## 6 Observability of the graph
### Constrained Nonlinear Programming Model
Thereby, the solution is a global optimum point. The optimal result \(\{2,\quad 10,\quad 11,\quad 13\}\) is interpreted as follows: From Equation (33), we found the bus combination set solutions: \(\{2,1,\;3,\;4,\;5\},\{6,\;5,\;11,\;12,\;13\},\{7,4,\;8,\;9\},\)\(\{9,\;4,\;7,\;10,\;14\}\). Each first entry in each subset is the center bus, which is the dominant bus, and that bus is installed as a PMU [60], i.e., the PMU set solution \(\{2,\;6,\;7,9\}\)[60]. We definitely achieved a set solution with the maximum observability index for a fixed channel capacity \(L=4\). The branch assignments are observed by optimal placement of PMUs at power nodes and are as follows:

\[\left\{ \begin{aligned} & 2-1,\;2-3,\;2-4,\;2-5\\ & 6-5,\;6-11,\;6-12,\;6-13\\ & 7-4,\;7-8,\;7-9\\ & 9-4,\;9-7,\;9-10,\;9-14\end{aligned} \right\}\]

When \(L=4\), solving the \(0-1\) ILP model for the IEEE-14-bus network in Figure 3, four combinations are derived, that is, \(\{2,\;10,\;11,\;13\}\). The algorithmic scheme delivers PMUs installed at buses \(\{2,\;6,\;7,\;9\}\) of subsets \(\{2,1,3,4,5\}\), \(\{6,5,11,12,13\}\), \(\{7,4,8,9\}\), \(\{9,4,7,10,14\}\). The PMU locations are \(\{2,\;6,\;7,\;9\}\), where the branch assignments are displayed in Table 2[64]. The \(0-1\) ILP model with a single line is optimized with the intlinprog optimizer function [37; 38; 41; 51]. Optimal set solutions are derived and presented in Table 3 for the IEEE-14-bus, 30-bus, 57-bus, and 118-bus systems. Each number reflects a line where a single-line PMU is assigned across the two-points of the line [61]. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**L** & **nPMU** & **PMU Locations** & **PMU Channels** \\ \hline  & & & \{2-1,\;2-3,\;2-4,\;2-5\} \\  & & & \{6-5,\;6-11,\;6-12,\;6-13\} \\
4 & 4 & 2,\;6,\;7,\;9 & \{7-4,\;7-8,\;7-9\} \\  & & & \{9-4,\;9-7,\;9-10,\;9-14\} \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Channel capacity, PMU number, PMU locations, and PMU channels. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**IEEE Power Systems** & **Branches** & **PMU Number** & \multicolumn{2}{c}{**Solution Algorithm**} \\ \cline{3-4}  & & **Primal Simplex** & **Dual Simplex** \\ \hline
14-bus & 20 & 7 & 1, 6, 10, 14, 17, 18, 19 & 2, 3, 9, 12, 14, 18, 20 \\ \hline
30-bus & 41 & 15 & 2, 3, 8, 10, 13, 16, 20, 21, 23, 25, 29, 32, 34, 36, 39 & 1, 4, 8, 10, 13, 16, 20, 21, 23, 25, 29, 32, 34, 36, 39 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: **Algorithm BBA:** Optimizing the \(0-1\) ILP model with a single line. The second produces results by optimizing a b-objective function under a set of inequality constraints [41; 51]. The b-objective is minimized under a certain number of constraints so that an optimal point can be determined [38; 41; 51]. Intlinprog removes the decision bound integrality and permits all variables to be temporarily real [17; 18]. The solution was found by solving the relaxed problems in the BBA construction tree and this solution is feasible and optimal for the initial MILP [13; 14; 15; 21]. Each instance was derived within the pre-specified stopping and tolerance criterion [13; 14; 15; 16; 19; 21]. BBA utilizes the optimality gap as a stopping criterion to end the iterative scheme [13; 14; 15; 16]. It is calculated for all instances returning a zero value. We observe that the placement set gives an optimum point where each center observes the maximum number of power nodes [78]. Each subset is a connected dominating set with the characteristic that each center is a vertex reaching any other vertex inside that subset by a connectivity line [30; 60]. Therefore, the observability is attained even in a restricted channel capacity scenario, as noted in [56; 57; 58; 59; 60; 61; 62]. In the meantime, we suggest a mixed-integer nonlinear program that leads to optimality and works together with a similar formulation in the continuous domain. We perform the \(0-1\) integer nonlinear programming using a global BBA embedded in the YALMIP toolbox and the log files produce the desired output [37; 38; 75; 77].