# Optimal Exploitation of Urban Water Supply Networks Based on Pressure Management with the Nondominated Sorting Differential Evolution (NSDE) Algorithm

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. The Optimization Algorithm

- NSDE exhibits a good balance between convergence (finding solutions close to the true Pareto front) and diversity (exploring different regions of the Pareto front). It has been observed to maintain a diverse set of high-quality solutions throughout the optimization process.
- NSDE is known for its efficiency in terms of computational time and resource utilization. It can effectively handle large-scale optimization problems and converge to near-optimal solutions within a reasonable number of iterations.
- NSDE is robust against noisy or uncertain objective functions. It can handle objective functions with stochastic variations or noise, making it suitable for real-world optimization scenarios where objective values might be subject to variability.
- NSDE tends to produce a well-distributed set of solutions that cover different regions of the Pareto front. This feature allows decision-makers to gain insights into the trade-offs between conflicting objectives and make informed decisions based on their preferences.
- NSDE can be easily adapted and applied to various domains, including water supply network optimization. It can handle different types of decision variables, constraints, and objective functions commonly encountered in such applications.

- Generate the initial population based on constraints and scale of the problem,
- Evaluate the population based on the defined objective functions,
- Apply the non-superior sorting method to categorize the population based on their performance,
- Calculate the control parameter called Crowding Distance for each member in each group, which represents the closeness of the target sample to other members of the population in that group (target group),
- Select the parent population for reproduction,
- Perform jump and intersection.

_{j}(k) = the distance of chromosome j crowding in procedure k, f

_{i}

^{max}, f

_{i}

^{min}respectively the maximum and minimum values of the objective function of the ith in procedure k, f

_{i}(k + 1), f

_{i}(k − 1), equal to j’s objective functions for the upper and lower chromosomes (ascending order) compared to j’s chromosome in order k [19]. One of the selection mechanisms is the selection based on a double tournament between two randomly selected members from the population.

#### 2.2. The EPANET Model

#### 2.3. Algorithm Implementation and Coding

#### 2.4. Equations and Laws in Water Distribution Networks

^{2}/2 g kinetic energy, and Z is the height energy of the fluid [28]. The energy loss equations are used to describe the relationship between pressure drop, output flow, and the geometric properties of the pipes in the water distribution network. There are two primary reasons for head loss within the network [29]:

- Losses due to friction along the walls of the pipes;
- Losses due to disturbances in the flow caused by equipment and other factors that alter the flow conditions (known as local losses).

_{f}: friction loss rate. The Hazen–Williams relation is considered as the most common relationship for calculating the pressure difference of two fluid flow parts, the Darcy–Weissbach relationship, which is given below [30]:

_{f}: unit conversion factor, h

_{f}: frictional loss. According to the Manning’s equation, we have the following [30]:

_{L}: frictional head loss, C

_{f}: unit conversion factor. When fluid flows through equipment such as valves, elbows, or changes in pipe diameter, it can cause disturbance to the flow lines, resulting in a local drop. Typically, these drops are minor and insignificant compared to frictional drops. They are often expressed as a factor of the head velocity. The equation for local drops is given by the following relation [30]:

_{m}: local head drop, K

_{L}: local drop coefficient, V: flow velocity, g: gravity acceleration.

#### 2.5. Formulation of the Optimization

#### 2.6. Alternative Techniques

- Alternative optimization techniques offer different search strategies and exploration-exploitation balances. By considering multiple approaches, you can explore different regions of the search space and potentially discover better solutions that may have been missed by a single technique. Each technique has its strengths and weaknesses, so combining them can provide a more comprehensive search.
- Different optimization techniques may have specific adaptations or variations designed for particular problem characteristics. By exploring alternative techniques, you may find an approach that is better suited for the specific problem at hand. For example, some techniques may excel in handling discrete variables or constraints, while others may be more efficient for continuous optimization or multi-objective problems.
- Comparing the performance of different optimization techniques can provide valuable insights into their strengths and limitations. It allows you to assess factors such as convergence speed, solution quality, robustness, and scalability. Comparative analysis helps in selecting the most suitable optimization technique for a given problem and understanding the trade-offs involved.
- By using alternative optimization techniques, you can benchmark and verify the results obtained using NSDE. This process ensures the reliability and accuracy of the optimization outcomes. Verification through different techniques provides confidence that the solutions achieved are robust and not biased due to a particular algorithm’s limitations.

## 3. Results and Discussion

- The final point can be determined based on a fixed and predetermined number of pressure-relief valves in the system,
- The minimum required pressure reduction in the desired network can be used to determine the final point (in general, the minimum required improvement in the objective function),
- The final point can be determined based on the location on the curve where increasing the number of pressure-relief valves does not significantly decrease the pressure and leakage of the entire system network,
- The maximum approved budget for the pressure management plan can be used to determine the end point.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

NSDE | Nondominated Sorting Differential Evolution |

PSO | Particle swarm optimization |

EPANET | US Environmental Protection Agency’s hydraulic design software |

GA | Genetic algorithm |

NSGA-II | Non-Dominated Sorting Genetic Algorithm II |

CP | Crossover probability |

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**Figure 1.**How to calculate the control parameter called Crowding Distance (Reprinted from Ref. [19]).

**Figure 2.**General working process and Appling mechanism of NSDE algorithm Distance (Reprinted from Ref. [20]).

**Figure 3.**The coding of the pressure management plans in distribution network via chromosome (Reprinted from Ref. [27]).

**Figure 8.**Water network relief valve pressures distribution zoning: (

**a**) in no pressure, (

**b**) in mode 1, (

**c**) in mode 2, (

**d**) in mode 3, (

**e**) in mode 4, (

**f**) in mode 5, (

**g**) in mode 6, (

**h**) in mode 7, (

**i**) in mode 8.

**Figure 9.**Optimal position of five pressure-relief valves for the studied water distribution network.

Parameter | Population Size | Max Iteration | Crossover Pop. | Scaling Factor |
---|---|---|---|---|

Value | 150 | 100 | 0.70 | 0.50 |

No. | Active Valve Number | Ave. Network Pressure (Water Meters) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

0 | Active valve number | 63.96 | ||||||||

1 | Active valve number | 1 | 52.71 | |||||||

2 | Active valve number | 1 | 25 | 40.86 | ||||||

3 | Active valve number | 1 | 4 | 13 | 36.55 | |||||

4 | Active valve number | 1 | 4 | 6 | 15 | 32.25 | ||||

5 | Active valve number | 1 | 4 | 11 | 13 | 26 | 28.06 | |||

6 | Active valve number | 1 | 4 | 11 | 13 | 26 | 38 | 27.71 | ||

7 | Active valve number | 1 | 4 | 13 | 14 | 26 | 33 | 20 | 27.36 | |

8 | Active valve number | 1 | 4 | 6 | 15 | 26 | 33 | 20 | 13 | 27.05 |

**Table 3.**Percentage reduction in pressure changes in the network for different pressure-relief valves.

No. | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|---|

Percentage reduction | 0 | 17.589 | 36.116 | 42.855 | 49.578 | 56.129 | 56.676 | 57.223 | 57.708 |

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**MDPI and ACS Style**

Cemiloglu, A.; Licai, Z.; Ugurenver, A.; Nanehkaran, Y.A.
Optimal Exploitation of Urban Water Supply Networks Based on Pressure Management with the Nondominated Sorting Differential Evolution (NSDE) Algorithm. *Water* **2023**, *15*, 2583.
https://doi.org/10.3390/w15142583

**AMA Style**

Cemiloglu A, Licai Z, Ugurenver A, Nanehkaran YA.
Optimal Exploitation of Urban Water Supply Networks Based on Pressure Management with the Nondominated Sorting Differential Evolution (NSDE) Algorithm. *Water*. 2023; 15(14):2583.
https://doi.org/10.3390/w15142583

**Chicago/Turabian Style**

Cemiloglu, Ahmed, Zhu Licai, Abbas Ugurenver, and Yaser A. Nanehkaran.
2023. "Optimal Exploitation of Urban Water Supply Networks Based on Pressure Management with the Nondominated Sorting Differential Evolution (NSDE) Algorithm" *Water* 15, no. 14: 2583.
https://doi.org/10.3390/w15142583