# On the Evolution of the Optimal Design of WDS: Shifting towards the Use of a Fractal Criterion

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Background

## 3. Methodology

#### 3.1. OPUS

- (1)
- Uniform distribution: It is assumed that all pipes have the same flow, Therefore the total demand of each node is divided into the number of upstream pipes connected to it.
- (2)
- Proportional distribution: the flow of each pipe is proportional to H/L
^{2}, where H are the head losses in the pipe and L is its length. - (3)

#### 3.2. NSGA-II

- Minimum pressure at all demand nodes.
- Maximum pressure and maximum velocity for avoidance of operational inconveniences.
- Mass and energy conservation.
- Discrete pipe diameters.

#### 3.3. NSGA-II Methodology with OPUS Intermittent Feedback

#### 3.3.1. Calibration Process

#### 3.3.2. Preprocessing of OPUS Results

#### 3.3.3. NSGA-II Intermittent Retrofitting through OPUS

_{i}represents a pipe diameter of an individual obtained through NSGA-II optimization and ${f}_{3}^{-1}$(Q

_{i}) represents a pipe diameter of an individual obtained through OPUS with flow rates extracted from the EPANET hydraulic simulation with former individuals having diameters d

_{i}[5]. Finally, Paez et al. [5] have set a feedback frequency of m $\in $ (5, 50) for balancing a trade-off between PF quality and the algorithm’s convergence rate.

#### 3.4. Optimal WDSs Selection

- (1)
- A min(C), min(NRI) point corresponding to the simultaneously minimum cost (C) and minimum reliability (NRI) WDS arrangement.
- (2)
- A knee(C), knee(NRI) point corresponding to the knee cost and knee reliability (NRI) WDS arrangement.
- (3)
- A max(C), max(NRI) point corresponding to the maximum cost (C) and maximum reliability (NRI) WDS arrangement.

#### 3.5. Obtention of the Fractal Dimension

_{j}is the piezometric head value obtained by EPANET using instantaneous hydraulic simulation, ${d}_{ij}$ is the diameter of the pipe i incoming or outgoing from node j, ${Q}_{ij}$ is the flow rate through the aforementioned pipe i incoming or outgoing from node j, and ${D}_{j}$ is the demand at node j. The criteria will be refered to as the HGL, Diamater, and Flow Rate criterion, respectively.

^{2}coefficient for the linear regression should be close to 1 [33].

^{2}value greater than 0.99 is achieved given that a minimum number of iterations is made.

^{2}value before flattening.

#### 3.6. Water Distribution System Classification

## 4. Study Cases

#### 4.1. Hanoi

#### 4.2. Fossolo

_{min}= 40 m and the maximum velocity in each pipe is 1 m/s. Figure 4 shows the HGL for the chosen WDS configurations for Fossolo (Table S4, Figure S4 in Supplementary Materials).

#### 4.3. Balerma

_{s}= 0.0025 mm. The network includes a minimum pressure constraint of P

_{min}$=$ 20 m [37]. Figure 5 shows the HGL for the chosen WDS configurations for Balerma (Table S2, Figure S2 in Supplementary Materials).

#### 4.4. Modena

## 5. Results and Discussion

## 6. Conclusions and Future Work

^{3}), which is polynomial and desirable. The results do show tendencies, but these are of a very small order; in general, it is safe to suggest that the fractal dimensions of optimal designs for a given network are almost the same. Although this study only analyzed optimal networks, Jaramillo [31] also found fractal dimensions, although with a slightly different methodology, of non-optimal networks and did not observe the patterns here exemplified. That suggests that the limiting value could be unique to the optimal networks. Therefore, this paper has successfully developed a consistent methodology with a low computational cost that can be integrated to more costly algorithms as an optimality criterion that can reduce the overall running time.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Water distribution network classification flowchart. $\overline{D}$ represents the length-weighted average pipe diameter, $BI$ is the Branch Index, and $M{C}_{O-R}$ is the Meshedness Coefficient for the reduced network. Source: Hwang and Lansey [34].

**Figure 2.**Selected individuals from retrofitted OPUS/NSGA-II PFs for the: (

**a**) Hanoi WDS; (

**b**) Fossolo WDS; (

**c**) Balerma WDS; and (

**d**) Modena WDS.

**Figure 3.**Hanoi WDS with the corresponding HGL surfaces for the: (

**a**) [min(C), min(NRI)]; (

**b**) [knee(C), knee(NRI)]; and (

**c**) [max(C), max(NRI)] configurations.

**Figure 4.**Fossolo WDSs with the corresponding HGL surfaces for the: (

**a**) [min(C), min(NRI)]; (

**b**) [knee(C), knee(NRI)]; and (

**c**) [max(C), max(NRI)] configurations.

**Figure 5.**Balerma WDS with the corresponding HGL surfaces for the: (

**a**) [min(C), min(NRI)]; (

**b**) [knee(C), knee(NRI)]; and (

**c**) [max(C), max(NRI)] configurations.

**Figure 6.**Modena WDS with the corresponding HGL surfaces for the: (

**a**) [min(C), min(NRI)]; (

**b**) [knee(C), knee(NRI)]; and (

**c**) [max(C), max(NRI)] configurations.

**Figure 7.**WDSs’ fractal analysis criteria (${w}_{j}$) applied to a retrofitted OPUS/NSGA-II Hanoi PF.

**Figure 8.**WDSs’ fractal analysis criteria (${w}_{j}$) applied to a retrofitted OPUS/NSGA-II Balerma PF.

**Table 1.**Values of each parameter for the implementation of the NSGA-II and retrofitted approach. Source: Paez et al. [5].

Network | Individuals | Generations | Mutation Distribution Index | Crossover Distribution Index | Retrofitted Frequency |
---|---|---|---|---|---|

Hanoi | 500 | 500 | 20 | 3 | 5 |

Fossolo | 500 | 500 | 100 | 10 | 20 |

Balerma | 2000 | 4500 | 100 | 2 | 10 |

Modena | 2000 | 4000 | 20 | 7 | 50 |

Network | Reservoirs | Size | Pipes | Pipe Diameter Options | Search Space | Pressure Constraint | Velocity Constraint |
---|---|---|---|---|---|---|---|

Hanoi | 1 | Medium | 34 | 6 | $2.8\times {10}^{6}$ | ${\mathrm{P}}_{\mathrm{min}}=30\mathrm{m}$ | No |

Fossolo | 1 | Intermediate | 58 | 22 | $7.25\times {10}^{77}$ | P_{min} $=$ 40 m | Yes |

Balerma | 4 | Large | 454 | 10 | $1.00\times {10}^{455}$ | P_{min} $=$ 20 m | No |

Modena | 4 | Large | 317 | 13 | $1.32\times {10}^{353}$ | P_{min} $=$ 20 m | Yes |

**Table 3.**Cost and NRI for the three selected points of each network: [min(C), min(NRI)], [knee(C), knee(NRI)], and [max(C), max(NRI)].

Network | Point | Cost ($) | NRI(-) |
---|---|---|---|

Hanoi | min(C), min(NRI) | 6,439,320.50 | 0.222 |

knee(C), knee(NRI) | 7,260,699.00 | 0.304 | |

max(C), max(NRI) | 10,969,798.00 | 0.354 | |

Balerma | min(C), min(NRI) | 2,288,460.00 | 0.431 |

knee(C), knee(NRI) | 2,807,625.00 | 0.731 | |

max(C), max(NRI) | 13,191,652.00 | 0.891 | |

Fossolo | min(C), min(NRI) | 23,046.98 | 0.295 |

knee(C), knee(NRI) | 43,330.23 | 0.715 | |

max(C), max(NRI) | 1,661,922.50 | 0.999 | |

Modena | min(C), min(NRI) | 2,613,550.00 | 0.361 |

knee(C), knee(NRI) | 3,089,496.75 | 0.569 | |

max(C), max(NRI) | 6,731,936.00 | 0.907 |

**Table 4.**WDS classification, according to Hwang and Lansey [36]. $\overline{D}$ represents the length-weighted average pipe diameter, $BI$ is the Branch Index, and $MC$ is the Meshedness Coefficient for the reduced network.

Network | Point | Class | $\overline{\mathit{D}}$ (mm) | BI (−) | MC (−) |
---|---|---|---|---|---|

Hanoi | min(C), min(NRI) | Transmission Dense-Loop (TDL) | 682.952 | 0.438 | 0.333 |

knee(C), knee(NRI) | Transmission Dense-Loop (TDL) | 750.144 | |||

max(C), max(NRI) | Transmission Dense-Loop (TDL) | 1016.000 | |||

Balerma | min(C), min(NRI) | Distribution Branch (DB) | 173.259 | 0.770 | 0.052 |

knee(C), knee(NRI) | Distribution Branch (DB) | 189.439 | |||

max(C), max(NRI) | Transmission Branch (TB) | 446.902 | |||

Fossolo | min(C), min(NRI) | Distribution Dense-Grid (DDG) | 37.542 | 0.017 | 0.328 |

knee(C), knee(NRI) | Distribution Dense-Grid (DDG) | 57.947 | |||

max(C), max(NRI) | Transmission Dense-Loop (TDL) | 409.200 | |||

Modena | min(C), min(NRI) | Distribution Dense-Grid (DDG) | 145.779 | 0.033 | 0.331 |

knee(C), knee(NRI) | Distribution Dense-Grid (DDG) | 150.102 | |||

max(C), max(NRI) | Distribution Dense-Grid (DDG) | 264.884 |

Classification | Network | ${{\mathit{w}}_{\mathit{j}}}_{}$ | min(C), min(NRI) | knee(C), knee(NRI) | max(C), max(NRI) |
---|---|---|---|---|---|

Transmission Dense-Loop (TDL) | Hanoi | $\sum {d}_{ij}$ | 1.929 | 1.934 | 1.891 |

$HG{L}_{j}$ | 1.829 | 1.826 | 1.946 | ||

$\left(\sum {Q}_{ij}\right)-{D}_{j}$ | 1.020 | 1.020 | 1.020 | ||

Distribution Branch (DB), Transmission Branch (TB) | Balerma | $\sum {d}_{ij}$ | 1.857 | 1.807 | 1.798 |

$HG{L}_{j}$ | 1.798 | 1.798 | 1.791 | ||

$\left(\sum {Q}_{ij}\right)-{D}_{j}$ | 1.798 | 1.798 | 1.798 | ||

Distribution Dense-Grid (DDG) | Fossolo | $\sum {d}_{ij}$ | 1.950 | 1.829 | 1.829 |

$HG{L}_{j}$ | 2.033 | 2.047 | 0.874 | ||

$\left(\sum {Q}_{ij}\right)-{D}_{j}$ | 1.831 | 1.835 | 1.839 | ||

Modena | $\sum {d}_{ij}$ | 1.936 | 1.941 | 1.875 | |

$HG{L}_{j}$ | 2.040 | 2.044 | 2.082 | ||

$\left(\sum {Q}_{ij}\right)-{D}_{j}$ | 1.976 | 1.966 | 1.940 |

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**MDPI and ACS Style**

Saldarriaga, J.; Salcedo, C.; González, M.A.; Ortiz, C.; Wiesner, F.; Gómez, S.
On the Evolution of the Optimal Design of WDS: Shifting towards the Use of a Fractal Criterion. *Water* **2022**, *14*, 3795.
https://doi.org/10.3390/w14233795

**AMA Style**

Saldarriaga J, Salcedo C, González MA, Ortiz C, Wiesner F, Gómez S.
On the Evolution of the Optimal Design of WDS: Shifting towards the Use of a Fractal Criterion. *Water*. 2022; 14(23):3795.
https://doi.org/10.3390/w14233795

**Chicago/Turabian Style**

Saldarriaga, Juan, Camilo Salcedo, María Alejandra González, Catalina Ortiz, Federico Wiesner, and Santiago Gómez.
2022. "On the Evolution of the Optimal Design of WDS: Shifting towards the Use of a Fractal Criterion" *Water* 14, no. 23: 3795.
https://doi.org/10.3390/w14233795