## 4 Case Study and Discussion
Table 2 provides a comparison of the correlation coefficients between the wind speeds modeled by various uncertainty modeling techniques and real historical wind speed data, calculated using the \"corr2\" function in MATLAB (version R2024a). This table focuses on both one-month and three-month uncertainty durations, allowing for an evaluation of how well each technique aligns with actual wind conditions. Monte Carlo simulation combined with the K-Means method demonstrates the highest correlation with real wind speed data, particularly over extended uncertainty periods. This result highlights the combined strengths of Monte Carlo's robust coverage of uncertainty and K-Means' ability to minimize intra-cluster distances. However, this finding reflects the specific metrics and conditions of the case study and should not be interpreted as a universal assertion of superiority in risk modeling. Other methods, such as Markov Chain Monte Carlo, IGDT, and robust optimization, have unique strengths depending on the application and context. MCMC improves over longer durations, while IGDT and robust optimization provide moderate accuracy. The increase in the correlation factor with longer time horizons can be attributed to the smoothing effect of aggregation. Longer time horizons average out these fluctuations, emphasizing the more stable and consistent relationship between wind speed and wind generation. Table 3 presents the system performance without wind farm integration, comparing the effects of a 20% and 30% reduction in the generation of CPPs. The results indicate that \begin{table}
\begin{tabular}{p{113.8 pt} p{28.5 pt} p{28.5 pt} p{28.5 pt} p{28.5 pt} p{28.5 pt} p{28.5 pt} p{28.5 pt}} \hline
**Index Technique** & \multicolumn{2}{c}{**Monte Carlo and K-Means**} & \multicolumn{2}{c}{**MCMC**} & \multicolumn{2}{c}{**IGDT**} & \multicolumn{2}{c}{**Robust**} \\ \hline Uncertainty duration [month] & 1 & 3 & 1 & 3 & 1 & 3 & 1 & 3 \\ \hline Correlation coefficient between the wind speeds modeled by the uncertainty modeling technique and real historical data of wind speeds & **0.9682** & **0.9748** & 0.9191 & 0.9711 & 0.9508 & 0.9578 & 0.9272 & 0.9381 \\ \hline \end{tabular}
\end{table} 
Table 2: Comparison of the correlation between the results of different uncertainty modeling techniques (1-month uncertainty) with real historical data of wind speeds. Figure 11: Wind speeds modeled by different uncertainty modeling techniques. (**a**) Wind speeds modeled for the last month of the year. (**b**) Wind speeds modeled for the last 3 months of the year. the absence of wind farms significantly impacts the system's ability to maintain reliability, particularly as the reduction in CPP generation increases. The relationship between EENS and its cost is a critical metric in power system reliability. EENS represents the energy demand not met by the generation system, and its cost is calculated using the Value of Lost Load (VoLL), which is assumed to be USD 3500/MWh in this study. As Table 3 shows, reductions in CPP generation lead to an increase in EENS and a corresponding rise in costs, demonstrating the economic impact of reduced reliability. Wind farm integration helps mitigate these effects by contributing renewable energy to meet demand, reducing both EENS and its associated costs. In the 20% reduction scenario, the system's reliability remains relatively high but starts to show signs of deterioration and vulnerability. The LOLE increases to 14.77 h per year, which is a significant jump compared to systems with wind farm integration (as seen in Tables 4 and 5). Similarly, the LOLP rises to 0.00502, indicating a noticeable increase in the probability of losing load. The EENS reaches 13.86 MWh, and the associated cost escalates to USD 4.85 \(\times\) 10\({}^{4}\), reflecting the system's growing challenges in supplying energy without wind power as a backup. In the 30% reduction scenario, the system experiences a dramatic decline in performance without wind farm integration.