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

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## 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

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**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 |

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**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