## 6 Observability of the graph
### Constrained Binary-Integer Programming Model with a Single-Line PMU  
Instead of using multi-channel PMUs, single-line PMUs observe a single line by calculating the running current in the line and the phasors' voltage at the end point of that line [61]. The problem is referred to as the minimum edge cover problem in graph theory and it is directly related to the minimum set covering problem, which is a combinatorial optimization problem [28; 29; 30; 31]. An edge cover of a graph \(G\) is a subset of edges \(F\subseteq E(G)\) such that for any \(u\in V\), there exists \(e\in F\) such that \(u\) is an endpoint of \(e\). As given in the definition, the power edge set (PES) problem is as follows [28; 29; 30; 31; 37; 38; 41; 51]:  
Definition 6: Declaration of the edge cover problem: Given a \(G=(V,E)\), find an edge subset \(\Pi\subset E\) of minimum cardinality such that a placement of PMUs on edges over \(\Pi\) yields the full observability of the graph \(G\)[28; 29; 31]. It is noticed that the OPP finds a subset of nodes such that the whole graph is topologically observable similar to the \"set covering problem\" [28; 29; 30; 31]. The problem solving goal is to monitor a single branch, namely the sending end node voltage and the sending end current flow towards the receiving end bus [61]. We note that single-line PMUs are placed at either end of the line since, in either case, both end bus voltages will be calculated due to the availability of the line current [61]. Thus, the problem is declared in installing PMUs to lines and not to nodes. If a PMU is considered to be a line, it can be installed arbitrarily at either end of that line without changing the output of the observability-based model [61; 64]. Thereby, the optimal branch phasor current measurement problem is to discover the minimum set of line phasor currents such that a power node is monitored by, at least and at most, one line phasor current [28; 29; 30; 31; 61]. The problem is declared as a binary-integer program, that is, a set covering problem, or otherwise a minimum edge set problem [28; 29; 30; 31; 61]. An edge cover is a subset of edges determined similarly to the vertex cover, that is, a collection of graph edges so as a union of edge endpoints relating to the vertex set of the graph [28; 29; 30; 31; 61]. The solution of the edge cover problem gives the smallest possible number of edges for a specific graph [28; 29; 31; 61]. In this paragraph, we study the OPP model with one channel, such that to guarantee a complete observability of the whole graph \(G\)[28; 29; 30; 31; 32]. If a PMU is selected for a branch, it can be installed arbitrarily at either end of that branch without changing the outcome of network observability analysis [31; 60; 61]. Thus, it confirms the observation that the matching is perfect [28; 29]. This is true if it covers every vertex of the graph, that is, every vertex is within exactly one edge of the matching where an edge is shown as a set of two different vertices [31; 60; 61]. This case considers the PMU where the PMU installed at a bus monitors a single line by computing the voltage phasor and the phasor current running this line [60; 61]. A PMU number is computed to maintain the whole observability [60; 61]. Let us assume a system consists of \(n\) buses and \(L\) branches [78]. The OPP problem is declared in a binary-integer programming format [61]:  
\[ \begin{split}&\mathit{min}\{x\}=\sum_{i=1}^{L}c_{i}x_{i}\\ & s.t.\left\{\begin{array}{l}TX\geq\overset{\rightarrow}{1}\\ x_{i}\in\{0,1\}^{L}\end{array}\right.\end{split}  \tag{19}\]  
where [31; 60; 61]:  
\[ \begin{split}& L:\text{the number of branches}\\ & x_{i}=\left\{\begin{array}{l}1\text{ \emph{a PMU is placed on branch i}}\\ 0\text{ otherwise}\end{array}\right.\end{split}  \tag{20}\]  
\[ \begin{split}& c_{i}:\text{cost of installation of a PMU at branch i}\\ &\text{Tij}=\left\{\begin{array}{l}1\text{ if branch j incident to bus i}\\ 0\text{ otherwise}\end{array}\right.\end{split}