The structure of the penetrating co-phase traction direct power supply system is shown in Figure 1. The system is mainly composed of public power grid, traction substation, traction network, and electric locomotive. The three-phase alternating current of the public power grid outputs a single-phase alternating current with equal amplitude and same phase through rectifier operation and inverter operation of traction substation (Li, 2014). Usually, the length of the line between two traction substations is 30–35 km. In this study, 35 km is adopted.

Connection method of co-phase traction substation and traction network is shown in Figure 2. The capacitors at the outlet are connected in parallel with the traction network to reduce the harmonic content entering the traction network and improve the power quality of the traction network lines.

The capacitance connected in parallel with traction network shows low impedance to high frequency current. Thus, it is of certain boundary characteristics. When the fault current passes through the boundary of the traction network, a part of the fault current flows into the capacitors, which leads to a sharp difference with the fault current detected at the relay location where internal and external fault appears. According to the method from Song et al., 2014, the capacitance at the outlet of traction substation plus-2 meters contact line is set as the line boundary.

According to the composition of traction network boundary of co-phase traction power supply system, the frequency characteristics are analyzed, as shown in Figure 3.

It can be seen from Figure 3 that fault signals with different frequencies show unequal passage characteristics after passing through the boundary. When the signal frequency is greater than 600Hz, the amplitude of amplitude–frequency characteristic is far less than 1, which indicates that the boundary has a strong attenuation effect on high frequency signal (Shen et al., 2017; Yang et al., 2018; Song et al., 2020).

When the fault occurs in f 2 (in the zone), the transient current from the fault point flows to both ends of the line, and the traction network has an attenuation effect on the transient signal of the fault. The polarity changes of the transient current detected by the protection devices at both ends of the line are the same, and the transient current only passes through the line, so the attenuation characteristics, reflection process, and transmission sequence are basically consistent, and the waveform shape of the transient current is basically the same. However, when the fault point is relatively far from the midpoint of the line, the waveform shape is different due to the different arrival time sequence of the fault transient current; at the same time, due to the attenuation effect of the traction network on the fault transient signal, the attenuation degree of the transient current with different frequency is different in the transmission process, so the amplitude is different.

When a metallic short-circuit fault occurs, a distance of 15 km form M , its mode component is obtained by decoupling and transforming the fault transient current of M and N , as shown in Figure 5.

It can be seen from Figure 5 that the waveform of mode component of fault transient current at both ends of the line is basically the same when the fault occurs in the zone.

In case of fault occurring in f 1 (outside the reverse zone) and f 3 (outside the forward zone), the transient current will flow through the boundary of the traction network line, and the waveform of the transient current will be reflected, and the high frequency component will be strongly attenuated when passing through the boundary. The polarity change direction of the current waveform detected by the detecting point, M , N , will be opposite; the reflection and refraction process, attenuation degree, and transmission sequence of the transient current waveform will be completely different. Therefore, the waveform of fault transient current will be completely different.

When a metallic short-circuit fault occurs, a distance of 35 km forms the forward zone of N , that is f 3 , its mode component is obtained by decoupling and transforming the fault transient current of M and N , as shown in Figure 6A. When a metallic short-circuit fault occurs, a distance of 20 km forms the reverse zone of M , that is f 1 , its mode component is shown in Figure 6B.

It can be seen from Figure 6 that the waveform of mode component of fault transient current at both ends of the line is opposite and in sharp difference when the fault occurs outside the zone.

To sum up, the waveform of fault current is related to transmission characteristics (refraction and reflection process, attenuation degree, arrival time sequence, change direction) and transmission function. When the fault occurs in the zone, the refraction and reflection process and attenuation characteristics of the transient current are basically the same, the polarity change direction is the same, and the transmission sequence is slightly different, all resulting in local differences in waveform. But generally speaking, the fault transient current waveforms detected at both ends of the line are basically similar. When the fault occurs outside the area, the transient current attenuates through the boundary, the frequency components are different, the refraction and reflection process, the transmission sequence are completely different, and the polarity change direction is opposite. The waveform of fault transient current detected at both ends of the line is significantly different.

Cosine similarity is widely applied for information retrieval and data mining. In recent years, many scholars have studied, with cosine similarity, the fault line detection, fault location, and line protection (Li B. et al., 2019; Wang et al., 2019; Li Z. et al., 2018). The cosine value of the angle between two vector inner spaces is used to characterize their similarity, which is known as the follows: cos ( θ ) = a → • b → ‖ a ‖ × ‖ b ‖ (1)

From Eq. 1, it can be concluded that when the direction of two vectors, a   → and b → are same, cosine similarity is 1; when a   → and b → are vertical, cosine similarity is 0; when the direction of two vectors, a   → and b → , are opposite, cosine similarity is -1.

If x = { x 1 , x 2 , ⋯ , x n } and y = { y 1 , y 2 , ⋯ , y n } are two independent variables, and n is sampling point, their cosine similarity can be expressed as follows: R ( x , y ) = ∑ i = 1 n x i y i ∑ i = 1 n x i 2 ∑ i = 1 n y i 2 . (2)

In the formula, R ( x , y ) represents cosine similarity, and x i , y i ( i = 1,2 , ⋯ , n ) are the No. i element of independent variable, x and y , respectively.

The value range of R ( x , y ) is [- 1,1], and the sign indicates the relevant direction. For R ( x , y ) , the higher the value, higher the similarity of the waveform of the two signals. When R ( x , y ) = − 1 , it means that the two signals are completely negatively correlated; when R ( x , y ) = 1 , it means that the two signals are completely positively correlated; when R ( x , y ) = 0 , the two signals are quite different and uncorrelated (Deng et al., 2018; Li Z. et al., 2018).

The basic flow chart of traction network protection algorithm using synchronous squeezed wavelet transformation and waveform similarity is shown in Figure 7.

(1) Start

Considering the influence of test error, communication delay, noise and other factors, the constant is set as 0.6, that is, k s e t = 0.6 .

In order to simulate and analyze the effectiveness of the protection scheme at different fault locations, assuming metal grounding short-circuit faults occur at f 1 (outside the reverse zone), f 2 (in the zone), and f 3 (outside the forward zone), respectively, with the initial fault angle of 60 ∘ . Among them, for f 2 , starting from 0km, simulated fault points are set every 5 km away from the positive direction of M ; for f 3 , the right exit of N end is taken as the reference point outside the positive zone, and starting from 0km, simulated fault points are set every 5 km from the positive direction of N ; for f 1 , the left exit of M end is taken as the reference point outside the negative zone, and starting from 0km, simulated fault points are set every 5 km from the negative direction of M . After computing the current waveform similarity of M 、 N at both ends, the results are shown in Table 1.

As shown in Table 1, in case of fault in the zone, the waveform similarity of both sides of the line is close to 1, indicating that the current waveform on both sides of the line is highly correlated; when the fault occurs outside the zone, value of waveform similarity is close to -1, indicating that the current waveform on both sides is negatively correlated. It can be seen from Table 1 that the internal and external faults can be accurately identified by the calculation results of cosine similarity.

As simulated analysis of effectiveness of the protection scheme, the transition resistances are 0.1 Ω , 10 Ω , 10 Ω , and 100 Ω ,respectively, and the initial fault angle is θ = 60 ∘ . The setting of fault point is the same as that in Section 5.1. The fault current in and outside the zone is detected, and the mode component after phase-mode transformation is taken for synchronous squeezed wavelet transformation, and the similarity of the reconstructed signal can be calculated. Due to limited space, this article provides only the calculation results of waveform similarity at the beginning, midpoint, and end of outside the reverse zone, in the zone, and outside of forward zone under different transition resistance in Table 2.

The waveform similarity calculation results of different fault locations under different transition resistances are shown in Figure 8.

In Figure 8, the abscissa is the distance from the fault point to the protection device, M , and the negative sign indicates the reverse fault; the ordinate is the calculation result of the current waveform similarity of M 、 N at both ends; the calculation results under different transition resistances are represented by different line types; those parallel to the abscissa are the thresholds ( k s e t ) set in this article. It can be seen from Table 2 and Figure 8 that under different transition resistances, the similarity of current waveforms at both ends is greater than 0.6 in the case of internal fault, and much less than 0.6 in the case of external fault.

The effectiveness of the protection scheme is analyzed when the fault inception angles are 5 ∘ , 30 ∘ , 45 ∘ , and   90 ∘ , respectively, and the transition resistance is 5 Ω . Due to limited space, this article only gives the calculation results of waveform similarity at the beginning, midpoint, and end of outside the reverse zone, in the zone, and outside of forward zone at different fault inception angles, as shown in Table 3.

The waveform similarity calculation results of different fault locations at different fault inception angles are shown in Figure 9.

It can be seen from Table 3 and Figure 8 that, at different fault inception angles, the similarity of current waveforms at both ends is greater than 0.6 in the case of internal fault, and much less than 0.6 in the case of external fault.

It can be seen from Figures 8, 9 that the similarity value of the fault in the zone is greater than the threshold value, and that of the fault outside the zone is less than the threshold value. When the fault occurs at different initial fault angles, faults can be correctly identified by the protection scheme, which shows that the protection scheme is less affected by the initial fault angle.

From the above simulation results, it can be seen that this protection scheme based on synchronous squeezed wavelet and waveform similarity can accurately identify the internal and external faults when they occur at different fault locations, under different transition resistances, and at different fault initial angles, so as to reliably protect the line.