## 4 Case Study  
### Basic Information  
The main goal was to investigate the effects of the characteristics of uncertainty on flexibility. A test system is designed by adding the wind power plants of 1250 MW to the IEEE-RTS-96; an effective load-carrying capability of the plants is estimated at  
Figure 3: Procedure for the effect of NLFE characteristics on flexibility. \begin{table}
\begin{tabular}{c c c c} \hline
**Scenario \#** & **FNL, MW** & **Skewness, Dimensionless** & **Kurtosis, Dimensionless** \\ \hline S1 & High & Particular range & Fixed \\ S3 & & Fixed & Particular range \\ S4 & Medium & Particular range & Fixed \\ S6 & & Fixed & Particular range \\ S7 & Low & Particular range & Fixed \\ S9 & & Fixed & Particular range \\ \hline \end{tabular}
\end{table} 
Table 3: Scenarios for three parameters: FNL, skewness (\(P_{\text{s}\text{i}\text{d}}\)), kurtosis (\(P_{\text{s}\text{i}\text{b}}\)). 79.88 MW [33]. The test system is composed of 26 generating units of 3,105 MW. Figure 4 shows the FNL profile; a peak load of 2,531 MW occurs at hour 19. The hourly generation schedule is determined using dynamic programming. All generating units are assumed in the spinning state; this condition is required to estimate the maximum value of the flexibility of the system. The \(RSP_{t}\) was calculated using MATLAB programming (R2020a version) [34]. The simulation for the test system was performed on a PC with 64 GB RAM and a 3.49 GHz CPU. As shown in Figure 5, a seven-step approximation was applied to calculate the probability of the NLFE distribution. According to the solution procedure shown in Figure 3, the following steps were performed. Step 1: The results of the \(RSP_{t}\) calculation are shown in Figure 6. Hours 18-19 were determined as the most critical time interval because the highest value appears at hour 19. Steps 2 to 4: Table 4 shows the generated NLFE scenario. The class and level of FNL were selected such that the high class (low class) was set at 101% and 102% (99% and 98%) of the FNL of the base case. A day-ahead NLFE distribution of the California Independent System Operator (CAISO) in the USA is applied as the NLFE of the base case [22]. The values of the mean, standard deviation, skewness, and kurtosis were \(-0.002\), \(0.026\), \(0.715\), and \(4.725\), respectively, and the corresponding PDFs are represented in Figure 7. The standard deviation of the NLFE can be calculated as approximately 65.8 MW, i.e., multiplying 2,531 MW with 0.026 because the \"normalized\" NLFE (of the x-axis) is  
Figure 4: Forecasted net-load profile. Figure 5: \(NLFE_{t}\) distribution represented with a seven-step approximation. Step 5: The hourly generation schedules for every scenario, i.e., S1 to S10, are determined based on the given FNL. Flexibility evaluation was later performed for every scenario. Figures 9 and 10 show the \(RSP_{t}\) results according to the changes in skewness and kurtosis over a certain range, respectively. In both Figures, \(RSP_{t}\) decreases (increases) according to the decrease (increase) in the FNL. This is because the available reserve capacity of the generating unit varies with the FNL. In each Figure, a sudden change in \(RSP_{t}\) occurs, i.e., the points of change lie between S5 and S7, and S6 and S8 in Figures 9 and 10, respectively. Sensitivity analysis was performed for both skewness- and kurtosis-varying scenarios; the points of change were 99.7% of the FNL of the base case in both scenarios. If the \(RSP_{t}\) is evaluated for a wider range of FNL than that of the FNL of the scenario, more points of change may appear. The changes in the FNL can cause a sudden change in the \(RSP_{t}\) because the committed set and operating conditions of the generating units change before and after the point of change. When there is no abrupt change in the \(RSP_{t}\), the changes in skewness and kurtosis were more influential than the FNL.