# Short-Term Power Load Forecasting Based on an EPT-VMD-TCN-TPA Model
## 2 Related Theories and Methods
### Variational Modal Decomposition (VMD)  
In many cases, due to the volatility of the signal data sequence, we need to decompose the signal sequence into a series of smooth subsequences in order to better perform subsequent processing. There are many commonly used decomposition methods, for example, EMD and EEMD decomposition methods, but all these methods have some limitations. In 2014, [PERSON] et al. proposed a completely non-recursive variational modal decomposition method (VMD), which adaptively decomposes complex timing signals into several simple modal components [44]. The VMD algorithm is an effective method for applying the traditional Wiener filter to multiple adaptive bands, which has a broad theoretical basis and is easy to implement. VMD decomposition is a way to convert complex mathematical problems into simple mathematical expressions by building variational problems, finding the best solution, and improving the performance of the model using alternating direction multipliers to improve sampling accuracy. The calculation flow of VMD is shown in Figure 3. The specific steps are as follows. 1. Constructing variational problems. Suppose the original input signal \(f(t)\) is decomposed into \(k\) components \(u\). In order to ensure that the decomposed sequence is a modal component with a finite bandwidth of the central frequency, the sum of the estimated bandwidths of each modal component is required to be minimal, and the constraint is that the sum of all modal components should be equal to the original signal. The corresponding expression for the constrained variational problem is as follows:  
\[\min_{\{u_{k}\}\{u_{k}\}}\left\{\sum_{k}\left\|\partial_{t}\bigg{[}\delta(t)+ \frac{j}{\pi t}\bigg{]}\times u_{k}(t)e^{-j\mu_{k}t}\right\|_{2}^{2}\right\} \tag{8}\]  
\[s.t.\sum_{k}u_{k}=f(t) \tag{9}\]  
where \(f(t)\) denotes the original signal, \(k\) is the predetermined number of modal components, \(\partial_{t}\) is the Dirac distribution function, \(u_{k}=\{u_{1},u_{2},\cdots,u_{k}\}\) is the \(k\) modal components obtained from the adaptive decomposition of the original signal, and \(w_{k}=\{w_{1},w_{2},\cdots,w_{k}\}\) is the central frequency of each modal component. Figure 3: VMD calculation process. Transformation of variational problems. By using the quadratic penalty term \(\alpha\) and the Lagrangian operator \(\lambda(t)\), we can convert the variational problem with constraints into a problem with unconstrained variation by the following equation:  
\[L(\{u_{k}\},\{\omega_{k}\},\lambda):=\alpha\sum\limits_{k}\left\|\partial_{t} \bigg{[}(\delta(t)+\frac{j}{\pi t})\times u_{k}(t)\bigg{]}e^{-jn_{k}z}\right\|_ {k}^{2}+\left\|f(t)-\sum\limits_{k}u_{k}(t)\right\|_{2}^{2}+\left\langle\lambda( t),f(t)-\sum\limits_{k}u_{k}(t)\right\rangle \tag{10}\]  
Solve the variational problem. By using the alternating direction multiplier method (ADMM), we can solve the initial minimization problem. The Fourier isometric transform is used to obtain the results of the sequence decomposition in the frequency domain. \(u_{k}^{n+1}\), \(\omega_{k}^{n+1}\), and \(\lambda\) are solved iteratively by searching the saddle points of the increasing Lagrangian function with the following equation:  
\[\dot{u}_{k}^{n+1}(\omega)=\frac{\hat{f}(\omega)-\sum\limits_{i\
eq k}\hat{u}( \omega)+\frac{\hat{\lambda}(\omega)}{2}}{1+2\pi(\omega-\omega_{k})^{2}} \tag{11}\]  
\[\omega_{k}^{n+1}=\frac{\int_{0}^{\infty}\omega|\hat{u}(\omega)|^{2}d\omega}{ \int_{0}^{\infty}|\hat{u}_{k}(\omega)|^{2}d\omega} \tag{12}\]  
\[\hat{\lambda}^{n+1}(\omega)=\hat{\lambda}^{n}(\omega)+\tau(\hat{f}(\omega)- \sum\limits_{k}\hat{u}_{k}^{n+1}(\omega)) \tag{13}\]  
where \(\hat{u}_{k}^{n+1}(\omega)\) is the \(k\)th modal component with the center frequency of \(\omega\) at the \((n+1)\) th iteration. \(\omega_{k}^{n+1}\) is the center frequency of the \(k\)th modal component during the \((n+1)\) iteration. \(\tau\) is the noise tolerance of the signal.