# Burst Detection in Water Distribution Systems:
## 3 Application
### Apulian
Due to the presence of background leakages along the entire network, it is interesting to note that such water losses follow the average pressure of the network. Once the background losses was modeled, the bursts generator has randomly introduced local pipe breaks along the networks and then the WDS has been simulated by DFPDM producing the time series of the consumption displayed in Figure 4. It is worth noting that the type of bursts generated, their characteristics, and frequency have been set according to the WSD characteristics, following Section 2.2. To emphasize the robustness of the SHtsG by the generation of a high number of pipe failures, the average age of the pipes has been assumed to be 60 years, and also the state of all the pipes of the WDS has been classified as \"pipes with one or more previous breaks\" [43]. Hence, the \(c_{1}\) coefficient of Equation (19) has been fixed to 7.364. The other coefficient has been fixed by assuming the value of pit cast iron pipes, which are 0.02577 and 0.0207 for \(a\) and \(b\), respectively. Regarding the magnitude of the bursts, the geometric parameters of the crack in Equations (15)-(17) have been randomly generated to grant the range of lost flow rate between 10 L/s and 70 L/s. In addition, the repair time has also been random between 12 and 48 h. The detail of a random generated burst is given in Figure 5.  
Figure 4: Total consumption of Apulian network generated for a period of four years. The plot also displays the bursts, the background losses, and the average daily consumption. Figure 5a shows the behavior of the network consumption when a burst is generated. The outlet flow rate of a burst depends on two main reasons: firstly, the characteristic of the burst itself (for details, see Section 2.2) and, secondly, the pressure where the bursts happen. This punctual pressure derives from the network capacity in terms of both pressure and flow rate. This dependency is well displayed in Figure 5b, where the burst is displayed with the available pressure at each network nodes during the time. In particular, a slight pressure drop at the nodes close to the pipe burst are present during the burst occurrence. Because the outlet flow rate fully depends on the local pressure, the behavior of the outlet flow rate follows the local pressure. These considerations are fundamental to properly model the consumption of the time series for anomaly detection purposes. In fact, the pressure dependency of a burst is mandatory to be modeled to catch the physics of this phenomenon, even more, when the network has a lack of pressure. These notes are also highlighted in Figure 6.  
Figure 5: Detail of simulated bursts in the Apulian network. (**a**) shows a burst event with its consumption behavior, while (**b**) depicts the water losses (burst and background leaks) and the pressure of each network nodes with also the average network pressure. Figure 6 shows the distribution of the generated consumption for each daily hour in the Apulian network. In particular, the black boxes represent the quantiles of the hourly water consumption of normal data, i.e., not affected by bursts, and the black points represent their outliers. The abnormal data are instead labeled with the red color. It is noteworthy that a lot of abnormal data lie inside the hourly boxes of the normal data during the daytime, and, vice versa, several outliers belong to normal data. This behavior has to be imputed to the scarce operating pressure of the WDS. This means that the scarce pressure condition in the network does not allow for sufficing to both burst and consumption. In conclusion, both the satisfied water demand and the outlet flow rate of a burst are dependent on the pressure; specifically, both of them decrease with a pressure drop. Thereby, a burst occurrence causes a decrease in the network pressure and consequently a reduction into the satisfied water request, resulting in a moderate changing of the final WDS flow rate distribution.