## 4 Test Feeders
However, the maximum variation step in the \(\lambda\) increments was set as \(1\times 10^{-6}\) to ensure this equivalence; this implies that 2.9128 million power flow solutions are required in C2 with one dispersed generator, and 3.3091 million power flow solutions are required in C3 with three dispersed generators. Further, the following observations were noted concerning the processing times. For the IEEE 33-bus system, the solution of the benchmark case in GAMS software with the BONMIN solver took about 0.387 ms; in addition, the processing time was about 3.0632 s in the case of one dispersed generator, about 6.770 s for two generators, and about 3.423 s for three generators. Note that these processing times confirm the effectiveness and robustness of GAMS software in solving MINLP models, as demonstrated in Reference [32]. Figure 5 presents the voltage profiles in the benchmark case and the C2 and C3 cases 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 33-bus system. The voltage profiles shown in Figure 5 indicate that the minimum voltage profile in C1 was caused at node 18 with a magnitude of 0.3905 pu, which directly relates to the numerical results in Table 4. Accordingly, in all the simulation cases, node 18 was selected as the optimal location for the dispersed generators, particularly since the unique increase in the voltage profile at this node reduces the total power consumption through the injection of active power locally. Figure 6 depicts the voltage performance at the worst node, i.e., bus 18, for all the simulation cases and for different numbers of dispersed generators. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multirow{2}{*}{\(N_{aaa}^{dg}\)**} & **Location** & \multirow{2}{*}{**PG1 (pu)**} & \multirow{2}{*}{**PG2 (pu)**} & \multirow{2}{*}{**PG3 (pu)**} & **GAMS** & **MATLAB** \\  & & & & & & **(\(\lambda\))** & **DigSILENT** \\ \hline
1 & 0 & 0 & — & — & — & 2.4079 & 2.4079 \\ \hline  & 1 & 18 & 1.2000 & — & — & 2.9128 & 2.9128 \\
2 & 2 & (18,32) & 0.9316 & 0.5544 & — & 3.0590 & 3.0590 \\  & 3 & (17,18,32) & 0.5735 & 0.3663 & 0.5462 & 3.0635 & 3.0635 \\ \hline  & 1 & 18 & 1.2000 & - & — & 2.9128 & 2.9128 \\
3 & 2 & (17,32) & 1.2000 & 1.029 & — & 3.3068 & 3.3068 \\  & 3 & (17,18,32) & 0.8324 & 0.3885 & 1.0080 & 3.3091 & 3.3091 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Numerical performance of the \(\lambda\)-coefficient in the IEEE 33-bus system for all the simulation cases. From the results in Figure 6, we can note that (i) the worst voltage profile occurs in C1, when the grid does not have the penetration of dispersed generation, with a magnitude of 0.3904 pu when the loadability index assumes a value of 2.4079 (Figure 6a); however, when one dispersed generator was installed in C2 and C3, the voltage profile increased to 0.5240 pu with the \(\lambda\)-coefficient increasing to 2.9128, which is an improvement of approximately 20.96% with respect to the benchmark case; (ii) when two dispersed generators were installed for C2 and C3, it was observed that the \(\lambda\)-coefficient values became 3.0590 and 3.30968, respectively, which correspond to improvements of approximately 5.02% and 13.53% with respect to the same scenarios with one dispersed generator; however, the voltage collapse point does not present high variations; it was approximately 0.4489 pu and 0.4435 pu in C2 and C3, respectively; and (iii) when three dispersed generators were installed, the increments of the \(\lambda\)-coefficient were approximately 0.15 % and 0.07%, respectively. It can be concluded from these values that the limitation imposed by the \(\beta\)-coefficient saturates the total improvement of the loadability index when the possibility of installing dispersed generators increases.