## 3 Results and Discussion
### Robust Estimation of Arrival Time of PD Pulse  
In the online PD localization system, due to complex noises, the robustness and accuracy of the estimation of arrival time is not high, and it results in a large localization error. To this end, \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & **Evaluation Parameter** & **Pulse 1** & **Pulse 2** & **Total** \\ \hline \multirow{2}{*}{CEEMD} & NCC & 0.3847 & 0.4387 & 0.3908 \\  & SNR(dB) & 0.6752 & 0.8935 & 0.7166 \\ \hline \multirow{2}{*}{VMD wavelet threshold method} & NCC & 0.6677 & 0.8100 & 0.5509 \\  & SNR(dB) & \(-\)0.2551 & 3.0568 & 3.0239 \\ \hline \multirow{2}{*}{Proposed method} & NCC & 0.9816 & 0.9970 & 0.9886 \\  & SNR(dB) & 13.3906 & 19.4398 & 15.5064 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Quantitative results of different methods. Figure 8: De-noising results by different methods. (**a**) Result of the CEEMD-based method. (**b**) Result of VMD—wavelet packet method. (**c**) Result of the proposed method. the proposed two-step de-noising method in Section 2 is used to suppress the noise of the detected on-site PD signal, extract a relatively pure PD signal, and then accurately estimate the first arrival time via the time varying kurtosis algorithm [11; 28]. Among the sequences containing PD pulses, the steepness of the signal is the most serious at the moment of pulse onset. Kurtosis is a classical statistic based on high order statistics, which can provide more comprehensive information than low order statistics. It reflects the concentration degree of asymmetric and non-gaussian time sequence, and its value can indicate the steepness of the signal. Given the random variable \(X\), \(p(X)\) is the probability density of \(X\), then its kurtosis can be defined as:  
\[M=\frac{m_{4}}{\left(m_{2}\right)^{2}}-3 \tag{22}\]  
where \(m_{k}\) is the k-order statistic of random variable \(X\). According to formula (22), kurtosis value only represents the overall steepness of the signal, and cannot reflect the real-time change of the steepness of time-varying signal. Therefore, time-varying kurtosis algorithm should be further adopted. In the discrete sequence \(X(i)\) to be detected, a sub-time window with length \(b\) and sampling point \(i\) as the center is designed, the kurtosis value of the signal in the sub-time window is obtained according to formula (22), and time-varying kurtosis value is defined according to the following formula:  
\[M_{t}(i)=\frac{M(i)-M}{M} \tag{23}\]  
where \(M\) is the kurtosis of \(X(i)\), and \(M(i)\) is the kurtosis of sub-time window. In practical applications, the collected PD pulse is sparse. If time-varying kurtosis is obtained on all data sequences, it will not only take time, but also store too much data. The time window energy ratio method was adopted for pretreatment here. The time window \([x_{n}]\) where the PD event occurred was determined first, and then the time-varying kurtosis was calculated within the time window. For the time sequence \(X(i)\), take the sampling point \(i\) as the center, and take a time window with the neighborhood of \(\pm l\). The energy ratio of the rear window and the front window is defined as the energy ratio of the time window. \[R(i)=\left[\left(\sum_{k=i}^{i+l-1}x_{k}^{2}\right)^{1/2}\right]\left\|\left( \sum_{k=i-l}^{i-1}x_{k}^{2}\right)^{1/2}+\lambda\right\| \tag{24}\]  
The threshold value of PD event is set, and the time window energy ratio R is compared with the threshold value to determine whether there is PD event in the window. If \(R\) is less than the threshold, no PD event occurs in the window, and then cycle to the next moment. If it is larger than the threshold value, the time window with this time as the center and the length \(2l\) as the PD time window to pick up the first PD pulse by the time-varying kurtosis.