# Transformer Fault Diagnosis Method Based on SCA-VMD and Improved GoogLeNet
## 2 Vibration Signal Data Preprocessing and Feature Extraction Modelling
### VMD Based on SCA Parameter Optimisation  
In VMD, the sine-cosine algorithm can be used to optimise the parameters K and alpha for optimal decomposition. For this purpose, an adaptive function needs to be defined and the minimum average envelope entropy is defined as the minimum of the average of the envelope entropies of all envelope segments. The performance superiority of the algorithms can be assessed by comparing the MAEE values obtained using the different algorithms since a smaller MAEE value indicates that the algorithm captures the complexity and nonlinear features of the signal better. Therefore, the objective is to find the minimum MAEE value. The minimum mean envelope entropy is the minimum of the average value of the envelope entropy for each segment, i.e.,  
\[MAEE=\min\biggl{(}\frac{1}{\overline{N}}\sum_{i=1}^{N}H_{i}\biggr{)} \tag{8}\]  
where N is the number of envelope segments and \(H_{i}\) is the envelope entropy of the i-th envelope segment.  
In order to achieve adaptive determination of the optimal decomposition parameters K and alpha for the optimised variational modal decomposition (VMD) based on the sine-cosine algorithm (SCA), the algorithm is designed according to the following steps:  
1. Initialisation parameters: the initial parameters of the algorithm, including K and alpha, are set using a priori knowledge, experience or random selection.
2. Define the fitness function: define the fitness function as shown in Equation (8), i.e., the minimum average envelope entropy.
3. SCA optimisation: The sine-cosine algorithm (SCA) is used to optimise the fitness function. In each iteration, the fitness function (MAEE) is minimised by adjusting the parameters K and alpha.
4. Iterative process: The parameters K and alpha are progressively optimised over several iterations. In each iteration, the value of the fitness function is calculated, the parameters are updated and the stopping condition is checked to see whether it is met.
5. Performance evaluation: During the iteration process, the value of MAEE is monitored. If the MAEE value is found to be small enough and the MAEE no longer changes significantly after the number of iterations is reached, the iteration can be stopped.  
Figure 1: Principle of SCA optimisation.  
In this solution process, the initial population is randomly generated, and each individual represents a set of sine and cosine parameters. For each individual, its sine and cosine parameters are applied to the VMD decomposition, and the calculated envelope entropy is used as the fitness value. The individuals are selected according to their fitness values, and the individual with high fitness is selected as the parent and cross-transformed to generate new individuals. For the newly generated individuals, their sine-cosine parameters are applied to the VMD decomposition, the calculated envelope entropy is used as the fitness value and the newly generated individuals are merged into the current population. According to the iteration number or fitness, determine whether the termination conditions are met; if the termination conditions are not met, continue the selection, crossover, mutation, fitness calculation and population update processes. The optimised VMD decomposition result is finally obtained, and the SCA-VMD algorithm flow is shown in Figure 2.