## 3 Proposed Method
### PQD Feature Extraction  
For the recognition of different PQD patterns, it is essential to utilize a classifier to analyze the signal that is obtained from the voltage or current sensors. However, it is difficult to identify the time-series signal directly for common classification methods. As feature extraction plays an important role in the pattern recognition schemes, VMD is utilized to decompose the obtained signal into several IMFs for potential feature extraction in this paper. The sub-mode decomposed by VMD contains a specific spectrum, which can accurately trace the signal changes of the power quality. Thus, signal decomposition can effectively eliminate the impact of noise and separate the useful components in high level modes. As mentioned above, the decomposition results depend on the mode number, which is defined as \(K\). When considering the harmonic components that exist in the PQD signal, the value of \(K\) is selected as 5. Additionally, parameter \(\alpha\) of the data-fidelity constraint is selected as 2500 based on the experimental trial. Subsequently, by generating artificial PQD signals, the IMF is decomposed by VMD. After the decomposition, the statistical features are calculated and extracted from each mode in the time domain. Table 2 marks and lists all of the features and the corresponding labels.  
With the proposed feature extraction method, the PQD signals are decomposed into five IMFs. Subsequently, the statistical feature vectors are given as the following equations, where \(F_{1}\), \(F_{2}\),...,  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**No.** & **Statistical Feature** & **Expression** \\ \hline
1 & RMS & \(RMS=\sqrt{\frac{1}{K}\sum\limits_{k=1}^{K}\left[x(k)\right]^{2}}\) \\ \hline
2 & Mean & \(\mu=\sum\limits_{k=1}^{K}\left[x(k)\right]/K\) \\ \hline
3 & Standard Deviation & \(\sigma=\sqrt{\sum\limits_{k=1}^{K}\left[x(k)-\overline{x}\right]^{2}/(K-1)}\) \\ \hline
4 & Variance & \(\text{var}=\sum\limits_{k=1}^{K}\left[x(k)-\overline{x}\right]^{2}/K\) \\ \hline
5 & Range & \(range=\max(x(k))-\min(x(k))\) \\ \hline
6 & Kurtosis & \(KT=\sum_{k=1}^{K}\left[\left(x(k)-\overline{x}\right)/o\right]^{4}/K\) \\ \hline
7 & Skewness & \(SK=\sum_{k=1}^{K}\left[\left(x(k)-\overline{x}\right)/o\right]^{3}/K\) \\ \hline
8 & Average Deviation & \(\overline{\sigma}=\sum_{k=1}^{K}\left|x(k)-\overline{x}\right|/K\) \\ \hline
9 & Permutation Entropy & \(PE=-\sum\limits_{i=1}^{m!}p(\pi_{i})\log(p(\pi_{i}))\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Statistical parameters for feature extraction.  
represent the feature vectors of RMS, standard deviation, variance, range, skewness, kurtosis, mean, average deviation, and permutation entropy, respectively.  
\[\begin{array}{l}F_{1}=[RMS1\;RMS2\;RMS3\;RMS4\;RMS5]\\ F_{2}=[\mu_{1}\;\mu_{2}\;\mu_{3}\;\mu_{4}\;\mu_{5}]\\ F_{3}=[\sigma_{1}\;\sigma_{2}\;\sigma_{3}\;\sigma_{4}\;\sigma_{5}]\\ F_{4}=[\mathrm{var}_{1}\;\mathrm{var}_{2}\;\mathrm{var}_{3}\;\mathrm{var}_ {4}\;\mathrm{var}_{5}]\\ F_{5}=[range_{1}\;range_{2}\;range_{3}\;range_{4}\;range_{5}]\\ F_{6}=[KT_{1}\;KT_{2}\;KT_{3}\;KT_{4}\;KT_{5}]\\ F_{7}=[SK_{1}\;SK_{2}\;SK_{3}\;SK_{4}\;SK_{5}]\\ F_{8}=[\overline{\sigma}_{1}\;\overline{\sigma}_{2}\;\overline{\sigma}_{3}\; \overline{\sigma}_{4}\;\overline{\sigma}_{5}]\\ F_{9}=[PE_{1}\;PE_{2}\;PE_{3}\;PE_{4}\;PE_{5}]\end{array} \tag{11}\]  
The data must be normalized between 0 and 1 by using the min-max normalization method in Equation (12), where \(Z_{i}\) is the normalized data, \(F_{\mathrm{max}}\) is the maximum data of the vector, and \(F_{\mathrm{min}}\) is the minimum data, respectively, since the elements of the extracted feature sets have different units.  
\[Z_{i}=\frac{F_{i}-F_{\mathrm{min}}}{F_{\mathrm{max}}-F_{\mathrm{min}}} \tag{12}\]  
After normalization, the feature set \(F\) is obtained while using Equation (13).  
\[F=[F_{1}\;F_{2}\;F_{3}\;F_{4}\;F_{5}\;F_{6}\;F_{7}\;F_{8}\;F_{9}\;] \tag{13}\]