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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

This paper proposes a new sensor placement approach for leak location in water distribution networks (WDNs). The sensor placement problem is formulated as an integer optimization problem. The optimization criterion consists in minimizing the number of non-isolable leaks according to the isolability criteria introduced. Because of the large size and non-linear integer nature of the resulting optimization problem, genetic algorithms (GAs) are used as the solution approach. The obtained results are compared with a semi-exhaustive search method with higher computational effort, proving that GA allows one to find near-optimal solutions with less computational load. Moreover, three ways of increasing the robustness of the GA-based sensor placement method have been proposed using a time horizon analysis, a distance-based scoring and considering different leaks sizes. A great advantage of the proposed methodology is that it does not depend on the isolation method chosen by the user, as long as it is based on leak sensitivity analysis. Experiments in two networks allow us to evaluate the performance of the proposed approach.

Leaks in water distribution networks are an issue of great concern for water utilities, strongly linked with operational costs and water resources savings. Continuous improvements in water loss management are being applied, and new technologies are developed to achieve higher levels of efficiency [

The traditional approach to leakage control is a passive one, whereby the leak is repaired only when it becomes visible. Recently, developed acoustic instruments [

Regarding leak location methods for DMAs, several works have been proposed in the literature. For example, a review of transient-based leak detection methods is offered in [

Thus, the development of a sensor placement strategy has become an important research issue in recent years. Ideally, a sensor network should be configured to facilitate fault detection and maximize leak location performance. However, it is obvious that only a limited number of sensors can be installed inside a DMA, due to budget constraints. The main objectives of sensor placement are leak detectability, isolability and localization. Leak detectability is the ability of monitoring a variation in pressure due to a loss of water occurring in the network. Leak isolability concerns the capacity of distinguishing between two possible leak occurrences, whereas leak localization refers to finding the node where the leak is occurring. There are some works devoted to sensor placement for fault detection and isolation (FDI). Some approaches propose to locate sensors based on isolability criteria according to the study of structural matrices [

Each of the previously mentioned works is used in the general framework of FDI of dynamic systems. However, there are several contributions dedicated to sensor placement in water distribution networks. Most of the works have addressed the sensor placement problem regarding contamination monitoring. See, for example [

This paper proposes a new approach for sensor placement for leak location in DMAs that can be used with the projection-based location scheme proposed in [

The rest of the document is organized as follows: Section 2 presents the leak localization methodology in which our work is based. Section 3 describes the problem formulation. Sections 4 and 5 present the sensor placement algorithms proposed in this work, while in Section 6, we show the improvements performed to increase the robustness of the approach. Section 7 shows the application and the results obtained in a real water distribution network. Finally, Section 8 concludes this work.

The leak location methodology used in this paper has been introduced in [

The leak location methodology aims to detect and isolate leaks in a DMA using pressure measurements and their estimation using the hydraulic network model. Let us consider a DMA with _{1} … _{n}^{T}_{i}_{i}_{i}

The leak isolation method relies on analyzing the residual vector (1) using sensitivity analysis, which is determined from the different effects on every pressure measurement caused by each possible leak at a time. To perform such sensitivity analysis, the following sensitivity vectors are derived from simulated leak scenarios [_{i}_{j}_{j}^{t}^{h}

Let _{j}

The objective of this work is to develop an approach to place a given number of sensors, n, in a DMA of a water distribution networks (WDN) in order to obtain a sensor configuration with a maximized leak isolability performance for a given leak detection and isolated scheme. In this work, we use the method based on projections that has been presented in the previous section.

It should be noted that the length of the sensitivity and residual vectors that appear in _{i}_{k}_{j}_{k}

Note that the matrices,

To select a configuration with _{i}_{i}

Then, the corresponding sensitivity and residual vectors can be determined as:
_{j}_{k}_{j}_{k}^{T}

Now, we are able to compute the projection matrix, Ψ, as:

In order to evaluate the quality of a sensor configuration regarding its capacity to locate a leak at node

This means that the error index _{i}_{i}

As the objective is to maximize the isolability regarding leaks in all network nodes, the error index that takes into account all the nodes leaks is computed as:

We remark that

Based on the vector,

As stated in Section 3, the problem of sensor placement involves finding an n-sensor configuration among a set of

_{min}

_{min}

_{NL}

^{k}

^{k}

^{k}

^{k}

^{k}

^{k}

_{min}

The method is described in Algorithm 1. The goal of this algorithm is to find the optimal sensor configuration, taking into account all the possible combinations of sensors and considering the method that will be used to perform the leak location. First, the algorithm initiates the minimum number of non-localizable (NL) leaks, _{NL}^{k}^{k}^{k}

The semi-exhaustive approach was tested in the water network of Hanoi, Vietnam [_{min}_{NL}^{pexp}_{exp}

As the second test, we perform the same experiment, but with three sensors. The results are shown in

In order to choose an adequate combination of sensors, we count the occurrences of the configurations leading to the error indices in

Nodes {12, 21} with 16 occurrences

Nodes {12,13} with 13 occurrences

Nodes {7,12} with seven occurrences

Nodes {12,14, 21} with 22 occurrences

Nodes {12, 21, 27} with 22 occurrences

Nodes {12, 21, 29} with 18 occurrences

Genetic algorithms (GAs) are well-known search and optimization tools based on the principles of natural genetics and natural selection [

The GAs can be used in the context of sensor placement in WDN in order to find near-optimal placement for leak location. In that case, a chromosome corresponds to the possible presence or absence of a sensor at a given node.

Here, the sensor placement problem formulated as an optimization problem in Section 3 is solved using genetic algorithms and implemented using the GA Toolbox of MATLAB. The GA needs to establish a function whose output involves an index to be minimized. In our case, this function corresponds to the evaluation of the error index computed in

_{min}

_{min}

^{k}

^{k}

^{k}

^{k}

_{min}

_{min}

The pseudo-code of the algorithm is shown in Algorithm 2. First, we initialize the variables of the GA (line 1), including the number of generations, the bit string type population, the tolerance as 10^{−10} and the elite count as one, in order to save one of the previous results analyzed. Then, we declare the search restriction (line 2), being that the number of “ones” in the solution corresponds to the number of sensors to install, and a seed size,

The experiments performed in the Hanoi network presented in Section 4.2 and based on a semi-exhaustive search were computationally very demanding, despite the lazy evaluation mechanisms involved. For such a simple example, 465 configurations had to be tested for the case of two sensors with an average computation time of 3 s and 4, 495 configurations for the case of three sensors, which takes 15 s on average. This means that performing a semi-exhaustive search in a large network is not feasible, because of the computational complexity that would quickly lead to testing millions of possible combinations. This is the motivation for using GA. All the tests performed using semi-exhaustive search are reproduced using Algorithm 2 applied to the Hanoi network. The solutions found are the same as the ones obtained with the semi-exhaustive search, but with a computational time lower than 9 s per iteration and including three generations in each of them. All the experiments were performed in MATLAB, using a Windows 7 computer with a Pentium Dual Core processor of 2 GHz, memory (RAM) of 4 GB and a 64-bit operating system.

In our experiments, despite Algorithm 2 providing efficiently optimal solutions (since they are consistent with the semi-exhaustive search), we have seen that the algorithm requires a post-treatment analysis in order to make an adequate sensor placement decision when uncertainty (e.g., about the unknown leak magnitude) is considered. Moreover, we know that this placement represents a near-optimal solution that works only for the time instant evaluated. In the following, we present three improvements that avoid any post-treatment and increase the robustness of the GA-based sensor placement.

In the semi-exhaustive search and GA-based algorithm that we have presented, we took into account a single instant of time for the analysis. However, the configuration that gives the minimum error index in the leak isolation can vary when the demand changes along a given period of time. To address this problem, it is possible to improve the tasks of leak detection and isolation by considering a time horizon, as suggested in [

The GA evaluation function is modified in order to work with the mean projection along the time horizon instead of using a single instant of time. With such modification, the candidate leak node is obtained by looking at the maximum of the mean projection, Ψ̄(^{t}

In the optimization problem

We propose to rely on topological distances (_{max}_{i}_{ij}_{i}_{1} (_{im}

The cut-off distance, _{max}^{2} nodes. Then, the distance from the center node to the network border would be given by

The choice of sensor placement is affected by the leak magnitude taken to build the sensitivities. However, in real scenarios, this magnitude cannot be determined in advance. To improve the robustness of the results according to such parameter changes, we propose to incorporate sets of sensitivities and residuals in the evaluation function that are computed from different leak magnitudes. Assume that there are ^{l}^{l}^{l}^{1},^{l}^{2}}, with _{1},_{2} ∈ {1,… ,L} would be ^{2}. Among them, we discard couples {^{l}^{l}^{l}^{1},^{l}^{2}} with _{1} > _{2}, since Ψ(^{l}^{1},^{l}^{2}) = Ψ(^{l}^{1},^{l}^{2})^{T}

The three steps presented above increase the robustness of the sensor placement method with respect to the experimental variations. These new procedures shown in Algorithm 3 modify the iteration steps performed by GA that were previously presented in Algorithm 2.

The method basically consists of ^{c}^{c}^{k}^{k}

Algorithm 3 was applied to the Hanoi network. In our tests, we take

Since the network has 31 nodes,

Algorithm 3 is executed varying the number of sensors from two to ten.

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^{ct}

_{min}

_{min}

^{k}

^{ct}

^{ct})

^{ct}

^{ct}

^{ct}

^{ct}(

^{ct}

^{c}

^{c}

^{k}

^{k}

_{min}

_{min}

The methodologies presented in Algorithms 2 and 3 are applied to a real network. We used the Limassol network in Cyprus that has a total of 197 demand nodes and is represented in ^{6} possible combinations of nodes to be considered. The computation time consumed by the semi-exhaustive search was approximately 60 h for the combination of sensitivity and residual chosen. This means that testing all the possible combinations of sensitivities and residuals is not feasible. The sensor placement problem is set up with _{min}

We apply Algorithm 2 for different types of residual and sensitivity matrices that are computed by varying the leak magnitudes within a given range. Here, the robustness improvements presented in Section 6.1 are not applied (they will be applied in the next subsection). The parameters for the genetic algorithm were selected after several trial and error tests. The initialization matrix was set with a size of 50 rows, and five iterations were allowed in order to increase the efficiency of the method with a maximum of five generations in each of them. The computation time was about 60 min. Compared to the time spent in the semi-exhaustive search, we can conclude that GA significantly reduces the time required to find a solution.

Thus, we perform a post-treatment analysis to decide what is the best sensor configuration for the network. This is based on the following tests:

In order to select the adequate configuration of sensors, we propose performing the experiments described above and look for the combination with the smallest average error index along all the possible leak magnitudes and sensitivities to test. This criterion is analytically established by taking the minimum of the average error indices:
^{ij}

Such a method gives the sensor placement at nodes {76, 133, 152}, which provides the lowest average error among the sets of sensor placements computed from the different combinations of residual and sensitivities. However, there is no guarantee that other sets would not lead to better results. Furthermore, this configuration was found based on a single time instant; thus, it is not robust to changes in the demand that would occur when considering a period of time. For these reasons and to get a more reliable solution, we prefer the method that includes the robustness improvements that we proposed in Section 6.1.

We applied Algorithm 3 to the same Limassol, Cyprus network. We took

Since the network has 197 nodes,

After the iterations, the best result obtained corresponds to a sensor configuration of nodes {2, 75, 158}, with an error index of 0.302, which means an average distance of two nodes between the located node and that with the real leak. This placement is shown in

A second test increasing the noise level up to 2% gives the best configuration at nodes {2, 75, 100} (

It is important to note that these results comes from the integration within the GA of all the improvements of sensor placement robustness that we described in Section 6.1. Thus, contrary to what was done in the previous section when using Algorithm 2, there is no need to perform any post-treatment analysis to extract a robust solution. Integrating a time horizon, a more informative distance-based scoring and the possible variations of leak magnitude, our method provides a solution configuration for sensor placement with a higher level of confidence.

When moving the proposed sensor placement approach to the real network, the following practical considerations should be taken into account considering the previous experience in [

the DMA EPANET model should be recalibrated with real data obtained from the available sensors already installed in the network (typically at the flow entrance points) in order to minimize the errors due to model mismatch between the real and simulated network.

the configuration of the internal valves should also be verified in order to assure that their positions in the real and simulated network are the same.

the nodal demand should be estimated as well as possible using information from water consumption and tele-measurement devices, if available.

leak size range that is considered, as well as sensor noise and precision should be characterized so that the robust sensor placement approach presented in Section 6 could be used in such a way that the installed sensors guarantee the minimal isolation error that is possible.

In this paper, we proposed a new approach to sensor placement for water distribution networks that maximizes leak isolability. The sensor placement problem has been formulated as a non-linear integer optimization problem. The optimization criterion is based on minimizing the number of non-isolable leaks according to the isolability criteria introduced. This approach is combined with a projection-based leak location scheme, but it could be easily adapted to any other sensitivity-based isolation scheme.

The first semi-exhaustive search method has been proposed that searches for the best configuration and relies on lazy evaluation mechanisms to reduce the computation cost. However, the computational effort remains too demanding for most realistic scenarios. Thus, we proposed to solve the optimization problem with GAs, which are known to work well in large-sized problems of a non-linear integer nature. We have seen that such approach allows us to find near-optimal solutions in an efficient way. We also highlighted that leak magnitude changes were impacting the resulting best sensor placement found by the GA algorithm, requiring a post-treatment analysis to tackle such a problem. Finally, we proposed three improvements that avoid any post-treatment and increase the robustness of the GA-based sensor placement. Experiments on two types of networks were performed to compare the different methods proposed in this paper. They demonstrate the relevance of the robust GA-based approach.

For future work, we would like to combine our robust GA-based approach with other methods to perform the leak isolation task. Furthermore, other types of optimization methods that provide some guarantee regarding the solution optimality could be investigated in the future. Enhancing the robustness of the sensor placement algorithm against additional sources of uncertainty, as in the model parameters or in nodal demand, will also be considered in future research.

This work is supported by the Research Chair in Supervision and Advanced Control of Tecnol ógico de Monterrey, Campus Monterrey and by a CONACYTstudentship. This work has been partially grant-funded by CICYTSHERECSDPI-2011-26243 and CICYT WATMANDPI-2009-13744 of the Spanish Ministry of Education, by iSense grant FP7-ICT-2009-6-270428 and by EFFINETgrant FP7-ICT-2012-318556 of the European Commission. This work is also partially founded by the Fond National de la Recherche, Luxembourg (CO11/IS/1206050).

The authors declare no conflict of interest.

Minimum error index according to the number of sensors in the Hanoi network.

Water network in Limassol, Cyprus.

Near-optimal placement of three sensors in the Limassol network. (

Minimum error indices in the Hanoi network after placing two sensors. EC, emitter coefficient.

| |||||||||
---|---|---|---|---|---|---|---|---|---|

2 | 3 | 4 | 5 | 6 | 7 | 8 | |||

2 | 0.032 | 0.032 | 0.129 | 0.129 | 0.129 | 0.193 | |||

3 | 0.032 | 0.032 | 0.096 | 0.129 | 0.129 | 0.161 | |||

4 | 0.064 | 0.032 | 0 | 0.064 | 0.096 | 0.129 | |||

5 | 0.161 | 0.064 | 0.032 | 0.032 | 0.064 | 0.096 | |||

6 | 0.161 | 0.129 | 0.064 | 0 | 0.032 | 0.096 | |||

7 | 0.193 | 0.161 | 0.129 | 0.064 | 0.032 | 0 | |||

8 | 0.193 | 0.193 | 0.161 | 0.129 | 0.064 | 0 |

Minimum error indices in Hanoi network with three sensors.

| |||||||||
---|---|---|---|---|---|---|---|---|---|

2 | 3 | 4 | 5 | 6 | 7 | 8 | |||

2 | 0 | 0 | 0.032 | 0 | 0.032 | 0.032 | |||

3 | 0 | 0 | 0 | 0.032 | 0 | 0.032 | |||

4 | 0 | 0 | 0 | 0 | 0.032 | 0.032 | |||

5 | 0.032 | 0 | 0 | 0 | 0.032 | 0.032 | |||

6 | 0 | 0 | 0 | 0 | 0 | 0.032 | |||

7 | 0.032 | 0.032 | 0.032 | 0 | 0 | 0 | |||

8 | 0.032 | 0.032 | 0.032 | 0 | 0 | 0 |

Best configurations and corresponding error indices for different leak magnitudes in the Hanoi network.

( |
( | ||
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{12, 21} | 0.131 | {12, 14, 21} | 0.025 |

{12, 13} | 0.133 | {12, 21, 27} | 0.028 |

{7, 12} | 0.157 | {12, 21, 29} | 0.035 |

Sensor configurations in Limassol network with three sensors.

| |||||||
---|---|---|---|---|---|---|---|

0.15 | 0.2 | 0.25 | 0.3 | 0.35 | |||

0.15 | {40, 77 172} | {25, 77, 133} | {76, 133, 185} | {76, 133, 152} | |||

0.2 | {76, 133, 152} | {76, 86, 152} | {77, 124, 152} | {76, 110, 173} | |||

0.25 | {85, 156, 196} | {8, 76, 150} | {75, 116, 157} | {72, 115, 150} | |||

0.3 | {72, 118, 163} | {76, 133, 141} | {77, 111, 150} | {75, 23, 152} | |||

0.35 | {76, 128, 140} | {75, 120, 150} | {77, 115, 137} | {29, 112, 152} |

Minimum error indices in the Limassol network for the configurations of

| |||||||
---|---|---|---|---|---|---|---|

0.15 | 0.2 | 0.25 | 0.3 | 0.35 | |||

0.15 | 0.324 | 0.294 | 0.299 | 0.314 | |||

0.2 | 0.299 | 0.284 | 0.279 | 0.294 | |||

0.25 | 0.279 | 0.274 | 0.243 | 0.243 | |||

0.3 | 0.309 | 0.279 | 0.263 | 0.258 | |||

0.35 | 0.324 | 0.279 | 0.263 | 0.258 |

Averaged error indices for configurations of

| |||||
---|---|---|---|---|---|

1 | {75, 116, 157} | 0.336 | 0.412 | 0.576 | 0.442 |

2 | {85, 156, 196} | 0.362 | 0.455 | 0.597 | 0.471 |

3 | {72, 115, 150} | 0.345 | 0.429 | 0.583 | 0.452 |

4 | {76, 110, 173} | 0.340 | 0.409 | 0.556 | 0.435 |

5 | {77, 124, 152} | 0.348 | 0.444 | 0.581 | 0.457 |

6 | {76, 133, 152} | 0.318 | 0.403 | 0.558 | 0.426 |

7 | {76, 86, 152} | 0.335 | 0.421 | 0.572 | 0.443 |

8 | {25, 77, 133} | 0.336 | 0.420 | 0.569 | 0.441 |

9 | {76, 133, 185} | 0.334 | 0.420 | 0.564 | 0.439 |

10 | {40, 77, 172} | 0.368 | 0.448 | 0.613 | 0.477 |

11 | { |
||||

12 | {8, 76, 150} | 0.356 | 0.462 | 0.613 | 0.477 |

13 | {72, 118, 163} | 0.373 | 0.443 | 0.576 | 0.464 |

14 | {76, 133, 141} | 0.341 | 0.416 | 0.570 | 0.442 |

15 | {76, 128, 140} | 0.355 | 0.425 | 0.573 | 0.451 |

16 | {75, 120, 150} | 0.328 | 0.431 | 0.582 | 0.447 |

17 | {77, 111, 150} | 0.342 | 0.422 | 0.566 | 0.443 |

18 | {77, 115, 137} | 0.330 | 0.417 | 0.561 | 0.436 |

19 | {75, 123, 152} | 0.339 | 0.421 | 0.575 | 0.445 |

20 | {29, 112, 152} | 0.394 | 0.455 | 0.590 | 0.480 |