## 6 Observability of the graph
### Constrained Nonlinear Programming Model
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Interior-Point Methods**} \\ \hline
**L** & **npmu** & **PMU Locations** & **PMU Channels** \\ \hline \multirow{8}{*}{4} & \multirow{8}{*}{17} & \multirow{8}{*}{17} & \{1-2, 1-15, 1-16, 1-17 \} \\  & & & 4-3, 4-5, 4-6, 4-18 \} \\  & & & 6-4, 6-5, 6-7, 6-8 \\  & & & 9-8, 9-12, 9-13, 9-55 \\  & & & 15-1, 15-3, 15-14, 15-45 \\  & & & 20-19, 20-21 & \\  & & & 24-23, 24-25, 24-26 \\  & & & 1, 4, 6, 9, 15, 20, 24, 28, & 28-27, 28-29 \\  & & & 31-30, 31-32 \\  & & & 51, 53, 57 & 32-31, 32-33, 32-34 \\  & & & 36-35, 36-37, 36-40 \\  & & & 38-22, 38-44, 38-48, 38-49 \\  & & & 41-11, 41-42, 41-43, 41-56 \\  & & & 47-46, 47-48 \\  & & & 51-10, 51-50 \\  & & & 53-52, 53-57 \\  & & & 57-39, 57-56] \\ \hline \hline \end{tabular}
\end{table} 
Table 18: Optimal PMU number with channel capacity \(L=4\) with redundancy term equal to 68. Figure 9: Plots derived by the GA optimizer in the MATLAB optimization library. Subsets: {1, 5, 12, 48, 55, 58, 62, 65, 66, 72, 77, 79, 81, 86, 91, 94}. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{4}{c}{**Binary Swarm Optimization**} \\ \hline
**L** & **n\({}_{\text{PMU}}\)** & **PMU Locations** & **PMU Channels** \\ \hline
4 & 17 &
\begin{tabular}{c} \{1-2, 1-15, 1-17 \\ 5-4, 5-6 \\ 9-8, 9-10, 9-12, 9-13 \\ 19-18, 19-20, 22-21, 22-23, 22-38 \\ 26-24, 26-27 \\ 29-27, 29-28, 29-52 \\ 30-25, 30-31 \\ 32-33, 32-33, 32-34 \\ 36-35, 36-37, 36-40 \\ 38-22, 38-44, 38-48. 38-49 \\ 39-37, 39-57 \\ 41-11, 41-42, 41-43, 41-56 \\ 46-14, 46-47 \\ 51-10, 51-50 \\ 54-53, 54-55] \\ \end{tabular} } \\ \hline \hline \end{tabular}
\end{table} 
Table 19: Optimal PMU number with channel capacity \(L=4\) with redundancy term equal to 65. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{4}{c}{**Genetic Algoritiyms**} \\ \hline
**L** & **n\({}_{\text{PMU}}\)** & **PMU Locations** & **PMU Channels** \\ \hline
4 & 17 &
\begin{tabular}{c} \{1-2, 1-15, 1-17 \\ 5-4, 5-6 \\ 9-8, 9-10, 9-12, 9-13 \\ 15-1, 15-3, 15-13, 15-45 \\ 22-21, 22-23, 22-38 \\ 26-24, 26-27 \\ 29-27, 29-28, 29-52 \\ 30-25, 30-31 \\ 32-33, 32-33, 32-34 \\ 36-35, 36-37, 36-40 \\ 38-22, 38-44, 38-48. 38-49 \\ 39-37, 39-57 \\ 41-11, 41-42, 41-43, 41-56 \\ 46-14, 46-47 \\ 51-10, 51-50 \\ 51-50 \\ 54-53, 54-55] \\ \end{tabular}
\end{table} 
Table 20: Optimal PMU number with channel capacity \(L=4\) with redundancy term equal to 68. Figure 10: _Cont._

## 8 Discussion

Keeping in mind the latest advancements in mixed-integer programming, mixed-integer nonlinear programming, and nonlinear programming [13; 14; 15; 72; 73; 74; 75; 76; 79; 80; 84; 85], the targets of this study are (I) to declare the OPP problem model under the consideration of channel capacity, and (II) to compare the computational and simulation results derived by different solutions in order to reach a global solution [19; 20; 21; 22; 23; 24; 25; 72; 73; 74; 75; 76; 79; 80; 84; 85]. The key factor of this study is to identify the channel capacity's effect of a specified PMU type on the OPP based-model solution for wide-area system observability. Our goal is to achieve global solutions using BBA, SQP, IPMs, GAs, BPSO, and GPSA. Initially, a feasible and optimal solution is derived by utilizing the BBA [74; 75; 76; 79; 80]. Then, the binary-integer program is transformed into a heuristically solution through the utilization of GAs and BPSO [25; 26; 39; 47; 48; 50; 51]. GA utilizes a parameter penalty for approaching and handling the integer program's constraints [25; 26; 50; 51; 72]. Fewer restrictions are needed in the implementation of the BPSO to find optimality [39; 47; 48; 50; 51]. The binary-integer program is transformed into a mixed-integer nonlinear program towards optimality [51]. Thereby, optimum points are derived with sufficient accuracy; they are characterized as global points [16; 37; 38; 41; 51; 72; 74; 75; 76; 79; 80]. Finally, some nonlinear algorithms are used to minimize the pure nonlinear programming model.