## 2 Knowledge and Tools of Model Preparation
The fuzzy clustering of sample space \(\mathbf{X}\) is represented by fuzzy matrix \(\mathbf{U}=(u_{ik})_{n\times c}\), and \(\mathbf{U}\) satisfies the following conditions:  
\[\left\{ \begin{array}{l}u_{ik}\in[0,1]\\ \sum_{k=1}^{c}u_{ik}=1\end{array} \right. \tag{10}\]  
The objective function is defined as:  
\[\mathbf{J}_{m}(\mathbf{U},\mathbf{V})=\sum\
olimits^{n}\sum\
olimits^{c}_{k=1}u^{m}_{ik} \cdot d^{2}_{ik}(x_{i},v_{k}) \tag{11}\]  
In the formula, \(\mathbf{V}=[v_{1},\cdots,v_{k},\cdots,v_{c}]\), \(v_{k}\) is the \(k\)-th clustering center, \(d^{2}(x_{i},v_{k})\) is the distance measurement function between \(x_{i}\) and \(v_{k}\), and \(m\) is the fuzzy weighted index. In order to get the best fuzzy \(c\) partition of dataset \(\mathbf{X}\), the solution (\(\mathbf{U}\),\(\mathbf{V}\)) that makes \(J_{m}(\mathbf{U}\),\(\mathbf{V}\)) the smallest need to be obtained. This can be achieved by the following steps [43]:  
**Step 1:** Initialization. Input dataset \(\{x_{i}\), \(i=1,2,\ldots\)\(.n\}\), clustering number \(c\), fuzzy weighted index \(m(m\in R>1)\), maximum number of iterations \(T\) and threshold \(\varepsilon\). The membership matrix \(\mathbf{U}^{(t)}\) (\(t=0\), \(t\) is the iteration numbers) is initialized randomly and satisfies the Formula (10). ``` Input: \(\bullet\)  MOP-multi-objective optimization problem \(\bullet\) \(N\)-the number of the MOEA/D subproblems \(\bullet\) \(\lambda_{1}\), \(\ldots\), \(\lambda_{N}\)-a uniform distribution of N weight vectors \(\bullet\) \(T\)-the number of the weight vectors in the neighborhood of each weight vector \(\bullet\) \(max\_gen\)-the maximum number of generations Output: \(\bullet\) EP-external population Setup: \(\bullet\) Set EP = \(\varnothing



**Step 2:** Update the clustering center and membership matrix. \[v_{k}^{(t+1)}=\frac{\sum_{i=1}^{n}{(u_{k}^{(t)})}^{m}x_{i}}{\sum_{i=1}^{n}{(u_{k} ^{(t)})}^{m}},\quad\forall k \tag{12}\]

\[u_{k}^{(t+1)}=\frac{\left[1/d^{2}(x_{i},v_{k}^{(t+1)})\right]^{1/(m-1)}}{\sum_{ j=1}^{c}{\left[1/d^{2}(x_{i},v_{j}^{(t+1)})\right]^{1/(m-1)}}},\quad\forall k, \forall i \tag{13}\]

**Step 3:** If \(\underset{i\downarrow k}{\max}\|u_{ik}^{(t)}-u_{ik}^{(t+1)}\|<\varepsilon\) or \(t>T\), the category of \(x_{i}\) is signed as \(z_{i}=\underset{1\leq k\leq c}{\arg\max}(u_{ik})\). Otherwise, \(t=t+1\) and go to **step 2**. **3. Construction of Power Load Double Prediction System**

In this paper, a double forecasting system of power load is established. The double forecasting system means a forecasting system that integrates point forecasting and interval forecasting. The relationship between point forecasting and interval forecasting is similar to point estimation and interval estimation in statistics. The flow of the system we designed is to carry out point prediction first, get the result of point prediction, then construct a suitable confidence interval according to the result of point prediction to carry out interval prediction. The forecasting system includes two modules: point forecasting module and interval forecasting module. The following is the system construction process, and the system structure is shown in Figure 1.

### Point Forecasting Module

In this section, we successfully put forward a new type of nonlinear hybrid point forecasting model, using RBF, ELM, ENN and ARIMA, as well as BP network, MOEA/D algorithm and nonlinear combination mechanism to achieve high-precision and more stable load point prediction results. Figure 1: The flowchart of load forecasting by double hybrid model. The model construction steps are recorded in the upper right, and the methods displayed in the specific model are divided into five parts. D and E part show the results of point forecasting and interval forecasting, respectively. Considering the good prediction performance of BP network, this paper takes BP network as the method of nonlinear combination. The point prediction module of the designed system is composed of four steps. The details are as follows:

**Step 1:** Power load data preprocessing.