New energy transmission line protection method based on current characteristic comprehensive correction

Through the comprehensive correction method based on current characteristics, the problem of inaccurate traditional relay protection when the new energy power generation field is faulted is solved, and the accurate judgment of fault direction, phase and positioning is achieved, adapting to the unique fault characteristics of the new energy power generation field is improved. The sensitivity and accuracy of the protection device.

CN119253549APending Publication Date: 2025-01-03CHINA UNIV OF MINING & TECH
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Patent Information

Application Number
CN202411439641.7
Authority / Receiving Office
CN · China
Patent Type
Applications(China)
Current Assignee / Owner
Filing Date
2024-10-15
Publication Date
2025-01-03

AI Technical Summary

Technical Problem

The fault characteristics of the new energy power generation field are different from those of the traditional electric field, which leads to inaccurate operation of traditional relay protection devices during failure and difficulty in positioning, especially due to current frequency offset, unequal positive and negative sequence impedances and distributed capacitance. Reduced protection sensitivity.

Method used

Using a new method based on comprehensive correction of current characteristics, by collecting voltage and current information in real time, using current peak start protection, performing current signal smoothing processing and complex frequency domain feature extraction, adaptive frequency domain error correction, and constructing fault direction criteria and selection Phase criterion, combined with reverse weight correction points and dynamic mapping transformation, achieve accurate judgment of fault direction, phase and positioning.

Benefits of technology

Improve the accuracy of fault direction determination, fault phase determination and fault positioning of new energy output line, adapt to the unique fault characteristics of new energy power generation fields, reduce the influence of unequal positive and negative sequence impedances and distributed capacitance, and ensure the fault Accurate identification and positioning.

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Abstract

The invention discloses a new energy outgoing line protection method based on current characteristic comprehensive correction. The method mainly comprises the following steps: firstly, judging whether a system fails or not by using a current peak value; when a fault occurs, the amplitude and phase of fault current phasors at the two ends of the outgoing line are calculated, whether the obtained phasors are compensated or not is selected according to a low-voltage ride-through criterion, and then the fault direction is judged according to the amplitude and phase construction direction criterion of the current phasors; when the fault is in the outgoing line direction, the fault type and the fault phase are determined by correcting a current break variable to construct a phase selection factor; and finally, calculating a fault positioning criterion through mapping transformation to realize accurate fault positioning. According to the new energy sending-out wire protection method based on current characteristic comprehensive correction, fault direction misjudgment caused by fault current frequency deviation can be overcome, fault phase selection failure caused by unequal positive and negative sequence impedance of new energy is eliminated, and fault positioning errors caused by sending-out wire distributed capacitance are weakened.
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Description

Technical Field

[0001] The present invention relates to the technical field of relay protection of power systems, and in particular to a new method for protecting a new energy transmission line based on comprehensive correction of current characteristics. Background Art

[0002] As clean energy becomes an important direction for future energy development, new energy power plants have become a research hotspot due to their mature technology and high conversion rate. The proportion of domestic new energy power plants is increasing. Therefore, it is of great significance to conduct research on the impact of new energy grid connection on the power system.

[0003] However, the transient characteristics of faults in renewable energy power plants are very different from those in traditional power plants. When the voltage drops to a greater extent, a crowbar circuit will be used to achieve low voltage ride-through and reduce the impact of the voltage drop on the system. At this time, the frequency of the short-circuit current will shift. The waveform of the partial fault current on the renewable energy power plant side is as follows: Figure 3 As shown in the figure, the current frequency offset range is 35-65Hz, and the vector extraction based on the power frequency quantity is no longer accurate. At this stage, the traditional relay protection device based on the power frequency vector cannot operate accurately and stably. At the same time, because the positive and negative sequence impedances of the new energy field are quite different, the positive and negative sequence impedances cannot be approximately equal like the traditional electric field, and the positive and negative sequence impedances are much larger than the zero sequence impedance. Therefore, when a ground fault occurs, the proportion of zero sequence current is very large, and the proportion of positive and negative sequence components is very small. The traditional fault phase selection method is no longer applicable. Moreover, with the increase in the distance of high-voltage transmission lines, distributed capacitance can no longer be ignored, and its existence has a huge impact on traditional fault location technology. The unique fault characteristics of the new energy field reduce the sensitivity of traditional protection, and the directional elements, distance elements, and phase selection elements are all seriously affected.

[0004] The unique fault characteristics of renewable energy power plants have many impacts on traditional protection and fault location technologies. However, the relay protection configuration of high-voltage transmission lines in many renewable energy power plants currently does not take into account the unique fault characteristics of their grid-connected systems and still uses traditional relay protection schemes, resulting in frequent accidents. Therefore, it is very necessary to build a new protection method to maintain the safe and stable operation of the power grid. Summary of the invention

[0005] In view of the problem that the existing relay protection is difficult to operate correctly and the fault location is inaccurate when the new energy electric field transmission line fails, the present invention provides a new method for protecting the new energy transmission line based on comprehensive correction of current characteristics. It can eliminate the influence of fault current frequency deviation, unequal positive and negative sequence impedance, and large distributed capacitance of the line on the protection of the new energy transmission line.

[0006] To achieve the above object, the technical solution of the present invention is:

[0007] S1: Real-time collection of voltage and current information at both ends of the transmission line;

[0008] S2: Use the current peak as the protection start-up value. When the current peak exceeds the threshold, the outgoing line protection is started and the fault time is recorded.

[0009] S3: Smoothing the current signal within one power frequency cycle after the fault occurs at both ends of the transmission line;

[0010] S4: Calculate the amplitude and phase of the fault current phasor at both ends of the transmission line using the complex frequency domain feature extraction equation;

[0011] S5: Determine the low voltage ride-through strategy. If the crowbar circuit is used for mortgage ride-through, proceed to step S6. If the crowbar circuit is not used, skip step S6 and proceed to step S7.

[0012] S6: Correcting the current phasor amplitude and phase obtained in step S4 using an adaptive frequency domain error correction strategy;

[0013] S7: construct a direction criterion using the current phasor obtained in the previous step and determine the fault direction;

[0014] S8: When the fault direction is the outgoing line, the phase selection criterion is constructed by using the correction current mutation amount and the phase-to-phase correction current mutation amount to determine the fault type and the fault phase;

[0015] S9: define a reverse weight correction point, and construct a weight matrix based on the reverse weight correction point to reduce the influence of the reverse weight correction point and its surrounding sampling points on the fault criterion calculation;

[0016] S10: Mapping and transforming the sampled current and sampled voltage at the protection M, and using the mapped voltage and mapped current to calculate the fault criterion and locate the fault.

[0017] Compared with the prior art, the technical solution of the present invention has the following beneficial technical effects:

[0018] The present invention improves the fault direction discrimination, fault phase discrimination and fault location of the new energy transmission line protection to adapt to the fault characteristics of the new energy transmission line. In terms of fault direction discrimination, it is considered whether the voltage drop degree meets the input conditions of the crowbar circuit. When the crowbar circuit is put into use, the amplitude and phase of the obtained fault current phasor are corrected by the adaptive frequency domain error correction strategy to ensure the correct judgment of the fault direction; in terms of fault phase selection, the corrected current mutation amount is obtained through the dynamic mutation amount compensation mechanism, and the corrected current mutation amount and the inter-phase corrected current mutation amount are used to construct a new fault phase selection criterion, so as to determine the fault type and fault phase, and eliminate the fault phase misselection caused by unequal positive and negative impedances; in terms of fault location, the reverse weight correction point and the weight matrix are defined to reduce the weights of some sampling points, and the line distributed capacitance is weakened in combination with the dynamic mapping transformation, so as to achieve more accurate fault location. BRIEF DESCRIPTION OF THE DRAWINGS

[0019] Figure 1 It is a flow chart of the implementation of the present invention;

[0020] Figure 2 This is the model diagram of the new energy transmission line;

[0021] Figure 3 It is the fault current waveform on the new energy electric field side; DETAILED DESCRIPTION

[0022] Step S1: Collect the voltage and current information at both ends of the transmission line in real time, and record the kth sampled current as i k , the kth sampling voltage is u k .

[0023] Step S2: Using the peak current i at the M end of the transmission line k·max As the starting quantity of protection, when the current peak value exceeds the set value, the outgoing line protection is started, as shown in the following formula, and the fault time is recorded.

[0024] i k·max >i M·set

[0025] In the formula, i M·set It is the setting value for the M-measurement protection startup.

[0026] Step S3: Smoothing the current signal within one power frequency cycle after the fault occurs at both ends of the transmission line, as shown in the following formula:

[0027]

[0028] In the formula, β is the smoothing coefficient, which is 0.8; are the first-order, second-order, and third-order smoothed values ​​of the current sampling point (the first sampling point is not smoothed, and the smoothed values ​​are replaced by themselves); are the first-order, second-order, and third-order smoothing values ​​of the previous sampling point.

[0029] Step S4: Calculate the amplitude and phase of the fault current phasor at both ends of the transmission line using the complex frequency domain feature extraction equation; the method is shown in the following formula:

[0030]

[0031] Where A1 is the amplitude of the fault current phasor; is the phase of the fault current phasor; f s is the sampling frequency; N is the number of sampling points in one power frequency cycle.

[0032] Step S5: Use the maximum value of the sampled voltage to determine the low voltage ride-through strategy, and the method is as shown in the following formula. If the following formula is satisfied, proceed to step S6, and if the following formula is not satisfied, skip step S6 and proceed to step S7.

[0033] u k·max <u crowbar

[0034] In the formula, u crowbar The voltage threshold of the crowbar circuit; u k·max It is the maximum value of the three-phase voltage sampling points within one power frequency cycle after the fault moment.

[0035] Step S6: Correct the current phasor amplitude and phase obtained in step S4 by using an adaptive frequency domain error correction strategy, as shown in the following formula:

[0036]

[0037] Where T is the sampling period (one power frequency period); τ is the decay time constant; s is the generator slip rate in the new energy field; ω s is the angular velocity of the power frequency of the generator in the new energy field.

[0038] Step S7: Fault direction judgment based on frequency domain error correction, the judgment formula is as shown in the following formula. If the following formula is satisfied, the fault occurs in the transmission line, and step S8 is executed to judge the fault type and determine the fault phase; if the fault occurs outside the transmission line area, return to step S1.

[0039]

[0040] In the formula, is the phase of the fault current phasor at the M and N terminals obtained in step 6 or step 4; A M , AN It is the amplitude of the fault current phasor at the M and N terminals obtained in step 6 or step 4.

[0041] Step S8: When it is determined that there is a fault in the outgoing line, the fault type and fault phase are determined according to the fault characteristics.

[0042] Taking phase A as the reference, the relationship between the current mutation at the protection installation location and the sequence current at the fault occurrence location is shown in the following formula:

[0043]

[0044] In the formula, Δi mA , Δi mB , Δi mC is the phase current mutation at the M side protection installation location; i f1 、i f2 、i f0 are the positive, negative and zero sequence currents of phase A where the fault occurs; C1, C2 and C0 are the positive, negative and zero sequence current distribution coefficients at the protection installation location; α is the rotation factor.

[0045] The current mutation in the above formula is compensated by the dynamic mutation compensation mechanism based on the current distribution coefficient shown in the following formula to obtain the corrected current mutation of each phase.

[0046]

[0047] In the formula, i mA2 、i mA0 is the negative sequence current and zero sequence current of phase A at the M terminal protection installation; Δi mA·comp , Δi mB·comp , Δi mC·comp is the correction current mutation amount; M2 and M0 are the negative sequence compensation factor and the zero sequence compensation factor, where

[0048] According to the ratio of the correction current mutation of each phase to the correction current mutation between the other two phases, the phase selection factor shown in the following formula is constructed: K A&BC is the phase selection factor of phase A, K B&CA is the phase selection factor of phase B, K C&AB is the phase selection factor of phase C.

[0049]

[0050] In the formula, It is the correction current mutation of one phase among the three phases; is the inter-phase correction current mutation amount of the other two phases (the inter-phase correction current mutation amount can be obtained by subtracting the correction current mutation amounts of the two phases).

[0051] When the following equation is satisfied, it is determined to be a single-phase fault. At this time, the phase selection factor of the faulty phase is the largest, and the phase selection factors of the non-faulty phases are equal and approximately equal to zero.

[0052]

[0053] When the following equation is satisfied, it is determined to be a two-phase short circuit fault. At this time, the phase selection factor of the non-fault phase is approximately zero, and the phase selection factors of the two fault phases are equal.

[0054]

[0055] If the following equations are satisfied at the same time, it is judged as a two-phase ground short circuit fault.

[0056] I0>I 0·zd

[0057] Where, I0 is the zero-sequence current detection value of the fault phase; I 0·zd It is the zero-sequence current setting value.

[0058] When none of the above equations are satisfied, it is judged as a three-phase short circuit fault.

[0059] Step S9: define a reverse weight correction point, and construct a weight matrix based on the reverse weight correction point to reduce the influence of the reverse weight correction point and its surrounding sampling points on the fault criterion calculation.

[0060] The sampling points satisfying the set Q are defined as the reverse weight correction points. The expression of the set Q is shown as follows:

[0061] Q={k|i k+2 -2i k+1 +i k =0, k=1,2,3,...,N}

[0062] Where N is the number of sampling points in one power frequency cycle.

[0063] When the sampling point belongs to the set Q, the weight matrix shown in the following formula is constructed to reduce the weight of the reverse weight correction point and its nearby points:

[0064]

[0065] Where C is the reverse weight correction coefficient (can be 1000).

[0066] Step S10: mapping and transforming the current and voltage at the protection M, and calculating the fault criterion through the mapped voltage and the mapped current.

[0067] For voltage u k and current i k Perform mapping transformation to obtain the mapping voltage u x·k and the mapped current i x·k, the formula is as follows:

[0068]

[0069] The specific expressions of A(x), B(x), C(x), and D(x) are as follows:

[0070]

[0071] In the formula, γ is γ = x / v, where x is the distance from the fault point to M, and v is the current wave speed; z c For z c =l1 / c1, where l1 is the inductance per unit distance, c1 is the capacitance per unit distance, and r1 is the resistance per unit distance.

[0072] Using the mapping voltage matrix U x and the mapping current matrix I x The impedance criterion is calculated as shown in the following formula, through which the criterion matrix β can be obtained, and the distance x from the fault point to the protection M can be obtained, thereby determining the fault location.

[0073]

[0074] Where U x ,I x is the mapping voltage matrix and the mapping current matrix; R F is the transition resistance; i' x·k is the derivative of sampling point k, (The first and last sampling points use their own values ​​instead of derivative values).

[0075] The following is a specific implementation case of the present invention:

[0076] Build a simulation model of the new energy transmission line on the MATLAB / SIMULINK platform, such as Figure 2 The sampling frequency of the simulation is 4kHz, where the N side is the system side, the M side is the new energy electric field side, the busbar M is the new energy field outlet busbar after the collector line converges, the length of the transmission line is 200km, the line unit length impedance is 0.08+j0.406(Ω / km), the main transformer ratio is 110kV / 35kV, and the system side voltage is 110kv.

[0077] (1) When a ground fault occurs on phase A at 80 km from the M side of the transmission line, the M side protection detects that the current peak exceeds the threshold and the protection is activated. Step S5 detects that the crowbar circuit is not activated. Enter step S6, the current phasor satisfies the judgment criterion The fault occurs in the outgoing line. The phase selection factor in step S8 satisfies K A&BC =0.784>KB&CA =K C&AB =0.071, it is determined to be a single-phase grounding fault, and the fault phase is phase A. Step S10 uses the voltage and current of the fault phase to calculate the impedance value from the M side to the fault point, which is 6.327+j32.447Ω, approximately 40% of the total impedance of the transmission line. The fault is located at 80 km from the M side, and the fault location is accurate.

[0078] (2) When a two-phase short circuit occurs in the transmission line 140 km away from the M side, the M side protection detects that the current peak exceeds the threshold and the protection is activated. Step S5 detects that the crowbar circuit is activated. Step S6 corrects the amplitude and phase of the current phasor. In step S7, the corrected current phasor satisfies the criterion The fault occurs in the outgoing line. The phase selection factor in step S8 satisfies K B&CA =K C&AB =0.763>0.1≥K A&BC =0.061, it is determined that there is a two-phase short circuit fault, and the fault phase is BC phase. Step S10 uses the voltage and current of the fault phase to calculate the impedance value from the M side to the fault point, which is 10.841+j52.658Ω, approximately 70% of the total impedance of the transmission line. The fault location is 140km away from the M side, and the fault location is accurate.

[0079] (3) When a three-phase short circuit occurs at 180 km from the M side of the transmission line, the M side protection detects that the current peak exceeds the threshold and the protection is activated. Step S5 detects the crowbar circuit connection. Step S6 corrects the amplitude and phase of the current phasor. In step S7, the corrected current phasor satisfies the criterion The fault occurs in the outgoing line. The phase selection factor in step S8 satisfies K B&CA =K C&AB =K A&BC , it is determined to be a three-phase short circuit fault. Step S10 uses the voltage and current of the fault phase to calculate the impedance value from the M side to the fault point, which is 13.164+j71.38Ω, approximately 90% of the total impedance of the transmission line. The fault is located 180km away from the M side, and the fault location is accurate.

[0080] (4) When the collector line on the right side of point M fails, the protection on the M side detects that the current peak exceeds the threshold and the protection is activated. Step S4 detects that the crowbar circuit is connected. Step S5 corrects the amplitude and phase of the current phasor. In step S6, the corrected current phasor does not meet the criterion The fault occurs outside the transmission line.

[0081] Multiple groups of experiments were conducted by changing the fault point and fault type. The calculated values ​​of phase selection factors, fault direction determination results, fault phase selection results, and fault location results are shown in Table 1.

[0082] Table 1 Protection and fault location of new energy transmission lines under different fault points and fault types

[0083]

[0084]

Claims

1. A new method for protecting new energy transmission lines based on comprehensive correction of current characteristics, characterized in that The steps include: S1: Real-time collection of voltage and current information at both ends of the transmission line; S2: Use the current peak as the protection start-up value. When the current peak exceeds the threshold, the outgoing line protection is started and the fault time is recorded. S3: Smoothing the current signal within one power frequency cycle after the fault occurs at both ends of the transmission line; S4: Calculate the amplitude and phase of the fault current phasor at both ends of the transmission line using the complex frequency domain feature extraction equation; S5: Determine the low voltage ride-through strategy. If the crowbar circuit is used for mortgage ride-through, proceed to step S6. If the crowbar circuit is not used, skip step S6 and proceed to step S7. S6: Correcting the current phasor amplitude and phase obtained in step S4 using an adaptive frequency domain error correction strategy; S7: construct a direction criterion using the current phasor obtained in the previous step and determine the fault direction; S8: When the fault direction is the outgoing line, the phase selection criterion is constructed by using the correction current mutation amount and the phase-to-phase correction current mutation amount to determine the fault type and the fault phase; S9: define a reverse weight correction point, and construct a weight matrix based on the reverse weight correction point to reduce the influence of the reverse weight correction point and its surrounding sampling points on the fault criterion calculation; S10: Mapping and transforming the sampled current and sampled voltage at the protection M, and using the mapped voltage and mapped current to calculate the fault criterion and locate the fault.

2. According to claim 1, a new method for protecting new energy transmission lines based on comprehensive correction of current characteristics is characterized by: In step S4, the amplitude and phase of the fault current phasor at both ends of the transmission line are calculated: The amplitude and phase of the fault current phasor at both ends of the transmission line are calculated using the complex frequency domain feature extraction equation; the method is shown in the following formula: Where A1 is the amplitude of the fault current phasor; is the phase of the fault current phasor; f s is the sampling frequency; N is the number of sampling points in one power frequency cycle.

3. According to claim 1, a new method for protecting new energy transmission lines based on comprehensive correction of current characteristics is characterized in that: In step S6, the current phasor obtained in step S4 is compensated: The current phasor amplitude and phase obtained in step S4 are corrected by an adaptive frequency domain error correction strategy, and the method is shown in the following formula: Where, T is the sampling period (one power frequency period); τ is the decay time constant; s is the slip rate of the generator in the new energy field; ω s is the angular velocity of the power frequency of the generator in the new energy field.

4. According to claim 1, a new method for protecting new energy transmission lines based on comprehensive correction of current characteristics is characterized in that: The step S7 is to construct a direction criterion and determine the fault direction: The fault direction judgment based on frequency domain error correction is as follows: In the formula, —The phase of the fault current phasor at the M and N terminals obtained in step 6 or step 4; A M , A N —Amplitude of the fault current phasor at the M and N terminals obtained in step 6 or step 4; If the above equation is satisfied, the fault occurs in the outgoing line, and step S8 is executed to determine the fault type and the fault phase; if the above equation is not satisfied, the fault occurs outside the outgoing line area, and the process returns to step S1.

5. According to claim 1, a new method for protecting new energy transmission lines based on comprehensive correction of current characteristics is characterized in that: In step S8, when the fault direction is the transmission line, the phase selection criterion is constructed by using the correction current mutation amount and the phase-to-phase correction current mutation amount to determine the fault type and the fault phase: When it is judged to be a transmission line fault, the fault type and fault phase are determined according to the fault characteristics; Taking phase A as the reference, the relationship between the current mutation at the protection installation location and the sequence current at the fault occurrence location is shown in the following formula: In the formula, Δi mA , Δi mB , Δi mC is the phase current mutation at the M side protection installation location; i f1 、i f2 、i f0 are the positive, negative and zero sequence currents of phase A at the fault location; C1, C2 and C0 are the positive, negative and zero sequence current distribution coefficients at the protection installation location; α is the rotation factor; The current mutation in the above formula is compensated by the dynamic mutation compensation mechanism based on the current distribution coefficient shown in the following formula to obtain the corrected current mutation of each phase; In the formula, i mA2 、i mA0 is the negative sequence current and zero sequence current of phase A at the M terminal protection installation; Δi mA·comp , Δi mB·comp , Δi mC·comp is the correction current mutation amount; M2 and M0 are the negative sequence compensation factor and the zero sequence compensation factor, where According to the ratio of the correction current mutation of each phase to the correction current mutation between the other two phases, the phase selection factor shown in the following formula is constructed: K A&BC is the phase selection factor of phase A, K B&CA is the phase selection factor of phase B, K C&AB is the phase selection factor of phase C; In the formula, It is the correction current mutation of one phase among the three phases; is the interphase correction current mutation of the other two phases, which can be obtained by subtracting the correction current mutation of the two phases; When the following equation is satisfied, it is determined to be a single-phase fault. At this time, the phase selection factor of the faulty phase is the largest, and the phase selection factors of the non-faulty phases are equal and approximately equal to zero; When the following equation is satisfied, it is determined to be a two-phase short circuit fault; at this time, the phase selection factor of the non-fault phase is approximately zero, and the phase selection factors of the two fault phases are equal; If the following equations are satisfied at the same time, it is judged as a two-phase ground short circuit fault; I0>I 0·zd Where, I0 is the zero-sequence current detection value of the fault phase; I 0·zd is the zero-sequence current setting value; When none of the above equations are satisfied, it is judged as a three-phase short circuit fault.

6. According to claim 1, a new method for protecting new energy transmission lines based on comprehensive correction of current characteristics is characterized by: The step S9: defining a reverse weight correction point, and constructing a weight matrix according to the reverse weight correction point, so as to reduce the influence of the reverse weight correction point and its surrounding sampling points on the fault criterion calculation: The sampling points satisfying the set Q are defined as the reverse weight correction points. The expression of the set Q is shown in the following formula: Q={k|i k+2 -2i k+1 +and k =0,k=1,2,3,...,N} Where N is the number of sampling points in one power frequency cycle; When the sampling point belongs to the set Q, the weight matrix shown in the following formula is constructed to reduce the weight of the reverse weight correction point and its nearby points: Where C is the reverse weight correction coefficient, which can be 1000.

7. According to claim 1, a new method for protecting new energy transmission lines based on comprehensive correction of current characteristics is characterized by: The step S10 is to map and transform the sampled current and sampled voltage at the protection M, and calculate the fault criterion by using the mapped voltage and mapped current to locate the fault: Map the current and voltage at the protection M, and calculate the fault criterion through the mapped voltage and mapped current; For voltage u k and current i k Perform mapping transformation to obtain the mapping voltage u x·k and the mapped current i x·k , the formula is as follows: The specific expressions of A(x), B(x), C(x), and D(x) are as follows: In the formula, γ is γ = x / v, where x is the distance from the fault point to M, and v is the current wave speed; z c For z c =l1 / c1, where l1 is the inductance per unit distance, c1 is the capacitance per unit distance; r1 is the resistance per unit distance; Using the mapping voltage matrix U x and the mapping current matrix I x The impedance criterion is calculated as shown in the following formula. The criterion matrix β can be obtained by the following formula to obtain the distance x from the fault point to the protection M, thereby determining the fault location; Where U x ,I x is the mapping voltage matrix and the mapping current matrix; R F is the transition resistance; i' x·k is the derivative of sampling point k, The first sampling point and the last sampling point use their own values ​​instead of derivative values.