# Load Forecasting Based on LVMD-DBFCM Load Curve Clustering and the CNN-IVIA-BLSTM Model
## 3 Methodology
### LVMD-DBFCM Imbalanced Data Clustering
#### 3.2.2 DBFCM Double Weights Fuzzy C-Means Algorithm  
\(\bullet\) FCM Fuzzy C-means Algorithm  
FCM is a soft clustering algorithm and is unlike traditional hard clustering (HCM) algorithms, which allows the same object to belong to the same cluster [34]. By optimizing the objective function to obtain the affiliation of each sample point to all class centers, the affiliation range is [0, 1] to determine the class of the sample points to achieve the purpose of automatic classification of sample data [35]. The objective function and constraints \(s.t.\) of the FCM algorithm are given in Equation (21):  
\[ \begin{array}{l}\min\limits_{u_{ij}\dot{c}_{i}}J(u_{ij},c_{i})=\sum\limits_ {i=1}^{K}\sum\limits_{j=1}^{N}u_{ij}^{m}\left\|x_{j}-c_{i}\right\|^{2}\\ s.t.\sum\limits_{i=1}^{K}u_{ij}=1,\qquad j=1,2,\ldots\,N\end{array}  \tag{21}\]  
where \(u_{ij}\) denotes that the sample belongs to the cluster affiliation value, \(m\) represents the fuzziness, \(K\) is the number of clusters, \(c_{i}\) is the \(i\)-th clustering center, and \(\left\|x_{j}-c_{i}\right\|^{2}\) is the 2 parity of the Euclidean distance from each data point to the cluster center. The clustering center and affiliation update equations are shown in Equations (22) to (23):  
\[u_{ij}=\frac{1}{\sum\limits_{l=1}^{K}\left(\frac{\left\|x_{j}-c_{i}\right\|} {\left\|x_{j}-c_{i}\right\|}\right)^{\frac{2}{m-1}}} \tag{22}\]  
\[c_{i}=\frac{\sum\limits_{j=1}^{N}u_{ij}^{m}x_{j}}{\sum\limits_{j=1}^{N}u_{ij}^ {m}} \tag{23}\]* DBPCM Double Weights Fuzzy C-means Algorithm  
Traditional FCM clustering is inaccurate in classification for holidays in small categories. From the date type, the power load data are unbalanced data. FCM has a \"uniform effect of cluster size\" issue, which affects the recognition effect and causes the algorithm to not recognize the holiday load model implied in the historical dataset [36; 37]. In this paper, the DBPCM algorithm is proposed to improve the clustering performance of imbalanced data. The objective function of DBPCM utilizes the clustering volume as the weight and the weighted Euclidean distance as the metric distance. 1. The affiliation matrix defines the class volume, which is then introduced as a constraint _s.t._ into the traditional FCM algorithm objective function, as shown in Equation (24): \[J_{DBPCM}=\sum\limits_{i=1}^{K}\sum\limits_{j=1}^{N}\frac{u_{ij}^{m}\big{\|}x_{ j}-c_{i}\big{\|}^{2}}{v_{j}}\] (24) where \(v_{j}\) is the volume of the \(j\)-th class; see Equation (25). Constraint _s.t._ is shown in Equation (26): \[v_{j}=\frac{\sum\limits_{i=1}^{N}u_{ij}}{N}\] (25) \[s.t.\sum\limits_{i=1}^{K}u_{ij}=1,\quad j=1,\;2,\;\ldots,\;N\] (26)  
The objective function of DBPCM uses the clustering volume as weights, which can balance the volume of each class in the clustering process. Thus, it can compensate for the unequal interactions between classes and improve the clustering performance of traditional algorithms for unbalanced data. \[\frac{\partial f_{DBPCM}}{\partial c_{i}}=\frac{\partial\sum\limits_{i=1}^{N}u _{ij}^{m}\big{\|}x_{j}-c_{i}\big{\|}^{2}/v_{j}}{\partial c_{i}} \tag{27}\]  
The derivative of Equation (27) with respect to the affiliation degree is constantly positive. Therefore, the Lagrange multiplier method is used to solve the affiliation degree and clustering center by using a greedy strategy and introducing constraint variables.