## 2 Multi-Scenario Generation Based on Wind Power Forecasting Errors
### Error Correlation Modeling Based on Copula Functions  
In practical wind power scenarios, forecasting errors not only exhibit random fluctuations within individual time intervals but also demonstrate significant temporal correlation. Traditional single-period probabilistic modeling methods fail to capture the interdependence among forecasting errors across different time intervals, which in turn compromises the accuracy of subsequent scenario generation and system scheduling. To address this limitation, this study introduces the Copula function to construct a joint probabilistic model that captures the dependency structure among wind power forecasting errors across multiple time intervals [25]. A Copula function describes the dependency structure among multivariate random variables. It allows for the separation of marginal distributions from the joint distribution, thus avoiding rigid assumptions about the form of the data distribution and offering broad applicability. According to [PERSON]'s theorem, for any set of random variables, the joint distribution can be expressed as a combination of the marginal distributions and a Copula function:  
\[F(x_{1},x_{2},\ldots,x_{n})=C[F_{1}(x_{1}),F_{2}(x_{2}),\ldots,F_{n}(x_{n})] \tag{7}\]  
where \(F(x_{1},x_{2},\ldots,x_{n})\) is the joint distribution function of the \(n\)-dimensional random variables, \(F_{i}(x_{i})\) is the marginal distribution function of the \(i\)-th variable, and \(C(\cdot)\) denotes the Copula function. Among various Copula functions, the Gaussian Copula is widely used to characterize the dependence structure among complex random variables due to its advantages of simple parameter estimation, clear functional form, and strong adaptability [26]. Therefore, the Gaussian Copula is selected to model the temporal correlation of wind power forecasting errors. Its mathematical formulation is given as follows:  
\[C(u_{1},u_{2},\ldots,u_{3}|\Sigma)=\Phi_{\Sigma}(\Phi^{-1}(u_{1}),\Phi^{-1}(u_ {2}),\ldots,\Phi^{-1}(u_{n})) \tag{8}\]  
where \(\Phi_{\Sigma}\) is the joint cumulative distribution function (CDF) of an \(n\)-dimensional standard normal distribution with covariance matrix \(\Sigma\); \(\Phi^{-1}(\cdot)\) is the inverse of the standard normal  
Figure 2: Comparison of forecasting error fitting methods. CDF; and \(u_{i}\) denotes the marginal cumulative probability of the forecasting error at time interval \(i\). To accurately capture the temporal correlation characteristics of wind power forecasting errors, a joint probability distribution model is constructed using the Gaussian Copula function. The specific implementation steps are as follows:  
1. Marginal Distribution Fitting  
First, kernel density estimation is applied to the historical forecasting error data from the first 29 days for each time interval, yielding the empirical distribution function of forecasting errors at each time step. Based on this, the original forecasting error data are transformed into a unified [0,1] probability space using their respective empirical cumulative distribution functions, thereby forming a set of standardized random variables with consistent marginal distributions. This transformation eliminates the influence of scale differences among error data on the modeling of correlation structures;  
2. Gaussian Copula Parameter Estimation  
Next, to determine the parameters reflecting the temporal correlation of errors, the standardized data in the [0,1] space are further transformed into the standard normal space using the inverse of the standard normal distribution function. Then, the covariance matrix parameter of the Gaussian Copula function is estimated using the maximum likelihood estimation method. This allows the model to fully capture the intrinsic temporal dependence among forecasting errors across different time intervals;  
3. Forecasting Error Scenario Generation  
Finally, based on the parameters of the Gaussian Copula model, stochastic scenarios of wind power forecasting errors are generated. The generation process consists of two stages.