Two-Leak Isolation in Water Distribution Networks Based on k-NN and Linear Discriminant Classifiers
Abstract
:1. Introduction
2. Materials and Methods
2.1. Sensor Placement
2.2. Leak Isolation Strategy
2.2.1. Dataset Generation
2.2.2. Leak Classification
2.2.3. k-NN Classifier
- 1.
- The training of the k-NN classifier is an offline process. In this process, a set of residual samples corresponding to leaks of available classes given by (21) is stored and each residual is assigned to its class label. The dataset used to train the classifier is obtained by performing all possible leak scenarios according to the procedure described in Section 2.2.1.
- 2.
- Leak class prediction is an online process. Here, a continuous comparison of the most recent residual is performed with the labeled residuals from the training dataset (21). If the leak class is denoted by according to (13), and is the probability that the leak location corresponds to the class given the residual , the k-NN classifier assumes that
2.2.4. Discriminant Analysis Classifier
- 1.
- The training of the DA classifier is an offline process where a set of residual samples corresponding to all possible leakage scenarios are assigned to the corresponding class by means of (21), this stage being when the discriminant functions are generated. In the same way, the dataset to train this classifier is obtained by simulating the leakage scenarios according to the procedure described in Section 2.2.1.
- 2.
- Leak class prediction is an online process. In this process, predictions are made using the actual residual and the predictive model obtained in the training stage. If the leak class is denoted by according to (13), then is the probability that the leak corresponds to the class given the residual , and the DA classifier computesThe class with the highest probability is chosen as the output of the classifier:
3. Results
3.1. Hanoi WDN Case Study
3.1.1. Leak Scenario
3.1.2. Leak Scenario
3.1.3. Leak Scenario
3.1.4. Relaxation Node Analysis
3.2. Madrid’s DMA Case Study
Relaxation Node Analysis
3.3. Discussion
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
AI | Artificial Intelligence |
WDN | Water Distribution Network |
OECD | Organization for Economic Cooperation and Development |
FIR | Finite Impulse Response |
RBF | Radial Base Function |
DA | Discriminant Analysis |
k-NN | k Nearest Neighbors |
h | Hour |
L/s | Liters per second |
Leak flow rate used for estimation of residuals | |
Leak flow rate used for estimation of sensitivities | |
Set of real numbers | |
diag | Diagonal matrix |
max | Maximum Value |
arg max | Maximum argument |
Appendix A. Sensor-Placement-Methodology-Based Algorithm
Algorithm A1: Sensor-placement-methodology-based algorithm. |
|
Appendix B. Dataset-Generation-Methodology-Based Algorithm
Algorithm A2: Dataset-generation-methodology-based algorithm |
|
Appendix C. Leak-Localization-Strategy-Based Algorithm
Algorithm A3: Leak-localization-strategy-based algorithm |
|
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Sensor’s Number | Optimal Placement |
---|---|
2 sensors | 12, 21 |
3 sensors | 12, 15, 21 |
Zone | Node Set |
---|---|
1, 2, 3 | |
4, 5, 6 | |
7, 8, 9 | |
10, 11, 12 | |
13, 14 | |
16, 17, 18 | |
19, 20, 21 | |
22, 23, 24 | |
15, 25, 26 | |
27, 28 | |
29, 30, 31 |
Relaxation Nodes | Hanoi WDN | Madrid DMA | ||||||
---|---|---|---|---|---|---|---|---|
k-NN | DA | k-NN | DA | |||||
= 1 | = 24 | = 1 | = 24 | = 1 | = 24 | = 1 | = 24 | |
0 | 25.5% | 48.0% | 53.5% | 70.0% | 23.5% | 36.0% | 65.0% | 77.0% |
1 | 40.5% | 73.5% | 65.0% | 82.5% | 29.0% | 46.0% | 75.5% | 85.5% |
2 | 64.0% | 82.5% | 81.0% | 90.5% | 35.0% | 53.5% | 79.0% | 89.5% |
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Rodríguez-Argote, C.A.; Begovich-Mendoza, O.; Navarro-Díaz, A.; Santos-Ruiz, I.; Puig, V.; Delgado-Aguiñaga, J.A. Two-Leak Isolation in Water Distribution Networks Based on k-NN and Linear Discriminant Classifiers. Water 2023, 15, 3090. https://doi.org/10.3390/w15173090
Rodríguez-Argote CA, Begovich-Mendoza O, Navarro-Díaz A, Santos-Ruiz I, Puig V, Delgado-Aguiñaga JA. Two-Leak Isolation in Water Distribution Networks Based on k-NN and Linear Discriminant Classifiers. Water. 2023; 15(17):3090. https://doi.org/10.3390/w15173090
Chicago/Turabian StyleRodríguez-Argote, Carlos Andrés, Ofelia Begovich-Mendoza, Adrián Navarro-Díaz, Ildeberto Santos-Ruiz, Vicenç Puig, and Jorge Alejandro Delgado-Aguiñaga. 2023. "Two-Leak Isolation in Water Distribution Networks Based on k-NN and Linear Discriminant Classifiers" Water 15, no. 17: 3090. https://doi.org/10.3390/w15173090