# Calibration Procedure for Water Distribution Systems: Comparison among Hydraulic Models

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## Abstract

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## 1. Introduction

## 2. Methodology

#### 2.1. Non-Uniqueness of the Solutions

#### 2.2. Hydraulic Models

#### 2.3. Decision Variables

#### 2.4. Objective Functions

## 3. Test Case

#### 3.1. Data Generation and Sensor Placement

#### 3.2. Results and Discussion

## 4. Test Case 2

#### Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Apulian network: (

**a**) model network where the calibration processes are launched (number of nodes = 23, number of pipes = 34); (

**b**) reference network where the measurements are taken (number of nodes = 238, number of pipes = 268).

**Figure 3.**Behaviour of the Mean Absolute Errors (MAEs), calculated as the difference between average values and values of the reference network, of the distributed pressure driven (DPD) approach, with respect to the number of runs. In panel (

**a**) is displayed the MAE related to the pressures and in panel (

**b**) is displayed the MAE related to the flow rates.

**Figure 4.**Pressure errors (i.e., the difference between simulated and reference values) at each node of the 100 runs for each hydraulic approach: (

**a**) NDD; (

**b**) NPD; (

**c**) DDD; (

**d**) DPD. The error of the average values (i.e., average values line) is also displayed in each plot.

**Figure 5.**Flow rate errors (i.e., the difference between simulated and reference values) at each pipe of the 100 runs for each hydraulic approach: (

**a**) NDD; (

**b**) NPD; (

**c**) DDD; (

**d**) DPD. The error of the average values (i.e., average values line) is also displayed in each plot.

**Figure 6.**Comparison of the solutions. (

**a**) Pressure absolute errors; (

**b**) flow rate absolute errors. Error distribution of the best solution among the 100 runs (left box); average values (middle box); final solution (right box).

**Figure 7.**Resulting solution in the Apulian network with the DPD approach. In panel (

**a**) are displayed the pressure distribution of the 100 runs, the reference values, the average values and the final one with confidence intervals. The flow rate distribution of the 100 runs, the reference values, the average values and the final one with confidence intervals are displayed in panel (

**b**).

**Figure 8.**Modena network: (

**a**) model network where the calibration processes are launched; (

**b**) reference network where the measurements are taken.

**Figure 9.**Resulting calibration of the Modena network with the DPD approach. In panel (

**a**) is displayed the pressure distribution of the 100 runs, the reference values, the average values and the final one. The flow rate distribution of the 100 runs is displayed in panel (

**b**).

**Figure 10.**Absolute errors calculated as the absolute difference between reference values and simulated values in the final calibrated model of the Modena network. Panel (

**a**) is related to the pressure at each node and panel (

**b**) to the flow rate at each pipe.

**Table 1.**Measured flow rate (on the bottom) and measured pressure (on the top) data for the Apulian network.

Node (ID) | Pressure (m) |
---|---|

4 | 17.92 |

13 | 13.37 |

16 | 16.55 |

23 | 13.57 |

Pipe ID | Flow Rate (L/s) |

34 | 240.82 |

**Table 2.**MAE regarding the simulated pressure of the Best solution, the Average values and the Final solution with respect to the reference network.

Solution | Best Solution | Average Values | Final Solution | |
---|---|---|---|---|

Approach | (m) | (m) | (m) | |

NDD | 0.57 | 0.60 | 0.62 | |

NPD | 0.28 | 0.34 | 0.34 | |

DDD | 0.52 | 0.41 | 0.46 | |

DPD | 0.28 | 0.19 | 0.24 |

**Table 3.**MAE regarding the simulated flow rate of the Best solution, the Average values and the Final solution with respect to the reference network.

Solution | Best Solution | Average Values | Final Solution | |
---|---|---|---|---|

Approach | (L/s) | (L/s) | (L/s) | |

NDD | 4.60 | 4.66 | 4.76 | |

NPD | 4.29 | 4.33 | 4.34 | |

DDD | 4.95 | 2.98 | 3.63 | |

DPD | 3.57 | 2.49 | 2.55 |

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**MDPI and ACS Style**

Zanfei, A.; Menapace, A.; Santopietro, S.; Righetti, M.
Calibration Procedure for Water Distribution Systems: Comparison among Hydraulic Models. *Water* **2020**, *12*, 1421.
https://doi.org/10.3390/w12051421

**AMA Style**

Zanfei A, Menapace A, Santopietro S, Righetti M.
Calibration Procedure for Water Distribution Systems: Comparison among Hydraulic Models. *Water*. 2020; 12(5):1421.
https://doi.org/10.3390/w12051421

**Chicago/Turabian Style**

Zanfei, Ariele, Andrea Menapace, Simone Santopietro, and Maurizio Righetti.
2020. "Calibration Procedure for Water Distribution Systems: Comparison among Hydraulic Models" *Water* 12, no. 5: 1421.
https://doi.org/10.3390/w12051421