# Two-Stage Short-Term Power Load Forecasting Based on SSA-VMD and Feature Selection
## 3 Sparrow Search Algorithm Optimizes Variational Mode Decomposition
### Variational Mode Decomposition  
Variational mode decomposition (VMD) is a signal decomposition algorithm, proposed by [PERSON] (2014) in 2014, which was first applied in the field of communication. The algorithm uses a controlled bandwidth to avoid the problem of mode mixing. Unlike the principle of empirical mode decomposition (EMD), VMD is a completely non-recursive model, which uses iterative searching to obtain the optimal solution of the variable model to determine the center frequency and bandwidth of each component. The VMD method achieves a more robust performance in terms of sampling and noise handling compared to the EMD method. The decomposition process of the VMD can be divided into the following steps:  
Step 1--The objective of the VMD is to minimize the sum of the estimated bandwidths of each mode, with the constraint that the sum of all modes is equal to the original signal, whereby the constrained variational expression can be expressed as Equation (3). \[\left\{ \begin{array}{l}\min\limits_{\{u_{k}\},\{\omega_{k}\}}\left\{\sum \limits_{k=1}^{K}\left\|\hat{\vartheta}_{t}\Big{[}\Big{(}\delta(t)+\frac{j}{ \pi t}\Big{)}*u_{k}(t)\Big{]}e^{-j\omega_{k}t}\right\|_{2}^{2}\right\}\\ s.t.\sum\limits_{k=1}^{K}u_{k}(t)=f(t)\end{array} \right. \tag{3}\]  
where \(\{u_{k}\}\) is the set of intrinsic mode functions obtained by decomposing the original signal \(f(t)\); \(\{\omega_{k}\}\) is the set of central frequencies corresponding to \(\{u_{k}\}\); \(\delta(t)\) is the pulse signal; \(K\) is the number of IMFs obtained by decomposing, which can be set artificially, which is one of the advantages of variational mode decomposition; \(*\) is the convolution operation; \(f(t)\) is the original signal sequence, which in this paper represents the original electrical load data; \(t\) is the sampling moment. Step 2--Lagrangian transformation. Introducing \(\lambda\), \(\alpha\), the constrained problem of the constrained variational model is transformed into an unconstrained problem, which can be expressed as Equation (4). \[ \begin{array}{l}L(\{u_{k}\},\{\omega_{k}\},\lambda)=a\sum\limits_{k}^{K} \left\|\hat{\vartheta}_{t}\Big{[}\Big{(}\delta(t)+\frac{j}{\pi t}\Big{)}*u_{k} (t)\Big{]}e^{-j\omega_{k}t}\right\|_{2}^{2}\\ +\left\|f(t)-\sum\limits_{k}u_{k}(t)\right\|_{2}^{2}+\left\langle\lambda(t),f (t)-\sum\limits_{k}u_{k}(t)\right\rangle\end{array}  \tag{4}\]  
Figure 1: The power load dataset of Place 1. The load dataset is taken from a place in China, spanning the period from 1 January 2013 to 1 January 2015, with a sampling interval of 1 h and 24 sampling points per day, for a total of 17,760 pieces of data. where \(\lambda\) is the Lagrange multiplication operator, \(\lambda(t)\) is used to maintain the stringency of the constraints, and \(\alpha\) is the quadratic penalty factor used to ensure the accuracy of signal reconstruction. Step 3--Alternating update. Using the alternating direction multiplier method, the \"saddle point\" of the extended Lagrange is sought in the iterative optimization sequence to achieve effective signal separation.