## 4 Test Feeders
### Results in the IEEE 69-Bus System  
For the computational evaluation of the test feeder, the voltage regulation constraints were relaxed to \(V_{\max}=1.20\) pu and \(V_{\min}=0.45\) pu during the GAMS implementation of the model (1)-(9). It is worth recalling that these relaxations were only used to determine the maximum \(\lambda\)-coefficient. During normal operative conditions, these bounds must be restored to \(\pm 10\%\) to ensure the fulfillment of regulatory policies. Table 5 reports the numerical performance for all the simulation cases in the IEEE 69-bus system using the BONMIN solver in the GAMS environment and the verification in MATLAB and DigSILENT software. The numerical results listed in Table 5 yield the following observations: (i) For C2 and C3 with three units of generation, increments of approximately 28.55% and 41.98%, respectively, were observed in the \(\lambda\)-coefficient. Moreover, the verification processes in MATLAB and DigSILENT indicated that, for all the simulation cases, the \(\lambda\)-coefficient was the same as that provided by GAMS software, which confirms the efficacy of the recursive evaluation processes. In C2 with two distributed generators, an estimation error lower than \(4\times 10^{-3}\%\) was noted, which is attributable to convergence errors of the power flow methods used in the recursive validations owing to the near singularity behavior of the Jacobian matrix when approaching the voltage collapse point. (ii) In the case of one dispersed generator, when the \(\beta\)-coefficient was varied from 40% to 60%, it was observed  
Figure 5: Behavior of the voltage profiles in the benchmark case, i.e., C1, and simulation cases C2 and C3 with three dispersed generators for the IEEE 33-bus system. that the location of the generator changed from node 58 to 59. However, in both the cases, the \(\lambda\)-coefficient was higher than 2.84, which improved the loadability coefficient by approximately 28.43% and 30.32%, respectively. (iii) When two and three dispersed generators were installed, they always appeared at nodes 64 and 65; however, node 65 had minimum generation with values of 0.2274 pu and 0.2451 pu, respectively. This behavior can be explained as follows: since the active power injection in upstream nodes reduces the power requirement at this node, voltage profiles increase owing to the power injections at nodes 61 and 64, respectively. Figure 6: Voltage performance at node 18: (**a**) benchmark case and C2 and C3 with one dispersed generator, (**b**) C2 and C3 with two dispersed generators, and (**c**) C2 and C3 with three dispersed generators. It was observed that the total processing times required for GAMS software with the BONMIN solver in the IEEE 69-bus system was 648 ms for the base cases, 23.353 s for one dispersed source, 11.867 \(s\) in the case of two generators, and about 9.585 \(s\) for three generators. These processing times confirm the effectiveness and robustness of the GAMS in solving complex MINLP models [32]. Figure 7 presents the voltage profiles in the benchmark case and C2 and C3 with three dispersed generators installed. These profiles were obtained from DigSILENT software to illustrate the effect of the optimal placement and sizing of dispersed generators in the IEEE 69-bus system. Figure 7 presents the weakest node in the IEEE 69-bus system, which is node 65; it has a voltage magnitude of 0.4720 pu in C1. Moreover, the figure confirms that all the best nodes for the optimal placement and sizing of dispersed generators to improve the voltage stability index are in the region between nodes 58 and 65, as presented in Table 5.  
Figure 8 depicts the voltage performance at the worst node, i.e., bus 65, for all the simulation cases with different numbers of dispersed generators.