# An Inverse Transient-Based Optimization Approach to Fault Examination in Water Distribution Networks

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Background and Problem Statement

#### 1.2. Literature Review

#### 1.3. Objective

## 2. Methodology

#### 2.1. Pipe Network Simulation

_{ij}(t)) in the H–W equation for a pipe at used year t is defined as

_{ij}is the length (m) of the pipe, and D

_{ij}is the internal pipe diameter (m). The modified H–W coefficient ${C}_{ij}^{HW}(t)$ (for modeling the effect of pipe aging) is defined as [49]

_{0ij}(t) is the initial roughness (mm) of the pipe, and a

_{ij}(t) is the roughness growth rate (unique per year) in the pipe at year t. The following equations are used in the proposed approach to calculate the values of e

_{0ij}and a

_{ij}[49]:

_{ij}(t) (m

^{3}/s) in each pipe at year t could be expressed as

_{ij}is the frictional head loss in a pipe. The equation of mass conservation at node i could be written as

_{i}(t) is the demand or the source at node i. The flow rate is positive for flow out of node i and negative for flow into node i, while QI

_{i}is positive for inflow and negative for outflow. The objective function used in the PNSOS is defined as

#### 2.2. Hydraulic Transient Model and Faults in the Pipeline

_{O}is the volumetric flow rate through the orifice, C

_{dO}is the discharge coefficient of the orifice, A

_{O}is the orifice area, and ΔH

_{O}is the head loss across the orifice. The leaks represent the flow loss through the offline orifice with no head loss, while the blockages represent the head loss through the inline orifice with no flow loss.

_{L}through leakage is denoted as [53]

^{U}and Q

^{D}are the volumetric flow rates upstream and downstream of the leakage, respectively; C

_{dL}A

_{L}is the discharge coefficient of leakage times the leak area of the orifice; H

_{P}and H

_{out}are respectively the heads at the leak and outside the leak; z is the pipe elevation at the leak; and ${H}_{P}^{U}$ and ${H}_{P}^{D}$ are respectively the heads upstream and downstream of the leak. The outside head is generally considered to be the atmospheric pressure head and is hence set to zero [53]. The initial value of C

_{dL}is set to unity, and the elevation z is assumed to be zero.

_{B}through the blockage is expressed as [53,54]

_{dB}A

_{B}is the discharge coefficient times the orifice area of the blockage. Note that Equation (12) is a simple model to approximate a blockage of any shape and length [53].

_{i}, and A

_{i}are respectively the impedance, wave speed, and pipe cross-sectional area of ith reach. Their values are known in the MOC analysis.

#### 2.3. Ordinal Optimization Approach (OOA)

#### 2.4. Symbiotic Organism Search (SOS)

#### 2.5. Inverse Transient Analysis (ITA)

_{p}, L

_{L}, C

_{dL}A

_{L}, B

_{p}, B

_{L}, C

_{dB}A

_{B}, D

_{p}, D

_{L}, L

_{D}, a

_{D}, and A

_{D}(listed and defined in Table 1).

#### 2.6. Development of PEOS

- Import the network configurations;
- Randomly generate candidate solutions (CASes) with different fault information consisting of the unknown variables listed in Table 1;
- Rearrange the network configurations, since the new fault points (leaks and blockages) and/or new fault pipe reaches (deterioration parts) are added;
- Use PNSOS to calculate the optimal steady-state nodal heads and piping flow rates within a given WDN for each CAS;
- Generate hydraulic transient events and apply the MOC to obtain the transient head distribution of each CAS;
- Utilize Equation (14) to calculate the CASes’ objective function values (OFVs) and rank them. The top 5% of CASes with smaller OFVs are selected for the next step;
- Consider the selected CASes to be initial organisms for the ecosystem of the SOS used in the pipe examination;
- Execute the fault detection procedure, in which the organisms containing fault information continually move forward to the current best solution (X
_{best}), with optimal fault information due to the three states of the SOS; - Check whether the optimization process satisfies the stopping criterion. If so, the fault detection procedure is then terminated and moves to the next step. Otherwise, the searching process goes on.

^{−4}within four iterations. The second criterion for fault detection is the iteration reaching the specified maximum limit.

#### 2.7. Benchmark Evolutionary Algorithms

## 3. Laboratory Experiments and PEOS Simulations

#### 3.1. Experiment Configurations

^{−5}m

^{2}and 1.50 × 10

^{−5}m

^{2}occurred at the locations of 65.95 m and 146.32 m, respectively: This was measured from upstream. These two leak orifices were very small, and the discharge coefficient was considered to be one. Thus, the C

_{dL}A

_{L}s for the two leaks was respectively 1.21 × 10

^{−5}m

^{2}and 1.50 × 10

^{−5}m

^{2}. The initial flow rate downstream was 1 L/s.

_{dB}A

_{B}was 1.18 × 10

^{−3}m

^{2}. The initial flow rate downstream was 2.57 L/s.

#### 3.2. PEOS Simulation

_{d}As = 1.23 × 10

^{−5}m

^{2}, and at 146 m in segment 3, with C

_{d}As = 1.52 × 10

^{−5}m

^{2}. Blockage B1 in the WEL pipeline system was identified at 88 m in segment 2, with C

_{dB}A

_{B}= 1.20 × 10

^{−3}m

^{2}. The leak and blockage locations in both systems were accurately determined by the proposed approach. The largest relative difference (E) between the actual C

_{dL}A

_{L}s/C

_{dB}A

_{B}and the predicted one was 1.69% for detecting blockage B1 in the WEL pipeline system. The relative differences were insignificant in both systems. The success of PEOS in fault detection indicated that PEOS performs excellently in a pipeline system.

## 4. Fault Detection in a Synthetic Pipe Network

#### 4.1. Simulation Setup and Pipe Network Configuration

^{HW}(0) for each pipe was 130. The H–W coefficient considering the effect of pipe aging (C

^{HW}(t)) for each pipe was calculated through Equations (2)–(4) and is given in the last column of Table 4. The initial wave speed a

_{0}of all pipes was postulated as 1000 m/s [25], except for the faulty parts. The impedance of each pipe was calculated by Equation (13) and is given in Table 4. Node N1 was the water supply node, with a constant inflow rate of 400 L/s and a constant head of 120 m. In addition, continuous discharges at N2, N3, N4, N5, N6, N8, and N9 had rates of 80, 40, 35, 35, 40, 80, and 80 L/s, respectively. The leak L1 was located at P11, 300 m away from N3, with C

_{dL}A

_{L}= 2.50 × 10

^{−4}m

^{2}and Q

_{L}= 3.0 L/s. A partial blockage B1 was placed at P10, 200 m away from N9. It blocked about 20% of the cross-sectional area of P10, and thus the C

_{dB}A

_{B}was 5.6 × 10

^{−2}m

^{2}. In addition, a distributed deterioration reach, D1, occurred at a segment of P1 and was 200 m away from N2. The length and cross-sectional area of D1 were respectively designed to be 80 m and 0.071 m. Its wave speed was assumed to be 800 m/s, and thus the impedance was calculated as 1148.98 s/m

^{2}from Equation (13). In the simulation, N8 was treated as the transient generation and data measurement point for the simulation of a sudden closure of the valve. The total transient simulation time was considered to be 30 s, with a simulation time interval (Δt) selected as 0.01 s. Thus, the initial Δx was 10 m for the nondeterioration reach and further changed with the wave speed of the deterioration reach. The transient operation was fixed to 5 s for a simulation of the complete closure of the valve.

_{iter}) was 10,000. Notice that all of the results presented in the following sections were performed on a personal computer with an Intel 2.8 G i5-8400 CPU and 32 GB of RAM.

#### 4.2. Validation and Application of PEOS

_{P}displayed in Figure 4a–d, and the predicted results are given in Table 5. The figures show that the transient perturbations fluctuated between 20 and 140 m with similar oscillatory patterns over 30 s. Figure 4a,b shows that PEGA and PEPSO overestimated the transient perturbations for the case N

_{P}= 10 due to an overestimation of the leakage area size by both algorithms. Such results reflect that the WDN contained a larger total flow rate at the beginning of transient perturbations. Moreover, the blockage at P10 was not detected by either PEGA or PEPSO. Thus, the transmission of water and pressure may not have been affected by the blockage, resulting in the accumulated volumes of water at N8 being higher than other cases when the transient operation point was closed. The predicted head was also overestimated in the case of PEGA for N

_{P}= 20. The calculations in both PEGA and PEPSO were forced to stop because they reached the maximum iteration, M

_{iter}= 10,000, in the cases of N

_{P}= 10 and 20. In contrast, the temporal transient perturbations displayed in Figure 4c,d were precisely reconstructed by two SOS-based approaches for all cases of ecosystem size. Deterioration, a blockage, and a leak were detected at P1, P10, and P11, respectively. Table 5 shows that the deterioration, blockage, and leak information was also accurately predicted by two SOS-based approaches. The results prove that those two SOS-based approaches are capable of obtaining optimal fault information even after using fewer initial organisms, reflecting that PESOS and PEOS had great abilities in obtaining the best solution even when using less input data and guessing values. This may have greatly reduced the searching process and computation times. Moreover, Figure 5a,b displays the predicted results of PEOS for impedance and wave speed along P1 and P10, respectively. Both a partial blockage and a deteriorated section can also be identified from the plots of the predicted distributions of the impedance and wave speed in Figure 5. The successful numerical simulation validated the proposed approach to detecting various faults in WDNs.

_{P}was fixed at 50 for all algorithms, and thus all approaches were ensured to deliver accurate predictions, as the results demonstrate above. Table 6 delineates the performance of PEOS and other approaches (five times) in obtaining the optimal fault information of pipe network A. PEGA, PEPSO, and PESOS took about 331.2, 302.2, and 105.4 min and 8072, 7604, and 3882 iterations, respectively, to obtain optimal results over a five-time average. In contrast, PEOS took about 50.6 min and 1382 iterations to complete the searching process and obtain the optimal result. Apparently, PEOS outperformed PEGA and PEPSO, not only in computation time but also in convergence speed. The computational efficiency of PEOS was approximately 84.7% and 83.2% better than PEGA and PEPSO. The computational efficiency of PEOS in fault detection in the WDN significantly improved as a result of using the OOA and SOS. In addition, PEOS saved about 52.8% in computing time and 64% in iterations compared to PESOS, indicating that the OOA could significantly speed up optimization computation by reasonably avoiding blind searches and unnecessary objective function evaluations in the optimization process. PEOS had superiority over the other methods in its fast convergence and effective computation. It also gave more accurate results than the other evolutionary-based algorithms.

## 5. Faults Detection in Large-scale WDN

#### 5.1. Simulation Setup and Large-Scale WDN

^{HW}(0) and wave speed a

_{0}for all pipes in pipe network B were 130 and 1000 m/s, respectively. The C

^{HW}(t) for various pipes was also calculated by Equations (2)–(4) and is listed in the last column of Table 7.

_{dL}A

_{L}values for L1, L2, and L3 were respectively 2.00 × 10

^{−4}, 1.00 × 10

^{−4}, and 1.20 × 10

^{−4}m

^{2}. In addition, Q

_{L}s was 2.0, 1.0, and 1.5 L/s for L1, L2, and L3, respectively. Two partial blockages, B1 and B2, were respectively situated at P23 and P39. B1 was 200 m away from N33 and blocked 20% of the cross-sectional area of P23, while B2 was 600 m away from N10 and blocked 15% of the cross-sectional area of P39. Hence, the C

_{dB}A

_{B}values of B1 and B2 were 4.0 × 10

^{−1}m

^{2}and 6.0 × 10

^{−1}m

^{2}, respectively. Moreover, two distributed deterioration reaches, D1 and D2, occurred at P62 and P67, respectively. D1 was located at P62, 400 m away from N22, while D2 was located at P67, 600 m away from N19. The length, wave speed, impedance, and cross-sectional area of D1 were respectively 40 m, 800 m/s, 163.2 s/m

^{2}, and 0.50 m

^{2}, while those of D2 were 30 m, 600 m/s, 122.4 s/m

^{2}, and 0.50 m

^{2}. The properties of the two deterioration reaches are shown in Figure 6 as well. The outflow node N17 was considered to be the transient operation and data collection point for pipe network B. The Δt was also selected to be 0.01 s. Thus, the initial Δx was also considered to be 10 m for the intact pipe reach and was further altered with different wave speeds in the deterioration reach. Because the WDN scale was large and complicated, the transient wave may have taken more time to arrive at the fault points/parts. The total simulation time increased to 60 s. A total of 6001 data points should be collected and used in a complete simulation. Two different cases with different data collection issues were considered to test the reliability of the proposed approach for fault detection in a large-scale WDN. N

_{P}was chosen to be 50, and M

_{iter}was updated to 20,000 for possible enormous iterations. The transient excitation period was chosen as 5 or 10 s for the simulation of the complete closure of the valve.

#### 5.2. Case Description and Error Criteria

#### 5.3. Results and Error Analysis

_{dL}A

_{L}s/C

_{dB}A

_{B}values and the predicted ones was insignificant. Table 9 shows the values of the ME and SEE, which for case 1 were 3.41 × 10

^{−6}m and 1.27 × 10

^{−4}m, respectively. The results denote that the predicted heads were not affected by the use of limited observations. The results for case 1 and the small ME and SEE values indicate that PEOS had the potential to deliver moderately good results in a field survey even when only a few observations were available. The success of using fewer measurements indicates that PEOS may not be restricted by instrument limitations. In addition, the data measurement period can therefore be reduced, and the system impact due to a transient event may be slight while using PEOS.

^{2}, with corresponding wave speeds of 794.3 and 595.8 m/s. For leak and blockage detection in case 2, the predicted locations of three leaks and two blockages were close to the real locations, implying that the measurement errors may not have affected location detection. There were errors in the predictions of C

_{dL}A

_{L}and C

_{dB}A

_{B}in case 2. The relative differences between the predicted C

_{dL}A

_{L}values and the actual ones were about 6%, 2%, and 5.83% for L1, L2, and L3, respectively. The relative differences between the determined C

_{dB}A

_{B}values and the real ones were about 5.25% for B1 and 4.17% for B2. The results showed that the predicted C

_{dL}A

_{L}values and C

_{dB}A

_{B}may have been more sensitive than location to measurement errors. This was due to the fact that the OFVs used in PEOS for fault detection were directly related to the head difference (i.e., Equation (14)), which may have been directly influenced by the change in leak area and blockage area. The MEs and SEEs for case 2 are listed in Table 9 and were respectively 1.73 × 10

^{−4}m and 6.35 × 10

^{−2}m, which were both two orders larger than those of case 1. Such a result indicates that measurement errors may have affected accuracy in determining the leak area and blockage area. Thus, data uncertainty should be of concern as an important issue in fault detection in a large-scale pipe network or in future field applications.

^{−6}m and 1.12 × 10

^{−4}m, as shown in Table 9. The results indicate that the predicted heads were not affected, while the transient operation was inadequate. Note that the concept of ITA is to minimize errors between the measured and calculated system state variables. Measurements with an unsuitable transient operation still work well based on the objective function of ITA. The results of case 3 reveal that PEOS can provide good predictions when using different transient operation durations. However, a rapid transient operation is recommended, because it produces large system response data, thus improving the performance of the ITA [31].

## 6. Conclusions

_{dL}A

_{L}and C

_{dB}A

_{B}had slight deviations compared to the actual ones, indicating that PEOS could achieve good results if the measurements were well collected. Moreover, the results revealed that inappropriate transient operation may not have affected the performance of PEOS in predicting head distribution and fault information.

## Supplementary Materials

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**The simulated head distributions at the valve for (

**a**) the Imperial College (IC) pipeline and (

**b**) the Water Engineering Laboratory (WEL) pipeline.

**Figure 4.**Temporal transient perturbations at N8 of pipe network A predicted by (

**a**) PEGA, (

**b**) PEPSO, (

**c**) PESOS, and (

**d**) PEOS with various N

_{P}.

Variable | Description |
---|---|

Leak | |

L_{p} | Leak pipe number |

L_{L} | Leak location |

C_{dL}A_{L} | Discharge coefficient times the leak area of the orifice |

Blockage | |

B_{p} | Blockage pipe number |

B_{L} | Blockage location |

C_{dB}A_{B} | Discharge coefficient times the open orifice area of the blockage |

Deterioration | |

D_{p} | Deterioration pipe number |

D_{L} | Deterioration location |

L_{Di} | Length of ith distributed deterioration reach |

a_{Di} | Wave speed of ith distributed deterioration reach |

A_{Di} | Pipe cross-sectional area of ith distributed deterioration reach |

**Table 2.**Specific parameters for each algorithm, with N

_{P}= 10, 20, or 50, and M

_{iter}= 10,000 or 20,000.

PEGA | PEPSO | PESOS and PEOS |
---|---|---|

m = 0.01 | w = 0.9~0.7 | No specific parameters required |

c = 0.8 | v = X_{min}/10~X_{max}/10 | |

g = 0.9 | - |

_{P}= population size/ecosystem size; M

_{iter}= maximum iteration; m = mutation rate; c = crossover rate; g = generation gap; w = inertia weight; v = limit of velocity.

IC pipeline | L1 | L2 | |||||||||

L_{p} | L_{L} (m) | C_{dL}A_{L} (m^{2}) | E (%) | L_{p} | L_{L} (m) | C_{dL}A_{L} (m^{2}) | E (%) | ||||

Actual | 2 | 15.95 | 1.21 × 10^{−5} | - | 3 | 46.32 | 1.50 × 10^{−5} | - | |||

PEOS | 2 | 16 | 1.23 × 10^{−5} | 1.65 | 3 | 46 | 1.52 × 10^{−5} | 1.33 | |||

WEL pipeline | B1 | ||||||||||

B_{p} | B_{L} (m) | C_{dB}A_{B} (m^{2}) | E (%) | ||||||||

Actual | 2 | 38.96 | 1.18 × 10^{−3} | - | |||||||

PEOS | 2 | 38 | 1.20 × 10^{−3} | 1.69 | |||||||

_{dL}A

_{L}/C

_{dB}A

_{B}and the actual one.

Pipe | Node | Diameter (mm) | Length (m) | Impedance (s/m^{2}) | Year Used (year) | C^{HW}(t) | |
---|---|---|---|---|---|---|---|

From | To | ||||||

P1 | N1 | N2 | 300.0 | 1000.0 | 1442.60 | 10 | 108.2 |

P2 | N2 | N3 | 300.0 | 1000.0 | 1442.60 | 15 | 90.2 |

P3 | N3 | N4 | 250.0 | 1100.0 | 2077.35 | 10 | 105.7 |

P4 | N1 | N4 | 400.0 | 1250.0 | 811.47 | 15 | 92.2 |

P5 | N4 | N5 | 200.0 | 500.0 | 3245.86 | 5 | 112.1 |

P6 | N5 | N6 | 400.0 | 400.0 | 811.47 | 5 | 114.2 |

P7 | N7 | N6 | 200.0 | 500.0 | 3245.86 | 5 | 112.1 |

P8 | N4 | N7 | 350.0 | 400.0 | 1059.87 | 5 | 113.6 |

P9 | N7 | N8 | 350.0 | 600.0 | 1059.87 | 5 | 113.6 |

P10 | N8 | N9 | 300.0 | 1100.0 | 1442.60 | 10 | 108.2 |

P11 | N3 | N9 | 300.0 | 1250.0 | 1442.60 | 15 | 90.2 |

N_{P} | Method | L1 | B1 | D1 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

L_{p} | L_{L} (m) | C_{dL}A_{L} (m^{2}) | B_{P} | B_{L} (m) | C_{dB}A_{B} (m^{2}) | D_{P} | D_{L} (m) | LD (m) | a_{D} (m/s) | ${\mathit{B}}_{\mathit{D}\mathbf{1}}^{\mathit{im}}$ (s/m^{2}) | ||

Actual | 11 | 300 | 2.50 × 10^{−4} | 10 | 200 | 5.60 × 10^{−2} | 1 | 200 | 80 | 800 | 1148.98 | |

10 | PEGA | 2 | 650 | 3.27 × 10^{−4} | Not detected | Not detected | ||||||

PEPSO | 11 | 830 | 3.19 × 10^{−4} | Not detected | 3 | 510 | 100 | 805 | 1156.16 | |||

PESOS | 11 | 300 | 2.49 × 10^{−4} | 10 | 200 | 5.58 × 10^{−2} | 1 | 200 | 80 | 800 | 1148.98 | |

PEOS | 11 | 300 | 2.51 × 10^{−4} | 10 | 200 | 5.61 × 10^{−2} | 1 | 200 | 80 | 800 | 1148.98 | |

20 | PEGA | 11 | 510 | 3.34 × 10^{−4} | Not detected | 3 | 490 | 70 | 805 | 1156.16 | ||

PEPSO | 11 | 300 | 2.49 × 10^{−4} | 10 | 200 | 5.61 × 10^{−2} | 3 | 700 | 70 | 805 | 1156.16 | |

PESOS | 11 | 300 | 2.50 × 10^{−4} | 10 | 200 | 5.59 × 10^{−2} | 1 | 200 | 80 | 800 | 1148.98 | |

PEOS | 11 | 300 | 2.50 × 10^{−4} | 10 | 200 | 5.60 × 10^{−2} | 1 | 200 | 80 | 800 | 1148.98 | |

50 | PEGA | 11 | 300 | 2.49 × 10^{−4} | 10 | 200 | 5.60 × 10^{−2} | 1 | 200 | 80 | 800 | 1148.98 |

PEPSO | 11 | 300 | 2.49 × 10^{−4} | 10 | 200 | 5.60 × 10^{−2} | 1 | 200 | 80 | 800 | 1148.98 | |

PESOS | 11 | 300 | 2.50 × 10^{−4} | 10 | 200 | 5.59 × 10^{−2} | 1 | 200 | 80 | 800 | 1148.98 | |

PEOS | 11 | 300 | 2.50 × 10^{−4} | 10 | 200 | 5.60 × 10^{−2} | 1 | 200 | 80 | 800 | 1148.98 |

Method | Round | CPU Time (min) | Average Time (min) | Iterations | Average Iterations |
---|---|---|---|---|---|

PEGA | 1 | 325 | 331.2 | 8021 | 8072 |

2 | 346 | 8216 | |||

3 | 322 | 8124 | |||

4 | 324 | 7983 | |||

5 | 339 | 8016 | |||

PEPSO | 1 | 310 | 302.2 | 7502 | 7604 |

2 | 308 | 7551 | |||

3 | 312 | 7669 | |||

4 | 294 | 7606 | |||

5 | 287 | 7710 | |||

PESOS | 1 | 101 | 105.4 | 3789 | 3882 |

2 | 107 | 4012 | |||

3 | 108 | 3883 | |||

4 | 110 | 3810 | |||

5 | 101 | 3915 | |||

PEOS | 1 | 56 | 50.6 | 1415 | 1382 |

2 | 49 | 1371 | |||

3 | 46 | 1337 | |||

4 | 52 | 1396 | |||

5 | 50 | 1391 |

Pipe | Node | Diameter (mm) | Length (m) | Impedance (s/m^{2}) | Year Used (year) | C^{HW}(t) | |
---|---|---|---|---|---|---|---|

From | To | ||||||

P1 | N48 | N1 | 950.0 | 240.0 | 143.86 | 5 | 120.5 |

P2 | N34 | N33 | 900.0 | 60.0 | 160.29 | 10 | 113.5 |

P3 | N2 | N46 | 1450.0 | 1830.0 | 61.75 | 0 | 130.0 |

P4 | N43 | N2 | 1150.0 | 3550.0 | 98.17 | 0 | 130.0 |

P5 | N41 | N45 | 1450.0 | 1220.0 | 61.75 | 0 | 130.0 |

P6 | N45 | N46 | 1450.0 | 640.0 | 61.75 | 0 | 130.0 |

P7 | N42 | N43 | 900.0 | 60.0 | 160.29 | 10 | 113.5 |

P8 | N41 | N43 | 900.0 | 60.0 | 160.29 | 10 | 113.5 |

P9 | N44 | N43 | 1000.0 | 50.0 | 129.83 | 10 | 114.6 |

P10 | N42 | N2 | 900.0 | 3660.0 | 160.29 | 10 | 113.5 |

P11 | N41 | N42 | 900.0 | 60.0 | 160.29 | 10 | 113.5 |

P12 | N42 | N44 | 1000.0 | 60.0 | 129.83 | 10 | 114.6 |

P13 | N40 | N42 | 900.0 | 800.0 | 160.29 | 10 | 113.5 |

P14 | N37 | N41 | 1450.0 | 3140.0 | 61.75 | 0 | 130.0 |

P15 | N38 | N43 | 1150.0 | 3140.0 | 98.17 | 0 | 130.0 |

P16 | N39 | N44 | 1650.0 | 3140.0 | 47.69 | 0 | 130.0 |

P17 | N38 | N36 | 900.0 | 60.0 | 160.29 | 10 | 113.5 |

P18 | N38 | N39 | 1000.0 | 60.0 | 129.83 | 10 | 114.6 |

P19 | N36 | N40 | 800.0 | 2300.0 | 202.87 | 10 | 112.8 |

P20 | N38 | N37 | 900.0 | 60.0 | 160.29 | 10 | 113.5 |

P21 | N35 | N38 | 1150.0 | 4050.0 | 98.17 | 0 | 130.0 |

P22 | N36 | N37 | 900.0 | 60.0 | 160.29 | 10 | 113.5 |

P23 | N33 | N36 | 800.0 | 4050.0 | 202.87 | 10 | 112.8 |

P24 | N34 | N37 | 1150.0 | 4050.0 | 98.17 | 0 | 130.0 |

P25 | N33 | N35 | 900.0 | 60.0 | 160.29 | 10 | 113.5 |

P26 | N34 | N35 | 900.0 | 60.0 | 160.29 | 10 | 113.5 |

P27 | N25 | N32 | 800.0 | 2150.0 | 202.87 | 10 | 112.8 |

P28 | N32 | N33 | 800.0 | 180.0 | 202.87 | 10 | 112.8 |

P29 | N23 | N34 | 1450.0 | 2980.0 | 61.75 | 0 | 130.0 |

P30 | N25 | N35 | 1450.0 | 2980.0 | 61.75 | 0 | 130.0 |

P31 | N31 | N30 | 1650.0 | 12,000.0 | 47.69 | 0 | 130.0 |

P32 | N22 | N24 | 950.0 | 670.0 | 143.86 | 10 | 114.0 |

P33 | N29 | N28 | 1000.0 | 60.0 | 129.83 | 10 | 114.6 |

P34 | N30 | N29 | 1650.0 | 13400.0 | 47.69 | 0 | 130.0 |

P35 | N13 | N11 | 900.0 | 80.0 | 160.29 | 10 | 113.5 |

P36 | N11 | N15 | 950.0 | 4290.0 | 143.86 | 5 | 120.5 |

P37 | N12 | N14 | 900.0 | 4290.0 | 160.29 | 5 | 115.7 |

P38 | N13 | N12 | 50.0 | 60.0 | 51,933.76 | 10 | 102.6 |

P39 | N10 | N11 | 970.0 | 2590.0 | 137.99 | 5 | 120.5 |

P40 | N11 | N12 | 50.0 | 60.0 | 51,933.76 | 10 | 102.6 |

P41 | N6 | N12 | 900.0 | 2960.0 | 160.29 | 5 | 115.7 |

P42 | N7 | N13 | 1150.0 | 2960.0 | 98.17 | 0 | 130.0 |

P43 | N9 | N8 | 1150.0 | 2280.0 | 98.17 | 0 | 130.0 |

P44 | N8 | N10 | 950.0 | 370.0 | 143.86 | 5 | 120.5 |

P45 | N8 | N7 | 1000.0 | 90.0 | 129.83 | 0 | 130.0 |

P46 | N6 | N7 | 50.0 | 60.0 | 51,933.76 | 10 | 102.6 |

P47 | N5 | N6 | 900.0 | 1610.0 | 160.29 | 5 | 115.7 |

P48 | N6 | N8 | 50.0 | 60.0 | 51,933.76 | 10 | 102.6 |

P49 | N3 | N5 | 950.0 | 1350.0 | 143.86 | 5 | 120.5 |

P50 | N4 | N8 | 50.0 | 2960.0 | 51,933.76 | 10 | 102.6 |

P51 | N47 | N3 | 950.0 | 6530.0 | 143.86 | 5 | 120.5 |

P52 | N3 | N4 | 900.0 | 60.0 | 160.29 | 10 | 113.5 |

P53 | N48 | N47 | 950.0 | 230.0 | 143.86 | 5 | 120.5 |

P54 | N48 | N4 | 950.0 | 7200.0 | 143.86 | 5 | 120.5 |

P55 | N27 | N26 | 1000.0 | 60.0 | 129.83 | 10 | 114.6 |

P56 | N29 | N27 | 1150.0 | 3200.0 | 98.17 | 0 | 130.0 |

P57 | N26 | N25 | 1450.0 | 4300.0 | 61.75 | 0 | 130.0 |

P58 | N28 | N26 | 1150.0 | 3200.0 | 98.17 | 0 | 130.0 |

P59 | N22 | N23 | 800.0 | 80.0 | 202.87 | 10 | 112.8 |

P60 | N23 | N25 | 750.0 | 90.0 | 230.82 | 0 | 130.0 |

P61 | N18 | N23 | 950.0 | 2050.0 | 143.86 | 5 | 120.5 |

P62 | N21 | N22 | 800.0 | 2380.0 | 202.87 | 10 | 112.8 |

P63 | N20 | N23 | 1150.0 | 3050.0 | 98.17 | 0 | 130.0 |

P64 | N19 | N21 | 50.0 | 670.0 | 51,933.76 | 5 | 105.8 |

P65 | N18 | N19 | 50.0 | 60.0 | 51,933.76 | 10 | 102.6 |

P66 | N19 | N20 | 50.0 | 60.0 | 51,933.76 | 10 | 102.6 |

P67 | N17 | N19 | 800.0 | 1830.0 | 202.87 | 10 | 112.8 |

P68 | N18 | N20 | 900.0 | 60.0 | 160.29 | 10 | 113.5 |

P69 | N14 | N17 | 800.0 | 1950.0 | 202.87 | 10 | 112.8 |

P70 | N15 | N18 | 950.0 | 3780.0 | 143.86 | 5 | 120.5 |

P71 | N16 | N14 | 50.0 | 60.0 | 51,933.76 | 5 | 105.8 |

P72 | N16 | N15 | 900.0 | 60.0 | 160.29 | 10 | 113.5 |

P73 | N13 | N16 | 1150.0 | 4290.0 | 98.17 | 0 | 130.0 |

P74 | N14 | N15 | 50.0 | 60.0 | 51,933.76 | 5 | 105.8 |

Case | Leak | Blockage | Deterioration | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

No. | L_{p} | L_{L} (m) | C_{dL}A_{L} (m^{2}) | E (%) | No. | B_{P} | B_{L} (m) | C_{dB}A_{B} (m^{2}) | E (%) | No. | D_{P} | D_{L} (m) | L_{D} (m) | a_{D} (m/s) | ${\mathit{B}}_{\mathit{D}}^{\mathit{im}}$ (s/m^{2}) | |||

Actual | L1 | 19 | 1150 | 2.00 × 10^{−4} | - | B1 | 23 | 200 | 4.00 × 10^{−1} | - | D1 | 62 | 400 | 40 | 800 | 163.2 | ||

L2 | 32 | 0 | 1.00 × 10^{−4} | - | B2 | 39 | 600 | 6.00 × 10^{−1} | - | D2 | 67 | 600 | 30 | 600 | 122.4 | |||

L3 | 41 | 960 | 1.20 × 10^{−4} | - | - | - | - | |||||||||||

Case 1 | L1 | 19 | 1150 | 1.98 × 10^{−4} | 1.00 | B1 | 23 | 190 | 3.98 × 10^{−1} | 0.50 | D1 | 62 | 400 | 40 | 799.2 | 163.0 | ||

L2 | 32 | 0 | 1.01 × 10^{−4} | 1.00 | B2 | 39 | 600 | 6.04 × 10^{−1} | 0.67 | D2 | 67 | 600 | 30 | 603.1 | 123.0 | |||

L3 | 41 | 950 | 1.18 × 10^{−4} | 1.67 | - | - | - | |||||||||||

Case 2 | L1 | 19 | 1160 | 1.88 × 10^{−4} | 6.00 | B1 | 23 | 200 | 3.79 × 10^{−1} | 5.25 | D1 | 62 | 390 | 40 | 794.3 | 162.0 | ||

L2 | 32 | 0 | 0.98 × 10^{−4} | 2.00 | B2 | 39 | 610 | 5.75 × 10^{−1} | 4.17 | D2 | 67 | 610 | 30 | 595.8 | 121.5 | |||

L3 | 41 | 950 | 1.11 × 10^{−4} | 5.83 | - | - | ||||||||||||

Case 3 | L1 | 19 | 1150 | 1.96 × 10^{−4} | 2.00 | B1 | 23 | 190 | 3.94 × 10^{−4} | 1.50 | D1 | 62 | 400 | 40 | 798.5 | 162.9 | ||

L2 | 32 | 0 | 0.99 × 10^{−4} | 1.00 | B2 | 39 | 600 | 6.07 × 10^{−4} | 1.16 | D2 | 67 | 600 | 30 | 598.2 | 122.1 | |||

L3 | 41 | 950 | 1.17 × 10^{−4} | 2.50 | - | - | - |

_{dL}A

_{L}/C

_{dB}A

_{B}and the actual one.

**Table 9.**The prediction errors for three cases. ME: mean error; SEE: standard error of the estimate.

Case | Prediction Errors | |
---|---|---|

ME (m) | SEE (m) | |

1 | 3.41 × 10^{−6} | 1.27 × 10^{−4} |

2 | 1.73 × 10^{−4} | 6.35 × 10^{−2} |

3 | 3.29 × 10^{−6} | 1.12 × 10^{−4} |

© 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**

Lin, C.-C.; Yeh, H.-D. An Inverse Transient-Based Optimization Approach to Fault Examination in Water Distribution Networks. *Water* **2019**, *11*, 1154.
https://doi.org/10.3390/w11061154

**AMA Style**

Lin C-C, Yeh H-D. An Inverse Transient-Based Optimization Approach to Fault Examination in Water Distribution Networks. *Water*. 2019; 11(6):1154.
https://doi.org/10.3390/w11061154

**Chicago/Turabian Style**

Lin, Chao-Chih, and Hund-Der Yeh. 2019. "An Inverse Transient-Based Optimization Approach to Fault Examination in Water Distribution Networks" *Water* 11, no. 6: 1154.
https://doi.org/10.3390/w11061154