# Short-Term Photovoltaic Power Generation Based on MVMD Feature Extraction and Informer Model
## 2 Power Feature Extraction
### MVMD  
After obtaining the MIC correlation weather features, a thorough analysis of the PV power time series combined with weather factors is conducted. In traditional prediction methods based on VMD, VMD is often only used to decompose historical PV power sequences, while environmental factors such as meteorological sequences are often neglected. Even when considering meteorological sequences, they are usually decomposed separately and cannot achieve time-frequency synchronous analysis of multidimensional sequences. Therefore, this paper adopts MVMD to decompose PV power along with its weather features with the highest correlation, thereby achieving time-frequency synchronous analysis.  
The core idea of MVMD is to decompose the input multivariate time-series data into its intrinsic multivariate modulated oscillations. The model is as follows:  
\[\left\{\begin{array}{l}\underset{\{u_{k}\},\{\forall u_{k}\}}{\min}\left\{ \sum\limits_{k}\sum\limits_{c}\left\|\partial_{t}\left[u_{+}^{\text{k},c} \text{e}^{-iy_{k}t}\right]\right\|_{2}^{2}\right\}\\ \sum\limits_{k}u_{k,c}(t)=x_{c}(t),c=1,2,3,\ldots,c,\end{array}\right. \tag{2}\]  
In the formula, \(k\) is the number of separated modes set in the algorithm; \(c\) is the number of original signals input into the algorithm; \(w_{k}\) is the center frequency of each mode after MVMD decomposition; \(x_{c}(t)\) is the set of input multivariate time-series feature signals; and \(u_{k,c}(t)\) is the set of decomposed time-series feature signals.  
When performing calculations, unconstrained variational problems are easier to handle and optimize. Common optimization algorithms and mathematical tools are more mature and efficient for unconstrained problems. In contrast, solving constrained problems is relatively complex, requiring consideration of the degree to which constraints are met.  
Thus, converting a constrained problem into an unconstrained one can simplify the computation process. To convert a constrained variational problem into an unconstrained one, the Lagrange multiplier method is used. The form of its Lagrangian function is as follows:  
\[\begin{split} L(\{u_{k,c}\},\{W_{k}\},\lambda_{c})=\epsilon\sum \limits_{k}\sum\limits_{c}\left\|\hat{a}_{t}\left[\mathrm{u}_{+}^{k,c}e^{-j \mu_{k}t}\right]\right\|_{2}^{2}+\\ \sum\limits_{c}\left\|x_{c}(t)-\sum\limits_{k}u_{k,c}(t)\right\|_ {2}^{2}+\\ \sum\limits_{c}\left\langle\lambda_{c},x_{c}(t)-\sum\limits_{k}u_{ k,c}(t)\right\rangle\end{split} \tag{3}\]  
In the formula, \(\epsilon\) is the penalty factor, which is a constant; \(\lambda(t)\) is the time-dependent Lagrange multiplier.  
In the process of conversion, the introduction of penalty factors and Lagrange multipliers will affect the optimization process of the objective function, which may lead to deviations in the final variational mode decomposition results. Therefore, in the decomposition operation, it is essential to adjust the penalty factor or Lagrange multiplier appropriately in real time.  
By solving the above equation, the solutions for each variable are obtained:  
\[\hat{a}_{k,c}^{n+1}(w)=\frac{\hat{x}_{c}-\sum_{i\
eq k}\hat{u}_{i,c}(w)+\frac{ \hat{\lambda}_{c}(w)}{2}}{1+2\epsilon(w-w_{k})^{2}} \tag{4}\]  
\[w_{k}^{n+1}=\frac{\sum\limits_{c}\int_{0}^{\infty}w\left|\hat{u}_{k,c}(w) \right|^{2}\!\!dw}{\sum\limits_{c}\int_{0}^{\infty}\left|\hat{u}_{k,c}(w) \right|^{2}\!\!dw} \tag{5}\]  
\[\hat{\lambda}_{c}^{n+1}(w)=\hat{\lambda}_{c}^{n}(w)+\tau\left[\hat{x}_{c}(w)- \sum\limits_{k}\hat{u}_{k,c}^{n+1}(w)\right] \tag{6}\]  
In the formula, \(\wedge\) denotes the Fourier transform; \(\tau\) is the update coefficient. Through continuous iterative updates of the algorithm, the desired decomposition modes are obtained.