*2.2. Characteristics Analysis of PPF*

When a PPF occurs in an MMC-HVDC system, which is located at F1 in Figure 1, the fault process can be divided into three stages: the unblocked SM stage, the initial stage after the SM is blocked, and the steady stage after the SM is blocked. In order to detect a fault in a short time, it is necessary to use transient information at the beginning of the fault. Therefore, the fault transient process studied in this paper mainly focuses on the unblocked SM stage. The equivalent circuit of the unblocked SM stage is shown in Figure 3. The fault current *Idc* is mainly composed of the SM capacitor discharging current and AC source feeding the current. During this stage, SMs are switched to the normal operation mode, which means a total of *n* SMs of each phase unit are in the on-state at any time to maintain the DC bus voltage. Due to the SM capacitance voltage balance control principle, SMs are switched on and off sequentially in a short time frame. All SMs in each phase can be divided into two groups, and discharge alternately, as shown in Figure 3b. Because of the high control frequency, it can be approximated that the two groups of SMs alternately discharged are in parallel in each phase. Therefore, it can be equivalent to a second order RLC discharging circuit, as shown in Figure 3c.

**Figure 3.** MMC-HVDC system pole-to-pole fault (PPF) circuit. (**a**) Detailed fault circuit; (**b**) equivalent fault circuit; (**c**) simplified fault circuit.

The waveform and frequency spectrum of fault current *Idc* is shown in Figure 4. As shown in Figure 4a, the fault time is 0.01 s, and the fault current rises to around 14 kA after 10 ms. A fast isolation is needed to avoid the large-amplitude fault currents. If DC circuit breakers are used, the post-fault transients of only a few microseconds will be recorded. The maximum breaking capacity of existing DC circuit breakers is 25 kA. According to the breaking capability and the operation time of the DC breaker, there is only a 3 ms transient signal after the fault is studied in this article. Due to the coupling between the two transmission lines, the Karenbauer phase mode transformation matrix is used to decouple the currents in both positive and negative poles. When compared with the ground mode component, the line mode component of currents is relatively stable. Its wave speed varies relatively little with the frequency and geographic environment of the corridor. Thus, the line mode component is adopted. The waveform of the current line mode component is shown in Figure 4b. As shown in Figure 4c, the line mode component of the fault current is used for spectrum analysis. It can be seen that the amplitude of the 0 Hz component exceeds 5 kA. The content of each frequency band gradually decreases with increasing frequency. There are obvious differences between the content of different frequencies, both in the low-frequency band and the intermediate frequency band. However, from around 20 kHz to around 50 kHz, the frequency spectrum is evenly distributed and varies slightly.

**Figure 4.** MMC-HVDC system PPF characteristics. (**a**) Current waveform; (**b**) current line mode component waveform; (**c**) spectrum analysis.

## *2.3. Characteristics Analysis of PGF*

The characteristics of a PGF of MMC-HVDC are different from either traditional LCCbased HVDC converters or two-level VSC-based converters. When the PGF occurs at the location F2 in Figure 1, no closed loop can be formed between the faulted point and the SM capacitors. Only a slight transient signal is produced due to the deceasing bus potential of the faulted line. The capacitor will not discharge. Figure 5 illustrates the corresponding equivalent circuit. Therefore, the average SM capacitor voltage remains the same, and the DC voltage between two poles will be unchanged. The fault pole voltage becomes 0. There is no steady-state fault current in the DC line.

The PGF transients also have wave processes similar to the PPF. The difference is that a PGF forms a fault loop with the earth, which causes the transient wave to be severely attenuated during propagation. This transient process can be equivalent to the result of switching an additional voltage source at the fault point. The equivalent additional source *E*add is:

$$E\_{\text{add}}(t) = -E\_{\text{tra}}\left(t \ge 0\right),\tag{1}$$

where *E*tra represents the transient voltage source that generates the transient current waveform. The transient characteristics of the PGF current *Idc* are shown in Figure 6. The energy decreases with the increase in frequency. More sizeable differences can be found in the frequency band from 0 Hz to around 10 kHz than the other bands. From around 10 to 50 kHz, the differences gradually decrease.

**Figure 5.** MMC-HVDC system pole-to-ground fault (PGF) circuit.

**Figure 6.** MMC-HVDC system PGF characteristics. (**a**) Current waveform; (**b**) current line mode component waveform; (**c**) spectrum analysis.

#### *2.4. Influence Caused by Lightning Strikes*

Lightning strikes are the main cause of transmission line protection misoperation. For the extremely short post fault transients of MMC-based protection, it is necessary to consider how to distinguish between faults and disturbances, especially LDs. Lightning waves generated by thunder are pulse transient waves, whose shape is mainly determined by its steepness and peak value. The double exponential wave is the equivalent calculation wave which is the closest to the actual lightning current wave, and is widely used in simulation analysis [38]. The generation of the transient waveform can be equivalent to superimposing an additional current source. The equation of the additional source *E*add is:

$$E\_{\rm add}(t) = I\_0(e^{-t/\alpha} - e^{-t/\beta}) \ (t \ge 0),\tag{2}$$

Here, *I*0 represents the amplitude of the lightning current; α and β represent the correlation coefficients of the rise and fall of lightning current, respectively.

When an LD occurs, which is located at F2 in Figure 1, it can be equivalent to a single current source superimposed on the transmission line. The signal source is connected for a short time and generates high-frequency signals. The transient characteristics of an LD fault current *Idc* are shown in Figure 7. Most energy is contained in the lower frequency band, from 0 Hz to around 30 kHz, while only a small part of energy is included beyond 30 kHz. At the same time, large amplitude oscillations can be found in the frequency band

below 15 kHz. Those harmonic energies cause the oscillating waveforms of lightning in the time domain.

**Figure 7.** Transient characteristics of a lightning disturbance (LD). (**a**) Current waveform; (**b**) current line mode component waveform; (**c**) spectrum analysis.

When the lightning current amplitude is large, the voltage between the line and the tower might exceed the flashover voltage of the insulator. It is easy to cause an insulation breakdown, especially when the insulator is partially damaged, or flashover occurs along the surface. Then, it develops into a stable arc in a short time. The transmission line has a pole-to-ground fault through the tower, which is called a lightning fault (LF). According to its physical mechanism, the mathematical model of the additional source *E*add can be expressed as a piecewise function in Equation (3),

$$E\_{\rm add}(t) = -E\_{\rm tra}\left(t\_0 \le t \le \infty\right), \text{ and } E\_{\rm add}(t) = l\_0(e^{-t/\hbar} - e^{-t/\beta}) \text{ (} 0 \le t \le \infty\text{)},\tag{3}$$

Here, *t*0 is the moment of insulation breakdown, after which the LD evolves into an LF. The LF transient characteristics of *Idc* are shown in Figure 8. The frequency spectrum of LF looks similar to that of LD in Figure 7c. However, they are different. Due to the superposition of GF, whose energy is focused in the low-frequency band, the energy spectrum of LF reveals the features of both GF and LD. As demonstrated in Figure 8c, most energy is concentrated in the range from 0 to 30 kHz. The energy decreases gradually beyond 30 kHz. Although more harmonic-like energies are included in the frequency from around 15 to 50 kHz, less steepness is shown in LF than that in LD.

**Figure 8.** Transient characteristics of a lightning fault (LF). (**a**) Current waveform; (**b**) current line mode component waveform; (**c**) spectrum analysis.

#### *2.5. Characteristics Analysis of External Faults*

When an external fault occurs, line protection should not be activated. In MMC-HVDC systems, external faults mainly include valve faults and AC side faults. Among them, SMFs and AG-ACs are the most likely faults, respectively. For that reason, these two

faults are considered in this paper: an SMF located at F3 and an AG-AC located at F4 in Figure 1. Figures 9 and 10 show the fault transient characteristics of *Idc* of the SMF and the AG-AC, respectively. For these two types of external faults, most of their energies are found between 0 Hz and around 5 kHz, but the energies attenuate greatly beyond 5 kHz.

**Figure 9.** Fault characteristics of a sub-module short circuit fault (SMF) on the arm of a phase. (**a**) Current waveform; (**b**) current line mode component waveform; (**c**) spectrum analysis.

**Figure 10.** Fault characteristics of a single-phase grounding AC fault (AG-AC). (**a**) Current waveform; (**b**) current line mode component waveform; (**c**) spectrum analysis.

When an internal fault occurs, the transient current propagates from the fault point to both sides along the transmission line. Because the actual transmission line is basically a uniform line, which means that the resistor, inductance, and capacitance parameters are evenly distributed along the line, the high-frequency attenuations of the transient current at both ends of the line are not large. On the contrary, when an external fault occurs, the high-frequency component of the transient current attenuates significantly after passing through the boundary.

According to the analysis in this section, it can be found that different fault transients have different characteristics such as waveforms and frequency spectrum distributions. The high-frequency components of the transient current of the SMF and the AG-AC are obviously smaller than the ones of internal faults. The 0 Hz component of PPF is greater than those of PGF, LD, and LF. The spectra of PGF, LD, and LF in different frequency bands are obviously different, both the amplitude and the variations. If a reasonable signal processing and analyzing tool is used to describe these differences effectively and stably, various transients can be well separated, and the protection function can be realized.

#### **3. Wavelet Entropy Characterization of Spectrum Distribution**
