# A New Vulnerability Measure for Water Distribution Network

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Proposed Vulnerability Measure

#### 2.2. Study Networks

^{3}/h, the hydraulic grade line elevation at each tank is set at 70 m and the minimum pressure head threshold is taken as 15 m for all junctions of the network.

_{r}proposed by Todini [19]

_{int}is the amount of power dissipated internally in the network to satisfy the demands for all nodes while P

_{int,max}is the maximum power that could be dissipated internally to satisfy the constraints in terms of nodal demands and the nodal heads.

_{int}is defined as:

_{n}is the number of nodes of the network and P

_{tot}is the total power available at the entrance of the network, expressed as:

_{k}is the flow delivered by the reservoir k and H

_{k}is the head at the reservoir k and n

_{r}is the number of reservoirs of the network.

_{int,max}is calculated as follows:

_{j}

^{req}is the demand at node j, H

_{j}is the head at node j and H

_{min,j}is the minimum required head at node j at which the nodal demands are to be supplied.

_{S}[39]:

_{D,j}is the demand at node j, Q is the total flow delivered by the network through demand nodes, H

_{j}is the head at node j, and H

_{j}

^{*}is the minimum required head at node j.

## 3. Results and Discussion

_{r}, NRI and MIr, only the comparison between the proposed measure and the decrease in the values of I

_{r}is reported.

- test the vulnerability measure to assess its applicability and the computational burden;
- demonstrate the usefulness of this measure for managers to identify the most vulnerable pipelines to carry out proper management and maintenance of the network; and
- bring out any critical issues or interesting aspects of possible future developments.

_{r}, and NRI, while the value increase at about 0.73 for H

_{S}index.

_{r}, and NRI, and 0.55 for Hs.

## 4. Conclusions

- the consideration of the components of the network and the evaluation of the level of vulnerability of each pipe; and
- based on topological evaluations.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 5.**Correlation between vulnerability measure and the decrease in the Ir values in Walski’s network.

**Figure 6.**Correlation between vulnerability measure and the decrease in the H

_{S}values in Walski’s network.

**Figure 7.**Correlation between vulnerability measure and the decrease in the I

_{r}values in Ozger and Mays network.

**Figure 8.**Correlation between vulnerability measure and the decrease in the H

_{S}values in Ozger and Mays network.

**Figure 9.**Correlation between vulnerability measure and the decrease in the I

_{r}values in the modified Ozger and Mays network.

**Figure 10.**Correlation between vulnerability measure and the decrease in the H

_{S}values in the modified Ozger and Mays network.

References | Indices | Analysis Type |
---|---|---|

Tanyimboh and Templeman [16] | Entropy as surrogate for reliability | Entropy approach |

Tanyimboh and Templeman [17] | Entropy/Reliability | Entropy approach |

Setiadi et al. [18] | Entropy/Reliability | Entropy approach |

Todini [19] | Resilience | Hydraulic approach |

Prasad and Park [20] | Resilience | Hydraulic approach |

Jayaram and Srinivasan [21] | Resilience | Hydraulic approach |

Zhuang et al. [22] | Resilience | Hydraulic approach |

Jung et al. [23] | Robustness | Hydraulic approach |

Wright et al. [24] | Resilience | Hydraulic approach |

Cimellaro et al. [25] | Resilience | Hydraulic approach |

Pinto et al. [26] | Vulnerability | Topological approach |

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Pipe ID | Length (m) | D (mm) | C (H–W) |
---|---|---|---|

1 | 304.8 | 203.2 | 90 |

2 | 609.6 | 304.8 | 110 |

3 | 609.6 | 304.8 | 110 |

4 | 304.8 | 304.8 | 110 |

5 | 457.2 | 304.8 | 120 |

6 | 152.4 | 203.2 | 120 |

7 | 914.4 | 254.0 | 110 |

8 | 762.0 | 406.4 | 130 |

9 | 609.6 | 304.8 | 110 |

Node ID | Elevation (m) | Demand (l/s) |
---|---|---|

1 | 0 | 25.2 |

2 | 0 | 63.1 |

3 | 0 | 94.6 |

4 | 0 | 31.5 |

5 | 0 | 31.5 |

6 | 0 | 126.2 |

**Table 4.**Flow values for the simulation in full function condition without pipe failures for Walski’s network.

Pipe ID | Flow (CMH) |
---|---|

1 | 25.2 |

2 | 20.1 |

3 | 11.9 |

4 | 43.5 |

5 | 62.6 |

6 | 27.7 |

7 | 47.3 |

8 | 216.5 |

9 | 108.4 |

Pipe ID | Length (m) | D (mm) | C (H–W) |
---|---|---|---|

1 | 609.60 | 762 | 130 |

2 | 243.80 | 762 | 128 |

3 | 1524.00 | 609 | 126 |

4 | 1127.76 | 609 | 124 |

5 | 1188.72 | 406 | 122 |

6 | 640.08 | 406 | 120 |

7 | 762.00 | 254 | 118 |

8 | 944.88 | 254 | 116 |

9 | 1676.40 | 381 | 114 |

10 | 883.92 | 305 | 112 |

11 | 883.92 | 305 | 110 |

12 | 1371.60 | 381 | 108 |

13 | 762.00 | 254 | 106 |

14 | 822.96 | 254 | 104 |

15 | 944.88 | 305 | 102 |

16 | 579.00 | 305 | 100 |

17 | 487.68 | 203 | 98 |

18 | 457.20 | 152 | 96 |

19 | 502.92 | 203 | 94 |

20 | 883.92 | 203 | 92 |

21 | 944.88 | 305 | 90 |

Node ID | Elevation (m) | Demand (CMH) |
---|---|---|

1 | 27.43 | 0 |

2 | 33.53 | 212.4 |

3 | 28.96 | 212.4 |

4 | 32.00 | 640.8 |

5 | 30.48 | 212.4 |

6 | 31.39 | 684.0 |

7 | 29.56 | 640.8 |

8 | 31.39 | 327.6 |

9 | 32.61 | 0 |

10 | 34.14 | 0 |

11 | 35.05 | 108.0 |

12 | 36.58 | 108.0 |

13 | 33.53 | 0 |

**Table 7.**Flow values for the simulation in full function condition without pipe failures for the Ozger and Mays network.

Pipe ID | Flow (CMH) |
---|---|

1 | 2253.3 |

2 | 2253.3 |

3 | 1211.9 |

4 | 791.1 |

5 | 66.4 |

6 | 893.1 |

7 | 216.7 |

8 | 208.4 |

9 | 545.2 |

10 | 37 |

11 | 304.4 |

12 | 309.9 |

13 | 125.9 |

14 | 75.5 |

15 | 283.9 |

16 | 164.8 |

17 | 119.1 |

18 | 11.1 |

19 | 96.9 |

20 | 67.9 |

21 | 184.0 |

Pipe ID | Vulnerability Measure |
---|---|

8 | 0.7969 |

9 | 0.7510 |

7 | 0.3861 |

6 | 0.3792 |

1 | 0.3534 |

4 | 0.2637 |

5 | 0.1600 |

2 | 0.1449 |

3 | 0.1420 |

Pipe ID | Vulnerability Measure |
---|---|

1 | 0.7648 |

6 | 0.7243 |

2 | 0.4181 |

12 | 0.3615 |

3 | 0.2782 |

15 | 0.2378 |

9 | 0.1938 |

8 | 0.1872 |

16 | 0.1771 |

17 | 0.1748 |

4 | 0.1377 |

5 | 0.1235 |

19 | 0.1180 |

18 | 0.1161 |

11 | 0.0825 |

7 | 0.0807 |

13 | 0.0790 |

10 | 0.0772 |

14 | 0.0744 |

21 | 0.0724 |

20 | 0.0707 |

Condition | I_{r} | NRI | MIr | H_{s} |
---|---|---|---|---|

No pipe failures | 0.650 | 0.153 | 22.967 | 10.335 |

Failure pipe 1 | 0.624 | 0.509 | 16.330 | 10.400 |

Failure pipe 2 | 0.634 | 0.555 | 22.408 | 10.084 |

Failure pipe 3 | 0.647 | 0.566 | 22.860 | 10.287 |

Failure pipe 4 | 0.612 | 0.530 | 21.609 | 9.724 |

Failure pipe 5 | 0.560 | 0.477 | 19.771 | 8.897 |

Failure pipe 6 | 0.635 | 0.553 | 22.435 | 10.096 |

Failure pipe 7 | 0.542 | 0.472 | 19.141 | 8.614 |

Failure pipe 8 | −1.150 | −1.005 | −40.643 | 0.000 |

Failure pipe 9 | 0.194 | 0.153 | 6.844 | 3.676 |

Condition | I_{r} | NRI | MI_{r} | H_{S} |
---|---|---|---|---|

No pipe failures | 0.269 | 0.214 | 98.504 | 14.838 |

Failure pipe 1 | −1.155 | −0.923 | −423.644 | 0.000 |

Failure pipe 2 | −1.155 | −0.923 | −423.657 | 0.000 |

Failure pipe 3 | 0.016 | 0.011 | 5.743 | 2.239 |

Failure pipe 4 | 0.160 | 0.127 | 58.546 | 8.830 |

Failure pipe 5 | 0.268 | 0.214 | 98.294 | 14.799 |

Failure pipe 6 | 0.133 | 0.104 | 48.665 | 7.345 |

Failure pipe 7 | 0.252 | 0.200 | 92.320 | 13.907 |

Failure pipe 8 | 0.251 | 0.199 | 91.932 | 13.849 |

Failure pipe 9 | 0.172 | 0.136 | 62.979 | 9.497 |

Failure pipe 10 | 0.268 | 0.214 | 98.373 | 14.816 |

Failure pipe 11 | 0.240 | 0.191 | 87.965 | 13.251 |

Failure pipe 12 | 0.239 | 0.190 | 87.805 | 13.226 |

Failure pipe 13 | 0.263 | 0.209 | 96.384 | 14.516 |

Failure pipe 14 | 0.266 | 0.211 | 97.473 | 14.680 |

Failure pipe 15 | 0.178 | 0.137 | 65.353 | 12.190 |

Failure pipe 16 | 0.250 | 0.199 | 91.743 | 13.819 |

Failure pipe 17 | 0.241 | 0.190 | 88.292 | 14.256 |

Failure pipe 18 | 0.268 | 0.214 | 98.440 | 14.825 |

Failure pipe 19 | 0.248 | 0.196 | 91.057 | 14.392 |

Failure pipe 20 | 0.264 | 0.210 | 96.644 | 14.554 |

Failure pipe 21 | 0.250 | 0.198 | 91.574 | 13.794 |

Pipe ID | Vulnerability Measure |
---|---|

2 (1) | 0.7834 |

3 | 0.7343 |

9 | 0.4068 |

15 | 0.3663 |

4 | 0.3550 |

8 | 0.2574 |

7 | 0.2489 |

5 | 0.2359 |

11 | 0.2330 |

12 | 0.1671 |

10 | 0.1489 |

14 | 0.1419 |

20 | 0.1394 |

19 | 0.0937 |

16 | 0.0915 |

17 | 0.0892 |

18 | 0.0870 |

13 | 0.0836 |

21 | 0.0811 |

6 | 0.0789 |

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## Share and Cite

**MDPI and ACS Style**

Maiolo, M.; Pantusa, D.; Carini, M.; Capano, G.; Chiaravalloti, F.; Procopio, A.
A New Vulnerability Measure for Water Distribution Network. *Water* **2018**, *10*, 1005.
https://doi.org/10.3390/w10081005

**AMA Style**

Maiolo M, Pantusa D, Carini M, Capano G, Chiaravalloti F, Procopio A.
A New Vulnerability Measure for Water Distribution Network. *Water*. 2018; 10(8):1005.
https://doi.org/10.3390/w10081005

**Chicago/Turabian Style**

Maiolo, Mario, Daniela Pantusa, Manuela Carini, Gilda Capano, Francesco Chiaravalloti, and Antonio Procopio.
2018. "A New Vulnerability Measure for Water Distribution Network" *Water* 10, no. 8: 1005.
https://doi.org/10.3390/w10081005