# Fault Diagnosis of Universal Circuit Breakers Based on Variational Mode Decomposition and WOA-DBN
## 3 Methodology
### Deep Belief Networks Optimized by Whale Optimization Algorithm  
The hyperparameter settings of DBN have a direct impact on its learning capacity. Compared to single-hidden layer structures, multi-hidden layer structures have superior learning capabilities. Nevertheless, the model's capacity for generalization is diminished, and the precision of the model's classification is impacted when the hidden layer has more than four layers [38]. Consequently, the DBN in this research is composed of three hidden layers. The control variable approach is typically used to determine the DBN's hyperparameters. Nevertheless, the following shortcomings of the widely used approaches exist [39]:  
1. The abundance of search parameters makes it challenging to implement. 2. Narrow search scope. As a result, the found optimal value is not the genuine best value; instead, it is merely the outcome of this narrow range. 3. Lack of theoretical foundation in science. A method called WOA-DBN was proposed to optimize the hyperparameters of DBN using WOA, and its flowchart is illustrated in Figure 9. Thus, the number of neurons in the three hidden layers, the number of backward fine-tuning, and the network learning rate are the hyperparameters that need to be identified. The following are WOA-DBN's significant steps:  
Step 1: The DBN's search range for optimized hyperparameters is determined beforehand empirically: these hyperparameters include the number of neurons in the first hidden layer \(n_{1}\), second hidden layer \(n_{2}\), third hidden layer \(n_{3}\), the network learning rate \(\eta\), and the number of backward fine-tuning \(T_{1}\). The number of neurons in the first, second and third hidden layers (\(n_{1}\), \(n_{2}\) and \(n_{3}\)) ranges from 1 to 1000, the network learning rate \(\eta\) ranges from 0.000001 to 0.1, and the number of backward fine-tuning \(T_{1}\) is searched in the range of 1\(\sim\)100. The optimized DBN parameters are used to determine the whale position of the WOA. The whale position \(\overrightarrow{Z}\) is denoted as  
\[\overrightarrow{Z}=[n_{1},n_{2},n_{3},\eta,T_{1}] \tag{13}\]  
Step 2: Prior to this, several whale parameters were established by trials and evaluations, including the maximum number of iterations \(T_{\max}=20\), the size of the whale population \(H=10\), and the spiral size constant \(c=2\), The dimension \(D=5\) was selected based on the number of parameters optimized by DBN. Figure 9: WOA optimizes DBN flowcharts. Step 3: Based on the fitness value, the most excellent adaptable individual is selected as the prey, and its location is chosen as the target place. In other words, the fitness function is determined by obtaining the output value and the actual value based on the WOA-DBN model:  
\[f=\left(\frac{\sum\limits_{i=1}^{W}\left(y_{i}-y_{i}^{\prime}\right)^{2}}{W}\right) \tag{14}\]  
where \(W\) is the number of samples, \(y_{i}\) is the actual value of the \(i\)th sample, and \(y_{i}^{\prime}\) is the output value of the \(i\)th sample obtained from the WOA-DBN model. Step 4: The probability \(p\) and the coefficient vectors \(\overrightarrow{r}\) ought to be assigned at random. Individual whales' positions are updated. A shrink-wrap method is employed when \(\overrightarrow{|S|}<1\) and \(p<50\%\). The spiral model is applied when \(\overrightarrow{|S|}<1\) and \(p\geq 50\%\). When \(\overrightarrow{|S|}\geq 1\), a haphazard hunt for prey is conducted. Step 5: When the number of iterations approaches the maximum number of iterations \(T_{\text{max}}\), the optimal position of each individual whale is output. Otherwise, step 3 of the process is repeated. Step 6: Decide which whale has the best position, which is represented by the optimized DBN parameter. The ideal DBN model's parameters, following WOA optimization, are 216-471-668-456-6. The network learning rate \(\eta=9.06878\times 10^{-4}\), the number of backward fine-tuning \(T_{1}=95\), and the parameters of the WOA-optimized DBN network are illustrated in Table 5--parameters of the WOA-optimized DBN network.