# Optimization of Water Distribution Networks Using Genetic Algorithm Based SOP–WDN Program

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### Background and Related Work

## 2. Materials and Methods

#### 2.1. Problem Formulation

^{10}, i.e., 1,073,741,824 different pipe combinations. Hence, even for a relatively small pipe network, the search space is large. The design of an economically optimal water distribution network is a difficult task, because it involves solving many complex, non-linear, and discontinuous hydraulic equations, while simultaneously optimizing pipe sizes and other network components [53,54].

#### 2.2. Genetic Algorithm

- 1.
- Generation of initial population

- 2.
- Computation of network cost

- 3
- Hydraulic analysis of each network

- 4.
- Computation of penalty cost

- 5.
- Computation of total network cost

- 6.
- Computation of the fitness

- 7.
- Generation of a new population using the selection operator

- 8.
- The crossover operator

- 9.
- The mutation operator

- 10.
- Production of successive generations

#### 2.3. EPANET

#### 2.4. SOP–WDN

#### 2.4.1. Encoding Scheme, Interpretation and Redundancy

^{4}= 16 discrete values. With 13 diameter options, three of the substrings become redundant (do not represent any diameters). The challenges of dealing with redundant binary codes have been overlooked in the literature and in published methods [60]. Saleh and Tanyimboh proposed representing redundant codes with closed pipes of fictitious diameters and low fitness values, assuming their extinction through evolution and natural selection [61]. However, this approach was found to prematurely lose valuable genetic information.

#### 2.4.2. Genetic Algorithm Operators

- 1.
- Selection

- 2.
- Crossover

- 3.
- Mutation

- 4.
- Elitism

#### 2.4.3. Penalty Functions

_{P}the pressure penalty, ${P}_{j}$ is the pressure of node $j$, ${T}_{P}$ is the target pressure, ${P}_{P1}$ is the pressure penalty coefficient if the pressure at the node is above the target pressure, and ${P}_{P2}$ is the pressure penalty coefficient if the pressure at the node is below the target pressure.

#### 2.4.4. Sensitivity Analysis

- Population Size
- Crossover Probability
- Mutation Rate
- Velocity Penalty 1
- Velocity Penalty 2
- Pressure Penalty 1
- Pressure Penalty 2

## 3. Results and Discussion

#### 3.1. Example 1: Two-Loop Network

#### 3.2. Example 2: Hanoi Network

#### 3.3. Example 3: GoYang Network

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Node No. | Elevation (m) | Demand (m^{3}/h) |
---|---|---|

1 | 210 | Reservoir |

2 | 150 | 100 |

3 | 160 | 100 |

4 | 155 | 120 |

5 | 150 | 270 |

6 | 165 | 330 |

7 | 160 | 200 |

Pipe No. | Begin Node | End Node | Length (m) |
---|---|---|---|

1 | 1 | 2 | 1000 |

2 | 2 | 3 | 1000 |

3 | 2 | 4 | 1000 |

4 | 4 | 5 | 1000 |

5 | 4 | 6 | 1000 |

6 | 6 | 7 | 1000 |

7 | 3 | 5 | 1000 |

8 | 5 | 7 | 1000 |

Node No. | Demand (m^{3}/h) |
---|---|

1 | Reservoir |

2 | 890 |

3 | 850 |

4 | 130 |

5 | 725 |

6 | 1005 |

7 | 1350 |

8 | 550 |

9 | 525 |

10 | 525 |

11 | 500 |

12 | 560 |

13 | 940 |

14 | 615 |

15 | 280 |

16 | 310 |

17 | 865 |

18 | 1345 |

19 | 60 |

20 | 1275 |

21 | 930 |

22 | 485 |

23 | 1045 |

24 | 820 |

25 | 170 |

26 | 900 |

27 | 370 |

28 | 290 |

29 | 360 |

30 | 360 |

31 | 105 |

32 | 805 |

Pipe No. | Begin Node | End Node | Length (m) |
---|---|---|---|

1 | 1 | 2 | 100 |

2 | 2 | 3 | 1350 |

3 | 3 | 4 | 900 |

4 | 4 | 5 | 1150 |

5 | 5 | 6 | 1450 |

6 | 6 | 7 | 450 |

7 | 7 | 8 | 850 |

8 | 8 | 9 | 850 |

9 | 9 | 10 | 800 |

10 | 10 | 11 | 950 |

11 | 11 | 12 | 1200 |

12 | 12 | 13 | 3500 |

13 | 10 | 14 | 800 |

14 | 14 | 15 | 500 |

15 | 15 | 16 | 550 |

16 | 17 | 16 | 2730 |

17 | 18 | 17 | 1750 |

18 | 19 | 18 | 800 |

19 | 3 | 19 | 400 |

20 | 3 | 20 | 2200 |

21 | 20 | 21 | 1500 |

22 | 21 | 22 | 500 |

23 | 20 | 23 | 2650 |

24 | 23 | 24 | 1230 |

25 | 24 | 25 | 1300 |

26 | 26 | 25 | 850 |

27 | 27 | 26 | 300 |

28 | 16 | 27 | 750 |

29 | 23 | 28 | 1500 |

30 | 28 | 29 | 2000 |

31 | 29 | 30 | 1600 |

32 | 30 | 31 | 150 |

33 | 32 | 31 | 860 |

34 | 25 | 32 | 950 |

Node No. | Elevation (m) | Demand (m^{3}/d) |
---|---|---|

1 | 71.0 | Reservoir |

2 | 56.4 | 153.0 |

3 | 53.8 | 70.5 |

4 | 54.9 | 58.5 |

5 | 56.0 | 75.0 |

6 | 57.0 | 67.5 |

7 | 53.9 | 63.0 |

8 | 54.5 | 48.0 |

9 | 57.9 | 42.0 |

10 | 62.1 | 30.0 |

11 | 62.8 | 42.0 |

12 | 58.6 | 37.5 |

13 | 59.3 | 37.5 |

14 | 59.8 | 63.0 |

15 | 59.2 | 445.5 |

16 | 53.6 | 108.0 |

17 | 54.8 | 79.5 |

18 | 55.1 | 55.5 |

19 | 54.2 | 118.5 |

20 | 54.5 | 124.5 |

21 | 62.9 | 31.5 |

Pipe No. | Begin Node | End Node | Length (m) |
---|---|---|---|

1 | 1 | 2 | 165 |

2 | 2 | 3 | 124 |

3 | 3 | 4 | 118 |

4 | 4 | 5 | 81 |

5 | 5 | 6 | 134 |

6 | 6 | 12 | 135 |

7 | 12 | 15 | 202 |

8 | 2 | 22 | 135 |

9 | 2 | 21 | 170 |

10 | 21 | 22 | 113 |

11 | 22 | 20 | 335 |

12 | 20 | 19 | 115 |

13 | 2 | 19 | 345 |

14 | 19 | 17 | 114 |

15 | 3 | 16 | 103 |

16 | 16 | 17 | 261 |

17 | 17 | 18 | 72 |

18 | 7 | 18 | 373 |

19 | 3 | 7 | 98 |

20 | 7 | 8 | 110 |

21 | 4 | 8 | 98 |

22 | 8 | 9 | 246 |

23 | 5 | 11 | 174 |

24 | 10 | 11 | 102 |

25 | 6 | 10 | 92 |

26 | 6 | 9 | 100 |

27 | 10 | 13 | 130 |

28 | 12 | 13 | 90 |

29 | 13 | 14 | 185 |

30 | 15 | 14 | 90 |

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S.N. | Available Pipe Sizes (mm) | Unit Cost (per m) | 3-Bit (Binary) Representation | 3-Bit (Grey) Representation |
---|---|---|---|---|

1 | 8 | 100 | 000 | 000 |

2 | 10 | 120 | 001 | 001 |

3 | 12 | 150 | 010 | 011 |

4 | 14 | 180 | 011 | 010 |

5 | 16 | 200 | 100 | 110 |

6 | 18 | 250 | 101 | 111 |

7 | 20 | 300 | 110 | 101 |

8 | 24 | 350 | 111 | 100 |

Randomly Generated Binary String | 101110011001000111 | |||||
---|---|---|---|---|---|---|

6 Individual Pipes | 101, 110, 011, 001, 000, 111 | |||||

Pipe Position in WDN | Pipe No.1 | Pipe No.2 | Pipe No.3 | Pipe No.4 | Pipe No.5 | Pipe No.6 |

Chromosome (Binary) | 101 | 110 | 011 | 001 | 000 | 111 |

Pipe Diameter (mm) | 20 | 16 | 12 | 10 | 8 | 18 |

Unit Cost (per m) | 300 | 200 | 150 | 120 | 100 | 250 |

Length (m) | 100 | 100 | 100 | 100 | 100 | 100 |

Cost of individual pipe | 30,000 | 20,000 | 15,000 | 12,000 | 10,000 | 25,000 |

Total Cost of Network | 112,000 |

S.N. | GA Parameters | Values |
---|---|---|

1 | Population Size | 80–100 |

2 | Crossover Probability (%) | 85–90 |

3 | Mutation Rate (%) | 4–6 |

4 | Velocity Penalty 1 | 0.3 |

5 | Velocity Penalty 2 | 0.06 |

6 | Pressure Penalty 1 | 0.02 |

7 | Pressure Penalty 2 | 1.9 |

Diameter (in) | Diameter (mm) | Unit Cost (USD/m) |
---|---|---|

1 | 25.4 | 2 |

2 | 50.8 | 5 |

4 | 101.6 | 11 |

6 | 152.4 | 16 |

10 | 254.0 | 32 |

14 | 355.6 | 60 |

16 | 406.4 | 90 |

18 | 457.2 | 130 |

Pipe No. | Pipe Diameter (mm) | Pipe Length (m) | Cost (USD) | Node No. | Nodal Pressure (m) |
---|---|---|---|---|---|

1 | 457.2 | 1000 | 130,000 | 1 | Reservoir |

2 | 254.0 | 1000 | 32,000 | 2 | 53.25 |

3 | 406.4 | 1000 | 90,000 | 3 | 30.46 |

4 | 101.6 | 1000 | 11,000 | 4 | 43.45 |

5 | 406.4 | 1000 | 90,000 | 5 | 33.81 |

6 | 254.0 | 1000 | 32,000 | 6 | 30.44 |

7 | 254.0 | 1000 | 32,000 | 7 | 30.55 |

8 | 25.4 | 1000 | 2000 | - | - |

Total Cost: | 419,000 | Check | OK |

Studies | Alperovitz and Shamir | Savic and Walters | Geem | Van Dijk et al. | SOP–WDN |
---|---|---|---|---|---|

Least cost obtained (USD) | 479,525 | 420,000 | 419,000 | 419,000 | 419,000 |

Diameter (in) | Diameter (mm) | Unit Cost (USD/m) |
---|---|---|

12 | 304.8 | 45.73 |

16 | 406.4 | 70.40 |

20 | 508 | 98.38 |

24 | 609.6 | 129.33 |

30 | 762 | 180.75 |

40 | 1016 | 278.28 |

Pipe No. | Pipe Diameter (mm) | Pipe Length (m) | Cost (USD) | Node No. | Nodal Pressure (m) |
---|---|---|---|---|---|

1 | 1016 | 100 | 27,828 | 1 | 100 (Reservoir) |

2 | 1016 | 1350 | 375,678 | 2 | 97.14 |

3 | 1016 | 900 | 250,452 | 3 | 61.67 |

4 | 1016 | 1150 | 320,022 | 4 | 56.92 |

5 | 1016 | 1450 | 403,506 | 5 | 51.02 |

6 | 1016 | 450 | 125,226 | 6 | 44.81 |

7 | 1016 | 850 | 236,538 | 7 | 43.35 |

8 | 1016 | 850 | 236,538 | 8 | 41.61 |

9 | 1016 | 800 | 222,624 | 9 | 40.23 |

10 | 762 | 950 | 171,712.5 | 10 | 39.20 |

11 | 609.6 | 1200 | 155,196 | 11 | 37.64 |

12 | 609.6 | 3500 | 452,655 | 12 | 34.21 |

13 | 508 | 800 | 78,704 | 13 | 30.01 |

14 | 406.4 | 500 | 35,200 | 14 | 35.52 |

15 | 304.8 | 550 | 25,151.5 | 15 | 33.72 |

16 | 304.8 | 2730 | 124,842.9 | 16 | 31.30 |

17 | 406.4 | 1750 | 123,200 | 17 | 33.41 |

18 | 609.6 | 800 | 103,464 | 18 | 49.93 |

19 | 508 | 400 | 39,352 | 19 | 55.09 |

20 | 1016 | 2200 | 612,216 | 20 | 50.61 |

21 | 508 | 1500 | 147,570 | 21 | 41.26 |

22 | 304.8 | 500 | 22,865 | 22 | 36.10 |

23 | 1016 | 2650 | 737,442 | 23 | 44.52 |

24 | 762 | 1230 | 222,322.5 | 24 | 38.93 |

25 | 762 | 1300 | 234,975 | 25 | 35.34 |

26 | 508 | 850 | 83,623 | 26 | 31.70 |

27 | 304.8 | 300 | 13,719 | 27 | 30.76 |

28 | 304.8 | 750 | 34,297.5 | 28 | 38.94 |

29 | 406.4 | 1500 | 105,600 | 29 | 30.13 |

30 | 304.8 | 2000 | 91,460 | 30 | 30.42 |

31 | 304.8 | 1600 | 73,168 | 31 | 30.70 |

32 | 406.4 | 150 | 10,560 | 32 | 33.18 |

33 | 406.4 | 860 | 60,544 | - | - |

34 | 609.6 | 950 | 122,863.5 | - | - |

Total Cost: | 6,081,115.4 | Check | OK |

Studies | Savic and Walters | Liong and Atiquzzaman | Geem | Van Dijk et al. | SOP–WDN |
---|---|---|---|---|---|

Least cost obtained (Million USD) | 6.187 | 6.220 | 6.056 | 6.110 | 6.081 |

Diameter (mm) | Unit Cost (Won/m) |
---|---|

80 | 37,890 |

100 | 38,933 |

125 | 40,563 |

150 | 42,554 |

200 | 47,624 |

250 | 54,125 |

300 | 62,109 |

350 | 71,524 |

Pipe No. | Pipe Diameter (mm) | Pipe Length (m) | Cost (Won) | Node No. | Nodal Pressure (m) |
---|---|---|---|---|---|

1 | 200 | 165.0 | 7,857,960 | 1 | 15.62 |

2 | 125 | 124.0 | 5,029,812 | 2 | 29.33 |

3 | 125 | 118.0 | 4,786,434 | 3 | 28.73 |

4 | 100 | 81.0 | 3,153,573 | 4 | 26.58 |

5 | 80 | 134.0 | 5,077,260 | 5 | 24.20 |

6 | 80 | 135.0 | 5,115,150 | 6 | 21.51 |

7 | 80 | 202.0 | 7,653,780 | 7 | 27.72 |

8 | 80 | 135.0 | 5,115,150 | 8 | 26.70 |

9 | 80 | 170.0 | 6,441,300 | 9 | 21.20 |

10 | 80 | 113.0 | 4,281,570 | 10 | 16.17 |

11 | 80 | 335.0 | 12,693,150 | 11 | 16.03 |

12 | 80 | 115.0 | 4,357,350 | 12 | 18.16 |

13 | 80 | 345.0 | 13,072,050 | 13 | 17.46 |

14 | 80 | 114.0 | 4,319,460 | 14 | 15.33 |

15 | 80 | 103.0 | 3,902,670 | 15 | 15.48 |

16 | 80 | 261.0 | 9,889,290 | 16 | 28.31 |

17 | 80 | 72.0 | 2,728,080 | 17 | 26.75 |

18 | 80 | 373.0 | 14,132,970 | 18 | 26.44 |

19 | 80 | 98.0 | 3,713,220 | 19 | 27.36 |

20 | 80 | 110.0 | 4,167,900 | 20 | 26.68 |

21 | 80 | 98.0 | 3,713,220 | 21 | 19.74 |

22 | 80 | 246.0 | 9,320,940 | 22 | 19.36 |

23 | 80 | 174.0 | 6,592,860 | - | - |

24 | 80 | 102.0 | 3,864,780 | - | - |

25 | 80 | 92.0 | 3,485,880 | - | - |

26 | 80 | 100.0 | 3,789,000 | - | - |

27 | 80 | 130.0 | 4,925,700 | - | - |

28 | 80 | 90.0 | 3,410,100 | - | - |

29 | 80 | 185.0 | 7,009,650 | - | - |

30 | 80 | 90.0 | 3,410,100 | - | - |

Total Cost: | 177,010,359 | Check | OK |

Studies | Original Network | Kim et al. | Geem | Menon et al. [68] | SOP–WDN |
---|---|---|---|---|---|

Least cost obtained (Million Won) | 179.428 | 179.142 | 177.135 | 177.417 | 177.010 |

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## Share and Cite

**MDPI and ACS Style**

Sangroula, U.; Han, K.-H.; Koo, K.-M.; Gnawali, K.; Yum, K.-T.
Optimization of Water Distribution Networks Using Genetic Algorithm Based SOP–WDN Program. *Water* **2022**, *14*, 851.
https://doi.org/10.3390/w14060851

**AMA Style**

Sangroula U, Han K-H, Koo K-M, Gnawali K, Yum K-T.
Optimization of Water Distribution Networks Using Genetic Algorithm Based SOP–WDN Program. *Water*. 2022; 14(6):851.
https://doi.org/10.3390/w14060851

**Chicago/Turabian Style**

Sangroula, Uchit, Kuk-Heon Han, Kang-Min Koo, Kapil Gnawali, and Kyung-Taek Yum.
2022. "Optimization of Water Distribution Networks Using Genetic Algorithm Based SOP–WDN Program" *Water* 14, no. 6: 851.
https://doi.org/10.3390/w14060851