## 6 Observability of the graph
### Constrained Nonlinear Programming Model
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**L** & **nPMM** & **PMU Locations** & \multicolumn{3}{c}{**PMU Channels**} \\ \hline  & & & & & & & [2-1, 2-4, 2-5, 2-6] & & & [4-2, 4-3, 4-6, 4-12] & \\  & & & & & & [6-2, 6-4, 6-7, 6-8] & & & [9-6, 9-10, 9-11] & \\
4 & 10 & 2, 4, 6, 9, 10, 12, 15, 18, 25, 27 & & & & [10-17, 10-20, 10-21, 10-22] & & & [12-4, 12-13, 12-14, 12-16] & \\  & & & & & & [15-12, 15-14, 15-18, 15-23] & & & [18-15, 18-19] & \\  & & & & & & [25-24, 25-26, 25-27] & & & [27-25, 27-28, 27-29, 27-30] & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Channel capacity, PMU number, PMU locations, and PMU channels. For the 30-bus system, we found the set solutions for a fixed channel capacity [65]. Tables A6 and A8 illustrate the best PMU number and the branch assignments related to each entry included in the PMU placement vector. Considering this, we optimize the two mathematical models to achieve PMU locations and related branch assignments. The first optimizes a one product objective, whereas the second optimizes the b-objective function. We succeed in finding PMU solutions covering branches with higher redundancy related to previous results presented in [64; 65; 66; 67]. Thus, we achieve better results to those found in [57; 58; 59; 60; 61; 62; 63; 64; 65; 66; 67] for a fixed amount of channel capacity [65]. \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{2}{c}{Mixed-Integer Nonlinear Program (MINLP) Optimization} \\ \hline min f(x) & & \\ \hline s.t. lb \textless{}n x \textless{}= ub & \\ \hline cl \textless{}= c(x) \textless{}= cu & \\ \hline xi = Integer/Binary & \\ \hline \multicolumn{2}{c}{Problem Properties:} \\ \hline \# Decision Variables: & 82 \\ \hline \# Constraints: & 276 \\ \hline \# Bounds: & 164 \\ \hline \# Binary Variables: & 82 \\ \hline \# Nonlinear Equality: & 30 \\ \hline \multicolumn{2}{c}{Solver Parameters:} \\ \hline Solver: & SCIP \\ \hline Objective Gradient: & @(x)mkJJac(fun,x) \\ \hline Constraint Jacobian: & @(x)mkJJac(nlcon,x) \\ \hline Jacobian Structure: & Supplied \\ \hline time \(\mid\) node \(\mid\) left \(\mid\) LP iter iter \(\mid\) LP it/n1 mem \(\mid\) mdpt \(\mid\) frac \(\mid\



As observed in Tables A5 and A7, the primal bound on the objective function value is calculated for the minimization problem. SCIP computes the difference between the primal and dual bounds on the objective function in the BBA tree implementation [74]. \(Gap=U-L<=AbsoluteGapTolerance\) is found to be equal to zero [79; 80]. The solution status illustrates that \"Globally Integer Optimal\" has been detected [74; 79]. Tables A9 and A10 presents the optima points derived by mathematical and heuristic algorithms considering a fixed channel capacity \(L=3\), \(L=4\). Tables A7 and A8 present the PMU numbers, the PMU locations, and the observed branches obtained by SQP, IPMs, GPSA, GAs, BPSO, and BBA [41; 51; 72; 73; 74; 75; 76; 79; 80; 84; 85; 86; 87; 88]. We derive not only the PMU locations, but also the channels being observed that lead to the measurement of the redundancy, as illustrated in Table 15. BBA, GA, and BSO optimize the integer program, while SQP, IPMs, and GPSA optimize the nonlinear program to find the best solutions under this contingency [41; 51]. SQP and IPMs are local search algorithms detecting local minimizers at the desired objective function value for the specific minimization problem [19; 20; 21; 25; 41]. SQP and IPMs are theoretically efficient algorithms, and they are also computational methods that are not wasteful [19; 20; 41]. SQP, IPMs, and GPSA emphasize the search of a neighborhood of a local point, and the solution is not the global optimum in case of minimizing non-convex nonlinear problems [19; 20; 22; 25; 41; 70; 71]. All algorithms deliver solutions as the iterations go to the final stage, where the tolerance and feasible criteria are fulfilled so as an optimal result is reached [21].