## 3 System Model and Parameterization
### Graph-Theoretic Approach for SCN Modelling  
SCNs can be modelled using a graph, where PMU and PDC are represented as the nodes and the communication link between them is represented by an edge in the graph. Let us consider the SCN as shown in Figure 1, where 12 electrical buses are considered. The SCN comprises four PMUs and two PDCs. The buses equipped with a PMU are shown in green, and the buses equipped with a PDC are shown in red. The remaining buses that are neither equipped with a PMU nor a PDC are shown in black. The electrical buses are electrically connected using Electrical Connections (EC), whereas PMUs and PDCs are communicated using Network Connections (NCs).  
From the communication network perspective, the PMU, PDC, and their interconnections using NC can be used for analysing SCNs based on a graph-theoretical approach. Let us consider the SCN as a graph \(G\), which is denoted as \(G(V,E)\), where \(V\) and \(E\) denote nodes and edges in the graph \(G\). If the SCN has \(n\) PMUs and \(k\) PDCs, then the total number of nodes corresponding to PMUs and PDCs in the graph will be \(N=n+k\). Apart from PMUs and PDCs, there will be many intermediate nodes in the SCNs. If \(U\) represents the set of intermediate nodes, such that \(u\in U=(1,2,\ldots,\delta)\), then the total number of nodes  
Figure 1: Illustration of an SCN for graph-theoretical modelling.  
in the network would be \(\eta=N+\delta\). Hence, \(\forall\{v_{i},v_{j}\}\in V\); there exists an edge \(e_{i-j}\in E\) connecting nodes \(v_{i}\in V\) and \(v_{j}\in V\), such that \(\{i,j\}=\{1,2,3,\ldots,\eta|i\
eq j\}\). We denote the existence of a path between two nodes \(v_{i}\in V\) and \(v_{j}\in V\) as \(\gamma_{i-j}\) which is defined as an unbroken connection through which one can traverse from node \(v_{i}\in V\) to node \(v_{j}\in V\). Moreover, path \(\gamma_{i-j}\) may consist of one or more edges. Thus, a path is a set of all nodes connected by edges joining the source and destination. In a network with more than one path, the different paths will be represented as a set of nodes connected through edges. Path '\(l\)' between any source \(i\) and destination \(j\) can be represented by \(\gamma_{i-j}^{l}\), where \(l=1,2,3\ldots,L\). Further, the number of paths is given by cardinality i.e., \(|\gamma_{i-j}|=L\). It should be noted that the graph is assumed undirected to support bidirectional synchrophasor data flow.  
For an SCN, it is not necessary to locate the PMUs on all grid buses. The entire system can be observable with a few PMUs located at their optimal locations. However, for the time being, let us assume a simplified diagram where the PMUs, PDCs, and the interconnection between them are as shown in Figure 1.  
As shown in Figure 1, the PMU and PDC are connected with a single edge that may have many intermediate nodes and edges. For the illustration, the network shown in Figure 2 was constructed from Figure 1 in such a way that a pair of PMU (\(v_{1}\)) and PDC (\(v_{6}\)) is connected through many intermediate nodes and edges. In this, node \(v_{1}\) and \(v_{6}\) represent the PMU and the PDC, whereas nodes \(v_{2},v_{3},v_{4}\), and \(v_{5}\) represent intermediate nodes on the IP network. Furthermore, node \(v_{0}\) indicates another device that generates background traffic along with the PMU (\(v_{1}\)) traffic. It is worth noting that \(|\gamma_{1-6}|=2\).