## 5 Results and Discussion
### Validating the Proposed Methodology on a 69-Node MG
#### 5.3.2 PGA's Performance in the 69-Node MG During Islanded Operation  
Table 9 presents the results obtained for the islanded 69-node MG with regard to the objective functions \(FO_{1}\), \(FO_{2}\), and \(FO_{3}\). It is important to note that, as in the analysis performed for the 33-node MG during islanded operation, no base case is provided. This is due to the fact that such a scenario would lead to a violation of the power generation limits set for the diesel generator located at the slack node.  
Regarding \(FO_{1}\), \(FO_{2}\), and \(FO_{3}\) in islanded mode, PGA yields weekly average values of USD 7212.0110, 171.7586 kWh, and 6416.4515 \(kgCO_{2}\), respectively. For \(FO_{1}\), the absolute daily reductions relative to the base case range from USD 7203.0919 to USD 7217.1257, with a standard deviation of 1.7022%. In the case of \(FO_{2}\), the absolute daily reductions  
\begin{table}
\begin{tabular}{l l l l l l l l} \hline \hline
**Objective Function** & **Case** & **Monday** & **Tuesday** & **Wednesday** & **Thursday** & **Friday** & **Saturday** & **Sunday** \\ \hline \multirow{2}{*}{\(FO_{1}\) (USD)} & PGA & 7217.1257 & 7214.5956 & 7210.7371 & 7215.8648 & 7215.1790 & 7207.4827 & 7203.0919 \\  & Time & 412.2137 & 416.2815 & 412.8688 & 411.7072 & 412.5868 & 417.4851 & 411.8585 \\ \hline \multirow{2}{*}{\(FO_{2}\) (kWh)} & PGA & 173.9042 & 177.5300 & 163.1527 & 175.5045 & 178.0434 & 168.1620 & 166.0131 \\  & Time & 414.5456 & 412.9107 & 405.9653 & 410.2896 & 410.9603 & 412.7385 & 411.1176 \\ \hline \multirow{2}{*}{\(FO_{3}\) (\(kgCO_{2}\))} & PGA & 6417.1207 & 6416.6017 & 6417.2474 & 6415.3316 & 6415.8770 & 6415.9251 & 6417.0570 \\  & Time & 412.6335 & 412.3623 & 411.0532 & 407.2092 & 411.2700 & 413.6384 & 411.3984 \\ \hline \hline \end{tabular}
\end{table}
Table 9: Objective function values for the 69-node MG in islanded mode over a one-week operation period.  
range from 163.1527 kW to 178.0434 kW, with a standard deviation of 3.20%. Finally, for \(FO_{3}\), the absolute daily reductions range between 6415.3316 and 6417.2474 \(kgCO_{2}\), with a standard deviation of 0.0140%.  
While these values are not as favorable as those obtained in grid-connected mode, this outcome is consistent with our expectations, given the additional technical constraints imposed to ensure a safe and reliable operation in an islanded configuration. These constraints inherently limit the operational flexibility of the MG, thereby reducing the potential for cost savings, loss minimization, and emissions reduction. Nevertheless, the results demonstrate the strategy's effectiveness in maintaining acceptable performance even under more restrictive operating conditions.  
The average processing times required by PGA are 413.5717, 411.2182, and 411.3664 s for the objective functions \(FO_{1}\), \(FO_{2}\), and \(FO_{3}\), respectively, which represents a consistent overall average of 412.0521 s across all objective functions.  
When comparing processing times required during islanded operation, the 33-node MG reports an average value of 217.4016 s, whereas the 69-node system reaches 412.0521. This corresponds to an increase of approximately 89.53%, reflecting the higher computational effort required for larger systems under similar operating conditions.