## 6 Observability of the graph
### Constrained Nonlinear Programming Model
\begin{bmatrix}1&1&0&0&1&0&0&0&0&0&0&0&0&0\\ 1&1&1&1&1&0&0&0&0&0&0&0&0&0&0\\ 0&1&1&1&0&0&0&0&0&0&0&0&0&0\\ 0&1&1&1&1&0&0&1&0&0&0&0&0&0\\ 0&1&1&1&0&0&0&1&0&0&0&0&0\\ 0&1&1&1&0&0&1&0&1&0&0&0&0\\ 0&0&1&1&1&0&1&0&1&0&0&0&0\\ 1&1&0&1&1&1&0&0&0&0&0&0\\ 0&0&0&0&1&1&0&0&0&0&0&0\\ 0&0&0&0&0&1&0&0&0&1&1&1&0\\ 0&0&0&1&0&0&1&1&0&0&0&0\\ 0&0&0&0&0&1&0&0&1&1&0&0\\ 0&0&0&0&1&0&0&0&0&1&1&1\\ 0&0&0&0&0&0&0&1&0&0&0&1&1\\ 0&0&0&0&0&0&0&1&0&0&0&1&1\\ \end{bmatrix}  \tag{32}\]  
Figure 3: IEEE-14-bus system. Suppose that the channel limit is equal to four, that is \(L=4\), and the possible number combinations are equal to 18, \(m=\sum_{i=1}^{14}r^{L}=18\)[56; 57; 58; 59]. The set of all subsets in the case of considering four channels is the following [60]:  
\[ \begin{array}{l}S=\{1,2,5\},\{2,1,\,3,\,4,\,5\},\{3,\,2,\,4\},\{4,\,2,\,3,\,5,\,7\},\{4,\,2,\,3,\,5,\,9\},\{4,\,2,\,3,\,7,\,9\}\\ \{4,2,\,5,\,7,\,9\},\{4,3,\,5,\,7,\,9\},\{5,\,1,\,2,\,4,\,6\},\{6,\,5,\,11,\, 12,\,13\}\\ \{7,4,\,8,\,9\},\{8,\,7\},\{9,\,4,\,7,\,10,\,14\},\{10,\,9,\,11\},\{11,\,6,\,10 \}\\ \{12,\,6,\,13\},\{13,\,6,\,12,\,14\},\{14,\,9,\,13\}\end{array}  \tag{33}\]  
Each first entry of the subset is the center bus used to monitor the adjacent buses. For example, 2 is the center in the second subset, 8 is the center in the 12 th subset, 10 is the center in the 14 th subset, and 13 is the center in the 17 th set [60]. When \(L=4\), and solving the \(0-1\) ILP model for the IEEE-14-bus network in Figure 3, four combinations are derived, that is, [2; 12; 14; 17]. Assume that \(x_{s}\in\{0,1\}\); declare a binary decision variable such that \(x_{s}=1\) if a PMU is placed at the center of the subset of the family S, and \(x_{s}=0\) otherwise, where s is an index within the family of the subsets [60]. The algorithm has chosen as a solution the set {2, 12, 14, 17}. From that subset, and considering the yes or no decision [15], a PMU placement is obtained in which the center is declared as 1 in the placement vector and monitors all buses inside that subset [60]. The algorithmic scheme delivers PMUs installed at buses \(\{2,8,10,13\}\), which are centers of the subsets \(\{2,1,3,4,5\},\ \{8,7\},\{10,9,11\},\{13,6,12,14\}\), respectively [60]. Thus, the placement result is a binary vector with \"yes\" or \"no\" decisions [13; 14; 15; 16; 17; 18]. \[x_{s}=[0\;1\;0\;0\;0\;0\;0\;1\;0\;1\;0\;0\;1\;0]^{T} \tag{34}\]  
Furthermore, the algorithms identify not only the PMU location, but also the branch assignments considering the extra restriction, which is a fixed amount of capacity. The solution is adequate to decrease the entire cost without sacrificing observability. For example, the PMU at bus 2 monitors buses 2, 1, 3, 4, and 5. The PMU locations are {2, 8, 10, 13}, where the branch assignments are the PMU channels presented in Table 1.  
The above results are produced by executing YALMIP's BBA and the log file is presented in Algorithm 6[75; 77]. YALMIP's BBA optimizes the \(0-1\) integer model declared in Equations (33)-(35). The log file is presented in Algorithms 6 and 7[37; 38; 72; 74; 75; 76; 77; 79; 80]. The binary decision variable is displayed in Table 1 and it is a placement in which each element is the \(j-th\) combination related to the center bus \(i\)[56; 57; 58; 59; 60; 61]. The outcome presents a PMU set solution with all network buses observed. One bus serves as the center interconnected with neighboring buses through branches, such that all buses are reachable by synchronized measurements and make the power grid fully observable [56; 57; 58; 59; 60; 61; 62; 63; 64; 65; 66; 67]. The set {2, 12, 14, 17} corresponds to PMU locations, namely {2, 8, 10, 13} illustrated in Figure 3. A PMU placed at bus 2 will monitor buses 1, 2, 3, 4 and 5; the one at bus 8 will monitor buses 7 and 8; the one at bus 10 will monitor buses 9, 10, and 11; and the one at bus 13 will monitor buses 6, 12, 13, and 14. Then, the observability is preserved.