# Load Forecasting Based on LVMD-DBFCM Load Curve Clustering and the CNN-IVIA-BLSTM Model
## 3 Methodology
### LVMD-DBFCM Imbalanced Data Clustering
#### 3.2.1. LVMD Double Decomposition Modal Strategy  
\(\bullet\) VMD Variational Mode Decomposition  
VMD [31] is an iterative process that is used to search for the optimal solution of a variational model and to determine the mode \(u_{k}(t)\) and its corresponding central frequency \(w_{k}\) and bandwidth. VMD has good denoising capabilities. VMD denoising are as follows:  
1. Initialize parameters \(\hat{n}_{k}^{1}\), \(w_{k}^{1}\), \(\hat{\lambda}^{1}\), and let \(n=0\). 2. \(n=n+1\), update \(w_{k}\) and \(u_{k}\). 3. \(k=k+1\), repeat the previous step until \(k=K\). 4. Update \(\lambda\), and the equation is shown in Equation (12). 5. Repeat steps 2-5 until the end, when the condition of Equation (13) is satisfied. 6. The component that contains the minimum time information after decomposition is judged and considered to be random noise that is eliminated. Then, the data are reconstructed by the remaining modal component pairs, and the reconstructed data are the load data that do not contain noise. \[\hat{\lambda}^{n+1}(\omega)=\hat{\lambda}^{n}(\omega)+\tau\Bigg{[}\hat{f}( \omega)-\sum_{k}\hat{n}_{k}^{n+1}(\omega)\Bigg{]}\] (12) \[\sum_{k}\Big{\|}\hat{u}_{k}^{n+1}-\hat{u}_{k}^{n}\Big{\|}_{2}^{2}/\|\hat{u}_{ k}^{n}\|_{2}^{2}<e\] (13)  
\(\bullet\) LMD Local Mean Decomposition  
LMD is a new time-frequency analysis method that adaptively decomposes complex signals into a finite sum of product function (PF) components. Each of these PF components is actually a single component of the AM-FM information. LMD has good feature decomposition abilities [32,33]. The LMD principle is as follows. 1. First, find all the extreme points contained in the data series \(x(t)\), assuming that the distribution of extreme points is \(\{n_{1},\ n_{2},\ n_{3},\cdots\}\), and then calculate the mean \(m_{i}\) and envelope \(a_{i}\) of the adjacent extreme points according to Equations (14) and (15). \[m_{i}=\frac{n_{i}+n_{i+1}}{2}\] (14) \[a_{i}=\frac{|n_{i}-n_{i+1}|}{2}\] (15)  
The local mean function curve \(m_{11}(t)\) and the envelope function curve \(a_{11}(t)\) are obtained by connecting \(m_{i}\) and \(a_{i}\) with line segments and smoothing, respectively, and the sliding average formula is given in Equation (16):  
\[Y_{s}(i)=\frac{1}{2R+1}(Y(i+R)+Y(i+R-1)+\cdots+Y(i-R)) \tag{16}\]  
where \(Y(i)\) is the sequence to be smoothed, \(2R+1\) is the sliding span, and \(R\) is the distance from \(Y(i)\) to the starting point of the sliding process. 2. \(m_{11}(t)\) is removed from the original signal \(x(t)\) to obtain \(h_{11}(t)\), and \(s_{11}(t)\) is obtained by demodulating \(h_{11}(t)\) using Equations (17) and (18): \[h_{11}(t)=x(t)-m_{11}(t)\] (17)\[s_{11}(t)=\frac{h_{11}(t)}{a_{11}(t)} \tag{18}\]  
When \(a_{12}(t)=1\), \(s_{11}(t)\) decomposes into a pure FM function; if \(a_{12}(t)\
eq 1\), repeat the above process until \(a_{1n}(t)=1\) and \(s_{1(n-1)}(t)\) is the corresponding pure FM function; see Equation (19) for details:  
\[h_{1n}(t)=s_{1(n-1)}(t)-m_{1n}(t) \tag{19}\]  
3. The PF component of the envelope signal is the product of the envelope signal and the pure FM signal, and the above steps are repeated after stripping the PF component from the original signal to obtain the new signal until the residual component is a monotonic function. The final result of LMD is shown in Equation (20):  
\[x(t)=\sum\limits_{p=1}^{k}PF_{p}(t) \tag{20}\]  
where the last component is a monotonic function. LMD decomposes the original signal into multiple high-frequency and low-frequency components without distorting the original signal in the decomposition process. If the sliding step of the sliding average algorithm is not chosen properly in the first step, the decomposition result will be greatly affected. In this paper, the original sliding average process is replaced by the three-time Hermite interpolation method to improve the decomposition accuracy of the LMD algorithm while reducing the overshoot and undershoot of the envelope.