | 50 k€ | 15 | |

Truck transportation | Distance (${v}^{\mathrm{dist}}$) | 2 €/km | - |

Time, LNG (${v}_{\mathrm{LNG}}^{\mathrm{time}}$) | 200 €/h | - | |

CNG (${v}_{\mathrm{CNG}}^{\mathrm{time}}$) | 80 €/h | - |

Node | Latitude | Longitude | D_{i} (MW) |
---|---|---|---|

1. LNG terminal | 63.08 | 21.57 | 10.0 |

2. Biogas plant | 63.13 | 21.76 | 0.0 |

3. CHP plant | 63.09 | 21.55 | 262.9 |

4. Waste water treatment | 63.11 | 21.59 | 0.5 |

5. Gas station I | 63.07 | 21.67 | 2.0 |

6. Engine production | 63.10 | 21.61 | 23.1 |

7. Industry I | 63.06 | 21.55 | 0.7 |

8. Gas station II | 63.14 | 21.76 | 1.9 |

9. Hospital | 63.08 | 21.61 | 1.3 |

10. University campus | 63.11 | 21.59 | 157.8 |

11. Greenhouses I | 63.15 | 21.64 | 1.6 |

12. Vasa airport | 63.04 | 21.76 | 2.1 |

13. Vasa port | 63.09 | 21.56 | 3.2 |

14. Aquaparc | 63.09 | 21.59 | 15.8 |

15. Vasa school | 63.08 | 21.64 | 10.5 |

16. Industry II | 63.08 | 21.67 | 21.0 |

17. Industry III | 63.17 | 21.59 | 17.9 |

18. Industry IV | 63.03 | 21.76 | 0.7 |

19. Greenhouses II | 63.00 | 21.62 | 0.9 |

20. Industry V | 63.10 | 21.73 | 0.5 |

21. Industry VI | 63.09 | 21.75 | 42.1 |

22. Laihia | 62.98 | 22.00 | 1.5 |

23. Pörtom | 62.71 | 21.61 | 1.4 |

24. Kvevlax | 63.16 | 21.82 | 1.3 |

25. Replot | 63.23 | 21.41 | 1.2 |

26. CNG terminal | 63.08 | 21.57 | 0.0 |

Node | ${\mathit{N}}^{\mathbf{CNG}}$ (1/d) |
---|---|

7. Industry I | 0.44 |

18. Industry IV | 0.44 |

19. Greenhouses II | 0.57 |

22. Laihia | 0.90 |

23. Pörtom | 0.84 |

24. Kvevlax | 0.78 |

25. Replot | 0.72 |

Unit Cost Term | Case 1 | Case 2 | Case 3 | Case 4 |
---|---|---|---|---|

Local LNG & CNG | +25% | −25% | +25% | −25% |

Distant LNG | −25% | +25% | −25% | +25% |

Storage investment | −25% | −25% | +25% | +25% |

Pipe investment | +25% | +25% | −25% | −25% |

**Table 5.**Main results of optimization of the Base Case and four cases listed in Table 4.

Variables | Unit | Base Case | Case 1 | Case 2 | Case 3 | Case 4 |
---|---|---|---|---|---|---|

LNG supply, Vasa (pipe+truck) | GWh | 4469 | 0 | 5029 | 0 | 5029 |

LNG suppy, Pori (truck) | GWh | 0 | 5098 | 0 | 5098 | 0 |

Biogas supply (pipe) | GWh | 560 | 0 | 0 | 0 | 0 |

CNG supply (truck) | GWh | 69 | 0 | 69 | 0 | 69 |

Pipeline length | km | 31.6 | 10.1 | 16.2 | 46.9 | 35.4 |

Pipeline diameter | m | 0.15, 0.25 | 0.15 | 0.15, 0.25 | 0.15 | 0.15, 0.25 |

Max. compression pressure | bar | 13.0 | 7.0 | 8.4 | 11.6 | 8.7 |

LNG storage, S1 units | - | 0 | 17 | 4 | 9 | 0 |

LNG storage, S2 units | - | 0 | 0 | 0 | 0 | 0 |

LNG storage, S3 units | - | 0 | 2 | 0 | 2 | 0 |

LNG storage, total capacity | t | 0 | 14,995 | 2232 | 14,322 | 0 |

CNG containers | - | 9 | 0 | 9 | 0 | 0 |

LNG trucks, Vasa | 1/a | 0 | 0 | 866 | 0 | 0 |

LNG trucks, Pori | 1/a | 0 | 21,950 | 0 | 21,950 | 0 |

CNG trucks | 1/a | 1713 | 0 | 1730 | 0 | 1730 |

Total Cost | M€ | 445.4 | 370.8 | 336.5 | 374.8 | 335.1 |

Variables | Unit | Low | High |
---|---|---|---|

LNG supply, Vasa (pipe+trucks) | GWh | 1278 | 6544 |

LNG supply, Pori (trucks) | GWh | 0 | 2312 |

Biogas supply (pipe) | GWh | 1206 | 1314 |

CNG supply (truck) | GWh | 65 | 26 |

Pipeline length | km | 23.5 | 33.8 |

Pipeline diameter | m | 0.15, 0.25 | 0.15,0.25,0.4 |

Max. compression pressure | bar | 15.8 | 11.2 |

LNG storage, S1 unit | - | 0 | 6 |

LNG storage, S2 unit | - | 0 | 0 |

LNG storage, S3 unit | - | 0 | 1 |

LNG storage, total capacity | t | 0 | 7998 |

CNG containers | 1/a | 11 | 4 |

LNG trucks, Vasa | 1/a | 0 | 0 |

LNG trucks, Pori | 1/a | 0 | 117,480 |

CNG trucks | 1/a | 1616.9 | 646.6 |

Total Cost | M€ | 223.9 | 905.2 |

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## Share and Cite

**MDPI and ACS Style**

Mikolajková-Alifov, M.; Pettersson, F.; Björklund-Sänkiaho, M.; Saxén, H.
A Model of Optimal Gas Supply to a Set of Distributed Consumers. *Energies* **2019**, *12*, 351.
https://doi.org/10.3390/en12030351

**AMA Style**

Mikolajková-Alifov M, Pettersson F, Björklund-Sänkiaho M, Saxén H.
A Model of Optimal Gas Supply to a Set of Distributed Consumers. *Energies*. 2019; 12(3):351.
https://doi.org/10.3390/en12030351

**Chicago/Turabian Style**

Mikolajková-Alifov, Markéta, Frank Pettersson, Margareta Björklund-Sänkiaho, and Henrik Saxén.
2019. "A Model of Optimal Gas Supply to a Set of Distributed Consumers" *Energies* 12, no. 3: 351.
https://doi.org/10.3390/en12030351