# Identifying Critical Isolation Valves in a Water Distribution Network: A Socio-Technical Approach

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Segment-Valve Representation

#### 2.2. Gomory–Hu Tree of Segment Connectivity to Address Combinatorial Complexity

#### 2.3. Logical Network

#### 2.4. Multi-Criteria Optimization

#### 2.4.1. Formulation

#### Formulation One

- Social vulnerability:$$max\sum _{i\in S}^{}{a}_{i}{s}_{i}$$
- Segment flow volume:$$max\sum _{i\in S}^{}{b}_{i}{s}_{i}$$
- Reachability impact:$$max\sum _{i\in S}^{}{c}_{i}{s}_{i}$$

**Proof.**

#### Formulation Two

- Social Vulnerability:$$max\sum _{i\in S}^{}\underline{{a}_{i}}{s}_{i}+(\overline{{a}_{i}}-\underline{{a}_{i}}){r}_{i}$$
- Segment Flow Volume:$$max\sum _{i\in S}^{}\underline{{b}_{i}}{s}_{i}+(\overline{{b}_{i}}-\underline{{b}_{i}}){r}_{i}$$

#### Formulation Three

#### 2.4.2. Solution Approach

#### 2.5. Post-Optimization Analysis

## 3. Case Study

## 4. Results and Discussion

#### 4.1. Spatial Variation of Segment Attributes

#### 4.2. Critical Valves Based on the Optimization Models

#### 4.3. Differences between the Formulations

#### 4.4. Potential Implications of the Optimization Models

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Segment-valve represenation and Gomory–Hu tree of a network example. (

**a**) Segment-valve representation of a small example network. (

**b**) Gomory–Hu tree of small example, where edges that represent more than one valve are bolded.

**Figure 2.**Representation of the logical network, where valves that were represented by edges are modeled as nodes and segments are also modeled as nodes, and the edges between valves and segments represent logical implications.

**Figure 3.**Segment reachability values and the subsequently identified critical valves. (

**a**) Map of segment reachability values. (

**b**) Map of critical valves.

**Figure 5.**Map of social-vulnerability index, adapted from [35].

**Figure 7.**Maps of identified valves using the three formulations, where formulation 1 is deterministic, formulation 2 accounts for uncertainty, and formulation 3 accounts for uncertainty and likelihood of failure.

**Figure 8.**Maps of valves identified by formulations 1 and 3 at various percentiles for the frequency of occurrence in the solution set.

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**MDPI and ACS Style**

Abdel-Mottaleb, N.; Saghand, P.G.; Wakhungu, M.J.; Charkhgard, H.; Wells, E.C.; Zhang, Q.
Identifying Critical Isolation Valves in a Water Distribution Network: A Socio-Technical Approach. *Water* **2022**, *14*, 3587.
https://doi.org/10.3390/w14213587

**AMA Style**

Abdel-Mottaleb N, Saghand PG, Wakhungu MJ, Charkhgard H, Wells EC, Zhang Q.
Identifying Critical Isolation Valves in a Water Distribution Network: A Socio-Technical Approach. *Water*. 2022; 14(21):3587.
https://doi.org/10.3390/w14213587

**Chicago/Turabian Style**

Abdel-Mottaleb, Noha, Payman Ghasemi Saghand, Mathews J. Wakhungu, Hadi Charkhgard, E. Christian Wells, and Qiong Zhang.
2022. "Identifying Critical Isolation Valves in a Water Distribution Network: A Socio-Technical Approach" *Water* 14, no. 21: 3587.
https://doi.org/10.3390/w14213587