# Fault Diagnosis of Universal Circuit Breakers Based on Variational Mode Decomposition and WOA-DBN
## 2 Experiment Setup and Data Acquisition
### VMD Approach for the Vibration Signal
To analyze the universal circuit breaker's vibration signals more comprehensively, the extracted time-domain features include standard deviation, mean square root, peak-to-peak value, square root of the mean square value, kurtosis, skewness, waveform factor, crest factor, impulse factor, margin factor, generalized energy, variance, absolute average, maximum value, and shape factor, as shown in Table 3.  
Figure 4: VMD result of the vibration signal. Figure 5: Frequency domain results of IMFs. Performing a discrete Fourier transform on the vibration signal yields the spectrum \(S_{v}\) and the power spectrum \(p_{v}\), \(v\) = 1, 2, 3, \(\ldots\), \(V\), Here, \(V\) represents the number of frequency points, \(f_{v}\) represents the Kth frequency value, \(\overline{f}\) represents the weighted average of all frequency values, \(A_{v}\) represents the amplitude of the \(k\)th frequency value, and \(\overline{A}\) represents the weighted average of all frequency amplitudes. The spectrum and power spectrum are analyzed to obtain various frequency domain features. Among these features are spectrum mean, spectrum standard deviation, spectrum root mean square, power spectrum means, power spectrum variance, power spectrum entropy, power spectral centroid, centroid frequency, center frequency, standard deviation of frequency, spectral skewness, and kurtosis of frequency, as shown in Table 4. \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{2}{c}{**Frequency Domain Feature**} \\ \hline \(F_{18}(spectrum\ mean)=\sqrt{\frac{1}{N}\sum\limits_{i=1}^{N}{(x_{i}-\overline{ \pi})^{2}N^{-1}}}\) & \(F_{17}(spectrum\ standard\ deviation)=\sqrt{\frac{1}{V}\sum\limits_{i=1}^{V}{(S_{v}- \frac{1}{V}\sum\limits_{i=1}^{V}{S_{v}})}}\) \\ \(F_{20}(power\ spectrum\ mean)=\frac{1}{V}\sum\limits_{p=1}^{V}{(f_{v}-\frac{ \sum\limits_{i=1}^{V}{f_{v}p_{v}}}{V})}\) & \(F_{21}(power\ spectrum\ entropy)=-\sum\limits_{p=1}^{V}{f_{v}\ln p_{v}}\) \\ \(F_{22}(power\ spectral\ centroid)=\frac{\sum\limits_{i=1}^{V}{f_{v}p_{v}}}{ \sum\limits_{i=1}^{V}{p_{v}}}\) & \(F_{23}( centroid\ frequency)=\frac{\sum\limits_{i=1}^{V}{f_{v}p_{v}^{2}}}{ \sum\limits_{i=1}^{V}{p_{v}^{2}}}\) \\ \(F_{24}(center\ frequency)=\frac{\sum\limits_{i=1}^{V}{f_{v}A_{v}}}{ \sum\limits_{i=1}^{V}{A_{v}}}\) & \(F_{25}(standard\ deviation\ of\ frequency)=\sqrt{\frac{\sum\limits_{i=1}^{V}{(A_{v}- \overline{\pi})^{2}p_{v}}}{\sum\limits_{i=1}^{V}{p_{v}}}}\) \\ \(F_{26}(spectral\ skewness)=\frac{\sum\limits_{i=1}^{V}{(f_{v}-\overline{f})^{3} \cdot p_{v}}}{\sum\limits_{i=1}^{V}{((f_{v}-\overline{f})^{2}\cdot p_{v}) \frac{\overline{2}}{\overline{2}}}}\) & \(F_{27}(kurtosis\ of\ frequency)=\frac{\sum\limits_{i=1}^{V}{(f_{v}-\overline{f})^{4}p_{v}}}{ \sum\limits_{i=1}^{V}{((f_{v}-\overline{f})^{2}\cdot p_{v})^{2}}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Frequency domain feature calculation formula. \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{2}{c}{**Time-Domain Feature**} \\ \hline \(F_{1}(standard\ deviation)=\sqrt{\frac{1}{N}\sum\limits_{i=1}^{N}{(x_{i}- \overline{1}\frac{1}{N}\sum\limits_{i=1}^{N}{x_{i}})^{2}}}\) & \(F_{2}(mean\ square\ root)=\sqrt{\frac{1}{N}\sum\limits_{i=1}^{N}{x_{i}}^{2}}\) \\ \(F_{3}(peak-to-peak\ value)=\max(x_{i})-\min(x_{i})\) & \(F_{4}(square\ root\ of\ the\ mean\ square\ value)=(\frac{1}{N}\sum\limits_{i=1}^{N}{\sqrt{|x_{i}|}}) ^{2}\) \\ \(F_{5}(kurtosis)=\frac{\frac{k}{N}\sum\limits_{i=1}^{N}{(x_{i}-\overline{f})^{4} }}{\sum\limits_{i=1}^{N}{\frac{(x_{i}-\overline{f})^{3}}{\overline{f}_{i}^{4}}}}\) & \(F_{6}(skewness)=\frac{\frac{1}{N}\sum\limits_{i=1}^{N}{(x_{i}-\overline{f}_{i})^{3 }}}{\sum\limits_{i=1}^{N}{\frac{(x_{i}-\overline{f}_{i})^{3}}{\overline{f}_{i}^{ 4}}}}\) \\ \(F_{7}(waveform\ factor)=\frac{\frac{1}{N}\sum\limits_{i=1}^{N}{\frac{(x_{i}- \overline{f})^{2}}{\overline{f}_{i}^{4}}}}{\sum\limits_{i=1}^{N}{\frac{In summary, a set of signals is initially subjected to VMD, extracting eight modal components. Each modal component contains 27 time and frequency domain eigenvalues, resulting in 216 eigenvalues.