## 4 Test Feeders  
Two test feeders were employed to validate the proposed solution methodology for enhancing the voltage stability margin in AC distribution networks by considering the optimal placement and sizing of dispersed generators. The test feeders correspond to the  
Figure 3: Recursive validation of the maximum loadability coefficient using MATLAB and DigSILENT. IEEE 33-bus system and the IEEE 69-bus system, both with a radial structure and operated at the substation bus with 12.66 kV of nominal voltage output. The electrical configuration of both the test feeders is depicted in Figure 4.  
Parametric information related to the active and reactive power consumption and the line resistance and reactance parameters for both the test feeders are reported in Tables 2 and 3. This information was obtained from Reference [15]. In the case of dispersed generators for the IEEE 33-bus system, it is assumed that each dispersed generator can generate a maximum rating of 1200 kW, and for the IEEE 69-bus system, the threshold is extended to 2500 kW. Moreover, the numerical evaluation used 12.66 kV and 100 kVA as the voltage and power bases, respectively. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**Node \(i\)** & **Node \(j\)** & \(R_{ij}\) (\(\Omega\)) & \(X_{ij}\) (\(\Omega\)) & \(P_{j}\) (\(\mathbf{k}\)W) & \(Q_{j}\) & **Node \(i\)** & **Node \(j\)** & \(R_{ij}\) (\(\Omega\)) & \(X_{ij}\) (\(\Omega\)) & \(P_{j}\) (\(\mathbf{k}\)W) & \(Q_{j}\) (\(\mathbf{k}\)W) \\ \hline
1 & 2 & 0.0922 & 0.0477 & 100 & 60 & 17 & 18 & 0.7320 & 0.5740 & 90 & 40 \\
2 & 3 & 0.4930 & 0.2511 & 90 & 40 & 2 & 19 & 0.1640 & 0.1565 & 90 & 40 \\
3 & 4 & 0.3660 & 0.1864 & 120 & 80 & 19 & 20 & 1.5042 & 1.3554 & 90 & 40 \\
4 & 5 & 0.3811 & 0.1941 & 60 & 30 & 20 & 21 & 0.4095 & 0.4784 & 90 & 40 \\
5 & 6 & 0.8190 & 0.7070 & 60 & 20 & 21 & 22 & 0.7089 & 0.9373 & 90 & 40 \\
6 & 7 & 0.1872 & 0.6188 & 200 & 100 & 3 & 23 & 0.4512 & 0.3083 & 90 & 50 \\
7 & 8 & 1.7114 & 1.2351 & 200 & 100 & 23 & 24 & 0.8980 & 0.7091 & 420 & 200 \\
8 & 9 & 1.0300 & 0.7400 & 60 & 20 & 24 & 25 & 0.8960 & 0.7011 & 420 & 200 \\
9 & 10 & 1.0400 & 0.7400 & 60 & 20 & 6 & 26 & 0.2030 & 0.1034 & 60 & 25 \\
10 & 11 & 0.1966 & 0.0650 & 45 & 30 & 26 & 27 & 0.2842 & 0.1447 & 60 & 25 \\
11 & 12 & 0.3744 & 0.1238 & 60 & 35 & 27 & 28 & 1.0590 & 0.9337 & 60 & 20 \\
12 & 13 & 1.4680 & 1.1550 & 60 & 35 & 28 & 29 & 0.8042 & 0.7006 & 120 & 70 \\
13 & 14 & 0.5416 & 0.7129 & 120 & 80 & 29 & 30 & 0.5075 & 0.2585 & 200 & 600 \\
14 & 15 & 0.5910 & 0.5260 & 60 & 10 & 30 & 31 & 0.9744 & 0.9630 & 150 & 70 \\
15 & 16 & 0.7463 & 0.5450 & 60 & 20 & 31 & 32 & 0.3105 & 0.3619 & 210 & 100 \\
16 & 17 & 1.2860 & 1.7210 & 60 & 20 & 32 & 33 & 0.3410 & 0.5302 & 60 & 40 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: IEEE 33-bus system parameters. Figure 4: Test feeders: (**a**) IEEE 33-bus system and (**b**) IEEE 69-bus system. From the numerical results in Table 4, the following can be summarized: (i) the active power injection allowed the loadability factor to increase by approximately 27.22% and 37.43% for the simulation cases C2 and C3, respectively, with the maximum number of dispersed generators installed, i.e., \(N_{aa}^{dg}=3\); (ii) the optimization methodology identified an identical set of nodes for one to three dispersed generators independent of the percentage of generation penetration allowed, i.e., nodes 17, 18, and 32; however, depending on the \(\beta\)-coefficient, the final value of the loadability coefficient showed important variations, which was an expected behavior; as the percentage of generation increases, the capacity of the grid to support a load also increases; and (iii) the \(\lambda\)-coefficient identified by the BONMIN solver in GAMS was exactly the same as that provided by the recursive approach in the MATLAB/DigSILENT environments.