## 3 Evaluation of the Effect of Uncertainty Characteristics on Flexibility  
### Modeling NLFE Characteristics Using a Pearson Distribution  
The output of RERs is volatile and uncertain. The forecasted output involves a forecast error, which can be modeled as a random variable. Regardless of how well the output is predicted, the forecast error is inevitable, and the system operator cannot avoid it. Flexibility enhancement is thus necessary to deal with the forecast error, which indicates the risk of the system. The system operator should determine an adequate level of flexibility, which can be quantified using the RSP index. The forecast error of the renewable energy resource, i.e., VGFE, is reflected within the NLFE in the RSP calculation., i.e., NLFE is set as LFE minus VGFE. This RSP method was used to estimate the RC shortage, and the RC requirement was associated with the NLFE.  
The normal distribution has been used to model NLFE in the RSP calculation [28; 29]; however, it has a limitation in representing the distribution of NLFE characteristics such as skewness and kurtosis [30]. The Pearson distribution has been widely used to generate the PDF with skewness and kurtosis [31]. An advantage of the Pearson distribution over other distributions, such as the Weibull distribution, is the use of PDF moments to determine the distribution parameters. This distribution can be modeled as a function of skewness and kurtosis, and it allows the separate analysis of the effects of skewness and kurtosis. Thus, the Pearson distribution is applied to the NLFE model. It refers to a family of several probability distributions; it is made up of seven probability distributions, i.e., Type I to VII. If the values of the first four moments are known, i.e., mean, standard deviation, skewness, and kurtosis, the PDFs can be generated. Some of these types are listed in Table 1.  
Figure 1: Probability distribution for \(E_{t}\).  
In the last column, the parameter \(\gamma\) is defined as follows:  
\[\gamma=\frac{P_{sk}^{2}{(P_{k}+3)}^{2}}{4\big{(}2P_{k}-3P_{sk}^{2}-6\big{)}\big{(}4 P_{k}-3P_{sk}^{2}\big{)}} \tag{6}\]  
where \(P_{sk}\) and \(P_{k}\) are the skewness and kurtosis of the NLFE, respectively. The PDF type is determined according to \(P_{sk}\), \(P_{k}\), and \(\gamma\). For reference, parameters such as \(y_{0}\), \(A_{1}\), \(A_{2}\), \(m_{1}\), \(m_{2}\), \(a\), \(r\), \(v\), \(m\), \(q_{1}\), and \(q_{2}\) can be calculated using the following moments: mean, standard deviation, skewness (\(P_{s}\)), and kurtosis (\(P_{k}\)). For further details, please refer to [32]. Table 2 presents a simple example of the parameters of the Pearson distribution of NLFE, and the corresponding results are shown in Figure 1. The skewness and kurtosis are only varied, i.e., the mean and standard deviations are kept the same. The results for C11 to C17 and those of C21 to C27 correspond to Figure 2a,b, respectively.  
In Figure 2a, the PDF for the negative (positive) skewness is skewed to the left (right). In Figure 2b, the larger the kurtosis, the sharper the PDF at the center; however, the difference between each PDF according to the increase in kurtosis becomes increasingly smaller.