*3.1. Definition of Wavelet Entropy*

Wavelet transform has good time–frequency localization performance [27,28]. It is considered as a "mathematical microscope", because it can "focus" the analysis object to any detail. Benefitting from the high sensitivity of the wavelets to the singularity and mutation of signals, wavelet transform is considered an effective signal processing method in multi-resolution analysis. Power system faults appear as sudden changes in voltage and current signals. Therefore, the use of a wavelet to detect the fault singularities is effective [29].

Entropy is one of the tools to measure the disorder degree of the whole system and it can also be regarded as a description of the degree of system uncertainty. If we regard a signal source as a material system, the more messages we may output and the more random and uncertain the signal source is, the more disordered and the greater the entropy is [13,25].

Wavelet entropy can represent the change in signal complexity in the time domain, and also many features of signal in the frequency domain. Therefore, wavelet entropy has unique advantages in the representation of fault information of non-stationary timevarying signals. According to wavelet transform, the entire frequency band is divided into *m* levels to obtain frequency bands of different levels. The wavelet coefficients in the *i*th (1 ≤ *i* ≤ *m*) frequency band form a set *Xi* [39,40]. Taking the maximum and minimum values of *X*i as upper and lower limits, respectively, this range is equally divided into *n* intervals. The number of wavelet coefficients distributed in the *j*th (1 ≤ *j* ≤ *n*) interval is denoted as *xij*. Its probability is recorded as *p*(*xij*), which is obtained by dividing the number of coefficients in different intervals by the total number of coefficients in this frequency band *Xi*. The formula of wavelet entropy *Hi* of set *Xi* is:

$$H\_i = -\sum p(\mathbf{x}\_{\vec{\eta}}) \log p\_b(\mathbf{x}\_{\vec{\eta}}) \ (j = 1, 2, \dots, n), \tag{4}$$

#### *3.2. Characterization of Frequency Spectrum by Wavelet Entropy*

In MMC-HVDC systems, to ensure protection speed and sensitivity, the sampling frequency of the protection device is required to be no less than 50 kHz; a short time window requires a high sampling frequency to increase the acquisition of transient information. In most engineering applications, the sampling frequency is set as 100 kHz [17,18,20], and the largest is 1 MHz [22]. In this work, we choose 200 kHz as the sampling frequency. Then, the signal is decomposed by wavelet transform. Figure 11 shows the energy distribution of different wavelet frequency bands. Through spectrum analysis, there is a small amount of content of different frequencies from 50 to 100 kHz of various faults, and the change is insignificant. Therefore, the spectrum display range is from 0 to 50 kHz in this case. There are clear differences in the frequency spectrum of different faults in different frequency bands; in particular, the degrees of frequency fluctuation and disorder are different. For example, on the fourth level in the spectrum from 6250 Hz (200 kHz/24+1) to 12,500 Hz (200 kHz/24), the differences between the content of different frequencies of lightning strikes are larger than the PGF. These characteristics can be characterized by entropy.

**Figure 11.** Spectrum analysis of different faults.

## *3.3. Wavelet Entropy of Different Transients*

When using wavelet decomposition, the wavelet function and the number of decomposition levels need to be considered. The fourth Daubechies wavelet "db4" is used in the wavelet decomposition in this paper [41]. The number of wavelet decomposition levels should be less than nine levels [40]. Combined with the spectrum displayed in Figure 11, the number of decomposition levels is selected as eight. The wavelet entropy of different faults is shown in Figure 12.

**Figure 12.** Wavelet entropies of different transients.

The wavelet entropy of each wavelet decomposition level indicates the degree of confusion of the detailed coefficients of the corresponding frequency band. From Figures 11 and 12, it can be seen that the amplitude and variation of the spectrum from 3125 Hz (200 kHz/25+1) to 50 kHz (200 kHz/22) of SMF and AG-AC are small. The wavelet entropy of the 2nd to 5th levels are all 0. This means that after filtering by the boundary, from the fifth to the first level, the differences between the content of different frequencies are too small that the

calculated entropy value is 0. Its energy is mainly concentrated in the low-frequency band. The amplitude and change in the frequency spectrum of PGF are close to zero from 6250 Hz (200 kHz/24+1) to 50 kHz (200 kHz/22). The wavelet entropies of the 2nd to 4th levels are all 0. This means that its transient signal energy and changes are mainly concentrated in the mid-frequency and low-frequency parts. From 12,500 Hz (200 kHz/23+1) to 50 kHz (200 kHz/22), the amplitude and change of the frequency spectrum of LD are also small. Its wavelet entropy of the 2nd to 3rd levels is 0. Thus, its transient signal energy and changes are concentrated in the mid-frequency and high-frequency parts. The frequency spectrum of LF is close to zero from 25,000 Hz (200 kHz/22+1) to 50,000 Hz (200 kHz/22). Its wavelet entropy of the 2nd is 0. Its transient signal energy and changes still exist in the high-frequency part. The wavelet entropies of PPF, from the 4th to 8th levels, are relatively large. This indicates that the content of each frequency band changes uniformly.

Different transients have clear differences in wavelet entropy. Combined with the distribution of wavelet entropy, it is expected that the precise action of protection will be realized. Furthermore, the transient frequency spectrum is affected by many factors, and it is necessary to analyze the influence of different influencing factors on the wavelet entropy.
