Convolution Power Ratio Based on Single-Ended Protection Scheme for HVDC Transmission Lines

Abstract

In order to solve the problems of insufficient abilities to withstand transition resistance under remote faults and difficulties in identifying internal and external faults for HVDC transmission line protection, a new single-ended protection scheme based on time-domain convolutional power was proposed. In this scheme, the ratio of time-domain convolution power at different frequencies is used to detect internal and external faults, and the long window convolution power is used to form the pole selection criteria. Due to the integration of transient power fault characteristics at high and low frequencies, this scheme amplifies the characteristic differences between internal and external faults caused by DC line boundaries and has a strong ability to withstand transition resistance. Based on PSCAD/EMTDC, simulation verification was conducted on the Yunnan–Guangzhou ±800 kV HVDC project. The results show that the proposed single-ended protection scheme can effectively identify fault poles, as well as internal and external faults. It has strong resistance to transition resistance and certain anti-interference ability and has strong adaptability to DC line boundaries, which meets the protection requirements of HVDC transmission systems for high speed, selectivity and reliability.

1. Introduction

Due to remarkable advantages such as lower construction costs, lower losses and bulk and long-distance power transmission capabilities, line-commutated converter high-voltage direct current (LCC-HVDC) systems are widely used [1,2,3,4]. The transmission line is the component with the highest fault probability, and line faults affect the whole HVDC system. Therefore, the rapid removal of faults is crucial to the enhancement of the transient stability of power systems [5,6,7,8]. Currently, fault identification methods for HVDC transmission lines can be summarized into the following two schemes:

Time domain characteristics based on methods: Traditional time domain schemes typically utilize the transient characteristics of voltage traveling wave attenuation caused by boundaries composed of smoothing reactors at both ends of the line, such as traditional methods of voltage change rate [9,10]. These methods are simple and have low computational complexity and are widely used in practical engineering. However, they will significantly decrease with an increase in transition resistance, thereby affecting the sensitivity of protection. Ref. [11] uses the method of integrating the sum of positive and negative limit current reactance voltages to identify internal and external faults, which has strong anti-noise interference ability and is not affected by AC measurement faults; however, the scheme has a weak ability to withstand transition resistance, and it is easy to misjudge when high-impedance grounding faults occur at the end of the line.

Compared with the above methods based on a certain time domain, utilizing the waveform characteristics of local wave fronts under internal and external faults is also an effective approach. In Refs. [12,13], the Levenberg–Marquardt method was used to fit the initial traveling wave voltage of the measurement point fault, and the failure modes and exponential terms of different fault characteristics were obtained. Reference [14] calculates the concavity and convexity of the current-limiting reactance voltage for fault identification. Reference [15] utilizes a multi-resolution morphological gradient algorithm to extract the polarity information of fault traveling waves, which has low sampling frequency requirements. However, references [12,13,14,15] all require correct calibration of the initial fault traveling wave to avoid the influence of reflected traveling waves at the beginning and end of the line. In the event of near-end or terminal faults, the arrival time interval between the first and secondary traveling waves is very short, and there may be significant errors in correctly calibrating the traveling wave within a short time window, resulting in protection malfunction or rejection.

Frequency domain characteristics based on methods: The frequency domain method usually utilizes the high-frequency attenuation of fault traveling waves at the boundaries of DC lines and uses frequency domain extraction tools such as wavelet transforms, and Fourier transforms to extract high-frequency information. Traditional frequency domain extraction methods usually extract transient feature information under a single frequency band. For example, in reference [16], wavelet transforms were used to extract the high-frequency voltage component to identify the faults; reference [17] uses the measured impedance at resonant frequency for fault identification based on the amplitude–frequency characteristics of DC filters. Study [18] extracted the voltage and current data of specific frequency bands on both sides of a reactor, and then calculated the boundary energy to form longitudinal protection. However, the traditional method of extracting a single frequency band has the problem of having poor resistance to transition resistance.

To address the above problems, many methods based on multi-frequency-band transient characteristics are proposed. Reference [19] uses the energy of multiple voltage characteristic frequency bands to construct fault criteria. However, its calculation principle is complex due to voltage energy calculations in multiple frequency bands. The ratios of voltages at different frequency features used in references [20,21,22] serve as a criterion for fault identification. In [23,24], stationary wavelet transforms and Hilbert–Huang transforms were used to extract the features of multi-frequency-band voltage, which can effectively amplify the characteristic differences of internal and external faults. However, all these methods require higher sampling frequencies and have complex calculation problems.

To improve the above-mentioned problems such as poor resistance and high sampling frequency, a single-ended protection scheme was proposed for high-voltage DC transmission lines based on convolutional power.

• The differences in the mathematical analytical formulas for transient frequency domain power under internal and external faults in the HVDC transmission line are deduced. Through detailed analysis of the frequency characteristics of faults that are internal and external to the area, it can be concluded that convolutional power is beneficial for amplifying the characteristic differences between internal and external faults to the area caused by the boundary of the DC line.
• In this study, by introducing the convolutional power in the time domain as the power in the frequency domain, it is proposed that the convolutional power of the short time window is used as the main criterion, and the ratio of the convolutional power of the long and short windows is used as the auxiliary criterion for the identification of high-impedance faults. The two complement each other to realize the fault pole, fault direction and fault identification, internal and external to the area.
• The proposed scheme can extract high- and low-frequency information without frequency domain extraction tools; the calculation principle is simple; the convolution power ratio at different frequencies is less affected by the transition resistance; and the proposed protection scheme is still adaptable under weak boundary conditions.

2. Analysis of Frequency Domain Characteristics and Influencing Factors

2.1. Frequency Domain Characteristics of Internal and External Faults

Forward faults (FFs) include internal faults and external faults near the inverter side. Taking the line end fault f₁ and the external fault f₂ on the HVDC transmission line in Figure 1 as examples, protection measuring points (M) are installed at the outlet of the rectifier station. The reverse fault (RF) is f₃.

To eliminate the influences of positive and negative pole coupling, the double-pole line is decoupled into a line-model and ground-model network through the Klenberg transform. Moreover, according to Peterson’s law and superposition theorem, superimposed circuits for f₁ and f₂ can be obtained for FF, as shown in Figure 2. The black arrow is the current-related direction at M, and the red arrow is the direction of the fault current.

For Figure 2, the internal equivalent impedance at f_x (x = 1,2) is as follows:

$$Z_{f1k}=Z_{ck}//Z_{filter}//s2L_{sr}(k=l,e)$$

(1)

$$Z_{f2k}=(Z_{ck}//Z_{filter}+s{L}_{sr})//s{L}_{sr}(k=l,e)$$

(2)

where L_sr is the current-limiting reactance value on the DC side, Z_cx(x=l,e) is the wave impedance of the DC transmission line, R_fx(x=1,2) is the transition resistor and U₀ indicates the rated voltage of the line $Z_{filter}=1/s{C}_{f1}+s{L}_{f1}+1/s{C}_{f2}//s{L}_{f2}+1/s{C}_{f3}//s{L}_{f3}$.

Thus, the electrical quantity of the mold network at f_x (x = 1,2) is the following:

$$\left\{\begin{array}{c}{U}_{fxk}(s)=\frac{-\sqrt{2}{U}_0{Z}_{fxk}(s)}{s(Z_{fxl}(s)+Z_{fxe}(s)+2R_{fx}) }\\ I_{fxk}(s)=\frac{\sqrt{2}{U}_0}{s(Z_{fxl}(s)+Z_{fxe}(s)+2R_{fx}) }\end{array}\right.(k=l,e)$$

(3)

where U_fxk(k=l,e) and I_fxk(k=l,e) are the voltages and currents of the line mode and zero mode components at the fault locations, respectively.

We determined that the TW at f₂ to M, which needs to be multiplied by a partial voltage coefficient ${\lambda }_u$, is as follows:

$${\lambda }_u=\frac{Z_{ck}\left(s)//Z_{filter}\right(s)}{Z_{ck}\left(s)//Z_{filter}\right(s)+s{L}_{sr}}(k=l,e)$$

(4)

According to (1)–(4), the general expression of fault TW for f_x (x = 1,2) can be obtained at M as follows:

$$\left\{\begin{array}{c}{U}_{mf1k}(s)=U_{mf1k}(s);U_{mf2k}(s)={\lambda }_u{U}_{mf2k}(s)\\ I_{mf1k}(s)=U_{mf1k}(s)/Z_{ck}(s);I_{mf2k}(s)={\lambda }_u{U}_{mf2k}(s)/Z_{ck}(s)\end{array}\right.$$

(5)

where U_mf1k(k=l,e) and I_mf1k(k=l,e) are the voltages and currents of the line mode and zero mode components at the measurement point when a fault occurs at f₁, respectively.

For the RF, the superimposed circuit for f₃ outside the rectifier side is shown in Figure 3.

Since both ends are on the LCC side, the equivalent impedance at the outlet of the converter station is the same, and the superimposed circuits for f₃ and f₂ are the same, i.e., Z_f3k = Z_f2k. The difference lies in the direction of the fault current. Therefore, the expression of the fault TW at M for f₃ is the following:

$$\left\{\begin{array}{c}{U}_{mf3k}(s)=\frac{-{\lambda }_u\sqrt{2}{U}_0{Z}_{f3k}(s)}{s(Z_{f3l}(s)+Z_{f3e}(s)+2R_{f3})}\\ I_{mf3k}(s)=-U_{mf3k}(s)/Z_{ck}(s)\end{array}\right.(k=l,e)$$

(6)

where U_mf3k(k=l,e) and I_mf3k(k=l,e) are the voltages and currents of the line mode and zero mode components at the measurement point M when the fault occurs at the fault point f₃, respectively.

The convolution in the time domain is equal to the product in the frequency domain [21], the power in the frequency domain corresponding to f₁, f₂, and f₃ can be obtained as follows:

$$P_{mf1k}(s)=U_{mf1k}(s)I_{mf1\mathrm{k}}(s)=-\frac{2U_0{}^2{(Z_{f1k}(s))}^2}{s^2{(Z_{f1l}(s)+Z_{f1e}(s)+2R_{f1})}^2{Z}_{ck}(s)}$$

(7)

$$P_{mf2k}(s)=U_{mf2k}(s)I_{mf2k}(s)=-\frac{2U_0{}^2{(Z_{f2k}(s){\lambda }_u)}^2}{s^2{(Z_{f2l}(s)+Z_{f2e}(s)+2R_{f2})}^2{Z}_{ck}(s)}$$

(8)

$$P_{mf3\mathrm{k}}(s)=U_{mf3k}(s)I_{mf3k}(s)=\frac{2U_0{}^2{(Z_{f3k}(s){\lambda }_u)}^2}{s^2{(Z_{f3l}(s)+Z_{f3e}(s)+2R_{f3})}^2{Z}_{ck}(s)}$$

(9)

where P_mf1k(k=l,e), P_mf2k(k=l,e) and P_mf3k(k=l,e) are the power values of the line mode and zero mode components at the measurement point M when faults occur at f₁, f₂ and f₃, respectively.

To visually evaluate the characteristic differences of different electrical quantities at f₁ and f₂, for G_U(s) = U_f2(s)/U_f1(s), G_I(s) = I_f2(s)/I_f1(s) and G_P(s) = P_f2(s)/P_f1(s), the amplitude–frequency curve is shown in Figure 4.

As a result of the influence of the DC boundary, the above ratios are each less than 1. As shown in Figure 4, the amplitude–frequency curves of |G_U(jω)| and |G_I(jω)| are very similar, and |G_P(jω)| is not as different from these values in the frequency range of 0 to 340 Hz. However, when the frequency is greater than 340 Hz, the |G_P(jω)| curve becomes lower than these curves. The gap between |G_P(jω)| and either |G_U(jω)| or |G_I(jω)| increases significantly as the frequency increases. Therefore, in the high-frequency band, the attenuation of the frequency domain power by the DC boundary is more serious than that of a single voltage or current, and the degree of attenuation intensifies as the frequency increases. Hence, compared to using voltage or current alone, P_fi(s) can be used to more effectively identify internal and external faults.

2.2. Influencing Factors

The generation of fault TWs is mainly affected by three factors: dispersion, refraction and reflection and fault resistance.

2.2.1. The Dispersion of Traveling Waves

As traveling waves propagate, fault decay due to the influence of long-distance transmission lines occurs, as shown in Figure 5.

The process satisfies the following frequency domain relationship:

$$\begin{array}{l}{B}_{mx}(s){=\mathrm{e}}^{-{\gamma }_{k}L}{B}_{mfx}(s)\\ {\gamma }_k(s)=\sqrt{(R(\omega )+j\omega L(\omega ))(G(\omega )+j\omega C(\omega ))}\end{array}(x=1,2)$$

(10)

where R(ω), L(ω), C(ω) and G(ω) are the resistance, inductance, capacitance and conductivity of the transmission line per unit length, respectively, and L is the length of line propagation. B_mfx is the initial traveling wave at the fault point, B_mx is the reverse traveling wave transmitted from the fault point to the measurement point M, and e^-γkL represents the transmission formula of the fault traveling wave.

The amplitude–frequency characteristics of the line transfer function are shown in Figure 6.

It can be seen from Figure 6 that the transfer function e^-γL of the line increases with the increase in frequency and distance; the high-frequency component and the length of the line will cause serious attenuation of the voltage and current traveling waves at the measuring point. Therefore, when the fault occurs in a long-distance line, it will affect the protection sensitivity that uses only high-frequency components.

2.2.2. The Reflection and Refraction of Traveling Waves

Refraction and reflection occur when the fault TW reaches the node with discontinuous wave impedance, as shown in Figure 7. Considering only the forward TW, the discontinuity of surge impedance is mainly at the outlet of each converter station.

According to the Peterson principle, α_k and β_k are the refractive coefficient and the reflection coefficient, respectively:

$$\left\{\begin{array}{c}{\alpha }_k(s)=\frac{2(2s{L}_{sr}//Z_{filter}(s))}{2s{L}_{sr}//Z_{filter}(s)+Z_{ck}(s)}\\ {\beta }_k(s)=\frac{2s{L}_{sr}//Z_{filter}(s)-Z_{ck}(s)}{2s{L}_{sr}//Z_{filter}(s)+Z_{ck}(s)}\end{array}\right.(k=l,e).$$

(11)

According to the refraction and reflection principles of TWs, the first TW at M is the superposition of the incident wave and the reflected wave. Therefore, the first TW of the voltage and current for f_x (x = 1,2) at M are shown in (12).

$$\left\{\begin{array}{c}{U}_{Mfxk}(s)=(1+{\beta }_k{)\mathrm{e}}^{-\gamma L}{U}_{mfxk}(s)\\ I_{Mfxk}(s)=U_{Mfxk}(s)/Z_{ck}(s)\end{array}\right.$$

(12)

Taken together, the frequency domain power for f_x (x = 1,2,3) at M near the rectifier station are as follows:

$$P_{Mf1k}(s)=U_{Mf1k}(s)I_{Mf1\mathrm{k}}(s)=-\frac{2{((1+{\beta }_k)e^{-{\gamma }_{k}L})}^2{U}_0{}^2{(Z_{f1k}(s))}^2}{s^2{(Z_{f1l}(s)+Z_{f1e}(s)+2R_{f1})}^2{Z}_{ck}(s)}$$

(13)

$$P_{Mf2k}(s)=U_{Mf2k}(s)I_{Mf2k}(s)=-\frac{2{((1+{\beta }_k)e^{-{\gamma }_{k}L})}^2{U}_0{}^2{(Z_{f2k}(s){\lambda }_u)}^2}{s^2{(Z_{f2l}(s)+Z_{f2e}(s)+2R_{f2})}^2{Z}_{ck}(s)}$$

(14)

$$P_{Mf3}(s)=U_{Mf3}(s)I_{Mf3}(s)=\frac{2U_0{}^2{(Z_{f3k}(s){\lambda }_u)}^2}{s^2{(Z_{f3l}(s)+Z_{f3e}(s)+2R_{f3})}^2{Z}_{ck}(s)}$$

(15)

Comparing (8)–(9) and (14)–(15) shows that the frequency domain power retains the polarity of the current. For the FF, the power value is negative, and for the RF, it is positive. Hence, the power polarity can be used to effectively identify external faults on the rectifier side.

2.2.3. Influence of Fault Resistance

According to (13)–(14), the frequency domain is inversely proportional to the square of the fault resistance, meaning that it is more affected by fault resistance than by voltage and current. In the worst-case scenario, R_f2 = 0 Ω. By changing the value of the fault resistance R_f1, the G_P(jω) curves at different frequencies and different R_f1 values can be obtained, as shown in Figure 8.

G_P(jω) =P_f2(jω)/P_f1(jω), which indicates the ratio of the transient frequency of faults outside the zone to the transient frequency of faults inside the zone. It can be seen from Figure 8:

(1)

In the same frequency band, as R_f1 increases, P_f1(jω) decreases, resulting in |G_P(jω)| rapidly increasing; that is, the G_P(jω) characteristics are greatly affected by the transition resistance.

(2)

At the same transition resistor, as the frequency increases, |G_P(jω)| decreases, indicating that in the high-frequency band, the fault protection method based on frequency extraction can better detect internal and external faults.

Therefore, the protection scheme based on frequency extraction can identify low-resistance faults, but when a high-impedance grounding fault occurs in the DC line, this method is no longer applicable, and theoretically it is impossible to distinguish between internal and external faults based on the power in the frequency domain.

3. Ratio Characteristics of the Frequency Domain Power at Different Frequencies

Based on the above analysis, only the frequency domain power of a certain frequency (frequency band) is greatly affected by the fault resistance. To weaken this influence, a quantitative index P_ratio can be constructed, which is defined as the ratio of the frequency domain power values at different frequencies. We suppose that two different frequencies are ω₁ and ω₂, and that ω₁ > ω₂. U_f(jω) is the step signal. The ratios P_ratio1 and P_ratio2 can be obtained under internal and external faults, respectively, as follows:

$$P_{ratio1}=P_{f1}\left(j{\omega }_1\right)/P_{f1}\left(j{\omega }_2\right);P_{ratio2}=P_{f2}\left(j{\omega }_1\right)/P_{f2}\left(j{\omega }_2\right)$$

(16)

The line wave impedance also changes with the frequency; therefore, it is necessary to consider the relationship between the line wave impedance and the frequency and draw the waveform of the line wave impedance with frequency according to Equation (10), as shown in Figure 9.

As can be seen from Figure 9, the line wave impedance changes very little with frequency in the high-frequency band and can be approximated into a constant in the high-frequency band. In the same way, it can be seen from Figure 9 that the linear mode wave impedance changes little with frequency compared with the ground mode wave impedance, so the linear mode component is considered for analysis and discussion.

Therefore, the expressions of P_ratio1 and P_ratio2 are as follows:

$$\begin{array}{cc}{P}_{ratio1}\hfill & =U_{Mf1}(j{\omega }_1)I_{Mf1}(j{\omega }_1)/U_{Mf1}(j{\omega }_2)I_{Mf1}(j{\omega }_2)\hfill \\ \hfill & \approx {\omega }_2{}^2{(1+\beta (j{\omega }_1))}^2/{\omega }_1{}^2{(1+\beta (j{\omega }_2))}^2\hfill \\ \hfill & =\frac{2Z_{filter}(j{\omega }_2)j{\omega }_2{L}_{sr}+\left(2j{\omega }_2{L}_{sr}+Z_{{}_{filter}}(j{\omega }_2)\right)Z_{c1}}{2Z_{filter}(j{\omega }_1)j{\omega }_1{L}_{sr}+\left(2j{\omega }_1{L}_{sr}+Z_{{}_{filter}}(j{\omega }_1)\right)Z_{c1}}\hfill \end{array}$$

(17)

$$\begin{array}{cc}{P}_{ratio2}\hfill & =U_{Mf2}(j{\omega }_1)I_{Mf2}(j{\omega }_1)/U_{Mf2}(j{\omega }_2)I_{Mf2}(j{\omega }_2)\hfill \\ \hfill & \approx {\omega }_2{}^2{\lambda }_u{({\omega }_1)}^2{(1+\beta ({\omega }_1))}^2/{\omega }_1{}^2{\lambda }_u{({\omega }_2)}^2{(1+\beta ({\omega }_2))}^2\hfill \\ \hfill & =P_{ratio1}{\left(j{\omega }_2{L}_{sr}+Z_{c1}\right)}^2/{\left(j{\omega }_1{L}_{sr}+Z_{c1}\right)}^2\hfill \end{array}$$

(18)

As shown in (17) and (18), P_ratio1 and P_ratio2 are only related to the frequency, current-limiting reactance and wave impedance, all of which are basically not affected by the fault resistance.

Letting |G_Pratio| = P_ratio1/P_ratio2, we can obtain the following:

$$\left|G_{Pratio}\right|=\frac{P_{ratio2}}{P_{ratio1}}\approx \frac{{\left(j{\omega }_2{L}_{sr}+Z_{c1}\right)}^2}{{\left(j{\omega }_1{L}_{sr}+Z_{c1}\right)}^2}$$

(19)

Substituting the model project parameters of the Yun–Guang UHVDC and taking ω₂ = 600 rad/s, the |G_Pratio| curves at different frequencies ω₁ and fault resistances R_f1 can be obtained, as shown in Figure 10.

As can be seen from Figure 10:

(1)

In the case of ω₁ > ω₂ = 600 rad/s, regardless of the transition resistance and frequency ω₁, G_Pratio < 1, P_ratio1 < P_ratio2.

(2)

As ω₁ gradually increases, |G_Pratio| also further decreases, while |G_Pratio| denotes P_ratio2/P_ratio1, so the difference between the quantitative indicators P_ratio1 and P_ratio2 is more obvious, which further amplifies the difference between internal and external faults.

(3)

In the same way, when ω₁ is fixed, as the transition resistance R_f1 gradually increases, the value of |G_Pratio| changes slightly, indicating that the ability of |G_Pratio| to identify internal and external faults is less affected by the transition resistance ability R_f1 at different frequencies.

4. Single-Ended Protection Scheme of HVDC Transmission Lines

First, the CP P_fi(t) is defined as the convolution of two functions, U_fi(t) and I_fi(t), in the time domain [24]:

$$P_{fi}(t)=(u_{fi}\times i_{fi})(t)={\displaystyle {\int }_{t_0}^{t_0+T_s}{u}_{fi}(t-\tau )}{i}_{fi}(\tau )d\tau$$

(20)

where t₀ and T_s are the starting time of the data window and the length of the time window, respectively.

From the convolution theorem, the Laplace transform of function convolution is the product of the function Laplace transform [25]; that is, U(t) and I(t) undergo the Laplace transform to obtain U(jω) and I(jω) in the frequency domain, and the expression for this relationship is the following:

$$(U\times I)(t)\underset{\mathrm{Equivalent} }{\stackrel{\mathrm{Laplace} \mathrm{transform} }{\to }}U(j\omega )I(j\omega )=P_{fi}\left(s\right)$$

(21)

Therefore, P_fi(t) is the CP and P_fi(s) is the CP in the frequency domain.

Specifically, the CP represents the application of voltage as an input quantity to all current points within T_s, and is generated by varying the voltage quantity on the current. The convolution operation convolves the input data and can be regarded as a weighted summation, whereas the result of the convolution is the feature extraction and the amplification of the fault characteristics. The convolution calculation process is shown in Figure 11. The input signal u(t) is regarded as a discrete sequence u(n) = {u₁, u₂, u₃, …, u_n}; i(t) is regarded as a discrete sequence i(n) = {i₁, i₂, i₃, …, i_n}.

It can be seen from Figure 11 that the convolution power result under the sampling point can be obtained by adding each column in Figure 11, and the convolution result presents the arrangement results from small to large and then large to small, and the discrete signal convolution power solution is obtained through the convolution calculation process of Figure 11:

$$P_a(X)={\displaystyle \sum _{i=1}^N{U}_a(i)}{I}_a(X-i+1) (a=p,n)$$

(22)

where P_a is the CP; X is the Xth sampling point; and N is the total number of sampling points, which reflects the length of the time window. N is calculated as follows, where f_s is the sampling frequency:

$$N=T_s{f}_s$$

(23)

In fact, because the TW is a nonstationary time signal, its spectrum distribution is related to the time window. For different time windows, the spectrum distribution is not consistent. Therefore, the spectrum distribution of the CP is also affected by the time window. From the perspective of signal processing, when the sampling frequency is fixed, the length of the time window reflects different frequency resolutions. According to the Shannon sampling theorem, the fault information also contains many harmonic component frequencies, except the fundamental frequency signal in the long and short windows, but the frequency resolution of each time window is inversely proportional to the length of the time window. Taking the sampling frequency of 10 kHz as an example, when the short window is 1 ms, the minimum frequency interval on the frequency axis is 1000 Hz, whereas when the long window is 3 ms, the minimum frequency interval is 333.33 Hz. Therefore, it is clear that compared with that of the short window, the frequency component of the long window is greater [26]. In detail, the fault frequency information in the short window can be regarded as high-frequency information. Similarly, the long window can be regarded as the wide band of low-frequency and high-frequency information.

The single-ended protection scheme for HVDC transmission lines is described in the following sections.

4.1. Starting Criterion

When a fault occurs in the line, the pole line voltage suddenly changes. Therefore, the variation in polar line voltage ΔU_a can be used as the starting criterion of the protection:

$${\displaystyle \sum _{j=0,1,2}\left|\Delta U_a(j+i)-\Delta U_a(j+i-1)\right|}>k_v{U}_{ref}(a=p,n)$$

(24)

where U_ref is the rated voltage and a represents the fault pole. To avoid the influence of steady-state voltage fluctuations, k_v is the voltage fluctuation coefficient and is taken as 0.08.

4.2. Fault Identification Criteria of the Internal and External Faults

The maximum value of the CP sequence under long and short windows is selected as the characteristic quantity.

4.2.1. Main Criterion 1

Main criterion 1 adopts the CP with a short window of T₁. The main function of the criterion is to quickly identify serious faults, including the metallic and low-resistance faults. Main criterion 1 is given as follows:

$$P_{aT1max}>P_{aT1.set}=k_{rel.1}{P}_{f2T1max}(a=p,n)$$

(25)

where $P_{aT1max}=max(\left|P_{aT1}\right|)$, and P_aT1 is the discrete CP value obtained by convolution of the voltage and current within T₁, taken as 1 ms. P_f2T1max represents the metal external fault inverter side within T₁; k_rel.1 is the reliability coefficient, taken as 1.5.

4.2.2. Main Criterion 2

Main criterion 1 is based on the CP with a short window, and has the advantages of simplification and rapid tripping, but its sensitivity is reduced with increasing fault resistance. To address this issue, a criterion composed of the CP ratio (R_a) at different frequencies is added to assist in identifying high-resistance faults. Main criterion 2 is the following:

$$R_a>R_{a.set}=k_{rel.2}{R}_{f2a}(a=p,n)$$

(26)

where $\left\{\begin{array}{l}{R}_a=P_{aT1\max }/P_{aT2\max }\\ P_{aT2\max }=\max (\left|P_{aT2}\right|)\end{array}\right.$, R_a.set represents the setting value, R_f2a is the ratio of long- and short-window CPs of the metal external fault inverter side, P_aT2 is the discrete CP value obtained by convolution of the voltage and current within T₂, taken as 3 ms, and k_{rel. 2} is the reliability coefficient, taken as 2.

Compared to main criterion 1, the above criterion requires waiting for a long window and increasing the number of calculations. However, when a high-resistance fault occurs, the rising speed of the fault current is relatively slow, and the system allows a longer response time for the protection action.

4.3. Fault Direction Identification Criterion

As mentioned in Section 2, the polarity of the CP can be used to distinguish the RF and FF, so the fault direction can be identified by the polarity of the short-window CP, including Δ_th1 as the threshold value:

$$\left\{\begin{array}{cc}{A}_1=\min (P_{aT1})>{\Delta }_{th1}\hfill & \mathrm{RF}\\ A_1=\min (P_{aT1})<-{\Delta }_{th1}\hfill & \mathrm{FF}\end{array}\right.$$

(27)

4.4. Fault Pole Selection Identification Criterion

The coupling strength between DC lines is related to frequency. In the high-frequency band, the coupling strength is relatively large, so the high-frequency components of the sound pole and fault pole are very similar. However, in the low-frequency band, the coupling strength is weak, so the low-frequency component of the fault pole is more significant than that of the sound pole. Therefore, a pole selection criterion based on a low-frequency CP under a long window can be constructed as follows:

$$\left\{\begin{array}{cc}\frac{P_{pT2\max }}{P_{nT2\max }}>{\Delta }_{th2}\hfill & \mathrm{PGF}\\ \frac{P_{pT2\max }}{P_{nT2\max }}<\frac{1}{{\Delta }_{th2}}\hfill & \mathrm{NGF}\\ {\Delta }_{th2}>\frac{P_{pT2\max }}{P_{nT2\max }}>\frac{1}{{\Delta }_{th2}}\hfill & \mathrm{PPF}\end{array}\right.$$

(28)

where Δ_th2 is the threshold value and k_rel.3 is the reliability coefficient, which is 1.5. The rectifier station acts according to the protection result after fault identification, and then the DC system enters the fault isolation and recovery phase. PGF is a positive ground fault, NGF is a negative ground fault, and PPF is a fault between poles.

In summary, this study designed a single-ended protection scheme that has both advantages with the CP ratios of short and long windows. The specific flow chart is shown in Figure 12, which is composed of main criterion 1 and main criterion 2.

5. Case Analysis

A ±800 kV UHVDC transmission system model was established in PSCAD/EMTDC with respect to the parameters of the Yunnan to Guangdong UHVDC transmission project. The specific model parameters are shown in

Table 1. Model parameters.

Parameter	Rectifier Side	Inverter Side
Rated power	5000 MW	5000 MW
Rated DC current(I0)	3125 A	3125 A
Rated DC voltage(U0)	±800 kV	±800 kV
Boundary element (Lsr)	0.15 H	0.15 H
Filter	Cf1 = 1.2 μF Cf2 = 2.824 μF Cf3 = 2.647 μF Lf1 = 9.345 mH Lf2 = 15.919 mH Lf3 = 4.6556 mH

. The structures of the DC line are depicted in Figure 13. The sampling frequency is 10 kHz. The time windows of the CPs are 1 ms and 3 ms, so the numbers of sampling points N are 10 and 30, respectively. The protection algorithm was written in MATLAB software. According to the setting principle and simulation results, the criterion-setting value of the protection scheme is shown in

Table 2. Setting value of protection criterion.

Criterion	Δth1	Δth2	PpT1.set	RP
Setting value	20	1	900	0.08

. The steps in the following sections are used to verify the action of protection under different faults.

5.1. Identification of Internal and External Faults

The fault occurs at t = 0.2 s, and when a PGF occurs at point f₁ (internal fault), the fault voltage and fault current at the boundary measurement point after the fault are shown in Figure 13.

The transient convolution power graphs of the long and short windows calculated with Equations (20)–(22) in the case of internal faults are shown in Figure 14.

As shown in Figure 14, P_{pT1_MAX} = 2039 > P_{pT1_set} = 900 in main criterion I; similarly, the ratio of the convolution power calculated with main criterion II according to Formulas (25) and (26), is P_ratio = 0.32 > R_p = 0.08. Therefore, the protection is determined as an internal fault.

When a PGF occurs at point f₂ (external fault), the fault voltage and fault current at the boundary measurement point after the fault are shown in Figure 15.

The transient convolution power graphs of the fault for long and short windows calculated using Equations (20)–(22) during external faults are shown in Figure 16.

From Figure 16, it can be seen that P_{pT1_max} = 681 < P_{pT1_set} = 900 for main criterion I; similarly, it can be seen that the ratio of convolution power calculated via main criterion II according to Formulas (25) and (26), is P_ratio = 0.045 < R_p = 0.08. Therefore, the protection is determined as an external fault.

To verify the accuracy and applicability of the protection scheme, the simulation was set to have high-resistance faults at every 200 km interval of the lines, with their resistances varying from 0 to 600 Ω. Considering the most serious situation, f₂ and f₃ were set as metallic ground faults in the later analysis. Since the positive and negative poles are symmetrical, only the positive fault is discussed here. In the table below, √ represents the protection action; conversely, × represents no action. Limited by the article space, only some fault point simulation results are shown here. The simulation test results are shown in

Table 3. Fault identification results for internal and external faults.

Fault Distance/km	Fault Type	Transition Resistance/Ω	Fault Direction	A1	PpT2max	PpT1max	RP	Action of Main Criteria 1 and 2
0	PGF	0	FF	−5916	18,743.6	5915.6	0.316	(1√, 2√)
		160	FF	−1354.8	4235.4	1354.8	0.320	(1√, 2√)
		200	FF	−1051	3301.7	1050.7	0.318	(1√, 2×)
		400	FF	−396	1258.2	395.9	0.315	(1×, 2√)
		600	FF	−196	623.4	196.1	0.314	(1×, 2√)
	PPF	0	FF	−30,961	88,281.1	30,960.9	0.350	(1√, 2√)
		600	FF	−2899	9027.5	2898.8	0.321	(1√, 2√)
400	PGF	0	FF	−3871	15,410.5	3870.6	0.251	(1√, 2√)
		160	FF	−1111.5	3933.8	1111.5	0.283	(1√, 2√)
		200	FF	−876	3087.5	875.6	0.284	(1×, 2√)
		400	FF	−353	1212.4	352.7	0.291	(1×, 2√)
		600	FF	−186	615.9	186.3	0.303	(1×, 2√)
	PPF	0	FF	−27,464	116,074.2	27,464.2	0.237	(1√, 2√)
		600	FF	−2482	9400.4	2481.7	0.264	(1√, 2√)
800	PGF	0	FF	−3802	14,702.4	3801.7	0.259	(1√, 2√)
		160	FF	−952.9	3736.3	952.9	0.255	(1√, 2√)
		200	FF	−752	2936.1	751.9	0.256	(1×, 2√)
		400	FF	−358	1194.2	357.8	0.300	(1×, 2√)
		600	FF	−190	616.0	190.0	0.308	(1×, 2√)
	PPF	0	FF	−22,507	105,594.7	22,506.7	0.213	(1√, 2√)
		600	FF	−2227	9245.6	2227.2	0.241	(1√, 2√)
1418 (f1)	PGF	0	FF	−3795	17,621.5	3795.1	0.215	(1√, 2√)
		160	FF	−882.0	3985.7	882.0	0.221	(1×, 2√)
		200	FF	−719	3691.0	718.8	0.195	(1×, 2√)
		400	FF	−283	1520.2	283.1	0.186	(1×, 2√)
		600	FF	−167	808.6	166.5	0.206	(1×, 2√)
	PPF	0	FF	−16,061	15,331.7	16,061.0	0.179	(1√, 2√)
		600	FF	−1918	614.0	1918.3	0.178	(1√, 2√)
f2	PGF	0	FF	−601	15,429.1	600.9	0.039	(1×, 2×)
f3	PGF	0	RF	0.00345	/	2235.11	/	/

.

Table 3 shows the following:

(1)

In the case of low-impedance grounding faults (internal faults), the maximum value of the high-frequency convolution power of each fault type, P_pT1max, is much greater than that of P_pT1set, and the A1 value is obtained by the calculation using Equation (27) in order to determine the fault direction; therefore, main criterion 1 in the proposed protection scheme can quickly and reliably identify the low-impedance grounding faults in the region.

(2)

As shown in the shaded font in Table 3, when the transition resistance is set at a distance of 1414 km from the protection installation to a fault of 600 Ω, P_pT1max is equal to 882, which is less than the setting value P_pT1set, which may cause the effect of rejection.

(3)

In order to prevent the rejection phenomenon when the high-impedance grounding fault is made, this study proposes to use main criterion 2 to identify the high-impedance grounding fault inside and outside the region. From the observations in the table, it can be seen that the P_ratio under the internal fault (f₁) is greater than that under the external fault (f₂), and the value of the P_ratio is still greater than that of the metal fault outside the area on the inverter side, even if the transition resistance is 600 Ω, which ensures the identification of the high-impedance fault in the area via main criterion 2.

5.2. Faulty Pole Selection

When the protection method in this research determines that the fault is an internal fault, it is necessary to further select the fault pole selection criterion for the fault in the region. The waveforms of the long window convolution power P_pT2 for a PGF, NGF and PPF occurring at point f₁ are shown in Figure 17.

As shown in Figure 17:

(1)

When a positive pole fault occurs in the system, the ratio of transient convolution power between the positive and negative poles on the rectifier side is much greater than 1.5.

(2)

When a negative pole fault occurs in the system, the ratio of transient convolution power between the positive and negative poles on the rectifier side is less than 0.67.

(3)

When an interpole fault occurs in the system, the ratio of transient convolution power between the positive and negative poles on the rectifier side is between 0.67 and 1.5.

At the 400 km location in the DC line, the NGF and PGF are applied under different resistances, and the corresponding waveform of P_pT2 with a long window is shown in Figure 18.

Figure 18 shows that although the CP with a long window decreases with increasing fault resistance, the waveform of P_pT2 of the PGF is always above that of the NGF, within 600 Ω. Therefore, the CP with a long window can be used to identify the faulty pole, which is consistent with the description in Section 4. In the actual criterion, the fault pole is judged by calculating the ratio (P_pT2max/P_nT2max). The simulation test results of specific fault pole selection are shown in

Table 4. Faulty pole selection results.

Fault Type	Resistance/Ω	PpT2max/PnT2max
400 km PGF	0	64.710
	200	65.547
	400	52.141
	600	43.037
400 km NGF	0	0.0181
	200	0.0220
	400	0.0208
	600	0.0228
400 km PPG	0	0.9989
	200	0.9988
	400	0.9990
	600	0.9991

.

Table 4 shows that in various fault conditions, the proposed criterion can reliably distinguish the fault pole.

5.3. Adaptability Analysis of the Noise Interference

In practical engineering, the line may encounter noise interference, which affects the reliability of the line protection action. After sampling interference signals such as noise, the obtained digital signal contains more high-frequency components, which affect the action performance of the criterion based on transient variation. To verify the anti-interference ability of the protection scheme, Gaussian white noise based on the actual signal power value was added to the voltage and current measured at M₁, and the signal-to-noise ratio (SNR) was 40 dB. Taking the middle of the line as an example, the specific simulation results of the scheme proposed in [26] and the scheme proposed in this research are shown under severe Gaussian white noise interference in

Table 5. Action performance of main criteria 1 and 2 after adding noise.

Fault Distance	Fault Resistance	Noisy PpT1max	Noisy RP	Criterion Action	No-Noise TTV	Noisy TTV
600 km	0 Ω	3694.5	0.254	(1√, 2√)	0.413	0.396
	200 Ω	768.5	0.267	(1×, 2√)	0.357	0.365
	400 Ω	280.9	0.253	(1×, 2√)	0.383	0.435
	600 Ω	158.2	0.272	(1×, 2√)	0.318	0.350
f2	0 Ω	851.7	0.053	(1×, 2×)	0.168	0.265

.

The simulation results in Table 5 show that the protection criteria proposed in this study can withstand 40 dB of Gaussian white noise and maintain the original performance. However, the protection scheme based on the time-domain transient voltage in [22] is vulnerable to Gaussian white noise, especially in the case of external faults. It is necessary to recalculate the setting value to avoid protection misoperation. Therefore, the proposed protection scheme has a better ability to resist noise interference.

5.4. Adaptability Analysis of the Boundary Element

When the high-frequency component passes through the boundary element, such as a smoothing reactor, it is greatly attenuated, and the attenuation degree of the high-frequency component is different for smoothing reactors with different values, which affects the adaptability of the protection principle. Therefore, the simulation verifies the adaptability of the method to smoothing reactors with different values when a fault occurs. The specific simulation results are shown in

Table 6. Results under different values of smoothing reactor.

Fault Distance	Fault Resistance	Lsr /H	PpT2max	PpT1max	RP	Protection Margin (PpT1max/PpT1.set, RP/RP.set)
0 km	0 Ω	0.15	18,743	5915.6	0.316	(6.572, 3.950)
		0.10	2215	6682.4	0.302	(4.330, 2.649)
		0.05	26,628	7505.4	0.282	(2.414, 1.270)
	600 Ω	0.15	623.4	196.1	0.314	(0.218, 3.925)
		0.10	587.0	185.3	0.316	(0.120, 2.772)
		0.05	567.2	176.0	0.310	(0.056, 1.396)
1418 km (f1)	0 Ω	0.15	17,621	3795.1	0.286	(4.217, 2.687)
		0.10	17,145	3175.6	0.185	(2.058, 1.614)
		0.05	18,172	4062.3	0.224	(1.307, 1.010)
	600 Ω	0.15	808.6	166.5	0.290	(0.185, 2.575)
		0.10	698.6	132.2	0.292	(0.128, 2.561)
		0.05	663.7	138.1	0.295	(0.044, 1.329)
f2	0 Ω	0.15	15,429	600.9	0.039	/
		0.10	18,008	1028.8	0.057	/
		0.05	18,684	2072.7	0.111	/

.

As shown in Table 6, with the reduction in the smoothing reactor value, the CP value of the short window of an external fault continues to increase, resulting in a reduction in the discrimination degree of the out-of-area faults. Thus, protection criteria based on high frequencies, such as main criterion 1, are greatly affected by boundary elements. However, different smoothing reactor values have little effect on the action performance of main criterion 2 proposed in this study. When L_sr > 50 mH and the fault resistance is 600 Ω, main criterion 2 still has a sufficient margin for identifying faults. Therefore, the proposed protection scheme is also suitable for HVDC transmission systems with weak boundary characteristics, which are easy to implement and have good universality.

5.5. Comparison Studies

(1)

Time-domain method:

Using the rate of change of DC voltage (ROCOV)-based fault protection scheme proposed in reference [6], the superiority of the proposed scheme is verified through simulation comparison.

Set a positive grounding fault with a transition resistance of 400 Ω at f₁ (internal fault), and a metallic grounding fault (external fault) at f₂. The ROCOV waveform is shown in Figure 19.

As shown in Figure 18, the ROCOV at f₂ is greater than that at f₁, and the ROCOV’s ability to withstand transition resistance is less than 400 Ω. However, the protection scheme proposed in this research has a resistance to transition resistance of 600 Ω. Therefore, using ROCOV alone cannot identify high-resistance faults.

(2)

Frequency domain method

A protection scheme based on wavelet transforms for measuring high-frequency component values of point voltages was proposed in reference [12]. Similarly, a positive grounding fault with a transition resistance of 500 Ω was set at f₁ under a weak boundary (L_dc = 50 mH) (internal fault), and a metallic grounding fault was set at f₂ (external fault). The transient high-frequency voltage components obtained through wavelet transforms are shown in Figure 20.

By setting the metal grounding fault outside the zone, the maximum transient high-frequency voltage of the most severe external fault is 9.7485. Setting the reliability coefficient K_rel = 1.3 can obtain the setting value for internal and external faults in the zone. From Figure 20, the grounding fault with a transition resistance of 500 Ω in the area is less than the set value, so the protection will refuse to operate. As the smoothing reactor of this scheme adopts 50 mH, it is not suitable for weak boundary systems. Meanwhile, as a time–frequency analysis tool, wavelet transform has complex computational principles. And this article uses time-domain convolution power to extract high-frequency signals with a simple principle, and the proposed protection scheme has a resistance to transition resistance of 600 Ω.

6. Conclusions

To resolve the problem of the insufficiency of HVDC transmission systems to withstand fault resistance, a fault characteristic quantity based on the CP was constructed, and a novel single-ended protection scheme based on the CP ratio for HVDC transmission lines was proposed. The results of the mathematical analysis and numerical examples show the following achievements:

(1)

By analyzing the characteristic difference in frequency domain power after DC line fault in detail, a new protection criterion was proposed using the power ratio of different frequencies under long and short windows. The scheme is less affected by transition resistance and fault distance, and the principle of the proposed scheme is simple. It can obtain the frequency domain component of a fault traveling wave without relying on other methods, such as extracting frequency domain information.

(2)

The Yunnan–Guangzhou ±800 kV HVDC project was built to verify the performance of the proposed fault rapid identification scheme. In all cases, it can accurately identify faults, has the identification ability of up to 600 Ω transition resistance, and has low requirements for the sampling frequency of the protection device. Only a 10 kHz sampling frequency is required to identify faults inside and outside the area. Moreover, when the SNR is greater than 40 dB, noise has little effect on the proposed method.

(3)

Compared with other time-domain methods based on measuring point voltage change and frequency domain methods based on wavelet transform extraction, the proposed protection scheme can identify different types of short-circuit faults and has a strong ability to withstand excessive resistance. The proposed protection scheme can also be applied to HVDC transmission systems with weak boundary characteristics.

Future work plans to apply this method to flexible HVDC transmission systems under weak boundaries.