## 6 Observability of the graph
### Constrained Binary-Integer Programming Model with a Fixed Number of Channels  
The widespread adoption of PMUs is gradually advancing but faces impediments from technical limitations and economic factors, thus hindering its pace [56; 57; 58]. In actual conditions, every PMU has a limited number of channels, and therefore we need a more realistic installation of PMUs in a way to further decrease the cost of the investment [56; 57; 59; 60; 61; 62; 63; 64; 65; 66; 67]. Every PMU produced by the industry has a restricted number of channels for installation at a substation [52; 56; 57; 58; 59; 60]. The number of branches adjacent to a power grid node in which a PMU is placed is known as channel capacity [60]. Considering the channel limitation capacities there is a reasonable query: which is the appropriate set of topological constraints to lead to a least PMU number with an adequate number of observed branches without sacrificing the entire observability at all [52; 56; 57; 58; 60]. Here we implement the OPP-based model when the number of measurement channels is restricted and the zero-injection buses are not assumed in the formulation. We have taken into account a PMU which has an \(L\) number of channels and is placed at a bus \(k\) within the power grid [56; 57; 47; 58]. We also consider that the bus \(k\) is adjacent to \(Nk\) number of buses [56; 57; 58]. If the number of channels \(L\) is greater than the number of neighbors \(Nk\), then a single PMU installed at the bus will give the phasor voltages at all its neighbor buses [56; 57; 58]. Differently, there will be \(rk\) combinations of possible channel assignments to branches adjacent to bus \(k\)[56; 57; 58; 59; 60; 61; 62; 63; 64; 65; 66; 67]:  
\[r_{k=} \begin{cases}\begin{pmatrix}Nk\\ L\end{pmatrix}&if\ L\leq Nk\\ 1&if\ Nk<L\end{cases}  \tag{13}\]  
where \(Nk\) is the number of lines joined to bus k, and \(L\) is the pre-given channel limit for the PMUs [56; 57; 58; 69; 60; 61; 62; 63; 64; 65; 66; 67]. The number of these combinations is defined by [56; 59; 60; 61; 62; 63; 64; 65; 66; 67]:  
\[m=\sum_{i=1}^{n}r_{i}^{l} \tag{14}\]  
where:  
\[ci= \begin{pmatrix}Nk\\ L\end{pmatrix} =\frac{Nk!}{L!\times(Nk-l)!} \tag{15}\]  
\(Nk\) is the number of branches, \(L\) is the number of available channels, and \(ci\) is the number of combinations of possible channel arrangements to lines adjacent to the related bus \(i\)[56; 57; 58; 69; 60; 61; 62; 63; 64; 65; 66; 67]. Figure 2: The flow chart of particle swarm optimization. The channel limit constraint function can be adopted so that a PMU posed at a node will monitor its adjacent nodes by choosing a suitable combination of L [56; 57; 58]. The OPP is declared using the \(0-1\) integer linear programming model based on a restricted number of channels, and can be declared as follows [56; 57; 58]:  
\[minJ(x)=w^{T}\times Y=\sum_{i=1}^{m}w_{i}\times y_{m} \tag{16}\]  
\[f(Y)=B^{T}\times Y\geq\overset{\rightarrow}{1}_{n} \tag{17}\]  
\[Y\in\{0.1\}^{m} \tag{18}\]  
where \(w^{T}=|1,1,\ldots,1|^{m}\) and the matrix \(B_{m\times n}\) consists of rows representing the possible combinations [56; 57; 58]. \(Y_{m\times 1}\) represents a PMU placement design that gives a combination where one bus is dominant and observes the neighboring buses [56; 57; 58]. This optimization problem is equivalent to the set covering problem [28; 29; 31], that is, the problem of seeking out the smallest number of subsets in the family of sets that constitutes the rows of the matrix \(B\)[56; 57; 58]. Hence, a PMU placement is derived in which each element observes a subset of buses [60]. \(Y\) is a placement vector and is formulated by \(Y=\{y_{1},y_{2},\ldots,y_{m}\}^{T}\in\{0,1\}\), where m is the number of subsets [56; 57; 58]. Each subset is constituted by a set of buses where the center bus observes some of the neighboring buses. The center bus is the dominant bus and is the first entry in the subset, as noted in [60].