## Materials and Methods  
### Conditional Copula Function  
The copula function, also known as the connection function, is a type of function that connects multiple random variable joint distribution functions with their respective marginal distribution functions. It can describe the correlation between multiple variables more comprehensively and accurately. Due to this characteristic, it has a very wide application in risk prediction in the fields of finance and insurance. Using the copula function for load interval prediction mainly involves the calculation of conditional copula function construction and confidence intervals. The copula function in its general form (Gaussian copula function, t-copula function, and others) always reveals differences from the actual situation. At the same time, the use of the continuous copula function form in calculations is relatively difficult because it requires obtaining the inverse function in the calculation of the confidence interval. Therefore, this paper uses the empirical conditional copula function, which could more accurately describe the correlation between random variables and uses the conditional copula function in the discrete form [34; 35; 36] to calculate the confidence interval at a given confidence level more conveniently. The construction method of the discrete condition copula function is given as follows. It is assumed that \(N\) independent samples for \(t+1\) variables are known, as shown in Equation (1), where each row represents one sample, and each column represents the value of the marginal distribution function of one variable under different samples:  
\[[X_{1},\cdots,X_{t},X_{t+1}]=\left[ \begin{array}{cccc}F_{1}(X_{1}^{1})& \cdots&F_{t}(X_{t}^{1})&F_{t+1}(X_{t+1}^{1})\\ F_{1}(X_{1}^{2})&\cdots&F_{t}(X_{t}^{2})&F_{t+1}(X_{t+1}^{2})\\ \cdots&\cdots&\cdots&\cdots\\ F_{1}(X_{1}^{N})&\cdots&F_{t}(X_{t}^{N})&F_{t+1}(X_{t+1}^{N})\end{array} \right]. \tag{1}\]The copula function is connected to the edge distribution function between \(t+1\) variables. We obtain the value of the edge distribution function of the corresponding Equation (1) by the literature [32; 33]. As shown in Equation (2):  
\[\left[ \begin{array}{cccc}F_{1}(X_{1}^{1})&\cdots&F_{t}(X_{t}^{1})&F_{t+1}(X_{ t+1}^{1})\\ F_{1}(X_{1}^{2})&\cdots&F_{t}(X_{t}^{2})&F_{t+1}(X_{t+1}^{2})\\ \cdots&\cdots&\cdots&\cdots\\ F_{1}(X_{1}^{N})&\cdots&F_{t}(X_{t}^{N})&F_{t+1}(X_{t+1}^{N})\end{array} \right]. \tag{2}\]  
\(F_{1},\cdots,F_{t},F_{t+1}\) in Equation (2) are the edge distribution function of the variable 1,..., variable \(t\), variable \(t+1\), respectively. The value of each edge distribution function in the matrix is between 0 and 1. We divide the interval \([0,1]\) into \(K\) subintervals equally. Each interval ranges is as in Equation (3):  
\[\delta=1/K,S_{1}\in[0,\delta],S_{j}=[(j-1)\delta,j\delta],j=2,3,\cdots K. \tag{3}\]  
It can be seen that there are \(K\) subintervals for each variable, and \(t+1\) variables will together form \(K^{t+1}\) subspaces. Any line in Equation (2) must fall within a certain subspace. The condition copula function refers to the probability distribution function of the corresponding \(t+1\)th variable, with the former \(t\) variable values of a certain sample F being known. That is, the \(F_{t+1}\big{(}x_{t+1}^{F}\big{)}\) probability distribution function in Equation (4) is the conditional copula function that needs to be obtained:  
\[F_{1}(x_{1}^{F}),F_{2}(x_{2}^{F}),\cdots,F_{t}(x_{t}^{F}). \tag{4}\]  
In order to approximately describe a continuous distribution function using a set of discrete probability distributions, we adopted a scenario approach [32; 33]. The specific operation is as follows: in \(N\) samples of Equation (2), the samples in the former \(t\) variables that fall in the same subspace as Equation (4) are removed to form a conditional matrix. Suppose there are \(N_{1}\) samples in the matrix, all samples in the matrix are then classified using K-means clustering. The method consists of two steps.