## Materials and Methods
### Multi-Objective Prediction Method Based on the Conditional Copula Function
#### 2.2.1 Load Interval Ultra-Short-Term Prediction  
A relatively strong correlation exists between load sequences in short time intervals. Therefore, we can take the continuous load sequences collected from certain regions as historical data and use the copula function to describe this correlation between load sequences. We assume that the continuous load sequence of a certain region is \([y_{1},y_{2},\cdots,y_{n-1},y_{n}]\), and we need to predict the interval of the load \(y_{n+1}\) at the next moment. In using the discrete conditional copula function to make a load prediction, assuming that the condition number is \(t\), the \((t+1)\) variables and \((n-t)\) samples can be transformed from the historical data:  
\[\left[\begin{array}{cccc}y_{1}&\cdots&y_{t}&y_{t+1}\\ y_{2}&\cdots&y_{t+1}&y_{t+2}\\ \cdots&\cdots&\cdots&\cdots\\ y_{n-t+1}&\cdots&y_{n-1}&y_{n}\end{array}\right]. \tag{6}\]  
Figure 1: Process diagram of the multi-objective prediction method based on the conditional copula function.  
Each column in the matrix is a load sequence at certain time intervals, which is viewed as \([X_{1},X_{2},\cdots,X_{t},X_{t+1}]\). Thus, we establish the conditional copula distribution function of load to be predicted. In other words, we use the marginal distribution function value \(F_{1}(x_{1}^{F}),F_{2}(x_{2}^{F}),\cdots,F_{t}(x_{t}^{F})\) of the load \([y_{n-t+1},\cdots,y_{n-1},y_{n}]\) as a known condition and obtain the probability distribution function of \(y_{n+1}\).  
We assume that the conditional copula function obtained is \(\big{[}F_{t+1}\!^{j},p^{j}\big{]}_{j=1,2,\cdots,J}\), where \(p^{1}\) is the largest and \(p^{l}\) is the smallest. As all values of the marginal distribution functions fall into the interval [0, 1], we divide the interval into \(K\) subintervals, which are denoted as \(S_{1},S_{2},\cdots,S_{K}\) in turn. Thus \(p^{j}\) is accumulated from \(j=1\) until the sum is greater than or equal to the preset confidence level \(\beta\). The value of \(\beta\) needs to be preset by humans according to actual needs and has nothing to do with the other factors. In the process of grid dispatching and decision-making, too small a \(\beta\) value will make the interval coverage rate low and the scheduling risk large, while too big a \(\beta\) value will make the interval width large and the application effect not good. As a compromise, we set \(\beta=0.9\) to get a better prediction.  
We assume that when accumulating to \(j=\Phi\), \(\sum\limits_{j=1}^{j=\text{q}}p^{(j)}\geq\beta\), the subinterval corresponding to \(F_{t+1}^{1},F_{t+1}^{2},\cdots F_{t+1}^{q}\) is given as follows:  
\[F_{t+1}^{1}\in\big{(}S_{1}\!^{l},S_{1}\!^{u}\big{]},F_{t+1}^{2}\in\big{(}S_{2 }\!^{l},S_{2}\!^{u}\big{]}\cdots F_{t+1}^{q}\in\big{(}S_{q}\!^{l},S_{q}\!^{u} \big{]}, \tag{7}\]  
where \(l\) (lower) refers to the lower bound of the interval, and \(u\) (upper) refers to the upper bound of the interval. Under the confidence level \(\beta\), the prediction interval of the load at the \((t+1)\)th moment is the union of Equation (5), namely:  
\[\big{(}S_{1}\!^{l},S_{1}\!^{u}\big{]}\cup\big{(}S_{2}\!^{l},S_{2}\!^{u}\big{]} \cup\cdots\big{(}S_{q}\!^{l},S_{q}\!^{u}\big{]}=\big{(}S^{l},S^{u}\big{]}. \tag{8}\]  
Using the inverse operation of the marginal distribution function, we obtain the prediction interval \(F_{t+1}^{-1}(S^{l})\leq W_{t+1}\leq F_{t+1}^{-1}(S^{u})\) of the load at the \((t+1)\)th moment, and the load interval prediction at \((t+1)\)th moment is realized.