# Optimal Resilience Enhancement of Water Distribution Systems

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Resilience Assessment

#### 2.2. Resilience Optimization

## 3. Results

## 4. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Existing main line water distribution system marked by solid lines and possible adaptations marked by dashed lines. Consumer and junction nodes are marked by dots and the source is marked by the tank image.

**Figure 2.**Optimal main line water distribution system adaptations for different optimization instances, differing in the maximum length of added pipes ${L}_{\mathrm{max}}$ for (

**a**) 200 m, (

**b**) 500 m, (

**c**) 1000 m, and (

**d**) 1750 m. The existing main line WDS is marked by blue edges while all chosen edges to be added are marked in yellow.

**Figure 3.**Cost-benefit resilience analysis given as the Pareto front of the existing WDS embedded in its urban structure.

Variable | Domain | Description |
---|---|---|

${b}_{i,j}$ | $\left\{0,1\right\}$ | Decision variable for choosing the pipes to be added to the existing water distribution system (WDS). |

${x}_{s,c,k}$ | $\left\{0,1\right\}$ | Decision variable for choosing the possible feeding paths for the adapted WDS. |

Set | Description |
---|---|

$S$ | Set of source nodes. |

$C$ | Set of consumer nodes. |

$J$ | Set of junction nodes. |

$N$ | Unified set of nodes—source, consumer or junction nodes, $N=S\cup C\cup J$. |

${K}_{s,c}$ | Set of the number of maximum possible paths, $\forall \text{}s\in S,\text{}c\in C$. |

Parameter | Domain | Description |
---|---|---|

${T}_{i,j}^{0}$ | $\left\{0,1\right\}$ | Edges of the existing WDS, $\forall \text{}i,j\in N$. |

${T}_{i,j}^{\mathrm{max}}$ | $\left\{0,1\right\}$ | Edges in the maximum possible WDS restricted by the urban structure, $\forall \text{}i,j\in N$. |

${A}_{s,c,k,i,j}$ | $\left\{0,1\right\}$ | All possible feeding paths in the maximum possible WDS ${T}_{i,j}^{\mathrm{max}}$, $\forall \text{}s\in S,\text{}c\in C,\text{}k\in {K}_{s,c},\text{}i,j\in N$. |

${g}_{s,c,k}$ | ${\mathbb{R}}_{0}^{+}$ | Hydraulic conductance of each feeding path determined in the preprocessing, $\forall \text{}s\in S,\text{}c\in C,\text{}k\in {K}_{s,c}$. |

${q}_{c}$ | ${\mathbb{R}}_{0}^{+}$ | Demand of each consumer node, $\forall \text{}c\in C$. |

${K}_{\mathrm{max}}$ | $\mathbb{N}$ | Maximum number of paths to be considered for the resilience assessment. |

${L}_{i,j}$ | ${\mathbb{R}}_{0}^{+}$ | Length of each pipe in the maximum possible WDS ${T}_{i,j}^{\mathrm{max}}$, $\forall \text{}i,j\in N$. |

${L}_{\mathrm{max}}$ | ${\mathbb{R}}_{0}^{+}$ | Maximum allowed length for the sum of all added pipes. |

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**MDPI and ACS Style**

Lorenz, I.-S.; Pelz, P.F.
Optimal Resilience Enhancement of Water Distribution Systems. *Water* **2020**, *12*, 2602.
https://doi.org/10.3390/w12092602

**AMA Style**

Lorenz I-S, Pelz PF.
Optimal Resilience Enhancement of Water Distribution Systems. *Water*. 2020; 12(9):2602.
https://doi.org/10.3390/w12092602

**Chicago/Turabian Style**

Lorenz, Imke-Sophie, and Peter F. Pelz.
2020. "Optimal Resilience Enhancement of Water Distribution Systems" *Water* 12, no. 9: 2602.
https://doi.org/10.3390/w12092602