# Optimization Difficulty Indicator and Testing Framework for Water Distribution Network Complexity

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Optimization Difficulty Level

#### 2.2. Two-Phase ODL Examination Framework

#### 2.2.1. Phase 1: Layout Generation

#### Branch Index

_{b}denote the numbers of edges and branched edges, respectively. A branched edge means that it is not located in a loop. A larger BI corresponds to a more branched network. Hwang and Lansey [28] classified a WDN with a BI ≥ 0.5 as a branched network; otherwise, the network is identified as gridded, a hybrid (a mix between branch and grid layouts), or looped.

#### Meshedness Coefficient

#### Layout Generation Model

_{1}):

_{target}and MC

_{target}are the target BI and MC values, respectively; BI

_{i}and MC

_{i}are the ith layout solution’s BI and MC values, respectively; P

_{BI}and P

_{MC}are the penalty factors considered for BI and MC, respectively, which are assigned based on the target BI and MC values; N

_{disconnected}is the disconnection degree calculated by $\sum}_{j=1}^{n}b$, where b = 0 if the jth node is connected to water source (i.e., a reservoir) and b = 1 otherwise (j = 1, 2, …, n); and P

_{disconnected}is the penalty factor of the disconnection, which generally requires a relatively large value (e.g., more than 10,000) to prevent any disconnection in the network. The layout generation model seeks a feasible optimal layout with no penalty values (i.e., F

_{1}= 0) for BI

_{target}and MC

_{target}. Note that the pipe sizes are not determined in Phase 1; only pipe installation (whether or not to install a pipe at a link) is determined at each link. During the network generation and BI and MC calculations, a node-reduction algorithm [28] is used to remove nodes along a single pipe that do not affect the network structure and cause errors in the structural measure calculations. A thorough analysis might be needed for the penalty factors given to Equation (4).

_{target}and MC

_{target}. These are then input in Phase 2 to find the optimal pipe sizes for the layouts in Phase 1 (Figure 1).

#### 2.2.2. Phase 2: WDN Design Optimization

#### Pipe Size Optimization Model

**L**):

_{k}and L

_{k}are the unit pipe cost and length of pipe k (k = 1, 2, …, l), respectively;

**L**i is a k × 1 binary matrix that indicates the pipe locations (pipe layout) of the ith solution and has the element of 1 at a pipe location and 0 otherwise (no pipe); P

_{j}is the jth node’s pressure; and P

_{min}is the minimum allowable pressure. The unit pipe cost is calculated by using the pipe construction cost function from [33]. The objective function and pressure constraint are consistently applied to the optimization of every layout, and a penalty is applied to the objective function if the constraint is violated.

#### Global Parallel Genetic Algorithm

#### 2.3. Warm Initial Solution Approach

#### 2.4. Study Network

## 3. Application Results

#### 3.1. Optimization Difficulty Level and Two Topological Measures

_{BI}and P

_{MC}(Equation (4)) and summarized in Table 2 and Table 3 for the BI’s and MC’s sensitivities, respectively. As shown in both tables, absolute error tends to decrease when the higher penalty factor is assigned for each topological measure (right-up for BI (Table 2) and left-down for MC (Table 3)). Multiplying a higher penalty factor assigns greater weighting to the topological measure in the objective function F

_{1}in Equation (4). Therefore, the minimum absolute error overall could be obtained in cases where the two penalty factors P

_{BI}and P

_{MC}were identical (diagonal in Table 2 and Table 3). In this study, the P

_{BI}and P

_{MC}of 10

^{5}are used in Equation (4).

#### 3.2. Numerical Experiments on System Characteristics/Optimization Conditions

#### 3.3. Elevation Change

#### 3.4. Multiple Reservoirs

#### 3.5. Source Supply Condition

#### 3.6. Warm Initial Solutions

#### 3.7. Combined Effect Investigation

#### 3.8. Testing in the Large C Network

## 4. Summary and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Choi, Y.H.; Jung, D.; Lee, H.M.; Yoo, D.G.; Kim, J.H. Improving the quality of pareto optimal solutions in water distribution network design. J. Water Resour. Plan. Manag.
**2017**, 143, 04017036. [Google Scholar] [CrossRef] - 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] - 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] - Geem, Z.W.; Kim, J.H.; Loganathan, G. A new heuristic optimization algorithm: Harmony search. J. Simulat.
**2001**, 76, 60–68. [Google Scholar] [CrossRef] - Kim, J.H.; Geem, Z.W.; Kim, E.S. Parameter estimation of the nonlinear muskingum model using harmony search 1. JAWRA J. Am. Water Resour. Assoc.
**2001**, 37, 1131–1138. [Google Scholar] [CrossRef] - Maier, H.R.; Simpson, A.R.; Zecchin, A.C.; Foong, W.K.; Phang, K.Y.; Seah, H.Y.; Tan, C.L. Ant colony optimization for design of water distribution systems. J. Water Resour. Plan. Manag.
**2003**, 129, 200–209. [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] - Yang, X.-S. Engineering Optimization: An Introduction with Metaheuristic Applications; John Wiley & Sons: Hoboken, NJ, USA, 2010. [Google Scholar]
- Kim, J.H. Harmony search algorithm: A unique music-inspired algorithm. Procedia Eng.
**2016**, 154, 1401–1405. [Google Scholar] [CrossRef] - Sadollah, A.; Yoo, D.G.; Kim, J.H. Improved mine blast algorithm for optimal cost design of water distribution systems. Eng. Optim.
**2015**, 47, 1602–1618. [Google Scholar] [CrossRef] - Yazdi, J.; Choi, Y.H.; Kim, J.H. Non-dominated sorting Harmony Search Differential Evolution (NS-HS-DE): A hybrid algorithm for multi-objective design of water distribution networks. Water
**2017**, 9, 587. [Google Scholar] [CrossRef] - Zecchin, A.C.; Simpson, A.R.; Maier, H.R.; Marchi, A.; Nixon, J.B. Improved understanding of the searching behavior of ant colony optimization algorithms applied to the water distribution design problem. Water Resour. Res.
**2012**, 48. [Google Scholar] [CrossRef] - Jain, P.; Kar, P. Non-convex optimization for machine learning. Found. Trends Mach. Learn.
**2017**, 10, 142–336. [Google Scholar] [CrossRef] - Alperovits, E.; Shamir, U. Design of optimal water distribution systems. Water Resour. Res.
**1977**, 13, 885–900. [Google Scholar] [CrossRef] - Smith-Miles, K.; Lopes, L. Measuring instance difficulty for combinatorial optimization problems. Comput. Oper. Res.
**2012**, 39, 875–889. [Google Scholar] [CrossRef] - He, J.; Reeves, C.; Witt, C.; Yao, X. A note on problem difficulty measures in black-box optimization: Classification, realizations and predictability. Evol. Comput.
**2007**, 15, 435–443. [Google Scholar] [CrossRef] - Zitzler, E.; Deb, K.; Thiele, L. Comparison of multiobjective evolutionary algorithms: Empirical results. Evol. Comput.
**2000**, 8, 173–195. [Google Scholar] [CrossRef] - Zheng, F.; Zecchin, A.C.; Maier, H.R.; Simpson, A.R. Comparison of the searching behavior of NSGA-II, SAMODE, and Borg MOEAs applied to water distribution system design problems. J. Water Resour. Plan. Manag.
**2016**, 142, 04016017. [Google Scholar] [CrossRef] - Alpcan, T.; Everitt, T.; Hutter, M. Can we measure the difficulty of an optimization problem? In Proceedings of the 2014 IEEE Information Theory Workshop (ITW 2014), Hobart, Australia, 2–5 November 2014; pp. 356–360. [Google Scholar]
- Jung, D.; Yoo, D.G.; Kang, D.; Kim, J.H. Linear Model for Estimating Water Distribution System Reliability. J. Water Resour. Plan. Manag.
**2016**, 142, 04016022. [Google Scholar] [CrossRef] - Gheisi, A.; Naser, G. Multistate reliability of water-distribution systems: Comparison of surrogate measures. J. Water Resour. Plan. Manag.
**2015**, 141, 04015018. [Google Scholar] [CrossRef] - Tanyimboh, T.T.; Templeman, A.B. A quantified assessment of the relationship between the reliability and entropy of water distribution systems. Eng. Optim.
**2000**, 33, 179–199. [Google Scholar] [CrossRef] - Yoo, D.G.; Jung, D.; Kang, D.; Kim, J.H.; Lansey, K. Seismic hazard assessment model for urban water supply networks. J. Water Resour. Plan. Manag.
**2015**, 142, 04015055. [Google Scholar] [CrossRef] - Kim, J.H.; Lee, H.M.; Jung, D.; Sadollah, A. Engineering benchmark generation and performance measurement of evolutionary algorithms. In Proceedings of the 2017 IEEE Congress on Evolutionary Computation (CEC), San Sebastian, Spain, 5–8 June 2017; pp. 714–717. [Google Scholar]
- Kim, J.H.; Choi, Y.H.; Ngo, T.T.; Choi, J.; Lee, H.M.; Choo, Y.M.; Lee, E.H.; Yoo, D.G.; Sadollah, A.; Jung, D. KU battle of metaheuristic optimization algorithms 2: Performance test. In Harmony Search Algorithm; Springer: Berlin/Heidelberg, Germany, 2016; pp. 207–213. [Google Scholar]
- 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] - Zheng, F.; Simpson, A.R.; Zecchin, A.C. A combined NLP-differential evolution algorithm approach for the optimization of looped water distribution systems. Water Resour. Res.
**2011**, 47. [Google Scholar] [CrossRef] - Hwang, H.; Lansey, K. Water distribution system classification using system characteristics and graph-theory metrics. J. Water Resour. Plan. Manag.
**2017**, 143, 04017071. [Google Scholar] [CrossRef] - Jung, D.; Kim, J.H. Water Distribution System Design to Minimize Costs and Maximize Topological and Hydraulic Reliability. J. Water Resour. Plan. Manag.
**2018**, 144, 06018005. [Google Scholar] [CrossRef] - Buhl, J.; Gautrais, J.; Solé, R.V.; Kuntz, P.; Valverde, S.; Deneubourg, J.-L.; Theraulaz, G. Efficiency and robustness in ant networks of galleries. Eur. Phys. J. B Condens. Matter Complex Syst.
**2004**, 42, 123–129. [Google Scholar] [CrossRef] - Yazdani, A.; Jeffrey, P. Applying network theory to quantify the redundancy and structural robustness of water distribution systems. J. Water Resour. Plan. Manag.
**2011**, 138, 153–161. [Google Scholar] [CrossRef] - Yazdani, A.; Jeffrey, P. Water distribution system vulnerability analysis using weighted and directed network models. Water Resour. Res.
**2012**, 48. [Google Scholar] [CrossRef] - Clark, R.M.; Sivaganesan, M.; Selvakumar, A.; Sethi, V. Cost models for water supply distribution systems. J. Water Resour. Plan. Manag.
**2002**, 128, 312–321. [Google Scholar] [CrossRef] - Iglesias-Rey, P.; Martinez-Solano, F.; Mora-Melia, D.; Ribelles-Aguilar, J. The battle water networks II: Combination of meta-heuristc technics with the concept of setpoint function in water network optimization algorithms. In Proceedings of the WDSA 2012: 14th Water Distribution Systems Analysis Conference, Adelaide, Australia, 24–27 September 2012; p. 510. [Google Scholar]
- Marchi, A.; Salomons, E.; Ostfeld, A.; Kapelan, Z.; Simpson, A.R.; Zecchin, A.C.; Maier, H.R.; Wu, Z.Y.; Elsayed, S.M.; Song, Y. Battle of the water networks II. J. Water Resour. Plan. Manag.
**2013**, 140, 04014009. [Google Scholar] [CrossRef] - Guidolin, M.; Fu, G.; Reed, P. Parallel evolutionary multiobjective optimization of water distribution system design. In Proceedings of the WDSA 2012: 14th Water Distribution Systems Analysis Conference, Adelaide, Australia, 24–27 September 2012; p. 113. [Google Scholar]
- Kandiah, V.; Jasper, M.; Drake, K.; Shafiee, M.; Barandouzi, M.; Berglund, A.; Brill, E.; Mahinthakumar, G.; Ranjithan, S.; Zechman, E. Population-based search enabled by high performance computing for BWN-II design. In Proceedings of the WDSA 2012: 14th Water Distribution Systems Analysis Conference, Adelaide, Australia, 24–27 September 2012; p. 536. [Google Scholar]
- Elfeky, E.Z.; Sarker, R.A.; Essam, D.L. Partial decomposition and parallel GA (PD-PGA) for constrained optimization. In Proceedings of the 2008 IEEE International Conference on Systems, Man and Cybernetics, Singapore, 12–15 October 2008; pp. 220–227. [Google Scholar]
- Lopes, N.; Ribeiro, B. GPU implementation of the multiple back-propagation algorithm. In Proceedings of the International Conference on Intelligent Data Engineering and Automated Learning, Burgos, Spain, 20–23 September 2006; pp. 449–456. [Google Scholar]
- Guidolin, M.; Burovskiy, P.; Kapelan, Z.; Savić, D. CWSNET: An Object-Oriented Toolkit for Water Distribution System Simulations. In Proceedings of the Water Distribution Systems Analysis 2010, Tucson, AZ, USA, 12–15 September 2010; pp. 1–13. [Google Scholar]
- Wu, Z.Y.; Eftekharian, A. Parallel artificial neural network using CUDA-enabled GPU for extracting hydraulic domain knowledge of large water distribution systems. In Proceedings of the 2011 World Environmental and Water Resources Congress, Reston, VA, USA, 22–26 May 2011. [Google Scholar]
- Balla, M.; Lingireddy, S. Distributed genetic algorithm model on network of personal computers. J. Comput. Civ. Eng.
**2000**, 14, 199–205. [Google Scholar] [CrossRef] - Karamouz, M.; Zahmatkesh, Z.; Saad, T. Cloud computing in urban flood disaster management. In Proceedings of the World Environmental and Water Resources Congress 2013, Cincinnati, OH, USA, 19–23 May 2013; pp. 2747–2757. [Google Scholar]
- Kang, D.; Lansey, K. Revisiting optimal water-distribution system design: Issues and a heuristic hierarchical approach. J. Water Resour. Plan. Manag.
**2011**, 138, 208–217. [Google Scholar] [CrossRef] - U.S. EPA. EPANET 2: Users Manual; U.S. Environmental Protection Agency: Cincinnati, OH, USA, 2000.
- Andrade, M.A.; Choi, C.Y.; Lansey, K.; Jung, D. Enhanced artificial neural networks estimating water quality constraints for the optimal water distribution systems design. J. Water Resour. Plan. Manag.
**2016**, 142, 04016024. [Google Scholar] [CrossRef] - Jung, D.; Kang, D.; Kim, J.H.; Lansey, K. Robustness-based design of water distribution systems. J. Water Resour. Plan. Manag.
**2013**, 140, 04014033. [Google Scholar] [CrossRef] - Giustolisi, O.; Laucelli, D.; Colombo, A.F. Deterministic versus stochastic design of water distribution networks. J. Water Resour. Plan. Manag.
**2009**, 135, 117–127. [Google Scholar] [CrossRef] - Lansey, K.E.; Duan, N.; Mays, L.W.; Tung, Y.-K. Water distribution system design under uncertainties. J. Water Resour. Plan. Manag.
**1989**, 115, 630–645. [Google Scholar] [CrossRef] - Kapelan, Z.S.; Savic, D.A.; Walters, G.A. Multiobjective design of water distribution systems under uncertainty. Water Resour. Res.
**2005**, 41. [Google Scholar] [CrossRef]

**Figure 1.**Schematic of the proposed two-phase optimization difficulty level (ODL) examination framework.

**Figure 3.**Two sets of the study network layouts generated in Phase 1 (low and high branched index (BI)): (

**a**) BI = 0.41, meshedness coefficient (MC) = 0.2; (

**b**) BI = 0.4, MC = 0.4; (

**c**) BI = 0.81, MC = 0.29; and (

**d**) BI = 0.81, MC = 0.4.

**Figure 5.**Variance of the pipe diameters in the final solutions: (

**a**) BI = 0.2, MC = 0.4; (

**b**) BI = 0.41, MC = 0.2; and (

**c**) BI = 0.46, MC = 0.1. Thicker pipes have greater pipe diameter variance.

**Figure 8.**ODLs of four selected networks in three nodal elevation cases: plane elevation (Case 1), consistently increasing elevation (Case 2), and random elevation (Case 3). Network 1 with a pipe at all links (full network), Network 2 of BI = 0.2 and MC = 0.3, Network 3 of BI = 0.81 and MC = 0.29, and Network 4 of BI = 1.0 and MC = 0.

**Figure 10.**ODLs of the combined case with two reservoirs and consistently increasing elevation compared to individual cases.

**Figure 11.**ODLs and MCs of the 12 networks grouped by BI values (C network). Note that the dashed circles indicate the case including infeasible solution(s).

Pipe Size (mm) | Unit Cost (USD/m) | Pipe Size (mm) | Unit Cost (USD/m) |
---|---|---|---|

50 | 86.3 | 800 | 873.6 |

100 | 95.2 | 900 | 1230.1 |

200 | 117.1 | 1000 | 1698.8 |

300 | 150.6 | 1200 | 3049 |

400 | 204.6 | 1400 | 5098.9 |

500 | 290.2 | 1600 | 8043.4 |

600 | 420.4 | 1800 | 12,091.5 |

700 | 609.5 | 2000 | 17,469.2 |

P_{BI} | |||||||
---|---|---|---|---|---|---|---|

10^{0} | 10^{1} | 10^{2} | 10^{3} | 10^{4} | 10^{5} | ||

P_{MC} | 10^{0} | 0.119 | 0.106 | 0.102 | 0.102 | 0.103 | 0.105 |

10^{1} | 0.217 | 0.117 | 0.104 | 0.102 | 0.101 | 0.101 | |

10^{2} | 0.265 | 0.213 | 0.114 | 0.099 | 0.103 | 0.104 | |

10^{3} | 0.266 | 0.267 | 0.210 | 0.115 | 0.101 | 0.104 | |

10^{4} | 0.276 | 0.264 | 0.266 | 0.213 | 0.117 | 0.105 | |

10^{5} | 0.271 | 0.261 | 0.268 | 0.264 | 0.213 | 0.117 |

P_{BI} | |||||||
---|---|---|---|---|---|---|---|

10^{0} | 10^{1} | 10^{2} | 10^{3} | 10^{4} | 10^{5} | ||

P_{MC} | 10^{0} | 0.085 | 0.116 | 0.121 | 0.121 | 0.122 | 0.129 |

10^{1} | 0.053 | 0.085 | 0.116 | 0.120 | 0.123 | 0.124 | |

10^{2} | 0.051 | 0.054 | 0.084 | 0.119 | 0.122 | 0.117 | |

10^{3} | 0.052 | 0.051 | 0.052 | 0.085 | 0.114 | 0.119 | |

10^{4} | 0.052 | 0.051 | 0.052 | 0.054 | 0.084 | 0.116 | |

10^{5} | 0.051 | 0.051 | 0.052 | 0.052 | 0.054 | 0.087 |

Reservoir | Indicator | Network 1 (Full Network) | Network 2 (BI = 0.2, MC = 0.3) | Network 3 (BI = 0.81, MC = 0.29 | Network 4 (BI = 1.0, MC = 0) |
---|---|---|---|---|---|

Single reservoir | ODL | 0.0147 | 0.0317 | 0.0200 | 0.0121 |

PPSC (M USD/pipe) | 0.88 | 1.24 | 1.61 | 2.50 | |

Two reservoirs | ODL | 0.0239 | 0.0332 | 0.0577 | 0.0390 |

PPSC (M USD/pipe) | 0.77 | 1.13 | 2.55 | 1.05 |

© 2019 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**

Jung, D.; Lee, S.; Hwang, H. Optimization Difficulty Indicator and Testing Framework for Water Distribution Network Complexity. *Water* **2019**, *11*, 2132.
https://doi.org/10.3390/w11102132

**AMA Style**

Jung D, Lee S, Hwang H. Optimization Difficulty Indicator and Testing Framework for Water Distribution Network Complexity. *Water*. 2019; 11(10):2132.
https://doi.org/10.3390/w11102132

**Chicago/Turabian Style**

Jung, Donghwi, Seungyub Lee, and Hwee Hwang. 2019. "Optimization Difficulty Indicator and Testing Framework for Water Distribution Network Complexity" *Water* 11, no. 10: 2132.
https://doi.org/10.3390/w11102132