# Network Subsystems for Robust Design Optimization of Water Distribution Systems

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## Abstract

**:**

## 1. Introduction

## 2. Material and Methods

#### 2.1. Subsystem Enumeration

#### Subsystem Pair Optimization

#### 2.2. Robust Counterpart Optimization

#### 2.3. Least Cost Design of WDS Matrix Formulation

#### 2.3.1. Uncorrelated Model of Data Uncertainty

#### 2.3.2. Correlated Robust Counterpart

#### Linearization of the Operating Domain [Q1,Q2]

## 3. Results

#### 3.1. FOWM Distribution Network

#### 3.1.1. Uncorrelated Data Uncertainty

#### 3.1.2. Correlated Data Uncertainty

#### 3.2. Fossolo Distribution Network

#### 3.2.1. Uncorrelated Data Model

#### 3.2.2. Correlated Data Uncertainty

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Solution to a Subproblem

**Box model:**$U=\left\{\mathsf{\xi}\right|{\Vert \mathsf{\xi}\Vert}_{\infty}\le r\}$

**Ellipsoidal set:**

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**Figure 3.**Schematic of pair of backups: (

**a**) The first subsystem generated following decreasing order; (

**b**) The second subsystem generated following increasing order.

**Figure 4.**Results of uncorrelated optimization for FOWM network: (

**a**) cost vs. protection factor Ω; (

**b**) head constraint violation probability.

**Figure 9.**Results of uncorrelated optimization for Fossolo network: (

**a**) cost vs. protection factor Ω; (

**b**) head constraint violation probability.

**Figure 10.**Results of correlated optimization for Fossolo network: (

**a**) cost vs. protection factor Ω; (

**b**) head constraint violation probability.

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Hayelom, A.; Ostfeld, A.
Network Subsystems for Robust Design Optimization of Water Distribution Systems. *Water* **2022**, *14*, 2443.
https://doi.org/10.3390/w14152443

**AMA Style**

Hayelom A, Ostfeld A.
Network Subsystems for Robust Design Optimization of Water Distribution Systems. *Water*. 2022; 14(15):2443.
https://doi.org/10.3390/w14152443

**Chicago/Turabian Style**

Hayelom, Assefa, and Avi Ostfeld.
2022. "Network Subsystems for Robust Design Optimization of Water Distribution Systems" *Water* 14, no. 15: 2443.
https://doi.org/10.3390/w14152443