## 2 Methods
### Proposed De-Noising Method for Complex Noisy PD Signal  
PD signal is usually expressed by four mathematical models: Single exponential decay function, double exponential decay function, single exponential oscillation attenuation function, and double exponential oscillation attenuation function. Due to the great oscillation and attenuation of PD signal during the propagation along the cable, the model of single exponential oscillation attenuation and double exponential oscillation attenuation shown in (18) and (19) are employed to describe the PD signal in power cable. \[f(t)=Ae^{-(t-t_{0})/\tau}\sin(2\pi f_{c}(t-t_{0})) \tag{18}\]  
\[f(t)=A\Big{(}e^{-1.3(t-t_{0})/\tau}-e^{-2.2(t-t_{0})/\tau}\Big{)}\sin(2\pi f_{c }(t-t_{0})) \tag{19}\]  
where \(A\) is the signal amplitude, which is 1 mV and 5 mV, respectively, here, \(\tau\) is the attenuation coefficient, which is 1 \(\mu\)s here, \(f_{c}\) is oscillation frequency, taking 5 MHz, \(t_{0}\) is the delay time. The sample frequency is set as 100 MHz, sample time as 40 \(\mu\)s at this frequency. Two kinds of cable PD pulses mentioned above are shown in Figure 4a. To simulate the actual situation, two type noises are added in PD signal. White noise interference with a SNR of \(-\)4 dB is added. Since the narrowband interference is mainly sinusoidal or cosine waveform, the sinusoidal function of different frequencies can be used for superposition to simulate the periodic narrowband interference. The frequencies of narrowband interference are set to 1 MHz, 10 MHz, 15 MHz, and the amplitudes are 2 mV, 1 mV, and 0.5 mV, respectively. The PD signal polluted by the two kinds of noise is shown in Figure 4b. The synthetical SNR of the noisy PD signal is \(-\)21.4 dB, which can't be recognized in the time domain. For the noisy PD signal \(g(x)\) synthesized above, its DFT is defined as:  
\[G(i)=\sum_{n=0}^{N-1}g(n)e^{-j\frac{2\pi}{N}in}(i=0,1,\ldots,N-1) \tag{20}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Signal1** & **Signal2** & **Signal3** \\ \hline SNR of noisy signal (dB) & 0.0 & \(-\)2.5 & \(-\)13.0 \\ NCC & 0.9507 & 0.9397 & 0.8294 \\ SNR of de-noised signal (dB) & 10.2 & 9.4 & 4.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Quantitative evaluation results of adaptive wavelet packet for different levels of noise. Figure 4: Simulate cable partial discharge (PD) signals. (**a**) Clear cable PD signal. (**b**) Noisy cable PD signal. Its power spectrum can be calculated as:  
\[P(i)=\frac{1}{N}\big{|}G(i)\big{|}^{2} \tag{21}\]  
where \(N\) is the data length, \(g(n)\) is the sample point. The DFT power spectrum of PD signal, periodic narrowband component, and white noise component is shown in Figure 5. It's easy to find that the amplitude of the DFT power spectrum of narrowband interference is much larger than that of the real PD signal and the white noise. To deal with complex noises with different characteristics, a novel two-step de-noising framework is proposed in which each noise is rejected respectively by appropriate algorithm according to their specific advantages. Because the VMD can decompose an input signal into an ensemble of BLIMFs with the minimal sum of estimated bandwidths for each mode, we employ its instinctive advantage to separate narrowband noise from the input signal firstly. Decompose the noisy PD signals by fast adaptive VMD proposed in Section 2.2, then calculate the DFT power spectrum for each intrinsic mode. According to the fact that the amplitude of the narrowband interference is much larger than the distortion peak of the real PD signal and the white noise power, we remove the mode with narrowband periodic noise, then synthesize the remaining modes for further processing. This middle result is shown in Figure 6a. Secondly, adaptive wavelet packet decomposition proposed in Section 2.3 is employed to remove the component containing white noise according to the threshold method [24], and the remaining components are synthesized as the de-noising PD signal, which is shown in Figure 6b.