Abstract
In flexible DC grids, the rapid rise of fault current requires that the line protection must complete the fault identification within a few milliseconds. Dynamic state estimation based protection (DSEBP) provides a new idea for flexible DC line protection with good performance. However, the operating frequency in the DC grid is 0 Hz. When the DC grid is operating normally, it is difficult to identify the line parameters online to improve the performance of the protection method. This paper proposes a method to identify the frequency-dependent parameters of flexible DC grids based on the characteristic signal injection of half-bridge modular multilevel converter (HB-MMC). The characteristic signal is extracted by the Prony algorithm to calculate the line parameter under different frequencies. Afterwards, the number and position of residues and poles of frequency-dependent parameters are determined using the vector fitting method. Finally, an improved DSEBP is proposed. The simulation shows that the frequency-dependent parameters obtained by the proposed parameter identification method can be used in the improved DSEBP normally, and the identified parameters have better precision.
FLEXIBLE DC grid offers some advantages, including the absence of commutation failure, the ability to provide island power supply, and convenience for networking. These advantages have led to the widespread application of the flexible DC grid in power delivery and asynchronous network interconnection [
In the current flexible DC grid projects, the main protection methods usually consist of the voltage derivative-based protection method and traveling wave protection method [
To improve the protection performance, the dynamic state estimation method is used for flexible DC protection [
Parameter identification methods can be divided into offline measurement methods and online measurement calculation methods [
Currently, the majority of researchers are engaged in the study of online parameter identification methods. The classical theory method, based on Carson formula, is used to calculate the line parameters through geometrical and structural configurations of the towers, conductors, and ground wires [
Fault identification based on the characteristic signal injection in DC grids is one of the research highlights [
To solve the above issues, a frequency-dependent parameter identification method based on characteristic signal injection of half-bridge modular multilevel converter (HB-MMC) is proposed. First, by controlling the HB-MMC, the specific frequency signal is injected into the line, enabling the accurate calculation of the frequency-dependent parameter. Afterward, the residues and poles of the frequency-dependent parameter can be obtained by the vector fitting method. Finally, the performance of DSEBP is improved based on the available residues and poles. The key contributions are as follows.
1) The redundant submodules of HB-MMC are used for characteristic signal injection, which solves the difficulty in parameter identification caused by the operating frequency of 0 Hz in the DC grid.
2) The proposed parameter identification method offers a precise calculation of frequency-dependent parameters, thereby enhancing the accuracy of parameter identification.
3) The utilization of the available residues and poles associated with the frequency-dependent parameter contributes to the development of the improved DSEBP. This improved method is deemed to be more practical and reliable in its application.
The topology of HB-MMC is shown in

Fig. 1 Topology of HB-MMC.
HBSM has three operating statuses and can output two voltage levels Uc and 0. In

Fig. 2 Traditional control strategy for HB-MMC.
The HB-MMC employs a traditional double closed-loop vector control strategy, wherein half of the submodule capacitors are engaged to sustain the DC voltage. The outer-loop control strategy involves the regulation of active power and reactive power, while the inner-loop control strategy focuses on current control.
Without considering the redundant submodules of the converter station, the relationship among the series number of bridge arm submodules N, the DC voltage Udc, and the rated capacitance voltage UcN can be expressed as:
(1) |
In the design of MMC, the operating condition with modulation factor should be considered. At this time, the voltage variation range of the upper and lower bridge arms is 0-Udc, and in consequence, the number of submodules inserted by each bridge arm is 0-N. In this case, the additional submodules in series outside the N submodules are the redundant submodules that are considered in the design, and the number is Nde. However, in the actual operation of MMC, the modulation factor Km is less than 1, and the maximum number of submodules to be invested by each bridge arm Nop can be expressed as:
(2) |
According to the above analysis, when the modulation factor Km<1, the bridge arm submodules will not be fully utilized. At any given time, at least submodules are in the bypass state, which are called operational redundant submodules . Therefore, when MMC is in normal operation, the total number of redundant submodules Nre is expressed as:
(3) |
In the field of engineering, the most commonly employed MMC modulation strategies are those based on NLM, which allows for the theoretical control of the voltage difference between the MMC output voltage and the modulation wave within ±Udc/2. The reference value of DC voltage is defined as Udcref, then the number of submodules to be incorporated into the upper and lower bridge arms at time t can be expressed as:
(4) |
where round(x) is the function to take the integer closest to x; and is the reference sinusoidal modulated voltage.
In this paper, the injection of characteristic signals is achieved through the control of input and the removal of redundant submodules. In other words, Nop is employed to modify the number and state of equivalent HBSM inputs. For the same phase, if the input submodules are simultaneously increased or decreased for both the upper and lower bridge arms, the reference potential of the AC-side voltage will remain unchanged. Therefore, the AC side of the MMC will not be affected.
The additional control strategy of signal injection is shown in

Fig. 3 Additional control strategy of signal injection.
In
The square wave is injected as the characteristic signal. When the injected characteristic signal is at a high level, the upper and lower bridge arms of the MMC will input additional submodules. When the injected characteristic signal is at a low level, the upper and lower bridge arms of the MMC will remove additional submodules. Afterwards, the number of additional submodules is added to the number of normal operating submodules to be put into operation during the process of generating injected signals.
In actual projects, it is typical for the bridge arms of converters to contain redundant submodules. For instance, the Nan’ao three-terminal flexible DC project in China comprises (redundant) submodules in the bridge arm of the Sucheng converter station and (redundant) submodules in the bridge arm of the Jinniu and Qing’ao converter stations. The ±500kV Zhangbei flexible DC grid project in China comprises (redundant) submodules in the bridge arm. Therefore, when HB-MMC is in normal operation, the number of redundant submodules can be employed for characteristic signal injection.
However, the selection of the characteristic signal necessitates consideration of a variety of factors, including the submodule switching control period, the limitations of practical engineering, and the duration of characteristic signal injection.
The choice of the injected characteristic signal frequency should accurately reflect the alterations in both the line characteristic impedance and attenuation function. Additionally, it is crucial for the characteristic signal frequency to be less than the sampling frequency of the protection system, which is inherently constrained by the switching speed of the submodules. In practical applications of high-voltage direct current (HVDC) projects, the typical sampling frequency ranges from 10 to 50 kHz, with a control cycle of approximately 100 . For instance, considering a sampling frequency of 10 kHz, the protection can correctly distinguish frequencies up to 5000 Hz according to the Shannon sampling theorem.
The amplitude of the injected characteristic signal is constrained by the number of redundant submodules. Many factors should be considered, such as the performance of the voltage/current transformer, the potential impact on the power system, and the overcurrent tolerance capability of power electronics. Consequently, the amplitude of the injected characteristic signal is constrained by rigorous limitations. However, it is worth noting that a larger amplitude of the injected characteristic signal can enhance the effectiveness of signal extraction. For power grids operating at 35 kV and above, the permissible voltage deviation range is typically set at 10% of the rated voltage. Additionally, the redundant submodules in the bridge arm usually account for 10% to 20% of the total. In this study, the amplitude of the injected characteristic signal is chosen to be 5% of the rated DC voltage.
For the duration of characteristic signal injection, the accuracy of signal extraction should be considered. First, if the duration of the characteristic signal injection is less than one period, the accuracy of the signal extraction is reduced. Second, during the first few periods of characteristic signal injection, some small disturbances may occur due to the switching of the control strategy. It is possible to await the stabilization of the injected characteristic signal before the signal extraction. Therefore, the duration is set to be ten periods of the characteristic signal.
Various methods can be employed for the extraction of amplitude and phase information from characteristic signals, such as wavelet transform, fast Fourier transform (FFT), and Prony algorithm. It is important to note that while the wavelet transform is adept at capturing the time-varying characteristics of a signal, it is unable to provide accurate phase information. The FFT reduces the computational complexity by minimizing multiplication and addition operations. However, its accuracy is constrained by the factors such as the chosen data window and the presence of DC attenuation components. The Prony algorithm offers the ability to describe transient characteristics of a signal, encompassing amplitude, phase, and frequency information.
The Prony algorithm is adopted in this study to extract the amplitude and phase of characteristic signals based on the measurement data obtained from both ends of the transmission line. The mathematical model is expressed as:
(5) |
where is the actual measured signal; Ai is the amplitude; is the attenuation factor; fi is the frequency; θi is the phase; and q is number of orders. The calculation amount of the Prony algorithm will increase exponentially when the order is high. In order to address this issue, the measured data are pre-filtered using a
According to the extraction length of the injected characteristic signal, the estimation accuracy and calculation complexity should be considered. It is recommended that the extracted data length be at least one period to avoid errors in the estimation result. However, it is not advisable to extend this length excessively, as this will increase the complexity of the calculation and consequently the computational speed. Therefore, the extraction length is set to be three periods of the characteristic signal.
The traditional method for identifying transmission line parameters in AC grids encounters challenges when applied to DC grids due to the operating frequency in the DC grid being 0 Hz. In this subsection, a novel method for identifying frequency-dependent parameters of flexible DC lines is proposed, which considers the distributed capacitance of the transmission line. This method effectively avoids the interference caused by high-frequency signals, enabling accurate identification of frequency-dependent parameters.
The mathematical model of the transmission line with distributed parameters can be expressed as:
(6) |
(7) |
(8) |
where U(0,s) and I(0,s) are the voltage and current at the beginning of a transmission line, respectively; U(l,s) and I(l,s) are the voltage and current at the end of a transmission line, respectively; l is the length of a transmission line; Zc(s) is the characteristic impedance; is the propagation function; and R0, L0, G0, and C0 are the resistance, inductance, conductance, and capacitance per unit length, respectively.
(9) |
According to (6) and (9), we can obtain:
(10) |
(11) |
(12) |
By analyzing (10)-(12), the characteristic impedance, propagation function, and attenuation function at a certain frequency can be calculated as:
(13) |
(14) |
(15) |
Since the transmission line and the earth will produce the high-frequency skin effect under the influence of alternating electromagnetic fields, the distribution parameters of the transmission line will change with frequency. As a result, the characteristic impedance and propagation constant become frequency-dependent parameters.
It is a network synthesis problem to chose a linear network that has the frequency characteristics of Zc(s). Zc(s) can be expressed by the RLC element in the form of the sum of partial fractions to express the characteristic impedance. In the complex frequency domain, a characteristic impedance function can be expressed as:
(16) |
where zj and pj are the zeros and the poles of the characteristic impedance, respectively; j is the number of the zeros and the poles; and is the value of the characteristic impedance when the frequency is infinite.
(17) |
where kj is the residue; and is a constant. The constant k0 directly corresponds to a fixed resistance Rs0, while the remaining terms represent complex frequency-domain functions of the parallel resistance-capacitance loop. Then, the circuit can be expressed as:
(18) |
where Rsi and Csi () are the resistors and capacitors of each item in the component fraction, respectively.
(19) |
The characteristic impedance can be represented as an equivalent network by using resistors and capacitors that can satisfy the boundary conditions of the characteristic impedance:
1) At low frequencies, the capacitor branch is approximately open-circuit, and the characteristic impedance is represented by a series of resistors.
2) At high frequencies, the capacitor branch is approximately short-circuit, and the characteristic impedance is Rs0.
(20) |
From the above analysis, the characteristic impedance value presents a monotonically decreasing trend with the increase in frequency. Therefore, the characteristic impedance at different frequencies is calculated by injecting characteristic signals of different frequencies several times, and the characteristic impedance in the full frequency domain is fitted by the linear interpolation method. The analytical method for the propagation constant is the same as that for the characteristic impedance.
Through the calculation in the previous subsection, we can obtain the amplitude-frequency characteristic curve of characteristic impedance and attenuation function. For unknown coefficients, (16) is a nonlinear equation, which makes the problem more complex. The vector fitting method is considered an effective technique for fitting data. By locating poles to make them become known, (17) is linearized, and then the unknown quantity can be easily calculated by the least squares method.
The approximate formula for the rational function is:
(21) |
where ci and ai are the residues and the poles of the rational function, respectively; and d and h are the parameters. The poles and residues in the context of the vector fitting method are either real or complex conjugates, while the parameters d and h are of real values.
In (21), a set of initial poles is specified, and the function is multiplied by an unknown function .
(22) |
where is the residue of the function ; and , , d, and h are unknown.
In (22), the rational approximation for shares the same poles as the approximation for . It is worth noting that the uncertainty in the solution for has been eliminated due to the approximation of being approximately equal to 1 at high frequencies. By multiplying the second row in (22) with , we obtain the relationship as:
(23) |
(24) |
When (24) is rewritten for multiple frequency points, it leads to a linear problem:
(25) |
(26) |
(27) |
(28) |
The DSEBP utilizes the concept of model matching to identify faults through the comparison between measurements and estimated states [
The relationship among the forward wave F(s), the backward wave B(s), the voltage measurements V(s), and the current measurements I(s) can be expressed as:
(29) |
(30) |
(31) |
where , which represent the names of both ends of the transmission line.
From (31), we can observe that the accurate characteristic impedance and attenuation function are the key to the subsequent calculations. The characteristic impedance amplitude exhibits a rapid decrease in the low-frequency range, followed by a relatively stable behavior in the high-frequency range. While the attenuation function varies very little in the frequency range considered in this study. Therefore, by making A(s) approximately equal to 1, (31) will change as:
(32) |
To convert (32) from the frequency domain to the time domain, the recursive convolution theorem is employed. The improved dynamic model of the DC line in the time domain can be obtained as:
(33) |
where g1, c1, and q1 are the constants that can be calculated by recursive convolution theorem; and and are the intermediate state variables.
Combined with the improved DC line model, the system measurement equation containing the measurements, state variables, and historical values is established.
The measurements can be divided into two categories: actual measurements and virtual measurements. Actual measurements include the voltage and current values , , , and . Virtual measurements represent a physical law that must be satisfied. Note that the value of the virtual measurement (zero) is known with certainty.
The state variables include the voltage state variables and , current state variable , and intermediate state variables and . The historical values include the voltage and current values before time , , , and . The system measurement equations can be established as:
(34) |
where z(t) is the column vector of measurements; x(t) is the column vector of state variables; is the function related to the state variables; er(t) is the column vector of measurement error; H is a Jacobian matrix; and C(t) is the column vector of historical values.
The concept of dynamic state estimation is integrated into the protection relay by determining if the measurements can match the established model. The specific implementation method is weighted least squares.
(35) |
where is a weight matrix, and is the standard deviation, and num is the number of measurements.
When line parameters are known, is a linear function and the Jacobi matrix is a constant matrix. Therefore, the column vector of the optimal estimated state variables is obtained as:
(36) |
The column vector of the estimated measurements and the sum of square of the normalized residuals can be calculated as (37) and (38). And is applied to identify whether an internal fault occurs.
(37) |
(38) |
where is the
The deviation between measurements and estimated measurements is defined as the residual r, which is expressed as:
(39) |
If there are no internal faults, the measurements match the established model and satisfy the chi-square distribution. However, during internal faults, the measurements will no longer match the established model, and will not meet the chi-square distribution. Therefore, can be employed to determine whether the line is healthy. The protection criterion of improved DSEBP is:
(40) |
(41) |
(42) |
where Tset is the reset time; is the threshold for ; Tdelay is the trip delay; is the estimated value of ; and Slope_k(t) is the ratio of the slope of and from time to time .
In this paper, is set to be 6.63. When is greater than 6.63, it is known that there is a 99% probability that an internal fault occurs on the protected line by consulting the chi-square distribution table. Nevertheless, the aforementioned probability does not indicate that there is a 1% chance that the protection will malfunction. This is because the protection criterion contains a user-defined trip delay. When remains greater than the for Tdelay, the protection relay will issue a trip signal.
Meanwhile, Slope_k indicates the ratio of the slope of estimated measurements to the slope of actual measurements. When the external fault occurs, the estimated measurements are basically consistent with the actual measurements, and Slope_k is greater than 0. But when the internal fault occurs, there is a notable discrepancy between the estimated measurements and the actual measurements, and Slope_k is less than 0.
Therefore, by combining and Slope_k, the improved DSEBP can identify the faults correctly. Considering the reliability and sensitivity, the reset time Tset is set to be 1 ms, and the trip delay Tdelay is set to be 5. The standard deviation of actual measurements and virtual measurements are assigned values of 0.02 p.u. and 0.001 p.u., respectively. The flow chart of the proposed improved DSEBP is shown in

Fig. 4 Flow chart of proposed method.
Step 1: determine whether the line parameters need to be calculated. If the protection method does not require line parameter identification, it can proceed directly to Step 3. However, if the protection method is used for the first time, the line parameters must be calculated.
Alternatively, the protection method needs to update the line parameters to improve the protection performance, and then online identification of the line parameters is also required.
The control strategy of HB-MMC is changed from the conventional control strategy to the characteristic signal injection control strategy. Afterwards, the characteristic signals are injected by controlling the addition and removal of additional submodules . And the amplitude and phase of the characteristic signals are extracted by the Prony method.
Step 2: using the amplitude and phase of the characteristic signals at different frequencies, the corresponding characteristic impedance Zc(s) and attenuation function A(s) can be calculated according to (13)-(15). Subsequently, the poles and residues of the frequency-dependent parameters can be calculated by the vector fitting method.
Step 3: the system measurement
Step 4: by combining the and in the protection criterion, the improved DSEBP can identify the faults correctly.
The ±500 kV four-terminal flexible DC transmission system shown in

Fig. 5 Four-terminal flexible DC transmission system.
The characteristic signal is injected into the flexible DC system through MMC1. The green squares are the location of the protection devices. The protection sampling rate is set to be 10 kHz. The main parameters of the simulation model are shown in
Model | Parameter | Value |
---|---|---|
Transmission line | Length of line I (MMC1-MMC2) | 200 km |
Length of line II (MMC1-MMC3) | 50 km | |
Length of line III (MMC2-MMC4) | 190 km | |
Length of line IV (MMC3-MMC4) | 220 km | |
HB-MMC | Number of units | 200 |
Bridge inductance | 100 mH | |
Submodule capacitance | 10 mF | |
Rated capacity | 1500 MVA |
To verify the accuracy of the parameter identification method proposed in this paper, the line parameters are calculated for line I. The parameter identification results of the line characteristic impedance amplitudes and phase angles at different frequencies are shown in
Frequency (Hz) | Accurate value (Ω) | Fitted value (Ω) | Relative error (%) |
---|---|---|---|
1 | 337.845 | 338.885 | <1 |
10 | 225.456 | 225.459 | <1 |
100 | 219.172 | 219.196 | <1 |
1000 | 216.241 | 216.245 | <1 |
4000 | 216.194 | 216.220 | <1 |
Frequency (Hz) | Accurate value (rad) | Fitted value (rad) | Relative error (%) |
---|---|---|---|
1 | -0.507100 | -0.504200 | <1 |
10 | -0.098400 | -0.098300 | <1 |
100 | -0.025800 | -0.025700 | <1 |
1000 | -0.003699 | -0.003691 | <1 |
4000 | -0.000929 | -0.000933 | <1 |
It can be observed that the parameter identification method proposed in this paper exhibits excellent accuracy in the calculation of the line characteristic impedance amplitudes and phase angles across a wide range of frequencies, from low to high. The relative error can be controlled to within 1% for both the calculation of line characteristic impedance amplitudes and phase angles.
As shown in Tables
From (7), it can be observed that the line characteristic impedance is only related to the line parameters. Therefore, the length of the transmission line does not affect the line characteristic impedance magnitudes and phase angles.
In order to verify the accuracy of the aforementioned analysis, the line parameters are identified with the length of line I set to be 1000 km. The parameter identification results of the line characteristic impedance amplitudes and phase angles at different frequencies are shown in Tables
Frequency (Hz) | Accurate value (Ω) | Fitted value (Ω) | Relative error (%) |
---|---|---|---|
1 | 337.845 | 338.847 | <1 |
10 | 225.456 | 225.416 | <1 |
100 | 219.172 | 219.224 | <1 |
1000 | 216.241 | 216.219 | <1 |
4000 | 216.194 | 216.135 | <1 |
Frequency (Hz) | Accurate value (rad) | Fitted value (rad) | Relative error (%) |
---|---|---|---|
1 | -0.507100 | -0.504600 | <1 |
10 | -0.098400 | -0.098100 | <1 |
100 | -0.025800 | -0.025700 | <1 |
1000 | -0.003699 | -0.003695 | <1 |
4000 | -0.000929 | -0.000932 | <1 |
As evidenced by Tables
The identification results of poles and residues of the line characteristic impedance are shown in
Category | Accurate value | Fitted value |
---|---|---|
Poles | -1.16143 | -1.04139 |
-6.11132 | -4.91769 | |
-619.66040 | -301.97479 | |
Residues | 693.90621 | 592.66130 |
688.34288 | 728.90311 | |
2849.97961 | 2289.83266 | |
Constant | 216.58498 | 216.49197 |
The fitted results of the line characteristic impedance amplitudes and phase angles are shown in

Fig. 6 Fitted results of line characteristic impedance amplitudes and phase angles. (a) Amplitudes. (b) Phase angles.
The simulation results under the normal operation condition of the power system with the length of transmission line of 200 km, as shown in

Fig. 7 Simulation results under normal operation condition of power system with length of transmission line of 200 km. (a) Voltage curves. (b) Current curves.
In Section IV, it is assumed that the propagation function is equal to 1 to reduce the computational complexity. From
From (8) and (15), it can be observed that the attenuation function is affected by the line parameters and the length of transmission line. At the same frequency, the attenuation function of a long transmission line is less than that of a short transmission line. Therefore, this simplification applied to long transmission lines may introduce more modeling errors.
The simulation results under normal operation condition of power system with the length of transmission line of 1000 km are shown in

Fig. 8 Simulation results under normal operation condition of power system with length of transmission line of 1000 km. (a) Voltage curves. (b) Current curves.
As illustrated in
According to
The simulation results of protection performance under different fault locations and types are shown in

Fig. 9 Simulation results of protection performance under different fault locations and types. (a) f2, P-PTG. (b) f2, PTP. (c) f4, PTP.
As shown in
As shown in
The performance of the improved DSEBP under different fault locations and types is shown in
Fault location | Fault type | Behavior of protection | |
---|---|---|---|
k side | m side | ||
f1 | P-PTG | Trip at 2.1 ms | Trip at 2.7 ms |
N-PTG | Trip at 2.2 ms | Trip at 2.7 ms | |
PTP | Trip at 2.4 ms | Trip at 1.6 ms | |
f2 | P-PTG | Trip at 2.4 ms | Trip at 2.4 ms |
N-PTG | Trip at 2.3 ms | Trip at 2.4 ms | |
PTP | Trip at 2.0 ms | Trip at 2.0 ms | |
f3 | P-PTG | Trip at 2.6 ms | Trip at 2.0 ms |
N-PTG | Trip at 2.7 ms | Trip at 2.1 ms | |
PTP | Trip at 1.6 ms | Trip at 2.3 ms | |
f4 | P-PTG | ||
N-PTG | |||
PTP |
To test the ability of the improved DSEBP to withstand fault resistance, an internal P-PTG fault (f2) with a fault resistance of 300 Ω is set and the simulation results are shown in

Fig. 10 Simulation results of protection performance under internal P-PTG fault (f2) with fault resistance of 300 Ω.
It shows that there are significant differences between the estimated and the measured current curves during the internal fault.
According to the protection criterion of the improved DSEBP and fault phase selection criterion, exceeds , and Slope_k is -8.45. Therefore, the improved DSEBP can operate normally with a fault resistance of 300 .
The ability of the improved DSEBP to withstand noise disturbance is also tested. The 25 dB noise is added to the measurement data to verify the protection performance.
As shown in
Fault location | Fault type | Slope_k | Fault identification | |
---|---|---|---|---|
f1 | P-PTG | 37.6 | -6.40 | Internal fault |
N-PTG | 58.3 | -5.50 | Internal fault | |
PTP | 85.9 | -8.10 | Internal fault | |
f2 | P-PTG | 31.4 | -11.40 | Internal fault |
N-PTG | 37.6 | -6.30 | Internal fault | |
PTP | 52.6 | -8.00 | Internal fault | |
f3 | P-PTG | 42.8 | -5.80 | Internal fault |
N-PTG | 44.1 | -7.20 | Internal fault | |
PTP | 39.8 | -8.80 | Internal fault | |
f4 | P-PTG | 16.6 | 0.60 | External fault |
N-PTG | 24.2 | 0.90 | External fault | |
PTP | 32.8 | 0.06 | External fault |
For double-ended protection methods, the synchronization error may affect the correct fault identification. A synchronization error of 1 ms is added to the PTP fault (f4), and the simulation result is given in

Fig. 11 Simulation results of protection performance with synchronization error of 1 ms.
Due to the synchronization error, there is a certain deviation between the estimated and measured current curves during the external fault. However, the trend and amplitude of the current curve remain consistent. Meanwhile, Slope_k is 0.02, which does not satisfy the protection criterion. Therefore, the simulation results verify that the improved DSEBP can withstand the effect of the synchronization error.
The trip signal is activated only when the slopes of the measured and estimated currents are opposite. It is important to determine if the slopes of the measured and estimated currents are always opposite when an internal fault occurs.
From (34), it can be observed that the system measurement equation in the time domain is more complex and it contains numerous state variables. Therefore, we qualitatively analyze this problem from the frequency domain based on (32). The current on the k side of the line in (32) can be expressed as:
(41) |
When the fault occurs at f1, the voltage value at the k side is almost zero and the current value at the k side is:
(42) |
The current value before the fault is , where is the current value at the m side of the line before the fault occurs. If is required, must be less than . As shown in (42), Vm(s) should be as large as possible and Im(s) should be as small as possible. If the length of the transmission line increases, Vm(s) after the fault will become larger. If the fault resistance increases, Im(s) after the fault will become smaller.
Therefore, the length of line I is set to be 1000 km, and the following two tests are used to verify the performance of the proposed improved DSEBP.
Test 1: the internal PTG fault occurs at f1, and the simulation results are shown in

Fig. 12 Simulation results of protection performance in test 1.
According to the protection criterion of the improved DSEBP, exceeds , and Slope_k is -4.6. Therefore, the improved DSEBP correctly sends out the trip signal.
Test 2: the PTG fault with 300 fault resistance occurs at f1, and the simulation results are shown in

Fig. 13 Simulation results of protection performance in test 2.
According to the protection criterion of the improved DSEBP, exceeds , and Slope_k is -2.4. Therefore, the improved DSEBP correctly sends out the trip signal.
When the fault occurs at f3, the voltage value at the m side is almost zero and the current value at the m side is:
(43) |
The value of the current before the fault is . It is evident that the slopes of and are opposite. Therefore, the slopes between the measured current Ik(t) and the estimated current are opposite.
In summary, the qualitative analysis and simulation results show that the slopes between the measured current Ik(t) and the estimated current are opposite after the internal fault occurs. The proposed improved DSEBP can be applied to transmission lines with a length of 200-1000 km, which can meet the requirements of the majority of practical projects.
The operating frequency of the DC system during steady-state operation is 0 Hz. Obtaining accurate transmission line parameters utilized by the protection is a challenging endeavor. Therefore, a method to identify the frequency-dependent parameters of flexible DC grids based on the HB-MMC is proposed. And the improved DSEBP can operate normally based on the identified parameters. The conclusions are as follows.
1) The frequency-dependent parameter can be identified with high accuracy by injecting characteristic signals into the HB-MMC. In the context of protection methods that rely on transmission line parameters, the practicality is enhanced.
2) The available residues and poles of the frequency-dependent parameter can be obtained by the vector fitting method, which can be used in the improved DSEBP to effectively identify faults.
3) The improved DSEBP can operate within 3 ms during an internal fault. And it can operate normally under a fault resistance of 300 , exhibiting a high degree of sensitivity. In addition, it has high reliability to withstand the 25 dB noise disturbance and 1 ms synchronization error.
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