# Robust Multi-Objective Design Optimization of Water Distribution System under Uncertainty

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Optimization under Uncertainty

#### 1.2. Multi-Objective Optimization of WDS Design under Uncertainty

## 2. Methodology

#### 2.1. Robust Optimization

#### 2.2. Ellipsoidal Uncertainty Set

#### 2.3. Two-Objective Cost vs. Resilience Design of Water Distribution System

#### 2.4. Robust Optimization Formulation Considering Demand(q) as an Uncertain Variable

## 3. Case Study

#### 3.1. Multi-Objective Optimization Method

#### 3.2. Case Study 1-Hanoi WDS

^{3}/h), respectively. Four different protection levels are studied $\mathsf{\Gamma}=\left[0.5,1,1.5,2\right]$. Furthermore, the correlation between the nodes within the region and the correlation between the regions are also altered. The intraregional correlation values are set to be $\rho =0.8,$ and the interregional correlation varies between positive, no correlation and negative correlation $\rho =\left[0.6,0,-0.6\right]$. SAMOCSA algorithm is used to solve the outer design problem, and the fmincon algorithm is used to solve the nonlinear inner optimization problem for minimization of the resilience index within the demand search space.

#### 3.3. Case Study 2-Large Network

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**(

**a**) Graph showing the WDS design Pareto fronts for Hanoi WDS with demand variation with positive correlation; (

**b**) graph showing the WDS design Pareto fronts for Hanoi WDS with demand variation with negative correlation; (

**c**) graph showing the WDS design Pareto fronts for Hanoi WDS with demand variation with no correlation.

**Figure 5.**Pareto front obtained for different ellipsoidal sizes with demands related with co-variance matrix (1) shown adjacent.

**Figure 6.**Pareto front obtained for different ellipsoidal sizes with demands related with co-variance matrix (2) shown adjacent.

**Figure 7.**Pareto front obtained for different ellipsoidal sizes with demands related with co-variance matrix (3) shown adjacent.

Uncertain Parameters | PDF Assumed | Uncertainty Handling Techniques | Optimization Techniques | References | Type |
---|---|---|---|---|---|

q, H, RC | Normal | FORM | GRG2 | [16] | Hydraulic analysis |

q, H, RC | Normal | MCS | GRG2 | [15] | SO Design |

q, RC | Normal | MCS | SFLA | [24] | Hydraulic analysis |

q, RC | Normal | FOSM and MCS | -- | [25] | Hydraulic analysis |

q | Gaussian | LHS | GA | [17] | SO Design |

q, RC | Normal | FORM | GA | [26] | SO Design |

q | Gaussian | LHS | RNSGA-II | [21] | MO Design |

q | -- | Robust optimization | cross entropy | [19,20] | SO Design |

Diameter (in) | Unit Cost ($/m) | Diameter (in) | Unit Cost ($/m) | Diameter (in) | Unit Cost ($/m) |
---|---|---|---|---|---|

12.0 | 45.73 | 20 | 98.39 | 30 | 180.75 |

16.0 | 70.4 | 24 | 129.33 | 40 | 278.28 |

**Table 3.**Cost and resilience comparison of extreme points in the Pareto front for two different uncertainty set sizes.

Correlation | Uncertainty Set Size | Extreme Point-1 | Extreme Point-2 | ||
---|---|---|---|---|---|

Cost | Resilience | Cost | Resilience | ||

Positive | 2 | 7.9157 | 0.154 | 10.969 | 0.186 |

0.5 | 6.7115 | 0.2394 | 10.755 | 0.3118 | |

Negative | 2 | 7.154 | 0.207 | 10.121 | 0.2593 |

0.5 | 6.61 | 0.2516 | 10.305 | 0.3298 | |

Zero(No) | 2 | 6.656 | 0.2211 | 10.747 | 0.308 |

0.5 | 6.584 | 0.2232 | 10.769 | 0.342 |

Diameter | 1 | 2 | 3 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 |

Unit Cost | 2.0 | 5.0 | 8.0 | 11.0 | 16.0 | 24.0 | 32.0 | 50.0 | 60.0 | 90.0 | 130.0 | 170.0 |

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Boindala, S.P.; Ostfeld, A.
Robust Multi-Objective Design Optimization of Water Distribution System under Uncertainty. *Water* **2022**, *14*, 2199.
https://doi.org/10.3390/w14142199

**AMA Style**

Boindala SP, Ostfeld A.
Robust Multi-Objective Design Optimization of Water Distribution System under Uncertainty. *Water*. 2022; 14(14):2199.
https://doi.org/10.3390/w14142199

**Chicago/Turabian Style**

Boindala, Sriman Pankaj, and Avi Ostfeld.
2022. "Robust Multi-Objective Design Optimization of Water Distribution System under Uncertainty" *Water* 14, no. 14: 2199.
https://doi.org/10.3390/w14142199