# Sensitivity Analysis for Performance Evaluation of a Real Water Distribution System by a Pressure Driven Analysis Approach and Artificial Intelligence Method

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

_{i}. To obtain these values, an iterative procedure by using software Epanet has been adopted. This approach allows defining which nodes work in PDA conditions for each k* value assumed.

_{0.3}= 0.3), the average conditions (k*

_{1.0}= 1.0), and the peak conditions (k*

_{1.8}= 1.8). In the second step, for each value of k*

_{i}an analysis with a PDA approach has been carried out by varying k*

_{i}in a defined range.

_{j,i}are obtained calculating them as follows in Equation (1):

_{i}from the design value adopted and it is considered as j = 80%, 85%, 90%, 95%, 100%, 105%, 110%, 115%, and 120%. In this case, the procedure furnishes the number of nodes working in PDA conditions: in these nodes, the user request is partially satisfied.

_{0.3}= 0.3, k*

_{1.0}= 1.0, and k*

_{1.8}= 1.8.

#### 2.1. Pressure Driven Analysis (PDA)

_{BD}.

_{real}for each user depends on the real head value calculated as the totality of elevation (z) and piezometric height (p/γ), i.e., the ratio among pressure (p) and a specific weight (γ). Here H

_{s}, the service head, is defined as the sum of ground level and p/γ

_{min}based on Equation (2):

- Z is the elevation of ground level;
- p/γ
_{min}is the lowest piezometric head essential to deliver the demand to the users; it depends on the height of building H_{b}; - H
_{b}is the height of each supplied building; - P
_{ms}represents the minimum pressure essential in each point of the building, usually 5 m; - P
_{p}indicates the head losses along the riser column; - P
_{D}represents the head losses between a network node and the base of each building.

_{min}is the head necessary to serve users at a ground level based on Equation (3):

_{min}= z + P

_{ms},

_{s}, Q

_{real}is equal to Q

_{BD}. If the head is lower than H

_{min}there is no service and Q

_{real}= 0. In the other cases, Q

_{real}can be calculated by using Equation (4)

_{real}= α Q

_{BD},

_{real}and Q

_{BD}: it represents the percentage of supplied flow and it can be obtained using the relations indicated below [28] and according to Figure 2:

#### 2.2. Sensitivity Analysis

_{1}, …, x*

_{n}), i.e., F* = f(x*

_{1}, …, x*

_{n}), the variation of the function, depending on x

_{j}parameter, can be calculated by direct differentiation as shown in Equation (8):

_{1}, … x*

_{m}, …, x*

_{n}) and F* = f(x*

_{1}, … x*

_{m}, …, x*

_{n}) is the value of calculated function value obtained by the model. When a single parameter changes a new value of F = f(x*

_{1}, …, x*

_{m}+ Δx

_{m}, …, x*

_{n}) can be calculated: the parameter variation can be expressed as Δx

_{m}/x*

_{m}= (x

_{m}− x*

_{m})/x*

_{m}.

_{m}changes, can be calculated through the finite differences method according to Equation (9):

_{j}− x

_{j}*)/x

_{j}* which is the variation of the parameter x

_{j}* as a percentage of the estimate value x

_{j}* adopted to evaluate F*.

#### 2.3. Group Method of Data Handling Algorithm

## 3. Case Study

_{m}in the network in steady-state conditions is 29.00 L/s for about 8500 users, according to the real data acquired. The value of H

_{min}varies from 300 to 353 m and H

_{s}varies from about 315 to 368 m. In the analysis, the value of p/γ

_{min}= 20 m has been assumed for each node. These input data are related to winter conditions because during the summer Q can vary significantly with the seasonal fluctuant.

^{3}.

_{i}is the peak coefficient, which is the ratio between the real demand Q

_{real}in each single one-hour time step and the base demand Q

_{m}.

_{i}coefficient adopted in the analysis are shown in Table 1 and they are those of a typical pattern (Figure 4) proposed for a similar town: k*

_{0.3}= 0.3 describes night condition, k*

_{1.0}= 1.0 is characteristic of the average conditions and k*

_{1.8}= 1.8 is the peak condition.

_{i-Train}, F*

_{i-Test}, and F*

_{i-Total}can be calculated and their values for each value of k*

_{i}can be obtained.

_{j,i}value have been conducted for each scenario. Values of k*

_{j,i}are shown in Table 2:

_{j,i}value were conducted, and then the hydraulic parameters in the network including the base demand, the pressure, and alpha were measured.

_{i}values according to sensitivity analysis, the aim of the study is to evaluate if corresponding calculated GMDH results are good and stable and very similar to the ones obtained with the design adopted k*

_{i}values.

## 4. Modelling by GMDH

_{s}and H ≥ Hs, respectively. In each scenario, several binary classification models are developed and the best models for each scenario are selected. Figure 6, Figure 7 and Figure 8 indicate the results for k*

_{0.30}= 0.3, k*

_{1.0}= 1, and k*

_{1.8}= 1.8, respectively. It is worth mentioning that the accuracy of each class for output and target is shown in the extra columns and rows in the confusion matrices.

_{0.3}= 0.3 was determined when the SP, MNL, and MNNL are equal to 0.6, 15, and 5, respectively. According to the results of binary classification for the first scenario with k*

_{0.3}= 0.3, this developed model was able to identify and determine a very suitable mapping between input and output data. The model structure assumes the MNL and MNNL equal to 5 and 15, correspondingly. The developed model could correctly predict and classify 18 nodes H < H

_{s}with the label “0”. Additionally, 28 and 3 nodes could be predicted and classified correctly and wrongly, respectively. Finally, this model could predict and classify the total dataset with an accuracy equal to 93.9%.

_{1.0}= 1.0 was determined as the SP, MNL, and MNNL equal to 0.6, 15, and 10, respectively. Compared to the structure of the best model with k*

_{1.0}= 1.0, there is no significant change and only the value of MNNL has changed. The best-developed binary classification model could predict with 100% accuracy for training data and 83.3% accuracy for testing data. Consequently, the total accuracy of this model was 95.9% that 47 nodes were correctly predicted and only 2 nodes were wrongly predicted.

_{1.8}= 1.8 after many modelings, the results showed that the structure of the best model was similar to the structure of the best model for k*

_{1.8}= 1.8. The accuracies of training and testing data were 91.9% and 91.7%, respectively. This model could predict 22 nodes (H < H

_{s}) with the label “0” as correct with 100% accuracy in all data. Additionally, from 27 nodes (H ≥ H

_{s}) with the label “1”, 23 nodes were correctly predicted, and the rest was predicted incorrectly. Therefore, this model was able to predict the total amount of data with 91.8% accuracy. Finally, it was found that the binary classification approach can provide suitable performance capacity in predicting the performance of water distribution networks for k*

_{0.3}= 0.3, k*

_{1.0}= 1.0, and k*

_{1.8}= 1.8.

## 5. Results and Discussion

_{i}). Table 3 and Table 4 show the three structures of best classification models for the three scenarios and a comparison of their results, respectively. It is necessary to mention that in this section to conduct a sensitivity analysis, the F* is used to present the results of accuracy of Train, Test, and Total for modeling with k*

_{0.3}= 0.3, k*

_{1.0}= 1.0, and k*

_{1.8}= 1.8. Furthermore, the F is considered for Train, Test, and Total to show the results of accuracy for modeling with the different values of k*

_{j,i}.

_{0.3}= 0.3, k*

_{1.0}= 1.0, and k*

_{1.8}= 1.8, the best model for each k*

_{i}is modeled varying this value: the change in the percentage of k

_{i}* is between 80% and 120% with units of 5%

_{j,i}, F*

_{j,i-Train}, F*

_{j,i-Test}, F*

_{j,i-Total}were calculated. Furthermore, the values of sensitivity functions (F*

_{j,i}− F

_{i}*)/F

_{i}* for each of them (Train, Test, Total) were also calculated.

#### 5.1. Scenario k*_{0.3}

_{j,0.3}

_{-Train}, F*

_{j,0.3}

_{-Test}, F*

_{j,0.3}

_{-Total}, and sensitivity function (F*

_{j,0.3}− F*

_{0.3})/F*

_{0.3}for Train, Test and Total obtained varying k*

_{0.30}= 0.30 in the range are shown in Table 5 and in terms of sensitivity plot in the Figure 9a–f.

_{j,0.3}(Train, Test, Total) are good: all values are higher than 91.9% for Train, 83.3% for Test, and 89.8% for Total. This confirms the quality of the results and the model accuracy. The sensitivity function assumes values that confirm the stability of the model results because the sensitivity function is always lower than 10%. If the input value of k*

_{i}is slightly different, due to an error or variability in a real case, the models furnish very similar results.

#### 5.2. Scenario k*_{1.0}

_{j,1.0}

_{-Train}, F*

_{j,1.0}

_{-Test}, F*

_{j,1.0}

_{-Total}and sensitivity function (F*

_{j,1.0}− F*

_{1.0})/F*

_{1.0}for Train, Test and Total obtained varying k*

_{1.0}= 1.0 in the range are shown in Table 6 and in terms of sensitivity plot in Figure 10a–f.

_{1.0}the values of each F*

_{j,1.0}(Train, Test, Total) are also good. The F*

_{j,1.0}values are higher than 94.6% for Train, 83.3% for Test, and 91.8% for Total. This is in agreement with the k*

_{0.3}case and confirms the goodness of both the results and the model accuracy.

#### 5.3. Scenario k*_{1.8}

_{j,1.8}

_{-Train}, F*

_{j,1.8}

_{-Test}, F*

_{j,1.8}

_{-Total}and sensitivity function (F*

_{j,1.8}− F*

_{1.8})/F*

_{1.8}for Train, Test and Total obtained varying k*

_{1.8}= 1.8 in the range are shown in Table 7 and in terms of sensitivity plot in Figure 11a–f.

_{1.8}scenario. F*

_{j,1.8}values for Train, Test, and Total are all higher than 89.8%. These results do not differ from previous ones and confirm the quality of the results and the model accuracy.

## 6. Conclusions

_{0.3}= 0.3, k*

_{1.0}= 1.0, and k*

_{1.8}= 1.8 were 93.9%, 95.9%, and 91.8% for all data, respectively. It was found that the binary classification approach demonstrates its high capability in the prediction and evaluation of WDNs. The comparison between the best classification models for the different values of each k*

_{i}showed the high capability of the model in predicting the performance of water distribution networks and the sensitivity analysis confirms that the results are stable.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**The relation among supplied flow and head at each node [28].

**Figure 6.**Results of the confusion matrix for training data (

**a**), testing data (

**b**), and total data (

**c**) for k*

_{0.3}= 0.3.

**Figure 7.**Results of the confusion matrix for training data (

**a**), testing data (

**b**), and total data (

**c**) for k*

_{1.0}= 1.0.

**Figure 8.**Results of the confusion matrix for training data (

**a**), testing data (

**b**), and total data (

**c**) for k*

_{1.8}= 1.8.

**Figure 9.**Values of F*

_{j,0.30}

_{-Train}, F*

_{j,0.30}

_{-Test,}and F*

_{j,}

_{0.30}

_{-Total}and sensitivity function for the scenario with k*

_{0.30}.

**Figure 10.**Value of F*

_{j,}

_{1.0}

_{-Train}, F*

_{j,}

_{1.0}

_{-Test,}and F*

_{j,}

_{1.0}

_{-Total}and sensitivity function for the scenario with k*

_{1.0}.

**Figure 11.**Values of F*

_{j,1.8}

_{-Train}, F*

_{j,1.8}

_{-Test,}and F*

_{j,1.8}

_{-Total}and sensitivity function for the scenario with k*

_{1.8}.

Scenario | Q_{real} (L/s) |
---|---|

k*_{0.3} = 0.3 | 9.7 |

k*_{1.0} = 1.0 | 29.0 |

k*_{1.8} = 1.8 | 52.2 |

j (%) | k*_{j,i} = j × k*_{i} | k*_{j,0.30} | k*_{j,1.0} | k*_{j,1.8} |
---|---|---|---|---|

80% | k*_{0.8,i} | 0.240 | 0.800 | 1.440 |

85% | k*_{0.85,i} | 0.255 | 0.850 | 1.530 |

90% | k*_{0.9,i} | 0.270 | 0.900 | 1.620 |

95% | k*_{0.95,i} | 0.285 | 0.950 | 1.710 |

100% | k*_{1,i} | 0.300 | 1.000 | 1.800 |

105% | k*_{1.05,i} | 0.315 | 1.050 | 1.890 |

110% | k*_{1.1,i} | 0.330 | 1.100 | 1.980 |

115% | k*_{1.15,i} | 0.345 | 1.150 | 2.070 |

120% | k*_{1.2,i} | 0.360 | 1.200 | 2.160 |

No. Scenario | k*_{i} | Selection Pressure (SP) | Maximum Number of Layers (MNL) | Maximum Number of Neurons in a Layer (MNNL) |
---|---|---|---|---|

1 | 0.3 | 0.6 | 15 | 5 |

2 | 1 | 0.6 | 15 | 10 |

3 | 1.8 | 0.6 | 15 | 10 |

**Table 4.**Values of F*

_{i,Train}, F*

_{i,Test}and F*

_{i,Total}assuming k*

_{0.3}= 0.3, k*

_{1.0}= 1.0 and k*

_{1.8}= 1.8.

No. Scenario | k*_{i} | F*_{i-Train} | F*_{i-Test} | F*_{i-Total} |
---|---|---|---|---|

1 | k*_{0.3} = 0.3 | 94.6 | 91.7 | 93.9 |

2 | k*_{1.0} = 1.0 | 100 | 83.3 | 95.9 |

3 | k*_{1.8} = 1.8 | 91.9 | 91.7 | 91.8 |

**Table 5.**Values of F*

_{j,0.30-Train}, F*

_{j,0.30-Test}, and F*

_{j,0.30-Total}by varying k*

_{0.30}in the fixed range.

k*_{j,0.3} | % | F*_{j,0.3}_{-Train} | $\frac{{\mathbf{F}}_{\mathbf{j},0.3\text{-}\mathbf{Train}}^{*}-{\mathbf{F}}_{0.3\text{-}\mathbf{Train}}^{*}}{{\mathbf{F}}_{0.3\text{-}\mathbf{Train}}^{*}}$ | F*_{j,0.3}_{-Test} | $\frac{{\mathbf{F}}_{\mathbf{j},0.3\text{-}\mathbf{Test}}^{*}-{\mathbf{F}}_{0.3\text{-}\mathbf{Test}}^{*}}{{\mathbf{F}}_{0.3\text{-}\mathbf{Test}}^{*}}$ | F *_{j,0.3}_{-Total} | $\frac{{\mathbf{F}}_{\mathbf{j},0.3\text{-}\mathbf{Total}}^{*}-{\mathbf{F}}_{0.3\text{-}\mathbf{Total}}^{*}}{{\mathbf{F}}_{0.3\text{-}\mathbf{Total}}^{*}}$ |
---|---|---|---|---|---|---|---|

0.24 | 80 | 97.3 | 0.03 | 83.3 | −0.09 | 93.9 | 0.00 |

0.255 | 85 | 94.6 | 0.00 | 91.7 | 0.00 | 93.9 | 0.00 |

0.27 | 90 | 97.3 | 0.03 | 83.3 | −0.09 | 93.9 | 0.00 |

0.285 | 95 | 100 | 0.06 | 91.7 | 0.00 | 98.0 | 0.04 |

0.3 | 100 | 94.6 | 0.00 | 91.7 | 0.00 | 93.9 | 0.00 |

0.315 | 105 | 94.6 | 0.00 | 91.7 | 0.00 | 93.9 | 0.00 |

0.33 | 110 | 94.6 | 0.00 | 83.3 | −0.09 | 91.8 | −0.02 |

0.345 | 115 | 91.9 | −0.03 | 83.3 | −0.09 | 89.8 | −0.04 |

0.36 | 120 | 94.6 | 0.00 | 91.7 | 0.00 | 93.9 | 0.00 |

**Table 6.**Value of F*

_{j,}

_{1.0-Train}, F*

_{j,}

_{1.0}

_{−}

_{Test}, and F*

_{j,}

_{1.0-Total}by varying k*

_{1.0}in the fixed range.

k*_{j,1.0} | % | F*_{j,1.0}_{-Train} | $\frac{{\mathbf{F}}_{\mathbf{j},1.0\text{-}\mathbf{Train}}^{*}-{\mathbf{F}}_{1.0\text{-}\mathbf{Train}}^{*}}{{\mathbf{F}}_{1.0\text{-}\mathbf{Train}}^{*}}$ | F*_{j,1.0}_{-Test} | $\frac{{\mathbf{F}}_{\mathbf{j},1.0\text{-}\mathbf{Test}}^{*}-{\mathbf{F}}_{1.0\text{-}\mathbf{Test}}^{*}}{{\mathbf{F}}_{1.0\text{-}\mathbf{Test}}^{*}}$ | F*_{j,1.0}_{-Total} | $\frac{{\mathbf{F}}_{\mathbf{j},1.0\text{-}\mathbf{Total}}^{*}-{\mathbf{F}}_{1.0\text{-}\mathbf{Total}}^{*}}{{\mathbf{F}}_{1.0\text{-}\mathbf{Total}}^{*}}$ |
---|---|---|---|---|---|---|---|

0.8 | 80 | 94.6 | −0.05 | 91.7 | 0.10 | 93.9 | −0.02 |

0.85 | 85 | 100 | 0.00 | 91.7 | 0.10 | 98 | 0.02 |

0.9 | 90 | 97.3 | −0.03 | 91.7 | 0.10 | 95.9 | 0.00 |

0.95 | 95 | 100 | 0.00 | 91.7 | 0.10 | 98 | 0.02 |

1 | 100 | 100 | 0.00 | 83.3 | 0.00 | 95.9 | 0.00 |

1.05 | 105 | 97.3 | −0.03 | 91.7 | 0.10 | 95.9 | 0.00 |

1.1 | 110 | 94.6 | −0.05 | 83.3 | 0.00 | 91.8 | −0.04 |

1.15 | 115 | 97.3 | −0.03 | 91.7 | 0.10 | 95.9 | 0.00 |

1.2 | 120 | 97.3 | −0.03 | 83.3 | 0.00 | 93.9 | −0.02 |

**Table 7.**Values of F*

_{j,}

_{−0-Train}, F*

_{j,}

_{1.80-Test}, and F*

_{j,}

_{1.80-Total}by varying k*

_{1.80}in the fixed range.

k*_{j,1.8} | % | F*_{j,1.8}_{-Train} | $\frac{{\mathbf{F}}_{\mathbf{j},1.8\text{-}\mathbf{Train}}^{*}-{\mathbf{F}}_{1.8\text{-}\mathbf{Train}}^{*}}{{\mathbf{F}}_{1.8\text{-}\mathbf{Train}}^{*}}$ | F*_{j,1.8}_{-Test} | $\frac{{\mathbf{F}}_{\mathbf{j},1.8\text{-}\mathbf{Test}}^{*}-{\mathbf{F}}_{1.8\text{-}\mathbf{Test}}^{*}}{{\mathbf{F}}_{1.8\text{-}\mathbf{Test}}^{*}}$ | F*_{j,1.8}_{-Total} | $\frac{{\mathbf{F}}_{\mathbf{j},1.8\text{-}\mathbf{Total}}^{*}-{\mathbf{F}}_{1.8\text{-}\mathbf{Total}}^{*}}{{\mathbf{F}}_{1.8\text{-}\mathbf{Total}}^{*}}$ |
---|---|---|---|---|---|---|---|

1.44 | 80 | 94.6 | 0.03 | 91.7 | 0.00 | 93.9 | 0.02 |

1.53 | 85 | 94.6 | 0.03 | 83.3 | −0.09 | 91.8 | 0.00 |

1.62 | 90 | 97.3 | 0.06 | 91.7 | 0.00 | 95.5 | 0.04 |

1.71 | 95 | 97.3 | 0.06 | 83.3 | −0.09 | 93.9 | 0.02 |

1.8 | 100 | 91.9 | 0.00 | 91.7 | 0.00 | 91.8 | 0.00 |

1.89 | 105 | 97.3 | 0.06 | 83.3 | −0.09 | 93.9 | 0.02 |

1.98 | 110 | 97.3 | 0.06 | 83.3 | −0.09 | 93.9 | 0.02 |

2.07 | 115 | 91.9 | 0.00 | 83.3 | −0.09 | 89.8 | −0.02 |

2.16 | 120 | 91.9 | 0.00 | 83.3 | −0.09 | 89.8 | −0.02 |

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## Share and Cite

**MDPI and ACS Style**

Fiorini Morosini, A.; Shaffiee Haghshenas, S.; Shaffiee Haghshenas, S.; Choi, D.Y.; Geem, Z.W. Sensitivity Analysis for Performance Evaluation of a Real Water Distribution System by a Pressure Driven Analysis Approach and Artificial Intelligence Method. *Water* **2021**, *13*, 1116.
https://doi.org/10.3390/w13081116

**AMA Style**

Fiorini Morosini A, Shaffiee Haghshenas S, Shaffiee Haghshenas S, Choi DY, Geem ZW. Sensitivity Analysis for Performance Evaluation of a Real Water Distribution System by a Pressure Driven Analysis Approach and Artificial Intelligence Method. *Water*. 2021; 13(8):1116.
https://doi.org/10.3390/w13081116

**Chicago/Turabian Style**

Fiorini Morosini, Attilio, Sina Shaffiee Haghshenas, Sami Shaffiee Haghshenas, Doo Yong Choi, and Zong Woo Geem. 2021. "Sensitivity Analysis for Performance Evaluation of a Real Water Distribution System by a Pressure Driven Analysis Approach and Artificial Intelligence Method" *Water* 13, no. 8: 1116.
https://doi.org/10.3390/w13081116