# A Multi-Leak Identification Scheme Using Multi-Classification for Water Distribution Infrastructure
## 3 Performance Evaluation of Different Classifier Scenarios  
### Experimental Setup for Evaluating MC-MLI  
In real WDI, the underground pipeline network consists of multiple branches for water distribution. In this paper, a multi-leak scenario is considered, and thus a multi-branch pipe network is established. Recently, plastic pipe has been more common in practical WDIs due to its low-cost, light-weight, chemically and electrically neutral, anti-rusting and anti-corrosion properties. Therefore, the pipes utilized in the experimental setup are made of PVC. The proposed scheme is applicable to other WDIs using pipes made of other materials by adjusting several parameters in the hydraulic model. The experimental setup of the multi-branch pipe network (i.e., 3 branches: B1, B2, B3) is shown in Figure 3. Each sensor group contains a flow meter (Q) and a pressure sensor (P). All sensors are connected to Arduino broad for data transmission. In the middle of each branch, one adjustable valve is installed to act as a leak point (L). This facilitates the evaluation of different leak sizes/rates by controlling the degree of valve opening.  
Figure 2: The parallel detection structure of MC-MLI.  
Graph theory is the best candidate to represent pairwise linkages between entities. In this paper, the WDI is transformed into a directed graph using graph theory, thus identifying the behavior of WDI and exploring the correlation between a real-world model and small-scale model. An example of transforming a realistic WDI model and the developed small-scale WDI model is shown in Figure 3. Water supply zones can be clustered by deploying cut-line valves and boundary valves. The valves control the amount of water flowing into the zones, and thus they can be transformed into control nodes \(c_{h}\) in the directed graph. Denoted \(n_{i}\) and \(w_{i-j}\) are the \(i\)-th node and the edge between the \(i\)-th and \(j\)-th nodes in the directed graph, respectively. The node \(n\) represents the reference and/or conjunction point in a realistic WDI, whereas the edge represents the pipe connecting two points. The pairwise relations between nodes can be formulated as an adjacent matrix, i.e.,  
\[A=\left[\begin{array}{ccc}a_{11}&\cdots&a_{1M}\\ \vdots&\ddots&\vdots\\ a_{M1}&\cdots&a_{MM}\end{array}\right] \tag{9}\]  
where the entry \(a_{ij}\) represents the directed connection from the \(i\)-th node to the \(j\)-th node. If there is a connection between the \(i\)-th node and the \(j\)-th node, \(a_{ij}\) equals 1. Otherwise, there is no connection between them, and \(a_{ij}\) equals 0.  
The essential metrics in graph theory include average node degree K, average node betweenness NB, edge betweenness EB, and the distribution of edge weighting. In order to compare these metrics between realistic a WDI and the small-scale WDI, the developed small-scale model is transformed into a graph representation, as shown in Figure 3. The transformed small-scale model is configured the same as one of the water supply zones. The correlation between the two transformed models is evaluated as follows.  
\[R_{m}=\frac{n(\sum r_{m}\times p_{m})-(\sum r_{m})(\sum p_{m})}{\sqrt{[n\sum r_{ m}{}^{2}-(\sum r_{m})^{2}][n\sum p_{m}{}^{2}-(\sum p_{m})^{2}]}} \tag{10}\]  
where \(n\) denotes the amount of data, \(r_{m}\) denotes the \(m\)-th metric of the transformed realistic WDI model, and \(p_{m}\) denotes the \(m\)-th metric of the transformed small-scale model.  
Figure 3: The experimental setup for multi-leakage detection.  
The results reveal that the transformed realistic WDI model is highly correlated to the transformed small-scale WDI model, reflected on the resultant coefficient of correlation (close to 1).