# Water Distribution Network Partitioning Based on Complex Network Theory: The Udine Case Study

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Clustering

^{T}, and A is the adjacency matrix.

_{1}, G

_{2}, …, G

_{p}. Then,

- Intracluster edges: ${E}_{1}\cup {E}_{2}\cup \xb7\xb7\xb7\cup {E}_{p}$;
- Intercluster edges: $\partial \left({V}_{1}\right)\cup \partial \left({V}_{2}\right)\cup \dots \cup \partial \left({V}_{p}\right)$

- Clustering quality indices:

- Hydraulic indices:

#### 2.2. Water Network Dividing

#### 2.3. Methodology

## 3. Results

- n = 5793 nodes;
- m = 6466 pipes;
- Six sources;
- Five tanks;
- Five pumps.

#### 3.1. Clustering Analysis

#### 3.2. Dividing Phase

- Individuals = 150;
- Generations = 150.

- Tournament selection with tournament dimension equal to two for the single-objective optimization;
- Elitist non-dominated sorting genetic algorithm (NSGA-II) for the multi-objective function;
- Two-point crossover with crossover probability (${p}_{c})=0.9$;
- Flip-bit mutation with individual mutation probability ${p}_{m}=0.2$ and gene mutation probability ${p}_{mutation}=4\mathrm{\%}$.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**(

**a**) Number of DMA-balance index relation for clustering algorithms; (

**b**) number of DMA-number of edge-cuts relation for clustering algorithms; (

**c**) number of DMA-conductance relation for clustering algorithms; (

**d**) number of DMA-resistance relation for clustering algorithms.

**Figure 4.**(

**a**) Udine WDN clustering: spectral clustering algorithm—12 DMAs; (

**b**) Udine WDN clustering: Girvan–Newman algorithm—12 DMAs.

**Figure 5.**(

**a**) Udine WDNP: Spectral clustering algorithm and investing cost objective function; (

**b**) Udine WDNP: Girvan–Newman algorithm and investing cost objective function; (

**c**) Udine WDNP: Spectral clustering algorithm and MRI objective function; (

**d**) Udine WDNP: Girvan–Newman algorithm and MRI objective function.

**Figure 7.**(

**a**) Pareto frontier: Spectral clustering algorithm and MRI + investing cost objective function; (

**b**) Pareto frontier: Girvan–Newman algorithm and MRI + investing cost objective function.

Name | Volume [m^{3}] | Min Level [m] | Max Level [m] |
---|---|---|---|

San Bernardo | 400 | 158.0 | 163.0 |

Vat | 1750 | 150.5 | 156.0 |

Don Bosco | 2200 | 145.0 | 153.5 |

Cotonificio | 2200 | 145.0 | 153.5 |

Castello | 4570 | 136.5 | 139.2 |

**Table 2.**Number of edge-cuts, clustering, and hydraulic indices for spectral clustering and Girvan–Newman algorithm.

Spectral Clustering | Girvan–Newman | |||||||
---|---|---|---|---|---|---|---|---|

DMAs | I_{B} [-] | N_{EC} [-] | C_{EC} [-] | R_{EC} [m^{−4}] | I_{B} [-] | N_{EC} [-] | C_{EC} [-] | R_{EC} [m^{−4}] |

7 | 2.295 | 56 | 0.774 | 7.39 × 10^{8} | 1.895 | 37 | 0.605 | 1.83 × 10^{8} |

8 | 2.632 | 58 | 0.821 | 9.08 × 10^{8} | 2.026 | 48 | 0.892 | 2.02 × 10^{8} |

9 | 2.345 | 56 | 1.119 | 4.25 × 10^{8} | 1.493 | 57 | 0.955 | 2.59 × 10^{8} |

10 | 3.102 | 49 | 0.801 | 7.33 × 10^{8} | 1.659 | 61 | 1.017 | 2.70 × 10^{8} |

11 | 2.523 | 56 | 1.280 | 2.58 × 10^{8} | 1.718 | 64 | 1.082 | 2.77 × 10^{8} |

12 | 2.740 | 56 | 1.135 | 4.19 × 10^{8} | 1.588 | 72 | 1.462 | 2.35 × 10^{8} |

13 | 2.640 | 67 | 0.756 | 9.97 × 10^{8} | 1.603 | 76 | 1.694 | 3.49 × 10^{8} |

Spectral Clustering | Girvan–Newman | |||||
---|---|---|---|---|---|---|

DMA | Dimension | Length [km] | Mean Demand [l/s] | Dimension | Length [km] | Mean Demand [l/s] |

1 | 476 | 34.07 | 27.8 | 724 | 38.15 | 60.85 |

2 | 53 | 2.94 | 3.57 | 472 | 30.35 | 30.46 |

3 | 185 | 27.42 | 17.84 | 765 | 47.04 | 52.89 |

4 | 1287 | 60.57 | 107.99 | 374 | 26.04 | 26.12 |

5 | 109 | 8.98 | 4.27 | 517 | 31.78 | 28.63 |

6 | 546 | 37.19 | 38.74 | 454 | 35.52 | 28.26 |

7 | 577 | 37.49 | 33.42 | 613 | 36.86 | 40.01 |

8 | 276 | 19.99 | 19.12 | 402 | 29.27 | 34.19 |

9 | 295 | 21.23 | 24.58 | 192 | 38.22 | 13.17 |

10 | 601 | 36.32 | 38.57 | 505 | 28.49 | 33.17 |

11 | 1320 | 107.69 | 87.91 | 517 | 20.78 | 42.92 |

12 | 56 | 5.57 | 5.69 | 246 | 35.48 | 18.84 |

Spectral Clustering Algorithm | Girvan–Newman Algorithm | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Objective Function: Investing Cost | Objective Function: MRI | Objective Function: Investing Cost | Objective Function: MRI | ||||||||

DMA | Mean Pressure [m] | Minimum Pressure [m] | DMA | Mean Pressure [m] | Minimum Pressure [m] | DMA | Mean Pressure [m] | Minimum Pressure [m] | DMA | Mean Pressure [m] | Minimum Pressure [m] |

1 | 33.93 | 22.53 | 1 | 31.43 | 18.67 | 1 | 32.03 | 16.9 | 1 | 37.34 | 27.04 |

2 | 28.48 | 15.12 | 2 | 40.51 | 30.74 | 2 | 34.06 | 22.91 | 2 | 34.8 | 23.84 |

3 | 30.63 | 20.8 | 3 | 30.69 | 20.96 | 3 | 30.57 | 18.16 | 3 | 31.85 | 17.13 |

4 | 33.35 | 20.36 | 4 | 37.15 | 26.79 | 4 | 37.8 | 15.12 | 4 | 47.62 | 36.46 |

5 | 54.68 | 33.7 | 5 | 62.16 | 47.82 | 5 | 35.36 | 19.75 | 5 | 43.63 | 31.24 |

6 | 35.2 | 16.17 | 6 | 46.21 | 33.9 | 6 | 80.3 | 29.2 | 6 | 80.3 | 29.2 |

7 | 38.17 | 22.29 | 7 | 44.68 | 29.94 | 7 | 31.78 | 21.99 | 7 | 33.76 | 25.7 |

8 | 47.11 | 23.78 | 8 | 55.87 | 40.99 | 8 | 33.06 | 18.92 | 8 | 33.23 | 18.6 |

9 | 34.96 | 26 | 9 | 36.38 | 26.58 | 9 | 37.67 | 18.85 | 9 | 38.24 | 18.54 |

10 | 30.58 | 15.92 | 10 | 37.48 | 28.31 | 10 | 32.61 | 17.19 | 10 | 38.99 | 24.48 |

11 | 33.21 | 22.21 | 11 | 34.63 | 22.55 | 11 | 33.58 | 22.1 | 11 | 39.04 | 26.75 |

12 | 43.31 | 30.16 | 12 | 43.09 | 29.95 | 12 | 43.24 | 26.6 | 12 | 43.86 | 23.27 |

**Table 5.**Cost, MRI, water age, number of boundary valves, and flow meters for the different clustering algorithms and objective functions.

Spectral Clustering | Girvan–Newman | |||
---|---|---|---|---|

Investing Cost | MRI | Investing Cost | MRI | |

Cost [€] | 85,382.4 | 114,434 | 97,981.3 | 154,057.3 |

MRI [-] | 0.15724 | 0.19035 | 0.17781 | 0.20621 |

Water age [h] | 13.01 | 13.54 | 11.83 | 13.49 |

Mean pressure [m] | 34.84 | 38.96 | 37.36 | 40.95 |

boundary valves | 40 | 27 | 56 | 20 |

flow meters | 16 | 29 | 16 | 52 |

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**MDPI and ACS Style**

Spizzo, F.; Venaruzzo, G.; Nicolini, M.; Goi, D.
Water Distribution Network Partitioning Based on Complex Network Theory: The Udine Case Study. *Water* **2023**, *15*, 1621.
https://doi.org/10.3390/w15081621

**AMA Style**

Spizzo F, Venaruzzo G, Nicolini M, Goi D.
Water Distribution Network Partitioning Based on Complex Network Theory: The Udine Case Study. *Water*. 2023; 15(8):1621.
https://doi.org/10.3390/w15081621

**Chicago/Turabian Style**

Spizzo, Federico, Giovanni Venaruzzo, Matteo Nicolini, and Daniele Goi.
2023. "Water Distribution Network Partitioning Based on Complex Network Theory: The Udine Case Study" *Water* 15, no. 8: 1621.
https://doi.org/10.3390/w15081621