# Development of Practical Design Approaches for Water Distribution Systems

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Practical Design of Water Distribution Systems

#### 2.1. Pipe Continuity Search Approach

#### 2.2. Multi-Objective Optimization Framework for Water Distribution Systems Design

#### 2.2.1. Objective Functions

_{i}) is the cost function of the i-th pipe per unit length (m) of each pipe diameter, and L

_{i}is the length (m) of the i-th pipe. D

_{i}is the pipe diameter (mm) of the i-th pipe. PN is the total number of pipes. $Penalty$ represents the penalty function; if the solution cannot satisfy for the constraints, the penalty function is applied.

_{j}is a head at node j, h

_{j}

^{*}is the minimum required head at node j, H

_{K}is the water level of reservoir K, Q

_{K}is water flow of reservoir K, and Pu

_{i}is the power of pump i.

#### 2.2.2. Hydraulic Constraints

^{7}, 10

^{8}are used.

_{i}is the pressure head at node i (m), v

_{j}is the water velocity at pipe j (m/s), h

_{min}and h

_{max}are the minimum and maximum pressure heads (m), respectively, v

_{min}and v

_{max}are the minimum and maximum water velocity (m/s), respectively, and α and β are the penalty constants.

#### 2.3. Metaheuristic Optimization Algorithm

_{i}

^{New}denotes a new decision variable, and x

_{i}

^{Lower}, x

_{i}

^{Upper}are the boundary conditions of the decision variables. Rnd is the uniform random value, and Bw is the bandwidth, HMCR, PAR, Bw are parameters for optimization.

## 3. Model Formulation

#### 3.1. Performance Measures

_{con}is the number of pipes satisfying the pipe size continuity and NP

_{total}is the total number of pipes.

#### 3.2. Study Network

## 4. Application and Results

#### 4.1. Hanoi Network

#### 4.2. Fossolo Network

#### 4.3. Cycling Network

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Existing pipe continuity search approach [10]. (

**a**) abnormal pipe unevenness; (

**b**) satisfy pipe continuity.

**Figure 5.**Complex joint point with pipe size continuity problem{ TC “Figure 4.2.1 Complex joint point with pipe size continuity problem” \f C \l “1” } (J23) using (

**a**) the existing pipe size continuity search approach [10], (

**b**) the continuity search approach proposed in this study.

**Figure 8.**Pipe diameter configuration: (

**a**) non-application, (

**b**) application of pipe size continuity{ TC “Figure 4.3.3 Design of HSA applied cost optimization” \f C \l “1” }.

Existing Pipe Size Continuity Search Algorithm [10] | Pipe Size Continuity Search Algorithm of This Study | |
---|---|---|

Ability to handle complex joint point | X | O |

Alleviation of calculation time | X | O |

Overlapping part of search process | O | X |

**Table 2.**Pseudocode for the proposed practical water distribution systems (WDSs) design approach. HM: harmony memory, HMS: HM size, HMCR: harmony memory considering rate, PAR: pitch adjust rate, Bw: bandwitdth.

Input: Objective functions (i.e., Minimum Construction Cost, Maximum System Resilience), Algorithm parameters (i.e., HMCR, PAR, HMS, BW) Generate initial HM randomly while stopping criterion is not satisfied do Calculate Objective function values (Cost and Resilience) Constraints check (Hydraulic and Pipe continuity) [Hydraulic constraints] if Hydraulic constraints condition is not satisfied Apply the Penalty function end if [Continuity constraints] If Flow(upstream) > Flow(downstream) and Diameter(upstream) < Diameter(downstream) Apply the Penalty function else if Flow(upstream) < Flow(downstream) and Diameter(upstream) > Diameter(downstream) Apply the Penalty function end if Calculate the Pareto ranking of HM considering the non-dominated sorting method Apply the crowding-distance method if the ranking of new harmony is better than the worst solution in HM Update HM end if Generate a new HM if (rand < HMCR) choose an existing harmony randomly if (rand < PAR) adjust the pitch randomly within the limits end if else generate new harmony via randomization1 end if end while |

Problem | NP | NN | PD | PCD | SS | Iteration |
---|---|---|---|---|---|---|

Hanoi network | 34 | 32 | 304.8, 406.4, 508.0, 609.6, 762.0, 1016 | 45.72, 70.40, 98.37, 129.33, 180.74, 278.28 | 2.87 × 10^{26} | 500,000 |

Fossolo network | 36 | 58 | 16, 20.4, 26, 32.6, 40.8, 51.4, 61.4, 73.6, 90, 102.2, 114.6, 130.8, 147.2, 163.6, 184, 204.6, 229.2, 257.8, 290.6, 327.4, 368.2, 409.2 | 0.38, 0.56, 0.88, 1.35, 2.02, 3.21, 4.44, 6.45, 9.59, 11.98, 14.93, 19.61, 24.78, 30.55, 38.71, 47.63, 59.7, 75.61, 99.58, 126.48, 160.29, 197.71 | 6.87 × 10^{46} | |

Cycling network | 165 | 242 | 113, 126.6, 144.6, 162.8, 180.8, 226.2, 285.0, 361.8, 452.2, 581.8 | 7.22, 9.1, 11.92, 14.84, 18.38, 28.6, 45.39, 76.32, 124.64, 215.85 | 1.16 × 10^{178} |

Method | Existing Pipe Size Continuity Search Approach [10] | Continuity Search Approach Proposed in This Study |
---|---|---|

Design cost ($) | 6,093,181 | 6,093,181 |

Mean calculation time | 21 min 20 s (Iterations = 10,000) | 4 min 49 s (Iterations = 10,000) |

1 h 46 min 39 s (Iterations = 50,000) | 24 min 6 s (Iterations = 50,000) | |

17 h 47 min 33 s (Iterations = 500,000) | 4 h 1 min 13 s (Iterations = 500,000) |

Method | Existing Pipe Size Continuity Search Approach [10] | Continuity Search Approach Proposed in This Study |
---|---|---|

Design cost (€) | 28,070.78 | 28,971.63 |

Mean calculation time | 49 h 51 min (Iterations = 500,000) | 6 h 34 min (Iterations = 500,000) |

**Table 6.**Optimization results and mean continuity index for the Cycling network. PCI: pipe continuity index.

Existing Pipe Size Continuity Search Approach [10] | Continuity Search Approach Proposed in This Study | |
---|---|---|

Cost ($) | 1,230,844 | 1,230,385 |

Resilience | 0.1629 | 0.1634 |

Mean PCI | 0.732 | 1.00 |

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**MDPI and ACS Style**

Choi, Y.H.; Lee, H.M.; Choi, J.; Yoo, D.G.; Kim, J.H. Development of Practical Design Approaches for Water Distribution Systems. *Appl. Sci.* **2019**, *9*, 5117.
https://doi.org/10.3390/app9235117

**AMA Style**

Choi YH, Lee HM, Choi J, Yoo DG, Kim JH. Development of Practical Design Approaches for Water Distribution Systems. *Applied Sciences*. 2019; 9(23):5117.
https://doi.org/10.3390/app9235117

**Chicago/Turabian Style**

Choi, Young Hwan, Ho Min Lee, Jiho Choi, Do Guen Yoo, and Joong Hoon Kim. 2019. "Development of Practical Design Approaches for Water Distribution Systems" *Applied Sciences* 9, no. 23: 5117.
https://doi.org/10.3390/app9235117