## 1 Introduction  
Due to the advantage of high reliability, less space demand, and beautiful appearance, power cables have been widely used in power grid of city, railway, offshore wind system, etc. [1]. However, insulation faults directly affect the safe operation of the cable grid. Online partial discharge (PD) detection and localization is considered as the most effective means of detecting the insulation defects [2; 3; 4; 5]. The electromagnetic coupling detection method, which uses a high-frequency current sensor (HFCT) to couple the electromagnetic signal generated by the PD pulse without any damage of the cable body [6; 7; 8], is currently the most commonly used cable PD detection method. The multi-sensor travelling wave method, which can effectively eliminate the influence of wave velocity uncertainty [9; 10; 11], is a novel travelling wave based localization method using the coupled PD current pulse. In practice, the environment in which the cable is laid is complicated, and the real PD signal detected by the electromagnetic coupling component is seriously interfered by complex and strong noises. The effective extraction of real PD signal from noisy signal and the high-precision estimation of the arrival time of PD pulse are the key to reliable localization of the PD fault. The interference in cable PD detection mainly consists of periodic narrowband noise and white noise [12]. Narrowband noise is usually generated by system harmonics, carrier communications, and radio communications. The suppression methods mainly include fast Fourier transform (FFT) threshold method, wavelet transform (WT) based method, adaptive decomposition method, etc. Based on the characteristics of narrowband interference with limited bandwidth and sharp pulse in the frequency domain, the FFT threshold method [13; 14] removes the noise by setting a threshold value of the spectrum amplitude. However, due to the inherent defects of FFT, such as spectrum leakage, the application of this method is limited in practice. As an alternative, wavelet based de-noising method [15; 16] have shown strong advantages in recent years, but it suffers from the choosing of suitable wavelet basis. As an improvement, multi-wavelet de-noising method [17; 18] is less affected by waveform matching, requires less prior information of PD signals, but it is still essentially the waveform matching, which has limitations in application. A novel method based on blind signal separation technology is proposed [19] to suppress periodic narrowband noise. However, after blind separation, the amplitude and phase of the signal are uncertain and the algorithm is computationally intensive. Noise suppression based on empirical mode decomposition (EMD) [20; 21] can decompose multi-time-scale noisy PD signals into single time-scale intrinsic mode functions (IMF) adaptively, it is suitable for non-stationary and nonlinear PD signals. However, the following defects limit its further usage [22]. The central frequency and bandwidth of the noise interferences associated with the PD signal in the detected noisy PD signal are uncertain. If the frequency band of interference falls within the band of the first IMF, too much interference may be introduced. Additionally, if the frequency band falls within the higher order IMF component, some important PD characteristics may be lost because of the narrow frequency band of the IMF. Variational mode decomposition (VMD) based de-noising methods [23; 24] orthogonally decompose the signal into independent frequency bands without redundancy. They can decompose narrowband noise and PD signal into corresponding frequency bands by adjusting its penalty factor and achieve outstanding effect of suppressing the periodic narrowband interference. In practice, the proper setting of the mode number is the main problem of this method [25]. Too large a number usually causes over-decomposition, and increases the amount of calculation. On the other hand, too small a number means mode loss, which results in waveform distortion.