# A Direct Approach for the Near-Optimal Design of Water Distribution Networks Based on Power Use

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Optimal Hydraulic Grade Line (HGL) Criteria

_{Max}is the hydraulic grade line at the source, HGL

_{Min}is the HGL at the final node and represents the minimum value allowable, F is the predefined sag as a percentage of (HGL

_{Max}− HGL

_{Min}), and L is the total length of the series [26].

## 3. Methodology

#### 3.1. Sump Search or Spannin Tree Structure

^{2}) time complexity, where NN is the number of nodes. At the end, the status of the last nodes in each branch is set to ‘sumps’, which will be useful for the rest of the methodology [28].

#### 3.2. Optimal Power Use Surface (OPUS)

#### 3.3. Optimal Flow Distribution

^{+}(positive real numbers) [24]. Thus, this step is intended to find a unique flow distribution in the network that fits to the Optimal Power Use Surface (OPUS) previously defined, ensuring it satisfies the mass conservation at each node.

- (1)
**Uniform distribution:**It assumes that all pipes have the same flow, which is calculated by dividing the total flow demand of each node into the number of upstream pipes connected to it [26].- (2)
**Proportional distribution:**For each pipe, the flow distributes proportionally to H/L^{2}, where H stands for the head losses in the pipe and L for its length.- (3)
**All-in-one distribution:**The conveyance capacity for all the pipes upstream of the analyzed node is computed assuming they have the minimum diameter available (d_{min}), and they are assigned as the design flows. If this is insufficient to transport the total flow demand, the residual flow is assigned to the one pipe with highest hydraulic favorability.

#### 3.4. Continuous Diameter Calculation

#### 3.5. Diameter Round-off

#### 3.6. Optimization

## 4. Results and Discussion

^{+}), where $\mathit{D}$ is the diameter of each pipe, to be considered as a theoretical solution giving the best costs that may be found with the methodology without the restriction of continuous diameter values. The input files corresponding to the optimal designs reported below are available in the supplementary files of this research.

#### 4.1. Hanoi

#### 4.2. Balerma

#### 4.3. Taichung

#### 4.4. Pescara Network

#### 4.5. Sensitivity Analysis OPUS Methodology—Sag

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Ilaya-Ayza, A.E.; Martins, C.; Campbell, E.; Izquierdo, J. Implementation of DMAs in Intermittentwater supply networks based on equity criteria. Water
**2017**, 9, 851. [Google Scholar] [CrossRef] [Green Version] - Giudicianni, C.; Herrera, M.; di Nardo, A.; Carravetta, A.; Ramos, H.M.; Adeyeye, K. Zero-net energy management for the monitoring and control of dynamically-partitioned smart water systems. J. Clean. Prod.
**2020**, 252, 119745. [Google Scholar] [CrossRef] - Central Public Health and Environmental Engineering Organisation; Ministry of Urban Development; World Health Organisation. Manual on Operation and Maintenance of Water Supply Systems; International Water Association: New Delhi, India, 2005. [Google Scholar]
- Yates, D.F.; Templeman, A.B.; Boffey, T.B. The computational complexity of the problem of determining least capital cost designs for water supply networks. Eng. Optim.
**1984**, 7, 143–155. [Google Scholar] [CrossRef] - Wu, Z.Y.; Walski, T. Self-adaptive penalty approach compared with other constraint-handling techniques for pipeline optimization. J. Water Resour. Plan. Manag.
**2005**, 131, 181–192. [Google Scholar] [CrossRef] - Montalvo, I.; Izquierdó, J.; Peerez-Garcia, R.; Herrera, M. Improved performance of PSO with self-adaptive parameters for computing the optimal design of Water Supply Systems. Eng. Appl. Artif. Intell.
**2010**, 23, 727–735. [Google Scholar] [CrossRef] - Gessler, J. Pipe network optimization by enumeration. In Proceedings of the Specialty Conference on Computer Applications in Water, Buffalo, NY, USA, 10–12 June 1985. [Google Scholar]
- Alperovits, E.; Shamir, U. Design of optimal water distribution systems. Water Resour. Res.
**1977**, 13, 885–900. [Google Scholar] [CrossRef] - Storn, R.; Price, K. Differential Evolution: A Simple and Efficient Adaptive Scheme for Global Optimization Over Continuous Spaces. J. Glob. Optim.
**1997**, 11, 341–359. [Google Scholar] [CrossRef] - Savic, D.A.; Walters, G.A. Genetic Algorithms for Least-Cost Design of Water Distribution Networks. J. Water Resour. Plan. Manag.
**1997**, 123, 67–77. [Google Scholar] [CrossRef] - Cunha, M.C.; Sousa, J. Water Distribution Network Design Optimization: Simulated Annealing Approach. J. Water Resour. Plan. Manag.
**1999**, 125, 215–221. [Google Scholar] [CrossRef] - Reca, J.; Martínez, J.; Gil, C.; Baños, R. Application of several meta-heuristic techniques to the optimization of real looped water distribution networks. Water Resour. Manag.
**2007**, 22, 1367–1379. [Google Scholar] [CrossRef] - Geem, Z.W. Optimal cost design of water distribution networks using harmony search. Eng. Optim.
**2006**, 38, 259–277. [Google Scholar] [CrossRef] - Lin, M.-D.; Liu, Y.H.; Liu, G.F.; Chu, C.W. Scatter search heuristic for least-cost design of water distribution networks. Eng. Optim.
**2007**, 39, 857–876. [Google Scholar] [CrossRef] - Perelman, L.; Ostfeld, A. An adaptive heuristic cross-entropy algorithm for optimal design of water distribution systems. Eng. Optim.
**2007**, 39, 413–428. [Google Scholar] [CrossRef] - Geem, Z.W. Particle-swarm harmony search for water network design. Eng. Optim.
**2009**, 41, 297–311. [Google Scholar] [CrossRef] - Zheng, F.; Simpson, A.R.; Zecchin, A.C. Coupled binary linear programming-differential evolution algorithm approach for water distribution system optimization. J. Water Resour. Plan. Manag.
**2014**, 140, 585–597. [Google Scholar] [CrossRef] [Green Version] - 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. [Google Scholar] [CrossRef] [Green Version] - Zheng, F.; Zecchin, A.C.; Simpson, A.R. Investigating the run-time searching behavior of the differential evolution algorithm applied to water distribution system optimization. Environ. Model. Softw.
**2015**, 69, 292–307. [Google Scholar] [CrossRef] [Green Version] - Rossman, L.A. Epanet 2 User’s Manual; Environmental Protection Agency: Boston, MA, USA, 2000. [Google Scholar]
- Wu, I. Design of drip irrigation main lines. J. Irrig. Drain. Div.
**1975**, 101, 265–278. [Google Scholar] - Featherstone, R.E.; El-Jumaily, K.K. Optimal Diameter Selection for Pipe Networks. J. Hydraul. Eng.
**1983**, 109, 221–234. [Google Scholar] [CrossRef] - Takahashi, S.; Saldarriaga, J.; Hernández, F.; Díaz, D.; Ochoa, S. An energy methodology for the design of water distribution systems. In Proceedings of the World Environmental and Water Resources Congress (EWRI) 2010, Providence, RI, USA, 16–20 May 2010. [Google Scholar]
- Saldarriaga, J.; Takahashi, S.; Hernández, F.; Escovar, M. Predetermining pressure surfaces in water distribution system design. In Proceedings of the World Environmental and Water Resources Congress 2011: Bearing Knowledge for Sustainability, Palm Springs, CA, USA, 22–26 May 2011. [Google Scholar]
- Páez, D.; Saldarriaga, J.; López, L.; Salcedo, C. Optimal design of water distribution systems with pressure driven demands. Procedia Eng.
**2014**, 89, 839–847. [Google Scholar] [CrossRef] [Green Version] - Saldarriaga, J.; Páez, D.; León, N.; López, L.; Cuero, P. Power use methods for optimal design of WDS: History and their use as post-optimization warm starts. J. Hydroinf.
**2015**, 17, 404–421. [Google Scholar] [CrossRef] [Green Version] - Prim, R.C. Shortest Connection Networks and Some Generalizations. Bell Syst. Tech. J.
**1957**, 36, 1389–1401. [Google Scholar] [CrossRef] - Saldarriaga, J.; Páez, D.; Cuero, P.; Leön, N. Optimal design of water distribution networks using mock open tree topology. In Proceedings of the World Environmental and Water Resources Congress 2013: Showcasing the Future, Cincinnati, OH, USA, 19–23 May 2013. [Google Scholar]
- Bragalli, C.; D’Ambrosio, C.; Lee, J.; Lodi, A.; Toth, P. On the optimal design of water distribution networks: A practical MINLP approach. Optim. Eng.
**2012**, 13, 219–246. [Google Scholar] [CrossRef] - Prasad, T.D.; Park, N.S. Multiobjective genetic algorithms for design of water distribution networks. J. Water Resour. Plan. Manag.
**2004**, 130, 73–82. [Google Scholar] [CrossRef] - Fujiwara, O.; Khang, D.B. A two-phase decomposition method for optimal design of looped water distribution networks. Water Resour. Res.
**1990**, 26, 539–549. [Google Scholar] [CrossRef] - Reca, J.; Martínez, J. Genetic algorithms for the design of looped irrigation water distribution networks. Water Resour. Res.
**2006**, 42, 42. [Google Scholar] [CrossRef] - Sung, Y.-H.; Lin, M.-D.; Lin, Y.-H.; Liu, Y.-L. Tabu search solution of water distribution network optimization. J. Environ. Eng. Manag.
**2007**, 17, 177. [Google Scholar] - Mora-Melia, D.; Iglesias-Rey, P.L.; Martinez-Solano, F.J.; Fuertes-Miquel, V.S. Design of Water Distribution Networks using a Pseudo-Genetic Algorithm and Sensitivity of Genetic Operators. Water Resour. Manag.
**2013**, 27, 4149–4162. [Google Scholar] [CrossRef] - Geem, Z.W.; Kim, J.H.; Loganathan, G.V. Harmony search optimization: Application to pipe network design. Int. J. Model. Simul.
**2002**, 22, 125–133. [Google Scholar] [CrossRef] - Eusuff, M.M.; Lansey, K.E. Optimization of water distribution network design using the shuffled frog leaping algorithm. J. Water Resour. Plan. Manag.
**2003**, 129, 210–225. [Google Scholar] [CrossRef] - Liong, S.-Y.; Atiquzzaman, M. Optimal design of water distribution network using shuffled complex evolution. J. Inst. Eng.
**2004**, 44, 93–107. [Google Scholar] - Vairavamoorthy, K.; Ali, M. Pipe index vector: A method to improve genetic-algorithm-based pipe optimization. J. Hydraul. Eng.
**2005**, 131, 1117–1125. [Google Scholar] [CrossRef] - Zecchin, A.C.; Simpson, A.R.; Maier, H.R.; Leonard, M.; Roberts, A.J.; Berrisford, M.J. Application of two ant colony optimisation algorithms to water distribution system optimisation. Math. Comput. Model.
**2006**, 44, 451–468. [Google Scholar] [CrossRef] [Green Version] - Suribabu, C.R.; Neelakantan, T.R. Design of water distribution networks using particle swarm optimization. Urban Water J.
**2006**, 3, 111–120. [Google Scholar] [CrossRef] - Kadu, M.S.; Gupta, R.; Bhave, P.R. Optimal design of water networks using a modified genetic algorithm with reduction in search space. J. Water Resour. Plan. Manag.
**2008**, 134, 147–160. [Google Scholar] [CrossRef] - Mohan, S.; Babu, K.S.J. Water distribution network design using heuristics-based algorithm. J. Comput. Civ. Eng.
**2009**, 23, 249–257. [Google Scholar] [CrossRef] - Suribabu, C.R. Differential evolution algorithm for optimal design of water distribution networks. J. Hydroinf.
**2010**, 12, 66–82. [Google Scholar] [CrossRef] - Mohan, S.; Babu, K.S.J. Optimal water distribution network design with honey-bee mating optimization. J. Comput. Civ. Eng.
**2010**, 24, 117–126. [Google Scholar] [CrossRef] [Green Version] - Suribabu, C.R. Heuristic-based pipe dimensioning model for water distribution networks. J. Pipeline Syst. Eng. Pract.
**2012**, 3, 115–124. [Google Scholar] [CrossRef] - Zheng, F.; Zecchin, A.C.; Simpson, A.R. Self-adaptive differential evolution algorithm applied to water distribution system optimization. J. Comput. Civ. Eng.
**2013**, 27, 148–158. [Google Scholar] [CrossRef] [Green Version] - Ochoa, S. Optimal Design of Water Distribution Systems Based on the Optimal Hydraulic Gradient Surface Concept. Master’s degree Thesis, Universidad de los Andes, Bogotá, Colombia, 2009. Unpublished (In Spanish). [Google Scholar]
- Tolson, B.A.; Asadzadeh, M.; Maier, H.R.; Zecchin, A. Hybrid discrete dynamically dimensioned search (HD-DDS) algorithm for water distribution system design optimization. Water Resour. Res.
**2009**. [Google Scholar] [CrossRef] - Baños, R.; Gil, C.; Reca, J.; Montoya, F.G. A memetic algorithm applied to the design of water distribution networks. Appl. Soft Comput. J.
**2010**, 10, 261–266. [Google Scholar] [CrossRef] - Bolognesi, A.; Bragalli, C.; Marchi, A.; Artina, S. Genetic Heritage Evolution by Stochastic Transmission in the optimal design of water distribution networks. Adv. Eng. Softw.
**2010**, 41, 792–801. [Google Scholar] [CrossRef] - Todini, E. Looped water distribution networks design using a resilience index based heuristic approach. Urban Water
**2000**, 2, 115–122. [Google Scholar] [CrossRef] - Reca, J.; Martinez, J.; Baños, R.; Gil, C. Optimal Design of Gravity-Fed Looped Water Distribution Networks Considering the Resilience Index. J. Water Resour. Plan. Manag.
**2008**, 134, 234–238. [Google Scholar] [CrossRef] - Monsef, H.; Naghashzadegan, M.; Farmani, R.; Jamali, A. Deficiency of Reliability Indicators in Water Distribution Networks. J. Water Resour. Plan. Manag.
**2019**, 145, 04019022. [Google Scholar] [CrossRef] - Wang, Q.; Savić, D.A.; Kapelan, Z. Hybrid metaheuristics for multi-objective design of water distribution systems. J. Hydroinf.
**2013**, 16, 165–177. [Google Scholar] [CrossRef] [Green Version] - Yazdi, J. Decomposition based Multi Objective Evolutionary Algorithms for Design of Large-Scale Water Distribution Networks. Water Resour. Manag.
**2016**, 30, 2749–2766. [Google Scholar] [CrossRef] - Giudicianni, C.; Di Nardo, A.; Di Natale, M.; Greco, R.; Santonastaso, G.F.; Scala, A. Topological taxonomy of water distribution networks. Water
**2018**, 10, 444. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Comparison of Hydraulic Grade Lines (HGL) resulting from Different Demand Configurations and Minimum HGL.

**Figure 4.**(

**a**) Layout of Pescara Water Distribution System (WDS). Pipe ID shown in the labels above the pipes (

**b**) Minimum-Cost Spanning Tree. Node ID shown in the labels. Dashed links correspond to the pipes that do not form part of the spanning tree.

**Figure 6.**(

**a**) Topographic profile for Pescara. (

**b**) Level curve for the HGL. (

**c**) Assigned surface for the Pescara network.

**Figure 8.**(

**a**) General layout and Spanning Tree for Hanoi Network. Dashed lines are pipes from the original networks that are not included in the Spanning Tree (

**b**) OPUS design considering continuous diameters (in inches) (

**c**) OPUS design considering discrete diameters (in inches).

**Figure 9.**Comparison between the ideal HGL and that obtained by OPUS considering the additional 50 in. diameter.

**Figure 10.**(

**a**) General layout for Balerma network and the spanning tree obtained for each reservoir, including the removed pipes. Dashed lines are pipes from the original networks that are not included in the Spanning Tree (

**b**) OPUS design for Balerma Network. The diameters are shown in millimeters.

**Figure 11.**(

**a**) Spanning Tree for Taichung Network. Dashed lines are pipes from the original networks that are not included in the Spanning Tree. (

**b**) OPUS design for Taichung Network. The diameters are shown in millimeters.

**Figure 12.**(

**a**) OPUS design for Pescara Network. The diameters are shown in millimeters. The resulting spanning tree is shown in Figure 4b. (

**b**) Zoom in to the location of Pipe 84. (

**c**) Zoom in to the location of Pipe 103.

**Figure 13.**Sensitivity Results for Cost and Network Resilience Index (NRI) for Different Values of Sag (

**a**) Hanoi Network. (

**b**) Balerma Network. (

**c**) Taichung Network (

**d**) Pescara Network.

Diameter (in.) | 12.0 | 16.0 | 20.0 | 24.0 | 30.0 | 40.0 |

Unit Cost ($/m) | 45.73 | 70.40 | 98.39 | 129.33 | 180.75 | 278.28 |

Algorithm | Cost (millions) | Number of Iterations |
---|---|---|

Genetic Algorithm [10] | $6.073 | 1,000,000 |

Simulated annealing [11] | $6.056 | 53,000 |

Harmony search [35] | $6.056 | 200,000 |

Shuffled frog leaping [36] | $6.073 | 26,987 |

Shuffled complex evolution [37] | $6.220 | 25,402 |

Genetic Algorithm [38] | $6.056 | 18,300 |

Ant colony optimization [39] | $6.134 | 35,433 |

Genetic Algorithms [32] | $6.081 | 50,000 |

Particle Swarm Optimization [40] | $6.093 | 6600 |

Genetic Algorithms [12] | $6.173 | 26,457 |

Simulated annealing [12] | $6.333 | 26,457 |

Simulated annealing with tabu search [12] | $6.353 | 26,457 |

Local search with simulated annealing [12] | $6.308 | 26,457 |

Harmony search [13] | $6.081 | 27,721 |

Cross entropy [15] | $6.081 | 97,000 |

Scatter search [33] | $6.081 | 43,149 |

Modified GA 1 [41] | $6.056 | 18,000 |

Modified GA 2 [41] | $6.190 | 18,000 |

Particle swarm harmony search [16] | $6.081 | 17,980 |

Heuristic based approach [42] | $6.701 | 70 |

Differential evolution [43] | $6.081 | 48,724 |

Honey-bee mating optimization [44] | $6.117 | 15,955 |

Heuristic based approach [45] | $6.232 | 259 |

Self-adaptive differential evolution [46] | $6.081 | 60,582 |

Pseudo-Genetic Algorithm [34] | $6.081 | 25,000 |

Linear Programming – Differential Evolution [17] | $6.081 | 33,148 |

B-GENOME (B-GA) [18] | $6.182 | 26,000 |

SOGH [47] | $6.337 | 94 |

OPUS (This research) | $6.374 | 106 |

Diameter (mm) | 113 | 126.6 | 144.6 | 162.8 | 180.8 | 226.2 | 285 | 361.8 | 452.2 | 581.8 |

Unit Cost (€/m) | 7.22 | 9.10 | 11.92 | 14.84 | 18.38 | 28.60 | 45.39 | 76.32 | 124.64 | 215.85 |

Algorithm | Cost (€ millions) | Number of Iterations |
---|---|---|

Genetic algorithm [32] | 2.302 | 10,000,000 |

Harmony search [13] | 2.601 | 45,400 |

Harmony search [13] | 2.018 | 10,000,000 |

Genetic algorithm [12] | 3.738 | 45,400 |

Simulated annealing [12] | 3.476 | 45,400 |

Simulated annealing with tabu search [12] | 3.298 | 45,400 |

Local search with simulated annealing [12] | 4.310 | 45,400 |

Hybrid discrete dynamically dimensioned search [48] | 1.940 | 30,000,000 |

Harmony search with particle swarm [16] | 2.633 | 45,400 |

SOGH [47] | 2.100 | 1779 |

Memetic algorithm [49] | 3.120 | 45,400 |

Genetic heritage evolution by stochastic transmission [50] | 2.002 | 250,000 |

Differential evolution [46] | 1.998 | 2,400,000 |

Self-adaptive differential evolution [46] | 1.983 | 1,300,000 |

OPUS (this study) | 2.015 | 1165 |

Diameter (mm) | Unit Cost (NT Dollar/m) |
---|---|

100 | 860 |

150 | 1160 |

200 | 1470 |

250 | 1700 |

300 | 2080 |

350 | 2640 |

400 | 3240 |

450 | 3810 |

500 | 4400 |

600 | 5580 |

700 | 8360 |

800 | 10,400 |

900 | 12,800 |

Algorithm | Cost (NT Dollar) | Number of iterations |
---|---|---|

Tabu search [33] | 8,774,900 | Not Available |

OPUS (this study) | 8,914,400 | 47 |

Diameter (mm) | Unit Cost (€/m) |
---|---|

100 | 27.7 |

125 | 38 |

150 | 40.5 |

200 | 55.4 |

250 | 75 |

300 | 92.4 |

350 | 123.1 |

400 | 141.9 |

450 | 169.3 |

500 | 191.5 |

600 | 246 |

700 | 319.6 |

800 | 391.1 |

Algorithm | Cost (€ millions) | Number of Iterations |
---|---|---|

Mixed Integer Linear Programming—MILP [29] | 1.820 | Not available—Time limit: 7200 s |

OPUS (This research) | 2.161 | 206 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Saldarriaga, J.; Páez, D.; Salcedo, C.; Cuero, P.; López, L.L.; León, N.; Celeita, D.
A Direct Approach for the Near-Optimal Design of Water Distribution Networks Based on Power Use. *Water* **2020**, *12*, 1037.
https://doi.org/10.3390/w12041037

**AMA Style**

Saldarriaga J, Páez D, Salcedo C, Cuero P, López LL, León N, Celeita D.
A Direct Approach for the Near-Optimal Design of Water Distribution Networks Based on Power Use. *Water*. 2020; 12(4):1037.
https://doi.org/10.3390/w12041037

**Chicago/Turabian Style**

Saldarriaga, Juan, Diego Páez, Camilo Salcedo, Paula Cuero, Laura Lunita López, Natalia León, and David Celeita.
2020. "A Direct Approach for the Near-Optimal Design of Water Distribution Networks Based on Power Use" *Water* 12, no. 4: 1037.
https://doi.org/10.3390/w12041037