# 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  
VMD is used to analyze the signal and extract fault characteristics. [PERSON] et al. proposed the VMD hypothesis in 2014 [37]. The essence of this model is to decompose \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Description** & **Amount** & **Database 1** & **Database 2** \\ \hline Normal & 100 sets & 80 sets & 20 sets \\ Mechanical structure stuck & 100 sets & 80 sets & 20 sets \\ Insufficient core travel & 100 sets & 80 sets & 20 sets \\ Core stuck & 100 sets & 80 sets & 20 sets \\ High operating voltage & 100 sets & 80 sets & 20 sets \\ Low operating voltage & 100 sets & 80 sets & 20 sets \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Data samples. Figure 3: Time-domain vibration signals for various situations. the original signal into \(K\) time series \(o_{k}(t)\) by iteratively searching a variational model and finally obtaining the constrained variational model as  
\[\left\{ \begin{aligned} \min\limits_{\{o_{k}\},\{\omega_{k}\}}\left\{ \sum\limits_{k=1}^{K}\left\|\partial_{t}\right[\left(\delta(t)+\frac{j}{\pi t} \right)\ast o_{k}(t)\right]\mathrm{e}^{-\mathrm{j}\omega_{k}t}\right\|_{2}^{2} \right\}\\ \sum\limits_{k=1}^{K}o_{k}(t)=f\end{aligned} \right. \tag{1}\]  
where \(K\) is the number of decomposed modes; \(o_{k}\) is the \(K\) modal components after decomposition; \(\omega_{k}\) is the center frequency of each modal component after decomposition; \(\ast\) is the convolution operation; \(f\) is the original signal. By introducing the quadratic penalty factor \(\alpha\) and Lagrange operator \(\lambda(t)\), the VMD algorithm transforms it into an unconstrained variational problem whose expression is  
\[L(\{o_{k}\},\{\omega_{k}\},\lambda)=\alpha\sum\limits_{k=1}^{K}\left\|\partial _{t}\right[\left(\delta(t)+\frac{j}{\pi t}\right)\ast o_{k}(t)\left]\mathrm{e }^{-\mathrm{j}\omega_{k}t}\right\|_{2}^{2}+\left\|f(t)-\sum\limits_{k=1}^{K} o_{k}(t)\right\|_{2}^{2}+\left\langle\lambda(t),f(t)-\sum\limits_{k=1}^{K}o_{k}(t) \right\rangle \tag{2}\]  
The alternating direction multiplier algorithm is used to iteratively update \(\sigma_{k}^{n+1},\omega_{k}^{n+1}\), \(\lambda^{n+1}\) to find the \"saddle point\" of Equation (2) and obtain the optimal solution:  
Step 1: Initializes \(\delta_{k}^{n+1}\), \(\omega_{k}^{1}\), \(\lambda^{1}\), \(n=0\). Step 2: Make \(n=n+1\), \(k=1:K\). Step 3: Update \(\delta_{k}^{1}\) iteratively for \(\omega>0\). Step 4: Update the Lagrange operator. \[\hat{\lambda}^{n+1}(\omega)=\hat{\lambda}^{n}(\omega)+\tau\left(\hat{f}( \omega)-\sum\limits_{k}\delta_{k}^{n+1}(\omega)\right) \tag{4}\]  
where \(\tau\) is the Lagrangian multiplier, \(\tau=0\) is usually set. Step 5: Repeat Step 2 to Step 4 until the conditions for terminating the iteration are met. \[\sum\limits_{k=1}^{K}\left(\left\|\partial_{k}^{n+1}-\delta_{k}^{n}\right\|_ {2}^{2}/\left\|\partial_{k}^{n}\right\|_{2}^{2}\right)<\varepsilon \tag{5}\]  
where \(\varepsilon\) is usually set to \(1\times 10^{-6}\). Based on a prior experience [14], the VMD parameter applicable to the vibration signals of universal circuit breakers derived from comparative experiments are \(K=8\) and \(\alpha=3000\), and the results of the comparison experiments are shown in the Section 3.4. This section focuses on showing the parameters that have been selected. Using the mechanical structure stuck fault as an example, Figure 4 displays the decomposition result following the signal's VMD decomposition. The same treatment is applied to the remaining five fault types. To validate the effectiveness of the VMD decomposition, the modal components obtained from the decomposition are subjected to spectral analysis. The results are presented in Figure 5. It is evident that the modal components are essentially separated from one another by virtue of their independent center frequencies.