## 5 Results and Discussion
### Results for the 33-Node AC MG Operating in Islanded Mode
#### 5.2.3 Statistical Analysis of the Objective Function \(FO_{3}\) (CO\({}_{2}\) Emissions)  
As for \(FO_{3}\) during islanded operation, all techniques yielded promising results; each of them reached optimal solutions, leading to an average minimum value of 11,117.5033 \(kgCO_{2}\). This demonstrates the overall effectiveness of the studied methods. Figure 26 illustrates the average solutions obtained by each metaheuristic technique, with PGA once again showing superior performance, exhibiting an average of 10,267.4865 \(kgCO_{2}\). Compared with MC, PSO, JAYA, GNDO, and MVO, this represents improvements of 28.5284%, 1.1314%, 2.3201%, 15.8819%, and 0.1453%, respectively.  
A detailed analysis of the standard deviations reveals PGA's exceptional repeatability, as shown in Figure 27, with a remarkably low standard deviation (0.0079%). This value is  
Figure 26: Reduction in emissions by PGA compared with other methodologies in the 33-node MG during islanded operation.  
Figure 25: Boxplot of the solutions for \(FO_{2}\) in the 33-node AC MG during islanded operation.  
lower by 0.6376%, 1.2086%, 1.9206%, 1.7986%, and 0.2322% than the deviations recorded by MC, PSO, JAYA, GNDO, and MVO for this specific optimization case.  
JAYA exhibited the shortest execution time (200.7937 s), closely followed by the proposed PGA, which completed the optimization in 218.1240 s. At the other end of the spectrum, MVO was the slowest approach, requiring 547.4273 s. The average processing time of all six methods was approximately 336.7343 s, i.e., less than six minutes. This confirms that all techniques, including PGA, can be executed efficiently within practical time constraints. Note that PGA once again demonstrates a consistently low computation time, reinforcing its suitability for real-time or near-real-time energy management in islanded MGs.  
To demonstrate compliance with technical constraints regarding voltage and load capacity in the islanded 33-node MG, Figures 28 and 29 are presented. These figures depict the optimal solutions obtained by each technique for \(FO_{1}\), \(FO_{2}\), and \(FO_{3}\). It is evident that the voltage limit (0.08 p.u.) and the maximum load capacity of the lines are not exceeded.  
Figure 30 presents the power injection of the diesel generators for each of the applied techniques, with the aim of demonstrating compliance with these units' dispatch limits in the islanded MG. Note that these limits range between 1600 kW and 3200 kW and that all strategies successfully maintained the generator outputs within the specified bounds at all times.  
Finally, it was essential to verify that the power injection limits of the WTs, as defined by their rated power capacity, were not violated when implementing the proposed PGA. To this effect, Figure 31 presents the active power injected by each WT within the framework of the best solutions obtained. In addition, the reactive power outputs are shown in Figure 32. Subsequently, using the information from both curves, the apparent power profile of the WTs is depicted in Figure 33 for each objective function as optimized by the best technique. The results clearly confirm that the nominal power limit of 1300 kW is not exceeded at any point.  
Figure 30: Power supplied by the diesel generator within the framework of each analyzed metaheuristic technique  
Figure 29: Line loadability for the 33-node AC MG operating in islanded mode.