# A Hybrid Water Distribution Networks Design Optimization Method Based on a Search Space Reduction Approach and a Genetic Algorithm

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Bounding Strategy

_{i}is the flow of the link i,

**A**is an (m × n) array, and m is the number of nodes, and

**q**is a vector of nodal demands.

_{ij}of array

**A**is 1 if the flow of link j goes into node i, −1 if it leaves the node, and 0 if link j is not connected to node i.

**A**is a (m × 2n) array. The solution of the minimization QPP problem provides the right flow directions and values that minimize the sum of network flows. The so called Maximum Dispersion (MD) flow distribution is obtained in this way.

_{MC}and Q

_{MD}), which bounds the range of possible flows within each network link. By imposing velocity restrictions, a pair of vectors defining the range of possible diameters between the minimum (D

_{m,i}) and the maximum (D

_{M,i}) for each link i can be calculated in the following way:

#### 2.2. Bounded Genetic Algorithm Formulation

^{©}(Microsoft, Redmond, Washington, DC, USA) spreadsheet environment.

_{d,i}(last possible diameter). This methodology has many advantages since there are no limitations on the number of possible diameter sizes that can be assigned to a specific pipe. In the classic formulation of GENOME, the number of possible diameter sizes was equal for each link and this value was equal to the total number of diameters in the pipe database. The same coding scheme has been adopted in B-GENOME, although some modifications have been made to allow for a variable number of possible diameters for each link. The new B-GENOME algorithm used in this work has an integer coding scheme and a variable number of alleles. The number of alleles depends on the number of possible diameters comprised within the velocity restrictions of each link.

_{i}is the pipe cost (€ m

^{−1}), which is a function of the diameter D

_{i}, L

_{i}is the length of the link i, p is a penalty multiplier, N is the number of nodes in the network, h

_{rj}is the required pressure head in the node j and h

_{j}is the actual pressure head computed by the hydraulic solver EPANET for the node j.

^{9}€/m) to avoid finding solutions that violate the pressure restrictions. In order to compute the pressure deficits, the nodal pressures for each individual in the population have been computed by using a network solver. The hydraulic solver EPANET has been used for this purpose [19]. The EPANET engine is used when needed by calling on the EPANET toolkit from the VBA software code developed in this work. B-GENOME implements all the different options to perform the three basic operations that were available in GENOME [2].

#### 2.3. Structure of B-GENOME

#### 2.4. Testing of the B-GENOME Model

_{cross}. In the uniform crossover, the parents’ chromosomes exchange their genetic information gene to gene. The probability of exchanging genes is defined by the gene crossing rate (r

_{cross}). Ten simulations were performed both for the new bounded algorithm and the classic GA algorithm.

## 3. Results

## 4. Discussion

^{8}= 1.48 × 10

^{9}possible network designs. The search space for the bounded problem is reduced to 4.61 × 10

^{7}, which means that the search space becomes approximately 3% of the total search space of the problem (see Table 2). In the case of the Hanoi network, the reduction is even higher. There are six possible diameters in the database and the number of links is equal to 34. The resulting number of alternative designs is 2.87 × 10

^{26}, whereas the size of the search space in the bounded problem is 4.35 × 10

^{16}(see Table 3). The reduction of the search space is expected to be higher for a larger number of links and the number of pipe diameters in a given problem. The velocity limits also play an important role as the search space reduction increases as the velocity limits range becomes narrower.

## 5. Conclusions

- A new approach based on bounding and reducing the total search space in a water distribution network design problem has been developed. This new approach reduces the search space by analyzing two opposite extreme flow distribution scenarios and then applying velocity restrictions to the pipes.
- This new approach has been coupled to a GA in order to improve its performance.
- The proposed B-GA algorithm considerably reduced the search space and provided a much faster and more accurate convergence than the classic GA formulation for a small and a medium benchmark network. It is expected that, for more complex networks, the advantages provided by the new B-GA approach could be even greater.
- This new approach could also be implemented in other types of heuristic methods. The improvements on the performance of these heuristics provided by the new approach are still to be investigated.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Layout of the Alperovits and Shamir network [3].

**Figure 4.**Evolution of the best fitness value for B-GENOME and GENOME algorithms (Alperovits and Shamir Network).

Parameter | A&S | Hanoi |
---|---|---|

Population (np) | 100 | 200 |

Generations (ng) | 200 | 300 |

Crossover prob. (p_{cross}) | 0.9 | 0.9 |

Mutation prob. (P_{mut}) | 0.05 | 0.05 |

Prob. of gene crossing (r_{cross}) | 0.5 | 0.5 |

Reproduction plan | steady-state-delete-worst plan | steady-state-delete-worst plan |

Crossover operator | uniform crossover | uniform crossover |

**Table 2.**Flow range, maximum and minimum diameters and number of possible diameters for each link obtained from the Quadratic Programming Problems QPPs for the Alperovits and Shamir network.

Link | Q_{MD} (L/h) | Q_{MC} (L/h) | D_{m} (mm) | D_{M} (mm) | N°D |
---|---|---|---|---|---|

1 | 311.1 | 311.1 | 356 | 610 | 6 |

2 | 117.0 | 27.8 | 102 | 559 | 10 |

3 | 166.3 | 255.6 | 254 | 559 | 8 |

4 | 40.0 | 75.0 | 102 | 457 | 8 |

5 | 93.0 | 147.2 | 152 | 610 | 10 |

6 | 1.3 | 55.6 | 25.4 | 406 | 10 |

7 | 89.3 | 0.0 | 25.4 | 508 | 12 |

8 | 54.3 | 0.0 | 25.4 | 406 | 10 |

**Table 3.**Flow range, maximum and minimum diameters and number of possible diameters for each link obtained from the QPPs for the Hanoi network.

Link | Q_{MD} (L/h) | Q_{MC} (L/h) | D_{m} (mm) | D_{M} (mm) | N°D |
---|---|---|---|---|---|

1 | 19,940 | 19,940 | 1016 | 1016 | 1 |

2 | 19,050 | 19,050 | 1016 | 1016 | 1 |

3 | 5326 | 6810 | 1016 | 1016 | 1 |

4 | 5196 | 6680 | 1016 | 1016 | 1 |

5 | 4471 | 5955 | 1016 | 1016 | 1 |

6 | 3466 | 4950 | 762 | 1016 | 2 |

7 | 2116 | 3600 | 609.6 | 1016 | 3 |

8 | 1566 | 3050 | 508 | 1016 | 4 |

9 | 1041 | 2525 | 406.4 | 1016 | 4 |

10 | 2000 | 2000 | 609.6 | 1016 | 3 |

11 | 1500 | 1500 | 508 | 1016 | 4 |

12 | 940 | 940 | 406.4 | 1016 | 5 |

13 | 1484 | 0 | 304.8 | 1016 | 6 |

14 | 2099 | 615 | 304.8 | 1016 | 6 |

15 | 2379 | 895 | 304.8 | 1016 | 5 |

16 | 2968 | 1205 | 508 | 1016 | 4 |

17 | 3833 | 2070 | 609.6 | 1016 | 3 |

18 | 5178 | 3415 | 762 | 1016 | 2 |

19 | 5238 | 3475 | 762 | 1016 | 2 |

20 | 7637 | 7915 | 1016 | 1016 | 1 |

21 | 1415 | 1415 | 508 | 1016 | 4 |

22 | 485 | 485 | 304.8 | 1016 | 6 |

23 | 4947 | 5225 | 1016 | 1016 | 1 |

24 | 2890 | 3065 | 609.6 | 1016 | 3 |

25 | 2070 | 2245 | 609.6 | 1016 | 3 |

26 | 992 | 1270 | 406.4 | 1016 | 5 |

27 | 92 | 370 | 304.8 | 762 | 5 |

28 | 278 | 0 | 304.8 | 762 | 5 |

29 | 1011 | 1115 | 406.4 | 1016 | 5 |

30 | 721 | 825 | 406.4 | 1016 | 5 |

31 | 361 | 465 | 304.8 | 1016 | 6 |

32 | 1 | 105 | 304.8 | 1016 | 6 |

33 | 104 | 0 | 304.8 | 1016 | 6 |

34 | 909 | 805 | 304.8 | 1016 | 6 |

Algorithm | Min Cost ($) | Max Cost ($) | Avrg. Cost ($) | Std. Dev. ($) | C. Var (%) |
---|---|---|---|---|---|

B-GA | 419,000 | 447,000 | 424,000 | 9099 | 2.15 |

GA | 420,000 | 448,000 | 430,900 | 11,344 | 2.63 |

Algorithm | Min Cost ($) | Max Cost ($) | Avrg. Cost ($) | Std. Dev. ($) | C. Var (%) |
---|---|---|---|---|---|

B-GA | 6,182,006 | 6,242,051 | 6,219,390 | 19,831 | 0.32 |

GA | 6,208,937 | 6,373,131 | 6,296,366 | 57,791 | 0.92 |

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**MDPI and ACS Style**

Reca, J.; Martínez, J.; López, R.
A Hybrid Water Distribution Networks Design Optimization Method Based on a Search Space Reduction Approach and a Genetic Algorithm. *Water* **2017**, *9*, 845.
https://doi.org/10.3390/w9110845

**AMA Style**

Reca J, Martínez J, López R.
A Hybrid Water Distribution Networks Design Optimization Method Based on a Search Space Reduction Approach and a Genetic Algorithm. *Water*. 2017; 9(11):845.
https://doi.org/10.3390/w9110845

**Chicago/Turabian Style**

Reca, Juan, Juan Martínez, and Rafael López.
2017. "A Hybrid Water Distribution Networks Design Optimization Method Based on a Search Space Reduction Approach and a Genetic Algorithm" *Water* 9, no. 11: 845.
https://doi.org/10.3390/w9110845