# Network Subsystems for Robust Design Optimization of Water Distribution Systems

^{*}

## 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:**

## References

- Ormsbee, L.; Kessler, A. Optimal upgrading of hydraulic-network reliability. J. Water Resour. Plan. Manag.
**1991**, 116, 784–802. [Google Scholar] [CrossRef] - Agrawal, M.L.; Gupta, R.; Bhave, P.R. Optimal Design of Level 1 Redundant Water Distribution Networks Considering Nodal Storage. J. Environ. Eng.
**2007**, 133, 319–330. [Google Scholar] [CrossRef] - Rathi, S.; Gupta, R. Genetic Algorithm for Minimization of Variance of Pipe Flow-Series for Looped Water Distribution Networks. In Hydrological Modeling; Springer: Cham, Switzerland, 2022; pp. 205–219. [Google Scholar] [CrossRef]
- Yazdani, A.; Jeffrey, P. Applying Network Theory to Quantify the Redundancy and Structural Robustness of Water Distribution Systems. J. Water Resour. Plan Manag.
**2012**, 138, 153–161. [Google Scholar] [CrossRef] - Rathi, S.; Gupta, R.; Palod, N.; Ormsbee, L. Minimizing the variance of flow series using a genetic algorithm for reliability-based design of looped water distribution networks. Eng. Optim.
**2020**, 52, 637–651. [Google Scholar] [CrossRef] - Gupta, R.; Kakwani, N.; Ormsbee, L. Optimal Upgrading of Water Distribution Network Redundancy. J. Water Resour. Plan Manag.
**2015**, 141, 04014043. [Google Scholar] [CrossRef] - Jung, D.; Kim, J.H. Water distribution system design to minimize costs and maximize topological and hydraulic reliability. J. Water Resour. Plan Manag.
**2018**, 144, 06018005. [Google Scholar] [CrossRef] - Ostfeld, A.; Shamir, U. Design of optimal reliable multiquality water-supply systems. J. Water Resour. Plan. Manag.
**1996**, 122, 322–333. [Google Scholar] [CrossRef] - Filion, Y.R.; Adams, B.J.; Karney, B.W. Stochastic design of water distribution systems with expected annual damages. J. Water Resour. Plan Manag.
**2007**, 133, 244–252. [Google Scholar] [CrossRef][Green Version] - Kapelan, Z.; Babayan, A.V.; Savic, D.A.; Walters, G.A.; Khu, S.T. Two new approaches for the stochastic least cost design of water distribution systems. Water Sci. Technol. Water Supply
**2004**, 4, 355–363. [Google Scholar] [CrossRef] - Guistolisi, O.; Laucelli, D.; Colombo, A.F. Deterministic versus stochastic design of water distribution networks. J. Water Resour. Plan Manag.
**2009**, 135, 117–127. [Google Scholar] [CrossRef] - Ben-Tal, A.; El Ghaoui, L.; Nemirovski, A. Robust Optimization; Princeton University Press: Princeton, NJ, USA, 2009. [Google Scholar] [CrossRef]
- Perelman, L.; Housh, M.; Ostfeld, A. Robust optimization for water distribution systems least cost design. Water Resour. Res.
**2013**, 49, 6795–6809. [Google Scholar] [CrossRef] - Perelman, L.; Housh, M.; Ostfeld, A. Explicit demand uncertainty formulation for robust design of water distribution systems. In Proceedings of the World Environmental and Water Resources Congress 2013: Showcasing the Future, Cincinnati, OH, USA, 19–23 May 2013; pp. 684–695. [Google Scholar] [CrossRef]
- Itai, A.; Rodeh, M. Multi-Tree Approach To Reliability in Distributed Networks. Annu. Symp. Comput. Sci.
**1984**, 79, 43–59. [Google Scholar] - Takeaki, U.N.O. An algorithm for enumerating all directed spanning trees in a directed graph. In International Symposium on Algorithms and Computation; Lect Notes Comput Sci (Including Subser Lect Notes Artif Intell Lect Notes Bioinformatics); Springer: Berlin/Heidelberg, Germany, 1996; Volume 1178, pp. 167–173. [Google Scholar] [CrossRef]
- Geen, Z.W. Multiobjective Optimization of Water Distribution Networks Using Fuzzy Theory and Harmony Search. Water
**2015**, 7, 3613–3625. [Google Scholar] [CrossRef]

**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