# Machine Learning and Simulation-Optimization Coupling for Water Distribution Network Contamination Source Detection

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Water Distribution Network Benchmarks

#### 2.2. Machine Learning and Simulation-Optimization Coupling Framework 1

#### 2.3. Machine Learning and Simulation-Optimization Coupling Framework 2

#### 2.4. Random Forests

#### 2.5. Stochastic Optimization Algorithms

#### 2.5.1. Fireworks Algorithm

#### 2.5.2. Particle Swarm Optimization

#### 2.5.3. Genetic Algorithms

#### 2.5.4. Preliminary Analysis

#### 2.6. Deterministic Optimization Algorithm

#### Mesh Adaptive Direct Search

## 3. Results and Discussion

#### 3.1. Random Forest Classifier Prediction

#### 3.2. Algorithmic Framework 1 Results

#### 3.3. Algorithmic Framework 2 Results

#### 3.4. Framework Comparison

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ANN | Artificial Neural Network |

CNN | Convolutional Neural Network |

DT | Decision Tree |

GA | Genetic Algorithms |

LVQNN | Learning Vector Quantization Neural Network |

MC | Monte Carlo |

ML | Machine Learning |

NM | Nelder-Mead |

NSGA-II | Non-dominated Sorting Genetic Algorithm-II |

PM | Powell’s method |

PNN | Probabilistic Neural Networks |

PSO | Particle Swam Optimization |

PSVM | Probabilistic Support Vector Machines |

RF | Random Forests |

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**Figure 9.**A comparison of sensor measurements through time for the NET3 benchmark network (framework 1).

**Figure 10.**Average contaminant mass flow at the source node 251 for the Richmond network over a 72 h period.

**Figure 11.**Network node locations of the two top candidate source nodes 251 and 260 (zoomed in part of the network).

**Figure 12.**A comparison of sensor measurements through time for the NET3 benchmark network (framework 2).

Source Node | Start Time | End Time | Contaminant Concentration |
---|---|---|---|

261 | 00:40 h | 06:30 h | 78.5 mg/L |

Algorithm | Successful Runs | Average Time per Run | Average f | Parameters |
---|---|---|---|---|

GA | 100/100 | 121.5 s | 0.015 | g: 100, p: 80, ${c}_{r}$: 0.8, m: 0.1 |

PSO | 98/100 | 178.6 s | 0.003 | i: 50, s: 100, ${c}_{1}$: 1, ${c}_{2}$: 1 |

FWA | 100/100 | 64.13 s | 0.009 | i: 80, n: 5, ${m}_{1}$: 5, ${m}_{2}$: 10 |

Network | Sensors | Inputs | Top Nodes | Accuracy | Time |
---|---|---|---|---|---|

NET3 | Perfect | 70,000 | 10 | 99% | 37 s |

Richmond | Fuzzy | 105,000 | 60 | 99% | 955 s |

Run | Source Node | Start Time | End Time | Contaminant Concentration | f | Time |
---|---|---|---|---|---|---|

Average | 261 | 0:39 h | 6:29 h | 78.49 mg/L | 0.024 | 142 s |

Most accurate | 261 | 0:40 h | 6:30 h | 78.5 mg/L | 0.002 | 133 s |

Least accurate | 263 | 0:20 h | 6:20 h | 78.7 mg/L | 0.436 | 117 s |

Source Node | Start Time | End Time | Contaminant Concentration |
---|---|---|---|

251 | 06:30 h | 21:30 h | 939.37 mg/L |

Run | Start Time | End Time | Contaminant Toncentration | f | Time |
---|---|---|---|---|---|

Average | 6:30 h | 21:00 h | 940.6 mg/L | 0.0 | 882 s |

Most accurate | 6:30 h | 21:30 h | 939.6 mg/L | 0.0 | 852 s |

Least accurate | 6.30 h | 11:40 h | 943.9 mg/L | 0.0 | 860 s |

Run | Start Time | End Time | Contaminant Concentration | f | Time |
---|---|---|---|---|---|

Average | 6:30 h | 20:09 h | 915.6 mg/L | 0.0 | 882 s |

Most accurate | 6:30 h | 21:30 h | 924.1 mg/L | 0.0 | 843 s |

Least accurate | 6.30 h | 14:50 h | 908.4 mg/L | 0.0 | 1002 s |

**Table 8.**NET3 RF regression average values and standard deviations for 100 runs with absolute error.

Average Prediction | Standard Deviation | Absolute Error | |
---|---|---|---|

Start time | 1:58 h | 0:079 h | 1:18 h |

End time | 6:56 h | 0:15 h | 0:16 h |

Contaminant concentration | 108.87 mg/L | 5.17 mg/L | 30.38 mg/L |

Run | Source Node | Start Time | End Time | Contaminant Concentration | f | Time |
---|---|---|---|---|---|---|

Average | 261 | 0:31 h | 6:30 h | 78.48 mg/L | 0.033 | 126 s |

Most accurate | 261 | 0:40 h | 6:30 h | 78.5 mg/L | 0.002 | 124 s |

Least accurate | 261 | 0:00 h | 6:30 h | 78.2 mg/L | 0.19 | 109 s |

**Table 10.**Richmond RF regression average values and standard deviations for 100 runs with an absolute error.

Average Prediction | Standard Deviation | Absolute Error | |
---|---|---|---|

Start time | 6:53 h | 0:015 h | 0:28 h |

End time | 17:57 h | 0:049 h | 3:73 h |

Contaminant concentration | 933.97 mg/L | 0.96 mg/L | 5.4 mg/L |

Run | Start Time | End Time | Contaminant Concentration | f | Time |
---|---|---|---|---|---|

Average | 6:30 h | 22:22 h | 942.2 mg/L | 0.0 | 632.7 s |

Most accurate | 6:30 h | 21:30 h | 939.7 mg/L | 0.0 | 602 s |

Least accurate | 6.30 h | 22:20 h | 944.3 mg/L | 0.0 | 591 s |

Run | Start Time | End Time | Contaminant Concentration | f | Time |
---|---|---|---|---|---|

Average | 6:30 h | 15:73 h | 917.8 mg/L | 0.0 | 632.7 s |

Most accurate | 6:30 h | 17:40 h | 924.3 mg/L | 0.0 | 570 s |

Least accurate | 6.30 h | 15:40 h | 908.5 mg/L | 0.0 | 596 s |

**Table 13.**A comparison of frameworks 1 and 2 in determining the true source node for both network benchmarks.

Framework | Network | Runs | True Source | False Source | Tie |
---|---|---|---|---|---|

1 | NET3 | 100 | 99 | 1 | 0 |

2 | NET3 | 100 | 100 | 0 | 0 |

1 | Richmond | 100 | 4 | 7 | 89 |

2 | Richmond | 100 | 63 | 0 | 37 |

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**MDPI and ACS Style**

Grbčić, L.; Kranjčević, L.; Družeta, S. Machine Learning and Simulation-Optimization Coupling for Water Distribution Network Contamination Source Detection. *Sensors* **2021**, *21*, 1157.
https://doi.org/10.3390/s21041157

**AMA Style**

Grbčić L, Kranjčević L, Družeta S. Machine Learning and Simulation-Optimization Coupling for Water Distribution Network Contamination Source Detection. *Sensors*. 2021; 21(4):1157.
https://doi.org/10.3390/s21041157

**Chicago/Turabian Style**

Grbčić, Luka, Lado Kranjčević, and Siniša Družeta. 2021. "Machine Learning and Simulation-Optimization Coupling for Water Distribution Network Contamination Source Detection" *Sensors* 21, no. 4: 1157.
https://doi.org/10.3390/s21041157