## 5 Results and Discussion
BESS operations are scheduled within a vast planning horizon (e.g., hourly or daily forecasts), allowing for more computationally intensive algorithms without compromising practical feasibility. The standard deviations (\(\sigma\)) reported quantify the variability of each algorithm's performance over 100 independent runs. A low \(\sigma\) value indicates consistent convergence behavior and high robustness of the optimization strategy, which is particularly important in microgrid applications where repeatable scheduling enhances operational reliability. The PMVO algorithm consistently achieved the lowest standard deviations in both energy losses and CO\({}_{2}\) emissions, confirming its repeatability and stability across multiple executions. For example, in the self-powered mode, PMVO achieved \(\sigma=0.2989\%\) for energy losses and \(\sigma=0.0113\%\) for emissions, outperforming PVSA and CBGA in both metrics. This stability reinforces the suitability of PMVO for planning applications where dependable outcomes are as important as optimal performance. In terms of overall ranking, PMVO ranked first in both scenarios, followed by PVSA and CBGA. This reinforces the robustness and reliability of the PMVO algorithm in addressing the multi-objective problem formulated for energy loss and emissions minimization. The consistent dominance of PMVO across both configurations demonstrates its adaptability and superior search capabilities under different operational conditions of the microgrid. Figures 5 and 6 illustrate the relative improvements achieved by each method in terms of energy loss and CO\({}_{2}\) emissions reduction for the 33-node test system under both grid-dependent and self-powered configurations over a 1-week evaluation horizon. In Figure 5, it is evident that PMVO consistently attains the highest percentage reductions in energy losses, both in the best and average cases. Under the grid-dependent mode, the PMVO algorithm achieves a best-case reduction of 4.416%, followed closely by  
Figure 5: Energy loss reduction: Best and average results for the 33-node system in grid-dependent and self-powered modes over a 24 h evaluation horizon. PVSA (4.396%) and CBGA (4.068%). A similar pattern is observed in the average results, with PMVO achieving 3.731%, outperforming PVSA (3.866%) and CBGA (3.669%). In the self-powered mode, the advantage of PMVO becomes even more pronounced, with a best-case reduction of 4.469%, surpassing PVSA (4.439%) and CBGA (3.543%). The average values reflect the same tendency, confirming the robustness of PMVO under varying conditions. Figure 6 presents the corresponding reductions in CO\({}_{2}\) emissions. Although the absolute values are smaller due to the nature of the emissions metric, the comparative performance remains consistent. In the grid-dependent case, PMVO leads with a best-case reduction of 0.183%, while PVSA and CBGA follow with 0.181% and 0.167%, respectively. The average values reinforce this trend, with PMVO, again, ahead at 0.17%. For the self-powered configuration, PMVO and PVSA tie in the best-case results at 0.183%, and PMVO achieves the highest average at 0.17%, compared to 0.166% for PVSA and 0.128% for CBGA. Figure 7 provides a complementary perspective on the performance of the evaluated algorithms through violin-scatter plots. These plots depict the distribution of the optimization outcomes for each method across multiple runs under both operational modes. The left side of each violin represents the grid-dependent case, and the right side shows the self-powered configuration. Figure 6: CO\({}_{2}\) emissions reduction: Best and average results for 33-node system in grid-dependent and self-powered modes over a 24 h evaluation horizon. Figure 7: Violin-scatter plot comparing the implemented methodologies for the 33-node system under grid-dependent and self-powered modes over a 24 h evaluation horizon. (**a**) Energy loss. (**b**) CO\({}_{2}\) emissions. Figure 7a illustrates the spread of energy loss values.