# Leak Signature Space: An Original Representation for Robust Leak Location in Water Distribution Networks

^{1}

^{2}

^{*}

## Abstract

**:**In this paper, an original model-based scheme for leak location using pressure sensors in water distribution networks is introduced. The proposed approach is based on a new representation called the Leak Signature Space (LSS) that associates a specific signature to each leak location being minimally affected by leak magnitude. The LSS considers a linear model approximation of the relation between pressure residuals and leaks that is projected onto a selected hyperplane. This new approach allows to infer the location of a given leak by comparing the position of its signature with other leak signatures. Moreover, two ways of improving the method’s robustness are proposed. First, by associating a domain of influence to each signature and second, through a time horizon analysis. The efficiency of the method is highlighted by means of a real network using several scenarios involving different number of sensors and considering the presence of noise in the measurements.

## 1. Introduction

## 2. Model Representation

#### 2.1. Water Network Model Solution

^{*}, q, d which are respectively the vectors of pressure in the junction nodes, pressure in reservoirs, flows trough the pipes and demands:

_{21}, A

_{01}the incidence matrices obtained when only junction and reservoir nodes are considered, respectively. In A

_{11}(

**q**), |q

_{i}| is the absolute value of the flow q

_{i}, c

_{i}is a constant parameter which depends on the diameter, the roughness and the length of the pipe, and γ

_{f}is the flow exponent parameter.

_{i}), i ∊ [1,⋯, f]. It is important to note that this resolution approach is commonly employed, as e.g., in the EPANET simulator [20] where large WDN can be simulated efficiently.

**p**

^{k}

^{+1}=

**p**

^{k}=

**p**and

**q**

^{k}+1 =

**q**

^{k}=

**q**. A representation of a leak in a WDN would theoretically involve a graph structure where each possible leak could be represented by a graph node. However, a leak could possibly appear at any point of a network pipe. For this reason, the exact modeling of any possible leak becomes unfeasible in practice. To mitigate this issue, it is usually assumed that leaks only appear at existing nodes [13]. With such assumption, we can add a leak written as a vector of extra demands Δ

**d**and the new demand d′ can be expressed such as:

**d**is a m dimensional vector with zeros everywhere except at the node’s index where the leak occurs. Now, assuming that in presence of a leak, the network flow equilibrium can also be reached, the pressure in case of leak can be expressed by:

**q**′ is the flow in leak case. Then, we propose to represent the residual

**r**(c.f. [15]) as the difference between the nominal pressure in the model without leaks and the pressure in the network in case of a leak. From Equations (3) and (5), the following approximate expression for the residual could be derived:

**q**=

**q**′). The proposed leak location approach is intended to be used for small leaks since large leaks are easily detected because usually reach the surface rapidly. In this way, it is possible to obtain an approximate linear relation between the residual (and consequently with the pressure measurement) and the leak through a matrix factor

**S**under equilibrium assumptions that is known as the sensitivity matrix. This approximation implies an error factor ε

_{q}when using the linear approximation

**r**=

**S**· Δ

**d**+ ε

_{q}. This error is not modeled when deriving Equation (6) but it will be taken into account as uncertainty in the realistic cases analyzed in the experiments. The

**S**matrix has been used in a variety of works [15,16,21] where it has been obtained by means of simulation. However, to our knowledge it is the first time that the analytical expression Equation (6) is proposed, opening new perspectives for developing analytical methods for model based leak location.

## 3. Leak Signature Space

**r**

_{1}and

**r**

_{2}corresponding to different leak magnitudes but occurring in the same node j, it can be stated that:

#### 3.1. Dealing with Realistic Cases

_{q}as mentioned previously. Also, the model representation used is always imperfect/incomplete with respect to what is happening in a real network and the simulation based on the Newton–Raphson scheme may introduce some numerical errors. In addition, some noise is typically associated with the sensor measurements and finally, the demand may vary along the time, whereas the leak in the presented model appears as an extra demand which follows a fixed demand pattern. The problem of changes in the water demand will be considered later on. For now, it is only explained how to deal with general noise and imperfect linear dependencies.

^{k}are simulated, with k ∊ [1,⋯, s]. Then, the associated s residual vectors ${\mathbf{r}}^{\{j,k\}}={[{r}_{1}^{\{j,k\}},\cdots ,\phantom{\rule{0.2em}{0ex}}{r}_{n}^{\{j,k\}}]}^{\mathbf{T}}$ are computed and if the projection is performed with respect to the last coordinate, their projection onto the LSS are given by the (n − 1)-dimensional points of coordinates ${\tilde{\mathbf{r}}}^{\{j,k\}}={[\frac{{r}_{1}^{\{j,k\}}}{{r}_{n}^{\{j,k\}}},\cdots ,\phantom{\rule{0.2em}{0ex}}\frac{{r}_{n-1}^{\{j,k\}}}{{r}_{n}^{\{j,k\}}}]}^{\mathbf{T}}$. Then, point ${\tilde{\mathbf{r}}}^{j}$, corresponding to the leak j, is taken as the barycenter of these s partial signatures ${\tilde{\mathbf{r}}}^{\{j,k\}}$ built from the different leak magnitudes:

#### 3.1.1. Leak Signature Domain

#### 3.1.2. Choice of the Projection

## 4. Small WDN Example

#### 4.1. Model Solution

_{11}, N and S (c.f. Equation (6)) have the following numerical values:

#### 4.2. Leak Signature

## 5. Leak Location Method

#### 5.1. Basic Approach

#### 5.2. Exploiting Leak Signature Domains

#### 5.3. Time Horizon Analysis

#### 5.4. Combining Time Horizon and Leak Signature Domains

## 6. Real Network Application

^{1}

^{/}

^{2}) based on the following expression:

_{exp}is the pressure exponent. Then, the EC will vary in a range going from 0.3 to 0.9 lps/m

^{1/2}to introduce leaks in the range specified above.

#### 6.1. Choice of Projection Hyperplane

#### 6.2. Leak Location: Single Node Analysis

#### 6.3. Leak Location: Full WDN Analysis

## 7. Test in a Real Leak Scenario

## 8. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**The linear model approximations of the residuals for the three possible leaks are projected onto a (n − 1)-dimensional hyperplane (here a plane).

**Figure 2.**Small Water Distribution Network (WDN) example. A reservoir is supplying a network composed of 3 demand nodes and 4 pipes.

**Figure 6.**View of a leak at node 69 in the LSS in case of 2 sensors (

**a,d**); 3 sensors (

**b,e**); and 4 sensors (

**c,f**). Lower pictures are zooms of the upper pictures.

**Figure 7.**View of a leak at node 150 in the LSS for a 24-hour time horizon analysis in case of 2 sensors (

**a**); 3 sensors (

**b**); and 4 sensors (

**c**). The lines correspond to connections between successive time samples. The leak is correctly located when 3 and 4 sensors are present.

Number of Sensors | Sensor Node Locations |
---|---|

2 | 2, 152 |

3 | 2, 146, 152 |

4 | 2, 76, 111, 152 |

5 | 2, 76, 111, 146, 152 |

**Table 2.**Overlapping leak signature domains for a single instant of time in function of the projection hyperplane and a given number of sensors.

Number of Sensors | Overlapping Domains | ||||
---|---|---|---|---|---|

2 | 282 | – | – | – | 282 |

3 | 248 | – | – | 241 | 213 |

4 | 152 | 160 | 186 | – | 148 |

5 | 150 | 175 | 179 | 173 | 147 |

Sensor Indexes | 2 | 76 | 111 | 146 | 152 |

**Table 3.**Overlapping leak signature domains, as a function of the projection hyperplane selected and the number of sensors when considering a time horizon.

Number of Sensors | Overlapping Domains | ||||
---|---|---|---|---|---|

2 | 260.7 | – | – | – | 260.7 |

3 | 245.0 | – | – | 266.7 | 262.4 |

4 | 152.0 | 165.2 | 187.3 | – | 157.1 |

5 | 151.4 | 177.4 | 183.8 | 170.44 | 150.9 |

Sensor Indexes | 2 | 76 | 111 | 146 | 152 |

Without Noise
| With Noise
| |||||
---|---|---|---|---|---|---|

Sensors | % Correct | Average topological distance | Mean Rank | % Correct | Average topological distance | Mean Rank |

2 | 61.9 | 2.8 | 1.6 | 58.4 | 3.1 | 2.0 |

3 | 75.6 | 1.8 | 1.4 | 65.0 | 1.9 | 1.7 |

4 | 85.8 | 1.3 | 1.2 | 73.1 | 1.3 | 1.5 |

5 | 86.3 | 1.3 | 1.2 | 74.6 | 1.4 | 1.4 |

Without Noise
| With Noise
| |||||
---|---|---|---|---|---|---|

Sensors | % Correct | Average topological distance | Mean Rank | % Correct | Average topological distance | Mean Rank |

2 | 88.3 | 1.52 | 1.7 | 68.5 | 2.2 | 2.0 |

3 | 91.9 | 1.44 | 1.6 | 74.1 | 2.0 | 1.8 |

4 | 95.4 | 1.27 | 1.4 | 76.1 | 1.5 | 1.5 |

5 | 96.4 | 1.30 | 1.3 | 78.2 | 1.4 | 1.4 |

Sensors | % Correct | Average Distance | Mean Rank to the Real Leak |
---|---|---|---|

2 | 31.8 | 4.4 | 5.0 |

3 | 41.2 | 4.1 | 4.6 |

4 | 60.4 | 1.31 | 2.5 |

5 | 64.3 | 1.22 | 1.4 |

© 2015 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 license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Casillas, M.V.; Garza-Castañón, L.E.; Puig, V.; Vargas-Martinez, A.
Leak Signature Space: An Original Representation for Robust Leak Location in Water Distribution Networks. *Water* **2015**, *7*, 1129-1148.
https://doi.org/10.3390/w7031129

**AMA Style**

Casillas MV, Garza-Castañón LE, Puig V, Vargas-Martinez A.
Leak Signature Space: An Original Representation for Robust Leak Location in Water Distribution Networks. *Water*. 2015; 7(3):1129-1148.
https://doi.org/10.3390/w7031129

**Chicago/Turabian Style**

Casillas, Myrna V., Luis E. Garza-Castañón, Vicenç Puig, and Adriana Vargas-Martinez.
2015. "Leak Signature Space: An Original Representation for Robust Leak Location in Water Distribution Networks" *Water* 7, no. 3: 1129-1148.
https://doi.org/10.3390/w7031129