# Robustness and Water Distribution System: State-of-the-Art Review

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## Abstract

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## 1. Introduction

## 2. Robustness-Based Approaches

#### 2.1. Design and Planning

#### 2.1.1. Hydraulic Robustness

_{i}is the reliability at node i, Prob() is a probability function of a given condition defined in the parenthesis, P

_{i}and P

_{min}are the random and minimum required pressure at node i, respectively, and n is the number of nodes.

_{w}) with deficit pressure (node with pressure lower than the allowable minimum pressure) under single pipe failure conditions. The ROB indicator is calculated by 1 minus N

_{w}as follows:

_{p}). However, the proposed index does not indicate the severity of failure and variation.

_{i}is the robustness at node i, and ${P}_{avg}$ and ${\sigma}_{P}$ are the average and standard deviations of stochastic pressure, respectively (COV = $\frac{{\sigma}_{P}}{{P}_{avg}}$). With regard to the proposed ROB scheme, a network yielding pressures in the range 23–33 m is preferred over that with pressures in the range 18–38 m given the same average pressure of 28 m (40 psi) (Figure 1). The system ROB was defined as the minimum nodal ROB value (=$\mathrm{min}\left({\mathrm{ROB}}_{i},i=1,\dots ,n\right)$). Therefore, increasing the system ROB decreases the variations in stochastic pressure while maximizing the average pressure at the critical node.

#### 2.1.2. Structural Robustness

#### 2.2. Operation

#### 2.3. Management

## 3. Recommendations

#### 3.1. Design and Planning

#### 3.2. Operation and Management

## 4. Summary and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Reference | Main Novelty | Study Network | Decision Variable | Methodology |
---|---|---|---|---|

Jung et al. [4] | Proposed a pressure-COV-based ROB indicator and ROB-based design approach | Anytown network | Pipe sizes and pump capacity | NSGA-II |

Puccini et al. [24] | Proposed a ROB indicator (based on the averaged ratio of the number of nodes with deficit pressure under single pipe failure conditions) | Two hypothetical networks and a real network | Pipe sizes | Multi-objective Simulated Annealing |

Jung et al. [15] | Investigated the correlation between different system performance indicators (including the ROB indicator proposed in Jung et al. [4]) | 16 real networks | Not considered | Pearson and Spearman rank correlation |

Jung and Kim [27] | Compared the Pareto optimal pipe sizes and layout obtained by four design approaches (including the ROB-based approach) | A large grid network | The installation of a pipe to each link and pipe sizes | NSGA-II |

Yazdani et al. [28] | Used graph theory indicators to measure structural ROB for WDS expansion | A large real network | Network layout | Graph theory (not based on optimization) |

Type | Metric | Definition | Reference |
---|---|---|---|

Statistical | Average node-degree | Average value of the node-degree distribution | Newman [41] |

Average path length | Average value of the geodesic distances between all pairs of nodes | Costa et al. [42] | |

Central-point dominance | Average difference in betweenness of the most central point and all others | Freeman [43] | |

Spectral | Algebraic connectivity | The second smallest eigenvalue of Laplacian matrix of the network | Fiedler [44] |

Spectral gap | The difference between first and second eigenvalues of graph’s adjacency matrix | Estrada [45] | |

Spectral Radius | The largest eigenvalue of the adjacency matrix | Bonacich [46] |

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Jung, D.; Lee, S.; Kim, J.H.
Robustness and Water Distribution System: State-of-the-Art Review. *Water* **2019**, *11*, 974.
https://doi.org/10.3390/w11050974

**AMA Style**

Jung D, Lee S, Kim JH.
Robustness and Water Distribution System: State-of-the-Art Review. *Water*. 2019; 11(5):974.
https://doi.org/10.3390/w11050974

**Chicago/Turabian Style**

Jung, Donghwi, Seungyub Lee, and Joong Hoon Kim.
2019. "Robustness and Water Distribution System: State-of-the-Art Review" *Water* 11, no. 5: 974.
https://doi.org/10.3390/w11050974