## 2 Load Characteristic Portrait
### Clustering with Warm Behavior Patterns  
The goal of users label clustering is to quickly grasp the heating characteristics of different thermal users to achieve different treatments of different heating groups. After combing and screening many times, the following six labels were obtained, as shown in Table 1. \begin{table}
\begin{tabular}{c c c} \hline \hline
**No.** & **Labels** & **Description** \\ \hline
1 & Type of heating & According to the characteristics of load timing sequence, it is divided into residents, education, commerce, and industry. \\ \hline
2 & Seasonal characteristics & According to users sensitivity to different seasons, it can be divided into summer sensitivity, winter sensitivity, summer-winter sensitivity and non-sensitivity. \\ \hline
3 & Temperature sensitivity: & According to the load changing trend of users at different temperatures, it is divided into low temperature sensitivity, high temperature sensitivity, high temperature sensitivity, high-low temperature sensitivity and non-sensitivity. \\ \hline
4 & Growth rate of heating & According to the heating growth of users, it can be divided into rising, falling, and flat. \\ \hline
5 & Peak and valley characteristics & According to the users sensitivity to the peak-valley electricity price system, it can be divided into peak-valley equilibrium, valley preference, and peak preference. \\ \hline
6 & Weekly rest & According to the characteristics of the users week, can be divided into five days, six days, seven days. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Users label description. In this paper, FCM clustering algorithm was used to calculate each thermal user's label data to be clustered, and four memberships of thermal user clustering centers to realize the clustering division of the sample thermal user's label data. During the iterative process, FCM takes the minimum intra-cluster distance and the maximum distance between clusters as the optimization goals, and constantly adjusts the categories of various thermal user data, so that it optimizes the clustering results continuously. The object to be clustered is the \(D\)-dimensional dataset of \(N\) samples:  
\[X=\left[ \begin{array}{cccc}x_{11}&\cdots&x_{1j}&\cdots&x_{1D}\\ &\vdots&\ddots&\vdots&\\ x_{i1}&\cdots&x_{ij}&\cdots&x_{iD}\\ \vdots&\ddots&\vdots&\ddots&\vdots\\ x_{N1}&\cdots&x_{Nj}&\cdots&x_{ND}\\ \end{array} \right]\]  
The data collection class is divided into \(K\) subsets, where the value of \(K\) is 4, the subset is represented as \(C=\{c_{1},\cdots,c_{i},\cdots,c_{4}\}\); and \(c_{1},\ c_{2},\ c_{3}\), and \(c_{4}\) represent residents, education, business and industry, respectively. Here, we set \(D=7\), which represents six items of user label data and one category data, respectively. Assuming that the probability of each thermal user \(j\) belonging to the \(ith\) clustering subset is \(u_{ij}\), then FCM can be defined as an optimization problem with equality constraints [15]:  
\[\left\{ \begin{array}{c}min\ J=\sum\limits_{i=1}^{K}\sum\limits_{j=1}^{N}u_{ ij}^{m}x_{j}-c_{i}^{2}\\ \\ \text{st.}\quad\sum\limits_{i=1}^{K}u_{ij}=1\\ \end{array} \right. \tag{1}\]  
where \(J\) is the value function of FCM. \(m\) is the membership factor, which is used to adjust the degree of influence of the membership indicator in the optimization target. In most cases, we chose \(m=2\). The iterative formulas for obtaining membership coefficients and cluster centers are shown in Equations (2) and (3) [15]:  
\[c_{i}=\frac{\sum\limits_{j=1}^{N}\left(x_{j}u_{ij}^{m}\right)}{\sum\limits_{j= 1}^{N}u_{ij}^{m}} \tag{2}\]  
\[u_{ij}=\frac{1}{\sum\limits_{k=1}^{K}\left(\frac{\left[\frac{x_{j}-c_{i}}{x_{ j}-c_{k}}\right]}{\left[x_{j}-c_{k}\right]}\right)^{\frac{2}{m-1}}} \tag{3}\]  
Clustering steps of the warm behavior patterns for thermal users based on the FCM algorithm:  
1. Initialization: Set the number of clustering \(K\) and membership factor \(m\), randomly initialize the membership matrix \(U\), \(U\) is a matrix made up of \(u_{ij}\), the center of the cluster. 2.