#
Optimization of Pipeline Network Layout for Multiple Heat Sources Distributed Energy Systems Considering Reliability Evaluation^{ †}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Problem Statement

#### 2.1.1. Assumption

- (1)
- The pipeline length is the distance between the two vertices that are connected.
- (2)
- The steam in the pipeline network system is an incompressible fluid.
- (3)
- The temperature of the fluid in each pipeline is constant.
- (4)
- The flow rate of the fluid in each pipeline is constant.

#### 2.1.2. Given

- (1)
- The number of consumers and heat sources.
- (2)
- The coordinates and heat demand of consumers.
- (3)
- The coordinates and the heat supply of heat source.

#### 2.1.3. Determine

- (1)
- The pipeline network topology of multiple heat sources DES.
- (2)
- Total annual cost and reliability of multiple heat sources DES.

#### 2.2. Mathematical Model

#### 2.2.1. Objective Function

#### 2.2.2. Linear Programming Model

**,**${b}_{x,y,z}^{Binary}$ is equal to 1, otherwise, it is equal to 0.

#### 2.2.3. Pipeline Cost Model

#### 2.2.4. Pressure Loss Cost Model

#### 2.2.5. Heat Loss Cost Model

#### 2.2.6. Reliability Assessment Model

## 3. Algorithm

#### 3.1. Clustering Algorithm

Algorithm 1: Clustering Algorithm | |

Input: The coordinates of consumers $\left[x,y\right]=\left\{\left({x}_{1},{y}_{1}\right),\left({x}_{2},{y}_{2}\right),\dots ,\left({x}_{{N}^{Consumer}},{y}_{{N}^{Consumer}}\right)\right\}$, the number of clusters ${N}^{Cluster}$, and the coordinates of heat sources $\left[\mu ,\lambda \right]=\left\{\left({\mu}_{1},{\lambda}_{1}\right),\left({\mu}_{2},{\lambda}_{2}\right),\dots ,\left({\mu}_{{N}^{Consumer}},{\lambda}_{{N}^{Consumer}}\right)\right\}$. | |

Output: A set of clusters $C=\left({C}_{1},{C}_{2},\cdots ,{C}_{{N}^{Cluster}}\right)$. | |

1: | Let ${C}_{j}=\left\{{\mathrm{Heat}\text{}\mathrm{source}}_{j}\right\},j=1,2,\cdots ,{N}^{Cluster}$ |

2: | for all $i=1,2,\cdots ,{N}^{Consumer}$ do |

3: | ${d}_{ij}=\sqrt{{({x}_{i}-{\mu}_{j})}^{2}+{({y}_{i}-{\lambda}_{j})}^{2}},j=1,2,\cdots ,{N}^{Cluster}$ |

4: | ${\xi}_{i}=argmin{d}_{ij},j=1,2,\cdots ,{N}^{Cluster}$ |

5: | ${C}_{{\xi}_{i}}={C}_{{\xi}_{i}}{{\displaystyle \cup}}^{\text{}}\left\{\left({x}_{i},{y}_{i}\right)\right\}$ |

6: | end for |

7: | return$C=\left({C}_{1},{C}_{2},\cdots ,{C}_{{N}^{Cluster}}\right)$ |

#### 3.2. Star Tree Algorithm

Algorithm 2: Star tree algorithm | |

Input: A graph $G=\left(V,E\right)$. | |

Output: A Star tree $T=\left({V}^{T},{E}^{T}\right)$. | |

1: | Let ${E}^{T}=\Phi $ |

2: | choose the heat source as the center of all consumers |

3: | for all $i=1,2,\cdots ,{N}^{V}$ do |

4: | generate ${e}_{i}^{T}$ by connecting consumer i and the center straightly |

5: | ${E}^{T}={E}^{T}{{\displaystyle \cup}}^{\text{}}\left\{{e}_{i}^{T}\right\}$ |

6: | end for |

7: | return $T=\left({V}^{T},{E}^{T}\right)$ |

#### 3.3. Kruskal Algorithm

Algorithm 3: Kruskal algorithm | |

Input: A connected graph $G=\left(V,E,W\right)$. | |

Output: A minimum spanning tree $T=\left({V}^{T},{E}^{T},{W}^{T}\right)$. | |

1: | Let ${E}^{T}=\Phi $ |

2: | Sort the edges such that $W\left({e}_{1}\right)\le W\left({e}_{2}\right)\le \cdots \le W\left({e}_{m}\right)$ |

3: | for all $i=1,2,\cdots ,{N}^{E}$ do |

4: | ${E}^{T}={E}^{T}{{\displaystyle \cup}}^{\text{}}\left\{{e}_{i}^{T}\right\}$ |

5: | if cycle is generated in $T$ then |

6: | delete ${e}_{i}^{T}$ from the ${E}^{T}$ |

7: | else |

8: | maintain the ${E}^{T}$ unchanged |

9: | end if |

10: | if ${N}^{{E}^{T}}={N}^{V}-1$ then |

11: | break |

12: | else |

13: | continue |

14: | end if |

15: | end for |

16: | return$T=\left({V}^{T},{E}^{T},{W}^{T}\right)$ |

#### 3.4. GeoSteiner Algorithm

_{x}, v

_{y}), which is obtained by intersecting the vertical line from point u and the horizontal line from point v. Additionally, the backbone, for Hwang topologies [34], (u, v) is consisted of line segments uw and vw. The points u, v, and w may be any combination of terminals, Steiner points, and corner points.

## 4. Case Study

#### 4.1. Data Acquisition

#### 4.2. Optimal Results and Analysis

#### 4.3. Analysis of Small-Scale DES

## 5. Conclusions

- Compared with the single heat source scenario, the multiple heat sources system will reduce the long-distance and high-flow pipelines in the system, so that both economy and reliability of the pipeline network system is improved.
- Compared with the traditional pipeline network obtained using MST, an ESMT pipeline network can reduce the total annual cost by 3% and increase reliability by 1%.
- When considering the actual path constraints, the RSMT pipeline network can be better adapted to the road layout.
- The geographically scale of the problem does not have a great impact on the relative performance of the four structures.
- By using the proposed method, both economic and reliability can be improved for the DES system.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Nomenclature

Abbreviations | |

DES | Distributed Energy System |

ESMT | Euclidean Steiner Minimum Tree |

FSTs | Full Steiner trees |

GA | Genetic algorithm |

LP | Linear programming |

MILP | Mixed Integer Linear Programming |

MIP | Mixed Integer Programming |

MST | Minimum Spanning Tree |

MSTHG | The MST in hypergraph |

NSGA-II | Non-dominated Sorting Genetic Algorithm II |

RSMT | Rectilinear Steiner Minimum Tree |

SMT | Steiner Minimum Tree |

STAR | Star tree |

TAC | Total annual cost, Ұ·a^{−1} |

Indices and Sets | |

$C$ | The set of clusters, denoted by index j |

$D$ | An index, which is used to control the positive and negative of ${w}_{x,y,z}$ by ${\left(-1\right)}^{D}$ |

$e$ | An index, referring to the edge in ${E}^{H}$ |

$E$ | The set of edges in a graph |

${E}^{H}$ | The set of edges in the hypergraph |

${E}^{T}$ | The set of edges in the connected tree, denoted by indices i,$\beta $ |

${E}_{z}^{T}$ | The set of edges in the connected tree which are connected directly to the vertex z. |

$G$ | The graph of empty set in Star tree algorithm, or the weighted connected graph without direction in Kruskal algorithm |

$H$ | A hypergraph |

$V$ | The set of all vertices in a graph, denoted by indices $x,y,z$ |

${V}^{H}$ | The set of vertices in the hypergraph, denoted by indices $x,y,z$ ${V}^{H}\subseteq V$ |

${V}^{T}$ | The set of vertices in MST, denoted by indices $x,y,z,\alpha $, ${V}^{T}\subseteq V$ |

$W$ | The set of weights of edges in a graph, denoted by indices $x,y,z$ |

${W}^{T}$ | The set of weights of edges in the connected tree, denoted by indices $x,y,z$, ${W}^{T}\subseteq W$ |

Variables | |

${a}^{E}$ | The electricity cost, Ұ·kW·h^{−1} |

${a}_{i,j}$ | The unit price of the $i$th pipeline of the cluster $j$, Ұ·m^{−1} |

${a}^{S}$ | The unit price of steam, Ұ·kg^{−1} |

${b}_{x,y,z}^{Binary}$ | Binary variables indicating whether the edge ($x$, $y$) in the connected tree is connected directly to the vertex z. |

${C}^{HeatLoss}$ | The heat loss cost of pipeline, Ұ·a^{−1} |

${C}_{i,j}^{HeatLoss}$ | The heat loss cost of the $i$th pipeline of the cluster $j$, Ұ·a^{−1} |

${C}^{Pipeline}$ | The construction cost of pipeline, Ұ·a^{−1} |

${C}_{i,j}^{Pipeline}$ | The construction cost of the $i$th pipeline of the cluster $j$, Ұ·a^{−1} |

${C}^{PressureLoss}$ | The pressure loss cost of pipeline, Ұ·a^{−1} |

${C}_{i,j}^{PressureLoss}$ | The pressure loss cost of the $i$th pipeline of the cluster $j$, Ұ·a^{−1} |

${d}_{i,j}$ | The Euclidean distance between vertex $i$ and vertex $j$ |

${D}_{i,j}^{inner,insulation}$ | The inner diameter of the insulation layer of the $i$th pipeline of the cluster $j$, m |

${D}_{i,j}^{inner,pipline}$ | The inner diameter of the $i$th pipeline of the cluster $j$, m |

${D}_{i,j}^{outer,insulation}$ | The outer diameter of the insulation layer of the $i$th pipeline of the cluster $j$, m |

${D}_{i,j}^{outer,pipline}$ | The outer diameter of the $i$th pipeline of the cluster $j$, m |

${e}_{i}^{H}$ | The $i$th edge in the hypergraph |

${e}_{i}^{T}$ | The $i$th edge in the connected tree |

g | Gravitational acceleration |

${H}_{i,j}^{f}$ | The head loss of the $i$th pipeline of the cluster $j$, m |

$I$ | The annual interest rate |

${L}_{i,j}$ | The length of the $i$th pipeline of the cluster $j$, m |

${L}_{\alpha ,\beta}$ | The length of the $\beta $th pipeline between consumer $\alpha $ and the heat source in the area where this consumer is located, m |

${N}^{Cluster}$ | The number of clusters of the pipeline network system |

${N}_{j}^{Cluster}$ | The number of branch pipelines in cluster $j$ |

${N}^{Consumer}$ | The number of consumers of the pipeline network system |

${N}^{{V}^{T}}$ | The number of vertices in the connected tree |

${N}^{{E}^{T}}$ | The number of vertices in the connected tree |

${N}^{year}$ | The life cycle of the pipeline network system, a |

${N}_{i,j}$ | The shaft power of the $i$th pipeline of the cluster $j$, W |

$N{e}_{i,j}$ | The effective power of the $i$th pipeline of the cluster $j$, W |

${p}_{\alpha ,\beta}^{Path}$ | The probability of connecting the $\beta $th pipeline in the path connected between the consumer $\alpha $ and the heat source in its area |

${P}_{\alpha}^{Probability}$ | The connected probability between the consumer α and the heat source of the area where this consumer is located |

$q$ | The latent heat of steam, kJ·kg^{−1} |

${Q}_{\alpha}^{HeatDemand}$ | The heat demand of consumer $\alpha $ |

${Q}_{i,j}$ | The heat loss of the $i$th pipeline of the cluster $j$, kJ·m^{−1}·s^{−1} |

${R}^{Reliability}$ | The reliability of the pipeline network system |

${S}_{i,j}^{inner,pipeline}$ | The head loss of the $i$th pipeline of the cluster $j$, m^{2} |

$T$ | The outer surface temperature of the pipeline, $\xb0\mathrm{C}$ |

${T}^{a}$ | The ambient temperature, $\xb0\mathrm{C}$ |

${T}^{Operating}$ | The number of annual operating hours of the device, h |

$u$ | The flow rate of the steam, m·s^{−1} |

${W}^{S}$ | The sum of the steam mass flow rate in all branch pipelines in the entire pipeline network |

${W}_{i,j}$ | The mass flow rate of the $i$th pipeline of the cluster $j$, kg·s^{−1} |

${W}_{z}^{S}$ | The mass flow rate of steam at vertex z, kg·s^{−1} |

${w}^{H}$ | A weight function of the edges in a hypergraph |

${w}^{H}\left(e\right)$ | The weight of edge $e$ in a spanning tree in the hypergraph |

${w}^{H}\left({T}^{H}\right)$ | The sum of weights of the edges in a spanning tree in the hypergraph |

${w}_{x,y,z}$ and ${w}_{y,x,z}$ | The mass flow rate within the branch pipeline connecting vertices x and $y$ when using vertex z as a reference for material accountancy, kg·s^{−1} |

$W{t}_{i,j}$ | The weight per unit length of the $i$th pipeline of the cluster $j$, kg·m^{−1} |

${x}_{e}$ | The solution vector in the hypergraph |

- Greek letters

$\epsilon $ | Heat transfer coefficient between the outer surface of the insulation and the atmosphere, W·m^{−2}·K^{−1} |

${\zeta}_{i,j}^{E}$ | The local resistance coefficient at the standard elbows (90°) |

${\zeta}_{i,j}^{Elbow}$ | The local resistance coefficient of the $i$th pipeline of the cluster $j$ |

${\zeta}_{i,j}$ | The resistance coefficient of the $i$th pipeline of the cluster $j$ |

$\eta $ | The efficiency of the conveying equipment |

$\lambda $ | The thermal conductivity of insulating material products at average temperature, W·m^{−1}·K^{−1} |

${\xi}_{i}$ | The index used to determine which heat source consumer $i$ is assigned to |

$\rho $ | The density of the steam, kg·m^{−3} |

$\sigma $ | The friction coefficient of the pipeline |

$\mathsf{\Phi}$ | $\mathrm{T}$he empty set |

## References

- Fichera, A.; Marrasso, E.; Sasso, M.; Volpe, R. Environmental and Economic Performance of an Urban Community Hybrid Distributed Energy System. Energy
**2020**, 13, 2545. [Google Scholar] - Ren, H.; Zhou, W.; Nakagami, K.I.; Gao, W.; Wu, Q. Multi-objective optimization for the operation of distributed energy systems considering economic and environmental aspects. Appl. Energy
**2010**, 87, 3642–3651. [Google Scholar] [CrossRef] - Wu, Q.; Ren, H.; Gao, W.; Ren, J. Multi-objective optimization of a distributed energy network integrated with heating interchange. Energy
**2016**, 109, 353–364. [Google Scholar] [CrossRef] - Jing, R.; Zhu, X.; Zhu, Z.; Wang, W.; Meng, C.; Shah, N.; Li, N.; Zhao, Y. A multi-objective optimization and multi-criteria evaluation integrated framework for distributed energy system optimal planning. Energy Convers. Manag.
**2018**, 166, 445–462. [Google Scholar] [CrossRef] - Yuan, J.; Xiao, Z.; Zhang, C.; Gang, W. A control strategy for distributed energy system considering the state of thermal energy storage. Sustain. Cities Soc.
**2020**, 63, 102492. [Google Scholar] [CrossRef] - Wang, X.; Tian, H.; Yan, F.; Feng, W.; Wang, R.; Pan, J. Optimization of a distributed energy system with multiple waste heat sources and heat storage of different temperatures based on the energy quality. Appl. Therm. Eng.
**2020**, 181, 115975. [Google Scholar] [CrossRef] - Buoro, D.; Casisi, M.; De Nardi, A.; Pinamonti, P.; Reini, M. Multicriteria optimization of a distributed energy supply system for an industrial area. Energy
**2013**, 58, 128–137. [Google Scholar] [CrossRef] - Wang, M.; Yu, H.; Lin, X.; Jing, R.; He, F.; Li, C. Comparing stochastic programming with posteriori approach for multi-objective optimization of distributed energy systems under uncertainty. Energy
**2020**, 210, 118571. [Google Scholar] [CrossRef] - Keçebaş, A.; Ali Alkan, M.; Bayhan, M. Thermo-economic analysis of pipe insulation for district heating piping systems. Appl. Therm. Eng.
**2011**, 31, 3929–3937. [Google Scholar] [CrossRef] - Salem, E.A.; Farid Khalil, M.; Sanhoury, A.S. Optimization of insulation thickness and emissions rate reduction during pipeline carrying hot oil. Alex. Eng. J.
**2021**, 60, 3429–3443. [Google Scholar] [CrossRef] - Li, X.-l.; Duanmu, L.; Shu, H.-w. Optimal design of district heating and cooling pipe network of seawater-source heat pump. Energy Build.
**2010**, 42, 100–104. [Google Scholar] [CrossRef] - Wang, Y.; Wang, J.; Gao, M.; Zhang, D.; Liu, Y.; Tan, Z.; Zhu, J. Cost-based siting and sizing of energy stations and pipeline networks in integrated energy system. Energy Convers. Manag.
**2021**, 235, 113958. [Google Scholar] [CrossRef] - Zeng, J.; Han, J.; Zhang, G. Diameter optimization of district heating and cooling piping network based on hourly load. Appl. Therm. Eng.
**2016**, 107, 750–757. [Google Scholar] [CrossRef] - Mehleri, E.D.; Sarimveis, H.; Markatos, N.C.; Papageorgiou, L.G. A mathematical programming approach for optimal design of distributed energy systems at the neighbourhood level. Energy
**2012**, 44, 96–104. [Google Scholar] [CrossRef] - Khir, R.; Haouari, M. Optimization models for a single-plant District Cooling System. Eur. J. Oper. Res.
**2015**, 247, 648–658. [Google Scholar] [CrossRef] - Chan, A.L.S.; Hanby, V.I.; Chow, T.T. Optimization of distribution piping network in district cooling system using genetic algorithm with local search. Energy Convers. Manag.
**2007**, 48, 2622–2629. [Google Scholar] [CrossRef] - Haikarainen, C.; Pettersson, F.; Saxén, H. A model for structural and operational optimization of distributed energy systems. Appl. Therm. Eng.
**2014**, 70, 211–218. [Google Scholar] [CrossRef] - Sanaye, S.; Mahmoudimehr, J. Optimal design of a natural gas transmission network layout. Chem. Eng. Res. Des.
**2013**, 91, 2465–2476. [Google Scholar] [CrossRef] - Su, H.; Zio, E.; Zhang, J.; Li, X.; Chi, L.; Fan, L.; Zhang, Z. A method for the multi-objective optimization of the operation of natural gas pipeline networks considering supply reliability and operation efficiency. Comput. Chem. Eng.
**2019**, 131, 106584. [Google Scholar] [CrossRef] - Sokolov, D.V.; Barakhtenko, E.A. Optimization of transmission capacity of energy water pipeline networks with a tree-shaped configuration and multiple sources. Energy
**2020**, 210, 118469. [Google Scholar] [CrossRef] - Liu, X. Energy stations and pipe network collaborative planning of integrated energy system based on load complementary characteristics. Sustain. Energy Grids Netw.
**2020**, 23, 100374. [Google Scholar] [CrossRef] - Rimkevicius, S.; Kaliatka, A.; Valincius, M.; Dundulis, G.; Janulionis, R.; Grybenas, A.; Zutautaite, I. Development of approach for reliability assessment of pipeline network systems. Appl. Energy
**2012**, 94, 22–33. [Google Scholar] [CrossRef] - Shan, X.; Wang, P.; Lu, W. The reliability and availability evaluation of repairable district heating networks under changeable external conditions. Appl. Energy
**2017**, 203, 686–695. [Google Scholar] [CrossRef] - Alsharqawi, M.; Zayed, T.; Parvizsedghy, L.; Senouci, A.; Al-Derham, H. Reliability Assessment Model for Water Distribution Networks. J. Pipeline Syst. Eng. Pract.
**2020**, 11, 04019059. [Google Scholar] [CrossRef] - Chen, Q.; Zuo, L.; Wu, C.; Cao, Y.; Bu, Y.; Chen, F.; Sadiq, R. Supply reliability assessment of a gas pipeline network under stochastic demands. Reliab. Eng. Syst. Saf.
**2021**, 209, 107482. [Google Scholar] [CrossRef] - Yu, W.; Huang, W.; Wen, Y.; Li, Y.; Liu, H.; Wen, K.; Gong, J.; Lu, Y. An integrated gas supply reliability evaluation method of the large-scale and complex natural gas pipeline network based on demand-side analysis. Reliab. Eng. Syst. Saf.
**2021**, 212, 107651. [Google Scholar] [CrossRef] - Zhu, Y.; Wang, P.; Wang, Y.; Tong, R.; Yu, B.; Qu, Z. Assessment method for gas supply reliability of natural gas pipeline networks considering failure and repair. J. Nat. Gas Sci. Eng.
**2021**, 88, 103817. [Google Scholar] [CrossRef] - Chang, C.; Chen, X.; Wang, Y.; Feng, X. An efficient optimization algorithm for waste Heat Integration using a heat recovery loop between two plants. Appl. Therm. Eng.
**2016**, 105, 799–806. [Google Scholar] [CrossRef] - Zhou, J.; Li, Z.; Liang, G.; Zhou, L.; Zhou, X. General Models for Optimal Design of Star–Star Gathering Pipeline Network. J. Pipeline Syst. Eng. Pract.
**2021**, 12, 04021024. [Google Scholar] [CrossRef] - Weiss, M.A. Data Structures and Algorithm Analysis in C++, 4th ed.; Pearson Education India: London, UK, 2014. [Google Scholar]
- Zachariasen, M. Rectilinear full Steiner tree generation. Networks
**1999**, 33, 125–143. [Google Scholar] [CrossRef] - Warme, D.M. Spanning Trees in Hypergraphs with Applications to Steiner Trees. Ph.D. Thesis, University of Virginia, Charlottesville, VA, USA, 1998. [Google Scholar]
- Juhl, D.; Warme, D.M.; Winter, P.; Zachariasen, M. The GeoSteiner software package for computing Steiner trees in the plane: An updated computational study. Math. Prog. Comp.
**2018**, 10, 487–532. [Google Scholar] [CrossRef] [Green Version] - Salowe, J.S.; Warme, D.M. Thirty-five-point rectilinear steiner minimal trees in a day. Networks
**1995**, 25, 69–87. [Google Scholar] [CrossRef]

**Figure 2.**Schematic of optimization of multiple heat sources DES network system: (

**a**) cluster consumers; (

**b**) generate heat source–heat source tree; (

**c**) generate heat source–consumers tree; (

**d**) obtain the layout of multiple heat sources DES pipeline network.

**Figure 6.**The relationship between the flow rate, the total pipeline cost and TAC of the four topologies.

DN (mm) | Pipeline Surface Temperature (°C) | ||||
---|---|---|---|---|---|

≤60 | ≤150 | ≤250 | ≤300 | ≤350 | |

Insulation Layer Thickness (mm) | |||||

15 | 30 | 30 | 40 | 50 | 50 |

20 | 30 | 30 | 40 | 50 | 50 |

25 | 30 | 30 | 50 | 50 | 60 |

40 | 30 | 50 | 50 | 60 | 60 |

50 | 30 | 50 | 50 | 60 | 70 |

80 | 30 | 50 | 60 | 70 | 70 |

100 | 30 | 50 | 60 | 70 | 80 |

150 | 30 | 60 | 70 | 70 | 80 |

200 | 30 | 60 | 70 | 80 | 90 |

250 | 30 | 60 | 70 | 80 | 90 |

300 | 30 | 60 | 70 | 80 | 90 |

350 | 30 | 50 | 70 | 80 | 90 |

400 | 30 | 50 | 70 | 80 | 90 |

450 | 30 | 50 | 70 | 80 | 90 |

500 | 30 | 50 | 70 | 80 | 90 |

600 | 30 | 50 | 70 | 80 | 90 |

700 | 30 | 50 | 70 | 80 | 90 |

800 | 30 | 50 | 70 | 80 | 90 |

900 | 30 | 50 | 70 | 80 | 100 |

1000 | 30 | 50 | 70 | 80 | 100 |

1100 | 30 | 50 | 70 | 80 | 100 |

1200 | 30 | 50 | 70 | 80 | 100 |

Name | Coordinate X (m) | Coordinate Y (m) | Heat Demand ( $\mathbf{t}\cdot {\mathit{h}}^{-1}$ Steam) |
---|---|---|---|

Consumer-01 | 7097 | 9542 | 15.0 |

Consumer-02 | 8800 | 4024 | 6.0 |

Consumer-03 | 9602 | 5124 | 0.5 |

Consumer-04 | 12,013 | 7072 | 4.0 |

Consumer-05 | 13,392 | 11,430 | 3.0 |

Consumer-06 | 13,949 | 14,857 | 5.0 |

Consumer-07 | 3384 | 24,093 | 10.0 |

Consumer-08 | 25,483 | 0 | 25.0 |

Consumer-09 | 11,914 | 2235 | 3.0 |

Consumer-10 | 11,369 | 3893 | 4.0 |

Consumer-11 | 6561 | 6546 | 5.0 |

Consumer-12 | 12,437 | 10,805 | 12.0 |

Consumer-13 | 12,454 | 11,538 | 0.3 |

Consumer-14 | 16,227 | 13,461 | 10.0 |

Consumer-15 | 18,914 | 14,341 | 1.2 |

Consumer-16 | 7171 | 12,650 | 0.3 |

Consumer-17 | 13,255 | 20,390 | 4.0 |

Consumer-18 | 15,489 | 16,736 | 7.0 |

Consumer-19 | 17,153 | 17,937 | 1.0 |

Consumer-20 | 22,789 | 24,165 | 2.0 |

Consumer-21 | 24,416 | 24,203 | 1.2 |

Consumer-22 | 6206 | 20,012 | 10.0 |

Consumer-23 | 5818 | 10,120 | 4.0 |

Consumer-24 | 3465 | 24,915 | 10.0 |

Heat source-01 | 0 | 3144 | −45.8 |

Heat source-02 | 12,200 | 21,944 | −45.2 |

Heat source-03 | 19,200 | 7544 | −52.5 |

Parameter | Number | Unit | Parameter | Number | Unit |
---|---|---|---|---|---|

${N}^{year}$ | 10 | a | ${a}^{S}$ | 0.1945 | Ұ·kg^{−1} |

$I$ | 0.02 | $q$ | 1999.9 | kJ·kg^{−1} | |

$\rho $ | 0.60 | kg·m^{−3} | ${T}^{a}$ | 3.5 | °C |

$u$ | 30.00 | m·s^{−1} | $\lambda $ | 0.06 | W·m^{−1}·K^{−1} |

${a}^{E}$ | 0.21 | Ұ·kW·h^{−1} | $v$ | 3.50 | m·s^{−1} |

${T}^{Operating}$ | 8760 | h | $\epsilon $ | 11.63 | W·m^{−2}·K^{−1} |

$\eta $ | 0.8 | ${p}_{\alpha ,\beta}^{Path}$ | 0.98 | km^{−1} | |

$\sigma $ | 0.015 |

Topology Type of Pipeline Networks | STAR | MST | RSMT | ESMT |
---|---|---|---|---|

Total pipeline cost (×10^{7} Ұ·a^{−1}) | 2.706 (100%) | 2.035 (75.2%) | 2.152 (79.5%) | 1.962 (72.5%) |

Pressure loss cost (×10^{7} Ұ·a^{−1}) | 0.668 (100%) | 0.504 (75.4%) | 0.533 (79.8%) | 0.486 (72.8%) |

Heat loss cost (×10^{7} Ұ·a^{−1}) | 0.990 (100%) | 0.623 (62.9%) | 0.654 (66.0%) | 0.605 (61.1%) |

Total annual cost (×10^{7} Ұ·a^{−1}) | 4.364 (100%) | 3.162 (72.5%) | 3.339 (76.5%) | 3.053 (70.0%) |

${R}^{Reliability}$ | 0.848 (100%) | 0.815 (96.1%) | 0.802 (94.6%) | 0.822 (96.9%) |

Topology Type of Pipeline Networks | STAR | MST | RSMT | ESMT |
---|---|---|---|---|

Total pipeline length (×10^{4} m) | 4.603 | 2.421 | 2.433 | 2.345 |

Total pipeline cost (×10^{6} Ұ·a^{−1}) | 5.412 | 4.069 | 4.304 | 3.925 |

Pressure loss cost (×10^{6} Ұ·a^{−1}) | 1.313 | 1.002 | 1.062 | 0.967 |

Heat loss cost (×10^{6} Ұ·a^{−1}) | 1.313 | 1.002 | 1.062 | 0.967 |

Total annual cost (×10^{6} Ұ·a^{−1}) | 6.725 | 5.072 | 5.366 | 4.892 |

${R}^{Reliability}$ | 0.967 | 0.959 | 0.956 | 0.961 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Cui, Z.; Lin, H.; Wu, Y.; Wang, Y.; Feng, X.
Optimization of Pipeline Network Layout for Multiple Heat Sources Distributed Energy Systems Considering Reliability Evaluation. *Processes* **2021**, *9*, 1308.
https://doi.org/10.3390/pr9081308

**AMA Style**

Cui Z, Lin H, Wu Y, Wang Y, Feng X.
Optimization of Pipeline Network Layout for Multiple Heat Sources Distributed Energy Systems Considering Reliability Evaluation. *Processes*. 2021; 9(8):1308.
https://doi.org/10.3390/pr9081308

**Chicago/Turabian Style**

Cui, Ziyuan, Hai Lin, Yan Wu, Yufei Wang, and Xiao Feng.
2021. "Optimization of Pipeline Network Layout for Multiple Heat Sources Distributed Energy Systems Considering Reliability Evaluation" *Processes* 9, no. 8: 1308.
https://doi.org/10.3390/pr9081308