# Transformer Fault Diagnosis Method Based on SCA-VMD and Improved GoogLeNet
## 2 Vibration Signal Data Preprocessing and Feature Extraction Modelling
### Vibration Signal Feature Extraction  
Accurate extraction of effective features in transformer vibration signals and the selection of an optimal subset are key aspects of fault identification. A periodic map spectrum, as a widely used tool for analysing the spectral characteristics of signals, can display the energy distribution of signals at different frequencies, which can provide a quick and intuitive understanding of the frequency distribution of signals, and the frequency filtering result, i.e., selecting or excluding certain frequency intervals in order to remove or retain some specific frequency components of the signal, based on the energy occupancy ratios of different frequency intervals [27]. Based on the SCA-VMD decomposition of the measured time-domain signals to obtain the periodic map spectrum, by comparing the periodic map spectrum under different working conditions, it is easier to find out the influence of the faults on the vibration signals, which is conducive to the diagnosis of the cause and location of the faults, and it has a high diagnostic accuracy and robustness.  
The following is the flow of generating feature maps after modal decomposition:  
1. Variational modal decomposition: the signal is decomposed using the VMD algorithm to obtain the kth order mode \(\mathrm{u_{k}}\), where \(\mathrm{u_{k}}\) can be expressed as  
Figure 2: SCA-VMD parameter optimisation flowchart.  
\[u_{k}(t)=\sum_{\omega}c_{k}(\omega)h_{k}(\omega,t) \tag{9}\]  
where \(c_{k}(\omega)\) denotes the amplitude of the mode at frequency \(\omega\) and \(h_{k}(\omega,t)\) is a complex amplitude denoting the amplitude of a sinusoidal wave with frequency \(\omega\) at time t.  
Calculate the periodogram: The short-time Fourier transform (STFT) is performed on u\({}_{\text{k}}\)(t) to obtain the short-time spectrum S\({}_{\text{k}}\)(\(\omega\),t):  
\[S_{k}(\omega,t)=\mid\sum_{T}w(\tau)u_{k}(t+\tau)e^{-j\omega\tau}\;\mid^{2} \tag{10}\]  
where w(\(\tau\)) is the window function, which is used to frame the signal. Substituting \(u_{k}(t)\) into the short-time Fourier transform equation and using [PERSON]'s formula yields  
\[S_{k}(\omega,t)=\mid\sum_{\omega^{\prime}}c_{k}(\omega^{\prime})\sum_{\tau}w( \tau)h_{k}(\omega^{\prime},t+\tau)e^{-j(\omega^{\prime}-\omega)\tau}\;\mid^{2} \tag{11}\]  
Since \(h_{k}(\omega^{\prime},t+\tau)\) is an amplitude and can be regarded as a constant, the above equation can be expressed as  
\[S_{k}(\omega,t)=\mid\sum_{\omega^{\prime}}c_{k}(\omega^{\prime})H_{k}(\omega ^{\prime},\omega)e^{-j\omega^{\prime}t}\;\mid^{2} \tag{12}\]  
\(H_{k}(\omega^{\prime},\omega)=\sum_{\tau}w(\tau)h_{k}(\omega^{\prime},\tau)e^ {-j(\omega^{\prime}-\omega)\tau}\) can be regarded as the frequency response of the filter and is used to measure the energy of the amplitude of frequency \(\omega^{\prime}\) at frequency \(\omega\). Thus, the amplitude energy periodic map spectrum S\({}_{\text{k}}\)(\(\omega\),t) can be derived at moment t with frequency \(\omega\).  
1. Summing by modal components: the total periodic map spectrum S(\(\omega\),t) is obtained by summing each modal component.
2. Plotting feature maps: After the periodic map spectrum of the vibration signal based on SCA-VMD decomposition is summed up using modal components, the modal components are used as the horizontal axis, the frequency is used as the vertical axis and the brightness of each point represents the energy magnitude of the modal component at the corresponding frequency to plot the feature maps so that we can obtain the energy distributions of the different modal components at different frequencies, which can better reflect the characteristics of the vibration signal faults and the information of the faults.