# Locating Leaks with TrustRank Algorithm Support

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## Abstract

**:**

## 1. Introduction

#### State of the Art

## 2. Methodology

#### 2.1. Methodology for Identification of Probable Leaky Pipes and Estimation of Their Leak Flow

#### 2.1.1. Link between Hydraulic Simulator and Optimization Model

#### 2.1.2. Optimization Model

#### 2.1.3. Hydraulic Simulation Model

^{3}/s); ${D}_{ij}$ = diameter of the pipe between nodes i and j (m);$CH{W}_{ij}$ = Hazen-Williams coefficient of the pipe between nodes i and j.

#### 2.1.4. Simulated Annealing Algorithm

Part of the algorithm | Implementation | |
---|---|---|

Initial solution (x_{0}) | The total leakage flow is split and each part is assigned to a selected pipe | |

Initial temperature (Tqt_initial) | Tqt_initial = −0.1 × F(x_{0})/Log(0.5) | |

Cooling schedule | If Pa > 80% Tqt_{i}_{+1} = 0.60 × Tqt_{i} | IA_{i+1} = 40 |

If Pa > 50% Tqt_{i}_{+1} = 0.75 × Tqt_{i} | IA_{i+1} =60 | |

If Pa > 20% Tqt_{i}_{+1} = 0.90 × Tqt_{i} | IA_{i+1} =80 | |

If Pa ≤ 20% Tqt_{i}_{+1} = 0.95 × Tqt_{i} | IA_{i+1} =100 | |

Number of evaluations at each temperature Tqt_{i} | IA_{i} × Number of pipes in the network | |

Stopping criteria | Pa < 5% and 2 temperatures without solution improvement |

_{0}), the initial temperature (Tqt_initial) and fix the number of solutions to be evaluated at each temperature (IA multiplied by the number of pipes in the WDN). The initial solution assumes the roles of current solution (x

_{current}) and best solution found so far (x

_{best});

_{candidate}) by applying one of the next two processes: (I) Transfer an elementary unit of the leakage flow from one pipe to one of its adjacent pipes (diversification mechanism); (II) Randomly select a leaky pipe and concentrate in it all the leaks from its adjacent pipes (concentration mechanism);

_{i}

_{+1}) and the number of solutions to be evaluated at the new temperature as a function of the percentage of accepted solutions (Pa) at the previous temperature (Tqt

_{i});

_{best}.

#### 2.2. Methodology for Pressure Transducers Location

#### Adaptation of the TrustRank Algorithm to Water Distribution Networks (WDN)

#### 2.3. Computer Application

_{0}) starts by assigning the water consumption to the junction nodes. Then, sequentially, each Δql is assigned to a pipe using the following procedure: assign one Δql to one of the pipes in the network (a half of each elementary unit of leakage flow is assigned to its end nodes), use the hydraulic simulation model to simulate the hydraulic behaviour of the WDN (considering both water consumption and leakage—Δql already assigned), assess the objective function (1), repeat the simulation for each of the pipes in the network and finally assign this Δql to the pipe that obtained the lowest value of the objective function (1); Repeat the process until all Δql are assigned. At this stage the total leakage flow is completely assigned (note that some pipes may have more than one Δql) and this is the initial solution to start solving the optimization model. The objective function value of the initial solution (F(x

_{0})) is then used to calculate the initial temperature (Tqt_initial) to start the simulated annealing algorithm.

## 3. Results

#### 3.1. Case Study

Situations—Pipe/Leakage Flow (L/s) | |||||||||
---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

5/0.5 | 54/0.1 | 39/0.1 | 10/0.1 | 28/0.1 | 3/0.5 | 4/0.1 | 22/0.1 | 32/0.1 | 6/0.5 |

25/0.3 | 55/0.2 | 41/0.2 | 30/0.2 | 44/0.2 | 66/0.3 | 17/0.2 | 62/0.2 | 34/0.2 | 10/0.2 |

35/0.4 | 57/0.5 | 44/0.4 | 55/0.3 | 92/0.4 | 71/0.4 | 43/0.5 | 72/0.4 | 61/0.3 | 63/0.1 |

37/0.2 | 74/0.3 | 69/0.5 | 75/0.4 | 96/0.5 | 85/0.2 | 87/0.3 | 107/0.5 | 64/0.5 | 87/0.4 |

82/0.1 | 86/0.4 | 98/0.3 | 95/0.5 | 99/0.3 | 91/0.1 | 111/0.4 | 108/0.3 | 68/0.4 | 98/0.3 |

#### 3.2. TrustRank Algorithm Application

Transducers | Placement (Nodes) |
---|---|

First 10 | 13, 17, 37, 49, 50, 63, 67, 85, 86, 99 |

11th and 12th | 12, 81 |

13th and 14th | 31, 82 |

15th and 16th | 1, 3 |

17th and 18th | 23, 40 |

19th and 20th | 44, 53 |

#### 3.3. Sensitivity Analysis of the Number of Pressure Transducers

#### 3.3.1. First WDN

Situation | Number of Pressure Transducers | |||||
---|---|---|---|---|---|---|

10 | 12 | 14 | 16 | 18 | 20 | |

1 | 7/5 | 7/5 | 7/5 | 5/5 | 5/5 | 5/5 |

2 | 14/5 | 14/5 | 6/5 | 6/5 | 7/5 | 10/5 |

3 | 18/6 | 10/6 | 12/5 | 12/5 | 8/5 | 8/5 |

4 | 16/14 | 13/16 | 11/9 | 12/9 | 12/9 | 17/9 |

5 | 11/5 | 9/5 | 8/5 | 7/5 | 7/5 | 7/5 |

6 | 19/7 | 14/6 | 15/12 | 12/5 | 5/5 | 5/5 |

7 | 17/5 | 9/5 | 8/5 | 7/5 | 8/5 | 8/5 |

8 | 16/5 | 14/5 | 13/5 | 9/5 | 6/5 | 6/5 |

9 | 13/12 | 17/11 | 16/7 | 15/7 | 16/7 | 10/7 |

10 | 19/11 | 12/5 | 11/5 | 9/5 | 12/5 | 10/5 |

Situation | Number of Pressure Transducers | |||||
---|---|---|---|---|---|---|

10 | 12 | 14 | 16 | 18 | 20 | |

1 | 7/5 | 7/5 | 7/5 | 5/5 | 5/5 | 5/5 |

2 | 24/19 | 26/17 | 24/9 | 18/9 | 20/5 | 22/11 |

3 | 36/24 | 31/27 | 16/5 | 12/5 | 10/5 | 8/5 |

4 | 31/25 | 28/25 | 18/14 | 20/11 | 22/17 | 23/14 |

5 | 34/5 | 14/5 | 8/5 | 12/5 | 11/5 | 11/5 |

6 | 68/45 | 47/28 | 50/30 | 32/25 | 18/7 | 5/5 |

7 | 49/34 | 21/6 | 8/6 | 7/5 | 11/5 | 12/5 |

8 | 32/5 | 27/5 | 20/5 | 20/5 | 6/5 | 6/5 |

9 | 42/34 | 45/35 | 33/13 | 26/16 | 25/9 | 19/7 |

10 | 63/54 | 46/38 | 21/30 | 22/24 | 20/5 | 14/5 |

#### 3.3.2. Second WDN

Situation | Number of Pressure Transducers | |||||
---|---|---|---|---|---|---|

10 | 12 | 14 | 16 | 18 | 20 | |

1 | 8/5 | 7/5 | 5/5 | 5/5 | 5/5 | 5/5 |

2 | 21/5 | 18/5 | 16/5 | 17/5 | 19/5 | 18/5 |

3 | 17/6 | 15/5 | 17/5 | 11/5 | 11/5 | 11/5 |

4 | 11/15 | 17/13 | 13/9 | 15/9 | 16/9 | 14/9 |

5 | 15/10 | 18/6 | 15/5 | 14/5 | 14/5 | 13/5 |

6 | 16/9 | 20/5 | 17/7 | 23/13 | 18/5 | 18/5 |

7 | 21/5 | 18/5 | 18/5 | 17/5 | 11/5 | 8/5 |

8 | 18/5 | 14/5 | 13/5 | 13/5 | 10/5 | 8/5 |

9 | 19/11 | 17/11 | 12/6 | 14/7 | 13/7 | 12/6 |

10 | 18/12 | 23/5 | 19/5 | 22/5 | 12/5 | 12/5 |

Situation | Number of Pressure Transducers | |||||
---|---|---|---|---|---|---|

10 | 12 | 14 | 16 | 18 | 20 | |

1 | 14/5 | 7/5 | 5/5 | 5/5 | 5/5 | 5/5 |

2 | 38/8 | 38/21 | 23/5 | 26/5 | 26/5 | 24/9 |

3 | 38/26 | 36/29 | 28/5 | 18/5 | 14/5 | 14/5 |

4 | 31/31 | 29/30 | 28/17 | 20/11 | 22/17 | 22/18 |

5 | 38/10 | 24/6 | 25/5 | 14/5 | 15/5 | 13/5 |

6 | 84/39 | 70/41 | 66/32 | 60/32 | 36/5 | 32/5 |

7 | 40/30 | 28/8 | 23/6 | 24/7 | 17/5 | 8/5 |

8 | 38/18 | 22/5 | 19/5 | 26/5 | 14/5 | 8/5 |

9 | 42/41 | 43/26 | 31/13 | 35/13 | 37/13 | 30/9 |

10 | 64/59 | 49/45 | 42/20 | 37/24 | 22/13 | 18/5 |

#### 3.4. Discussion of Results

**Figure 10.**Relationship between the number of pressure transducers and the number of total and reliable pipes.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Appendix

Node | Trust Score | Node | Trust Score | Node | Trust Score |
---|---|---|---|---|---|

1 | 0.04688 | 35 | 0.00781 | 69 | 0.00781 |

2 | 0.01563 | 36 | 0.06250 | 70 | 0.03125 |

3 | 0.03125 | 37 | 0.01565 | 71 | 0.06250 |

4 | 0.25000 | 38 | 0.06250 | 72 | 0.12500 |

5 | 0.25000 | 39 | 0.03125 | 73 | 0.00781 |

6 | 0.02344 | 40 | 0.06250 | 74 | 0.06250 |

7 | 0.06250 | 41 | 0.12500 | 75 | 0.50000 |

8 | 0.12500 | 42 | 0.06250 | 76 | 0.07813 |

9 | 0.12500 | 43 | 0.01563 | 77 | 0.02734 |

10 | 0.03125 | 44 | 0.06250 | 78 | 0.00781 |

11 | 0.25000 | 45 | 0.06250 | 79 | 0.03125 |

12 | 0.03125 | 46 | 0.25000 | 80 | 0.06250 |

13 | 0.02734 | 47 | 0.25000 | 81 | 0.03125 |

14 | 0.03125 | 48 | 0.06250 | 82 | 0.03125 |

15 | 0.25000 | 49 | 0.00391 | 83 | 0.12500 |

16 | 0.03125 | 50 | 0.02344 | 84 | 0.06250 |

17 | 0.03125 | 51 | 0.03125 | 85 | 0.03125 |

18 | 0.01563 | 52 | 0.06250 | 86 | 0.03125 |

19 | 0.12500 | 53 | 0.12500 | 87 | 0.06250 |

20 | 0.25000 | 54 | 0.06250 | 88 | 0.12500 |

21 | 0.01172 | 55 | 0.04688 | 89 | 0.25000 |

22 | 0.12500 | 56 | 0.12500 | 90 | 0.06250 |

23 | 0.06250 | 57 | 0.00781 | 91 | 0.06250 |

24 | 0.03125 | 58 | 0.03125 | 92 | 0.03125 |

25 | 0.06250 | 59 | 0.06250 | 93 | 0.12500 |

26 | 0.04688 | 60 | 0.06250 | 94 | 0.03125 |

27 | 0.01563 | 61 | 0.03125 | 95 | 0.12500 |

28 | 0.12500 | 62 | 0.04688 | 96 | 0.06250 |

29 | 0.25000 | 63 | 0.03125 | 97 | 0.00781 |

30 | 0.12500 | 64 | 0.00781 | 98 | 0.01563 |

31 | 0.03125 | 65 | 0.03125 | 99 | 0.00195 |

32 | 0.06250 | 66 | 1.00000 | 100 | 0.06250 |

33 | 0.06250 | 67 | 0.00781 | 101 | 1.00000 |

34 | 0.12500 | 68 | 0.50000 | – | – |

## Conflicts of Interest

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## Share and Cite

**MDPI and ACS Style**

Ribeiro, L.; Sousa, J.; Marques, A.S.; Simões, N.E.
Locating Leaks with TrustRank Algorithm Support. *Water* **2015**, *7*, 1378-1401.
https://doi.org/10.3390/w7041378

**AMA Style**

Ribeiro L, Sousa J, Marques AS, Simões NE.
Locating Leaks with TrustRank Algorithm Support. *Water*. 2015; 7(4):1378-1401.
https://doi.org/10.3390/w7041378

**Chicago/Turabian Style**

Ribeiro, Luísa, Joaquim Sousa, Alfeu Sá Marques, and Nuno E. Simões.
2015. "Locating Leaks with TrustRank Algorithm Support" *Water* 7, no. 4: 1378-1401.
https://doi.org/10.3390/w7041378