## 2 Automatic POD Diagnosis Method  
### Time-Frequency Analysis of Disturbance Signals  
Single and combined PQDs contain abundant time domain and frequency domain information. Hereby, different manually selected features are applied to extract as complete feature information as possible from the time-frequency analysis since the classification accuracy of PQDs is highly affected by the performance of time-frequency analysis [22]. The advantage of DRST is that it has high energy concentration at both low and high frequencies. In our work, we use the DRST, which has a double resolution, as a time-frequency decomposition algorithm for PQDs [16]. The DRST of input PQD signal \(x(t)\) can be defined as  
\[\text{DRST}(\mu,f)=\int_{-\infty}^{\infty}x(t)\frac{\sqrt{\beta_{1,2}|f|}}{ \sqrt{2\pi}}e^{-\frac{(y-t)^{2}\beta_{1,2}|f|}{2}}e^{-\beta_{2}\pi ft}dt \tag{1}\]  
where \(\beta_{1,2}\) denotes the adjustable parameters, \(f\) denotes the signal frequency, \(\mu\) is the time shift factor. Essentially, the time-frequency spectrum of PQDs is divided into two parts, where \(f\leq 1.5f_{0}\) and \(f>1.5f_{0}\) represent the low-frequency part and high-frequency part, respectively. It is worth mentioning that to avoid the spectrum leakage and provide more accurately signal analysis result, the separation value is set to 1.5. The \(f_{0}\) is the fundamental frequency. Besides, to improve the frequency resolution and noise reduction capability, the values of parameter \(\beta_{1,2}\) are set to 121 and 10, respectively, according to [16]. To verify the effectiveness, we compared the disturbance signal analysis result with different parameter \(\beta_{1,2}\). The disturbance signal is the harmonic disturbance with the harmonic level is the third and the fifth harmonic in Figure 1a-c show the signal analysis result with different parameters of \(\beta_{1,2}\). From Figure 1, the parameters \(\beta_{1,2}\) directly affect the time-frequency result. For example, when parameters \(\beta_{1,2}\) are set to 121 and 10, respectively, the third harmonic and the fifth harmonic can be identified accurately. When parameters \(\beta_{1,2}\) are set to 150 and 40, respectively, the time-frequency resolution is lower. Besides, there is also the spectrum leakage problem. Thus, to improve the frequency resolution and noise reduction capability, the \(\beta_{1,2}\) are set to 121 and 10, respectively. To further verify the detection performance, we compared the transients disturbance analysis result with different parameters of \(\beta_{1,2}\). Figure 2a shows the transients input signal. Figure 2b,c show the signal analysis result with different parameters of \(\beta_{1,2}\). Figure 1: Comparison result on harmonic disturbance with different parameter \(\beta_{1,2}\). (**a**) input signal, (**b**) \(\beta_{1}=150,\beta_{2}=40\), (**c**) \(\beta_{1}=121,\beta_{2}=10\). From Figure 2, when parameters \(\beta_{1,2}\) are set to 121 and 10, respectively, the transients disturbance can be identified accurately, indicating a suitable frequency resolution. When parameters \(\beta_{1,2}\) are set to 150 and 40, respectively, the frequency resolution is lower. Besides, there is also the spectrum leakage problem. Thus, to improve the time-frequency resolution, the \(\beta_{1,2}\) are set to 121 and 10, respectively. Different from the manually selected features perspective, the output of DRST, which is an amplitude time-frequency matrix, can be regarded as an image perspective instead of a conventional single amplitude value feature. At this time, the amplitude and time of the time-frequency can be considered as pixel value and location of the image, respectively. In this work, the sampling frequency is set to 2.56 kHz where the fundamental frequency \(f_{0}\) = 50 Hz and the length of the sampling period is 10 cycles of the fundamental component. Then the size of the amplitude time-frequency matrix is 256 \(\times\) 512 (height \(\times\) width).