## 6 Observability of the graph
### Constrained Nonlinear Programming Model
The \(0-1\) nonlinear integer programming model is declared using the IEEE-14-bus system. The polynomial constraint function is written. The objective function is minimized subject to a set of equality constraints within binary decision variables [19; 20; 21; 37; 38]. The instance is solved (\(gap=0\)) as an empty entry, when the \(Ub\) coincides with the best value whereas the \(lb\) closes the optimality gap [37; 38; 41; 51; 72; 73; 74; 75; 76; 79; 80]. Therefore, the \(0-1\) \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{L} & \multirow{2}{*}{\(\mathbf{N_{PMU}}\)} & \multicolumn{2}{c}{**Primal-Simplex**} & \multicolumn{2}{c}{**Dual-Simplex**} \\ \cline{3-5}  & & **PMU Locations** & **PMU Channels** & **PMU Locations** & **PMU Channels** \\ \hline \multirow{8}{*}{1} & \multirow{8}{*}{7} & \multirow{8}{*}{2, 5, 6, 7, 9, 10, 13} & \{2-3\} & \{2-1\} \\  & & & \{5-1\} & \{4-3\} \\  & & & \{6-12\} & \{6-5\} \\  & & & \{7-8\} & \{7-8\} & \{7-8\} \\  & & & \{9-4\} & \{9-4\} \\  & & & \{10-11\} & \{10-11\} \\  & & & \{13-14\} & \{13-12\} \\ \hline \multirow{8}{*}{2} & \multirow{8}{*}{5} & \{2-1, 2-3\} & \{2-1, 2-5\} \\  & & & \{6-5, 6-13\} & \{4-2, 4-3\} \\  & & & \{7-4, 7-8\} & \{7-8, 7-9\} \\  & & & \{7-4, 7-8\} & \{7-8, 7-9\} \\  & & & \{10-9, 10-11\} & \{11-6, 11-10\} \\  & & & \{13-6, 13-12\} & \{13-12, 13-14\} \\ \hline \multirow{8}{*}{3} & \multirow{8}{*}{4} & \{2-1, 2-3, 2-5\} & \{2-1, 2-3, 2-5\} \\  & & & \{6-11, 6-12, 6-13\} & \{6-11, 6-12, 6-13\} \\  & & & \{7-4, 7-8, 7-9\} & \{7-4, 7-8, 7-9\} \\  & & & \{9-4, 9-10, 9-14\} & \{9-7, 9-10, 9-14\} \\ \hline \multirow{8}{*}{4} & \multirow{8}{*}{4} & \{2-1, 2-3, 2-4, 2-5\} \\  & & & \{2-1, 2-3, 2-4, 2-5\} \\  & & & \{6-5, 6-11, 6-12, 6-13\} & \{9-7, 9-10, 9-14\} \\ \cline{1-1} \cline{2-5}  & & & \{2-1, 2-3, 2-4, 2-5\} \\ \cline{1-1} \cline{2-5}  & & & \{6-5, 6-11, 6-12, 6-13\} & \{6-5, 6-11, 6-12, 6-13\} \\ \cline{1-1} \cline{2-5}  & & & \{7-4, 7-8, 7-9\} & \{7-4, 7-8, 7-9\} \\ \cline{1-1} \cline{2-5}  & & & \{9-4, 9-7, 9-10, 9-14\} & \{9-4, 9-7, 9-10, 9-14\} \\ \hline \hline \end{tabular}
\end{table} 
Table 5: **Algorithm BBA:** Optimizing the \(0-1\) b-objective ILP with multiple lines for IEEE-14 bus. nonlinear integer programming model is exactly solved by calling the BBA embedded in the SCIP optimizer function [74]. \[minJ(y)=\sum_{i=1}^{m}w_{i}\times y_{i}\quad w_{i}=1,\;i=1,2,\ldots 18,\;m=18 \tag{38}\]

\[f(y)\] \[(1-y(1))\times(1-y(2))\times(1-y(9))=0\] \[(1-y(1))\times(1-y(2))\times(1-y(3))\times(1-y(4))\times(1-y(5)) \times(1-y(6))\times(1-y(7))\times(1-y(9))=0\] \[(1-y(2))\times(1-y(3))\times(1-y(4))\times(1-y(5))\times(1-y(6) )\times(1-y(8))=0\] \[(1-y(2))\times(1-y(3))\times(1-y(4))\times(1-y(5))\times(1-y(6) )\times(1-y(7))\times(1-y(8))\] \[\times(1-y(9))\times(1-y(11))\times(1-y(13))=0\] \[(1-y(1))\times(1-y(2))\times(1-y(4))\times(1-y(5))\times(1-y(7) )\times(1-y(8))\times(1-y(9))\times(1-y(10))=0\] \[(1-y(9))\times(1-y(10))\times(1-y(15))\times(1-y(16))\times(1-y( 17))=0\] \[minJ(y)=\sum_{i=1}^{m}w_{i}\times y_{i}\quad w_{i}=1,\;i=1,2,\ldots 18,\;m=18 \tag{38}\]0 \[minJ(y)=\sum_{i=1}^{m}w_{i}\times y_{i}\quad w_{i}=1,\;i=1,2,\ldots 18,\;m=18 \tag{38}\]1 \[minJ(y)=\sum_{i=1}^{m}w_{i}\times y_{i}\quad w_{i}=1,\;i=1,2,\ldots 18,\;m=18 \tag{38}\]2 \[minJ(y)=\sum_{i=1}^{m}w_{i}\times y_{i}\quad w_{i}=1,\;i=1,2,\ldots 18,\;m=18 \tag{38}\]3 \[minJ(y)=\sum_{i=1}^{m}w_{i}\times y_{i}\quad w_{i}=1,\;i=1,2,\ldots 18,\;m=18 \tag{38}\]4 \[minJ(y)=\sum_{i=1}^{m}w_{i}\times y_{i}\quad w_{i}=1,\;i=1,2,\ldots 18,\;m=18 \tag{38}\]5 \[minJ(y)=\sum_{i=1}^{m}w_{i}\times y_{i}\quad w_{i}=1,\;i=1,2,\ldots 18,\;m=18 \tag{38}\]6

\(y\) is an integer decision variable that takes two values, i.e., 0 or 1 [16]. The value 1 means the \(i\)th PMU combination is installed at the center bus [60]. The \(0-1\) nonlinear programming is well posed given that the number of equality constraints is fewer than the number of the decision variables [19; 20; 21; 22; 23; 24; 70; 82].