## 4 Case Study and Discussion  
This section presents a comprehensive case study to assess the performance of various uncertainty modeling techniques in a power system integrating wind farms. The studied system is the IEEE 24-Bus reliability test system, which is affected by fluctuations in wind speeds and a reduction in the generation of conventional power plants (CPPs). Details of the system configuration and data processing are provided in [12; 87]. First, two scenarios were considered for the levels of CPP generation reduction, combined with wind power variability. Next, reliability metrics were computed using probabilistic simulations. These simulations incorporated uncertainty modeling techniques to evaluate the system's performance under uncertain wind conditions. Finally, the economic impact of energy not supplied was calculated to quantify the cost implications of reliability challenges in different scenarios. This analysis compares the Monte Carlo simulation combined with the K-Means method, MCMC, RO, and IGDT in managing the uncertainty of wind farm generation over different time horizons. The number of Monte Carlo simulation scenarios is considered 300, the number of Markov chain states is considered 10, and the worst-case uncertainty level parameter for both IGDT and RO is considered 0.2. The cut-in wind speed, cut-out wind speed, and rated wind speed for wind turbines are considered 4 m/s, 20 m/s, and 16 m/s, respectively. The rated power of each wind turbine is 1 MW and the total number of wind turbines in the wind farm is 1000 (i.e., total capacity of wind farm is 1000 MW). This paper evaluates how uncertainty affects key reliability indices such as LOLE, LOLP, and EENS. Due to the extensive nature of the data analyzed, spanning an entire year in some cases, presenting all individual data points numerically would be impractical. Therefore, key results are summarized in figures, which effectively highlight trends, relationships, and comparative performances across methods. Real wind speed data were sourced from Milwaukee, USA, through Meteoblue's weather archive [88], covering an entire year. This combination of the IEEE test system and real-world wind data ensures the analysis reflects practical and realistic conditions. Figure 8 illustrates the annual system load profile, displaying fluctuations between 861 MW and 2850 MW. On the other hand, Figure 9 illustrates the generation output from 33 CPPs over the period of one year, fluctuating between 2624 MW and 4539 MW. Figure 8: The system load for one year [9]. The generation of conventional power plants (CPPs) does not correlate with wind speed because CPPs are dispatchable and their output is determined by the consideration of grid demand, market conditions, and reliability requirements. In contrast, wind farm generation depends directly on wind speed. Wind farms contribute renewable energy when wind conditions are favorable, while CPPs provide stability and reliability to balance the grid during periods of low wind generation or high load. Figure 10 presents the historical wind speed data over the period of one year. The wind speeds exhibit significant variability, ranging from near 0 to over 30 m/s, with occasional high spikes above 25 m/s. Figure 11a,b compare wind speed data modeled by different uncertainty modeling techniques, including Monte Carlo simulation combined with the K-Means method, MCMC, IGDT, and RO, for the last month of the year and last 3 months of the year, respectively, alongside the actual historical data. Monte Carlo provides more conservative estimates in instances of normal wind speeds compared to historical data, while IGDT and RO provide estimates with fewer spikes for wind speeds compared to Monte Carlo simulation and MCMC methods. Figure 10: Wind speeds’ historical data for one year. Figure 9: CPPs’ generation for one year [9].