## 2 Multi-Scenario Generation Based on Wind Power Forecasting Errors
### Kernel Density Estimation for Error Distribution Fitting  
To characterize the distribution of wind power forecasting errors across different time intervals, this study employs the kernel density estimation (KDE) method to fit the error samples derived from the historical data of the first 29 days [22]. Compared with traditional parametric distribution fitting methods, KDE--as a non-parametric technique--can adaptively capture the true underlying distribution of the data without assuming any specific distribution form in advance.  
\[\hat{f}_{t}(e)=\frac{1}{d\cdot h}\sum_{i=1}^{d}K(\frac{e-e_{i}(t)}{h}) \tag{4}\]  
where \(d\) is the number of historical sample days, and \(K(\cdot)\) is the kernel function. In this study, the Gaussian function is selected, which is defined as follows:  
\[K(u)=\frac{1}{\sqrt{2\pi}}\exp(-\frac{u^{2}}{2}) \tag{5}\]  
In Equation (4), \(h\) denotes the bandwidth parameter, which directly affects the smoothness of the kernel density estimation. To determine an appropriate bandwidth, this study adopts [PERSON]'s rule of thumb [23], which defines the bandwidth as follows:  
\[h=1.06\times\sigma\times d^{-1/5} \tag{6}\]  
where \(\sigma\) is the standard deviation of the forecasting error sample set, and \(d\) is the number of historical samples used for error distribution estimation. If the bandwidth is set too small, the KDE curve becomes overly steep and sensitive to noise; conversely, if the bandwidth is too large, the estimated distribution may become overly smooth, potentially resulting in the loss of important detail. Therefore, [PERSON]'s rule of thumb is adopted in this study to determine the bandwidth parameter.  
In addition, to further validate the fitting performance of the kernel density estimation method, two typical parametric approaches--namely, the normal distribution and the \(t\)-distribution--are selected for comparative analysis. The normal distribution fitting method derives the probability density function using the sample mean and standard deviation of the historical forecasting errors, estimated via the maximum likelihood method. The \(t\)-distribution fitting method accounts for the heavy-tailed characteristics of the data, making it more suitable for modeling extreme forecasting errors [24]. Figure 2 presents a comparison between the histogram of forecasting errors from the first 29 days and the fitted curves obtained from the three methods.  
As shown in Figure 2, the normal distribution fitting curve performs well near zero forecasting errors, but its accuracy deteriorates in regions with larger errors. Although the \(t\)-distribution curve improves the fitting performance to some extent compared to the normal distribution, its unimodal nature prevents it from accurately capturing multimodal patterns or local density fluctuations present in the actual data. In contrast, kernel density estimation, as a non-parametric method, is not constrained by any specific distributional assumptionsand can flexibly adapt the fitting curve according to the actual data characteristics. It effectively captures the skewness, multimodality, and heavy-tailed features of forecasting error data. The KDE fitting curve exhibits a significantly better match with the error histogram than both the normal and \(t\)-distribution fitting methods.