# Burst Detection in Water Distribution Systems:
## 3 Application
### Egna  
The WDS of Egna has been considered as the second test case to test the reliability of the proposed data generation methodology. Figure 7 reports the Egna aqueduct, which consists of around 28 km of pipes with diameter between 50 and 150 mm and two tanks, which supply water to the WDS. In particular, the network model consists of 169 pipes and 149 junctions. More information about the network can be found in [46]. Differently from the previous test case, this WDN does not exhibit a pressure deficit, providing consequently further scenarios for evaluating the proposed methodology. In addition, this test results in being more interesting due to the availability of a small dataset of measured hourly water consumption during March and April 2019. It is worthwhile pointing out that the water demand always matches the water request due to the high operating pressure in the Egna WDS. Figure 6: Box & Whiskers plot of the four-year Apulian consumption for each hour of the day. The red points represent hours where at least a burst is active, and black boxes and points stand for normal data. As a preliminary step, to generate a proper water consumption time series, which are close to the real Egraa WDS, the SHtsG has been set up with the data of March and then tested with the April data. Firstly, a data imputation has been performed adopting the Kalman filter jointly with Arima model according to [47] to fill \(13\,na\) values into the dataset. Once the dataset was fulfilled, the consumption time series of March has been decomposed into the different components as follows:  
\[\widetilde{\overline{W}t}=\widetilde{\overline{W}d}+\widetilde{\overline{W}l}+ \widetilde{\overline{W}b}, \tag{20}\]  
where \(\widetilde{\overline{W}t}\) represents the monthly average water consumption, \(\widetilde{\overline{W}d}\) is the monthly average water demand, \(\widetilde{\overline{W}l}\) represents the monthly average background losses and \(\widetilde{\overline{W}b}\) the monthly average bursts. By analysing the March consumption, it emerges that the minimum night flow (which ranges from 2:00 a.m. to 4:00 a.m.) is constant during March. This leads to the assumption that no bursts are present in this period. This follows that the \(\widetilde{\overline{W}b}\) term in Equation (20) is zero. In addition, the water demand \(\widetilde{\overline{W}d}\) of this month has been calculated given the water request of \(123\,\frac{l}{\overline{huditant\text{-}day}}\) and the number of habitants equal to 5290. Thus, the average background losses \(\widetilde{\overline{W}l}\) has also been directly calculated through Equation (20). To decompose the signal hour per hour, the behavior of the background losses during the days has been evaluated. The relationship for undetected leaks [48] has been adopted as follows:  
\[\frac{Wl_{t}}{\overline{W}l}=\left(\frac{P_{t}}{\overline{P}}\right)^{1.5}, \tag{21}\]  
where \(Wl_{t}\) is the background losses term at the \(t\)-\(th\) hour, the \(\overline{W}l\) is the average background losses of March, the \(P_{t}\) is the average network pressure at the \(t\)-\(th\) hour, and the \(\overline{P}\) term is the average network pressure in the month of March. The pressure of the network is known because of a measurement campaign made in the same month. Hence, the behavior of the background losses was estimated through Equation (21) and the time series has been decomposed. Differently from the March period, the time series of the consumption of April results in being not stationary during the night. This means that the term \(\widetilde{\overline{W}b}\) is different from zero during some days in April. To decompose the signal for April, the background losses have been assumed constant  
Figure 7: The water distribution system of Egraa and its six districts. in time assuming that the degradation of the network in a short period of a month is unchanged. Thus, the same values of \(Q_{l,i}\) of March were also used for April.