A High-Resolution Fault Location Method for Half-Wavelength Transmission Lines

Abstract

The present invention relates to a high-resolution fault location method for half-wavelength transmission lines, comprising: calculating a fault initiation criterion based on the three-phase voltage and current at the sending end and the three-phase voltage and current at the receiving end to determine whether a fault has occurred, and recording the fault initiation time when a fault occurs; synthesizing the sending end current fault reverse wave by taking the three-phase voltage and current at the receiving end within a time window before and after the initiation time; sequentially performing short-time Fourier transform and second-order transient extraction transform processing on the sending end current fault reverse wave, obtaining the arrival time of a preset frequency band, and determining whether to complete the arrival time sequence using the Lagrange interpolation method based on whether the arrival time has a sudden change, and then performing median filtering; listing an overdetermined system of equations and finding the optimal solution using the gradient descent method. The present invention achieves accurate fault location without the need for dual-end synchronous timing, and controls the ranging error to less than 1 km using the STET algorithm, overdetermined equations, and gradient descent method, while the ranging performance is not affected by transition resistance.

Description

Technical Field

[0001] The present invention relates to the field of ultra-high voltage long-distance power transmission, and in particular to a high-resolution fault location method for half-wavelength transmission lines. Background Art

[0002] Half-wavelength AC transmission technology is a point-to-point AC transmission method without intermediate substations. Its unit construction cost is significantly lower than traditional AC transmission lines, offering significant advantages in reactive power self-balancing and large-capacity power transmission. Compared to HVDC transmission, half-wavelength AC transmission is also considered more economical and efficient. Research has shown that the construction investment for half-wavelength AC transmission can be approximately 30% lower than that for HVDC transmission, making it particularly suitable for countries with vast territories. However, there are currently no real-world engineering applications of half-wavelength AC transmission. Firstly, half-wavelength AC transmission has significantly different operating and fault characteristics from traditional AC transmission, raising technical challenges in areas such as system stability and overvoltage. Secondly, the relay protection and fault location technologies used for traditional AC transmission will fail on half-wavelength transmission lines, posing safety risks.

[0003] Against this backdrop, traveling-wave-based relay protection and fault location technologies have gradually garnered widespread attention in academia. Some researchers have employed a notch filter with a center frequency of 50 Hz and a low-pass filter with a cutoff frequency of 2 kHz to identify peak values in measured voltage and current. Based on polarity comparison, faults are classified as internal or external, effectively distinguishing between fault types. However, this method relies on traveling waves with frequencies below 2 kHz, resulting in significant interference from non-periodic components. Other researchers have proposed traveling-wave differential protection schemes and optimized them to account for dispersion on long-distance lines, effectively avoiding protection misoperation and ranging errors caused by abnormal differential currents. Still others have utilized methods such as fault-induced high-frequency signals and capacitor current compensation to locate faults. A common problem with these methods is that the measurement signals at the sending and receiving ends must be transmitted in real time and synchronized with GPS timing, resulting in significant timing errors that can significantly affect ranging accuracy. Furthermore, real-time signal transmission significantly reduces the speed of protection and the real-time performance of ranging. For example, for a 3,000-km half-wavelength transmission line, under close-in fault conditions, the transmission delay includes 1) the time it takes for the fault wave to reach the other side; 2) the time it takes for the other side's protection to identify the fault; and 3) the communication delay for the remote trip command from the other side to be transmitted via optical fiber to the local busbar. During this period, the local busbar will experience prolonged low voltage and overcurrent, which is unbearable for the UHV system. Summary of the Invention

[0004] The present invention provides a high-resolution fault location method for half-wavelength transmission lines to solve at least one of the above technical problems.

[0005] The present invention solves the above-mentioned technical problem with the following technical solution: a high-resolution fault location method for half-wavelength transmission lines, comprising:

[0006] S1, within a first preset time window before the current moment, sampling the three-phase voltages and three-phase currents at the sending end and the receiving end of the half-wavelength transmission line at a first preset sampling frequency to obtain a first three-phase voltage and current matrix at the sending end and a first three-phase voltage and current matrix at the receiving end; calculating a fault initiation criterion based on the first three-phase voltage and current matrix at the sending end and the first three-phase voltage and current matrix at the receiving end; comparing the fault initiation criterion with a preset fault initiation threshold to determine whether a fault occurs in the half-wavelength transmission line; if not, repeatedly executing S1; if so, recording the fault initiation moment at the current moment and executing S2 to S4;

[0007] S2, sampling the three-phase voltage and three-phase current at the receiving end at a second preset sampling frequency within a second preset time window before and after the fault initiation moment to obtain a second three-phase voltage and current matrix at the receiving end; and synthesizing a line mode reverse traveling wave of the sending end current based on the second three-phase voltage and current matrix at the receiving end;

[0008] S3, performing a short-time Fourier transform on the line-mode reverse traveling wave of the sending-end current to obtain a time-frequency distribution of the line-mode reverse traveling wave of the sending-end current; performing a second-order transient extraction transformation on the time-frequency distribution to obtain a first time-frequency spectrum; intercepting a time-frequency spectrum of a preset frequency band from the first time-frequency spectrum to obtain a second time-frequency spectrum; judging whether there is a sudden change in the arrival time of the second time-frequency spectrum; if so, removing the sudden change in the arrival time of the second time-frequency spectrum, and using a Lagrange interpolation method to fill the removed arrival time in the second time-frequency spectrum to obtain a third time-frequency spectrum, and performing a median filter on the third time-frequency spectrum to obtain a fourth time-frequency spectrum; if not, performing a median filter on the second time-frequency spectrum to obtain a fourth time-frequency spectrum;

[0009] S4, extracting multiple frequency components from the fourth time-frequency spectrum, and finding the time component corresponding to each frequency component; freely combining multiple frequency components and their corresponding time components to obtain multiple pairs of heterofrequency components; writing overdetermined equations according to the wave velocity and time of the multiple pairs of heterofrequency components, and using the gradient descent method to find the optimal solution of the overdetermined equation to obtain the fault distance of the half-wavelength transmission line.

[0010] The beneficial effects of the present invention are as follows: a high-resolution fault ranging method for a half-wavelength transmission line can accurately locate the fault position without requiring dual-end synchronous timing; simultaneously, a second-order transient extraction transformation algorithm with high time-frequency resolution is used to process the time-frequency distribution of a short-time Fourier transform to obtain an energy-concentrated time-frequency spectrum, which can more accurately obtain the independent arrival moments of each frequency component of the traveling wave, thereby reducing the ranging error caused by the "uncertainty principle"; and an overdetermined equation and a gradient descent method are used to estimate the fault position, thereby further reducing random errors. Generally speaking, the ranging error can be controlled to below 1 km; in addition, the ranging performance of the present invention is not affected by transition resistance. BRIEF DESCRIPTION OF THE DRAWINGS

[0011] Figure 1 This is a flow chart of a high-resolution fault location method for half-wavelength transmission lines according to the present invention;

[0012] Figure 2 Schematic diagram of a half-wavelength transmission line in an embodiment;

[0013] Figure 3 The waveform diagram of the sampling points before and after the fault in the embodiment;

[0014] Figure 4 This is a waveform diagram of the startup criterion in the embodiment;

[0015] Figure 5 : This is a waveform diagram of the reverse traveling wave of the sending-end current line mode in the embodiment;

[0016] Figure 6 Schematic diagram of the time-frequency distribution of the line-mode reverse traveling wave of the sending-end current in the embodiment;

[0017] Figure 7 is a schematic diagram of a first time-frequency spectrum in an embodiment;

[0018] Figure 8 Schematic diagram of the low frequency band of the fourth time spectrum in the embodiment;

[0019] Figure 9 FIG. 4 is a schematic diagram of a high frequency band of a fourth time spectrum in the embodiment. DETAILED DESCRIPTION

[0020] The principles and features of the present invention are described below with reference to the accompanying drawings. The examples given are only used to explain the present invention and are not used to limit the scope of the present invention.

[0021] like Figure 1 As shown, a high-resolution fault location method for half-wavelength transmission lines includes:

[0022] S1, within a first preset time window before the current moment, sampling the three-phase voltages and three-phase currents at the sending end and the receiving end of the half-wavelength transmission line at a first preset sampling frequency to obtain a first three-phase voltage and current matrix at the sending end and a first three-phase voltage and current matrix at the receiving end; calculating a fault initiation criterion based on the first three-phase voltage and current matrix at the sending end and the first three-phase voltage and current matrix at the receiving end; comparing the fault initiation criterion with a preset fault initiation threshold to determine whether a fault occurs in the half-wavelength transmission line; if not, repeatedly executing S1; if so, recording the current moment as the fault initiation moment, and executing S2 to S4;

[0023] S2, sampling the three-phase voltage and three-phase current at the receiving end at a second preset sampling frequency within a second preset time window before and after the fault initiation moment to obtain a second three-phase voltage and current matrix at the receiving end; and synthesizing a line mode reverse traveling wave of the sending end current based on the second three-phase voltage and current matrix at the receiving end;

[0024] S3, performing a short-time Fourier transform on the line-mode reverse traveling wave of the sending-end current to obtain a time-frequency distribution of the line-mode reverse traveling wave of the sending-end current; performing a second-order transient extraction transformation on the time-frequency distribution to obtain a first time-frequency spectrum; intercepting a time-frequency spectrum of a preset frequency band from the first time-frequency spectrum to obtain a second time-frequency spectrum; judging whether there is a sudden change in the arrival time of the second time-frequency spectrum; if so, removing the sudden change in the arrival time of the second time-frequency spectrum, and using a Lagrange interpolation method to fill the removed arrival time in the second time-frequency spectrum to obtain a third time-frequency spectrum, and performing a median filter on the third time-frequency spectrum to obtain a fourth time-frequency spectrum; if not, performing a median filter on the second time-frequency spectrum to obtain a fourth time-frequency spectrum;

[0025] S4, extracting multiple frequency components from the fourth time-frequency spectrum, and finding the time component corresponding to each frequency component; freely combining multiple frequency components and their corresponding time components to obtain multiple pairs of heterofrequency components; writing overdetermined equations according to the wave velocity and time of the multiple pairs of heterofrequency components, and using the gradient descent method to find the optimal solution of the overdetermined equation to obtain the fault distance of the half-wavelength transmission line.

[0026] S1 to S4 are described in detail below with reference to examples.

[0027] Figure 2 Schematic diagram of a half-wavelength transmission line, specifically a half-wavelength AC transmission line. Both ends of the line are connected to the AC system. The three-phase voltage and three-phase current are measured at the sending end M and the receiving end N of the half-wavelength transmission line. In this embodiment, a single-phase grounding fault is assumed to occur at a position 50% of the distance from the sending end M.

[0028] In the S1, the three-phase voltage and three-phase current at the sending end M and the receiving end N are monitored in real time, and the first preset time window [tt s , t], in the first preset time window with the first preset sampling frequency f l The three-phase voltage and three-phase current at the sending end M and the receiving end N are sampled to obtain the first three-phase voltage and current matrix H1 at the sending end and the first three-phase voltage and current matrix H2 at the receiving end; wherein, t s is the first preset time window length; the first three-phase voltage and current matrix H1 at the sending end and the first three-phase voltage and current matrix H2 at the receiving end are respectively expressed as:

[0029]

[0030] Among them, u A 、u B 、u C and i A 、i B 、i C Indicates that the first preset sampling frequency f is used within the first preset time window. l The three-phase voltage sequence and three-phase current sequence collected at the sending end M; u AA 、u BB 、u CC and i AA 、i BB 、i CC Indicates that the first preset sampling frequency f is used within the first preset time window. l Three-phase voltage sequence and three-phase current sequence collected at the receiving end N;

[0031] and Indicates that tn / f l The three-phase voltage and three-phase current collected at the sending end M at all times; and Indicates that tn / f l The three-phase voltage and three-phase current collected at the receiving end N at all times;

[0032] n+1 represents the first preset sampling frequency f l The number of sampling points in the first preset time window, and n=t s f l In this embodiment, the first preset sampling frequency f l Set to 500kHz, the first preset time window length t s It is set to 1 ms, so the number of sampling points n+1 in the first preset time window is 501.

[0033] In S1, a fault initiation criterion is calculated based on the first three-phase voltage and current matrix H1 at the sending end and the first three-phase voltage and current matrix H2 at the receiving end, which specifically includes the following S11 to S13:

[0034] S11, performing phase mode transformation processing on the first three-phase voltage and current matrix H1 at the sending end and the first three-phase voltage and current matrix H2 at the receiving end, respectively, to obtain the first three-phase voltage and current phase mode transformation matrix H1′ at the sending end and the first three-phase voltage and current phase mode transformation matrix H2′ at the receiving end;

[0035] Specifically, the formulas for performing phase mode transformation processing on the first three-phase voltage and current matrix H1 at the sending end and the first three-phase voltage and current matrix H2 at the receiving end are respectively:

[0036] H1′=AH1, H2′=AH2; (3)

[0037] Among them, A represents the phase transformation matrix, and its specific expression is:

[0038]

[0039] In addition, the specific expressions of the first three-phase voltage and current phase model transformation matrix H1′ at the sending end and the first three-phase voltage and current phase model transformation matrix H2′ at the receiving end are:

[0040]

[0041] Wherein, u1, u2, u0, i1, i2, i0 represent the sending-end voltage α-mode component, the sending-end voltage β-mode component, the sending-end voltage 0-mode component, the sending-end current α-mode component, the sending-end current β-mode component, and the sending-end current 0-mode component within the first preset time window, respectively;

[0042] Represents tn / f respectively l The sending end α mode voltage, sending end β mode voltage, sending end 0 mode voltage, sending end α mode current, sending end β mode current, sending end 0 mode current at the time, that is, at tn / f l The three-phase voltage and current collected at the sending end M at the moment Phase mode transformation results;

[0043] u 11 、u 22 、u 00 、i 11 、i 22 、i 00 Respectively represent the α-mode component of the receiving-end voltage, the β-mode component of the receiving-end voltage, the 0-mode component of the receiving-end voltage, the α-mode component of the receiving-end current, the β-mode component of the receiving-end current, and the 0-mode component of the receiving-end current within the first preset time window;

[0044] Represents tn / f respectively l The receiving end α mode voltage, receiving end β mode voltage, receiving end 0 mode voltage, receiving end α mode current, receiving end β mode current and receiving end 0 mode current at the moment, that is, at tn / f l The three-phase voltage and current collected at the receiving end N at the moment Phase mode transformation results.

[0045] S12, summing up all elements of the α modulus component of the sending-end current and all elements of the β modulus component of the sending-end current in the first three-phase voltage and current phase modulus transformation matrix of the sending-end to obtain the sum of the αβ modulus components of the sending-end current; summing up all elements of the α modulus component of the receiving-end current and all elements of the β modulus component of the receiving-end current in the first three-phase voltage and current phase modulus transformation matrix of the receiving-end to obtain the sum of the αβ modulus components of the receiving-end current;

[0046] Specifically, the sending-end current α modulus component and the sending-end current β modulus component are extracted from the sending-end first three-phase voltage and current phase modulus transformation matrix H1′, that is, the row vectors of the fourth and fifth rows of the sending-end first three-phase voltage and current phase modulus transformation matrix H1′ are extracted; and the receiving-end current α modulus component and the receiving-end current β modulus component are extracted from the receiving-end first three-phase voltage and current phase modulus transformation matrix H2′, that is, the row vectors of the fourth and fifth rows of the receiving-end first three-phase voltage and current phase modulus transformation matrix H2′ are extracted. The extracted sending-end current α modulus component and the sending-end current β modulus component, as well as the receiving-end current α modulus component and the receiving-end current β modulus component are combined to form the sending-end and receiving-end current αβ modulus matrix G; wherein the sending-end and receiving-end current αβ modulus matrix G is expressed as:

[0047]

[0048] Then, the αβ modulus matrix G of the transmitting and receiving end currents is transformed according to formula (8) to obtain the first processing matrix G1 of the αβ modulus matrix of the transmitting and receiving end currents:

[0049] G1=BG; (8)

[0050] Wherein, B represents the first transformation matrix. The specific representation of the first transformation matrix B and the first processing matrix G1 of the current αβ modulo the transmitting and receiving ends is:

[0051]

[0052] Next, the first processing matrix G1 of the current αβ modulo the transmitting and receiving ends is transformed according to formula (11) to obtain the second processing matrix G2 of the current αβ modulo the transmitting and receiving ends:

[0053] G2=CG1; (11)

[0054] Where C represents the second transformation matrix. The specific representation of the second transformation matrix C and the second processing matrix G2 of the current αβ modulo the transmitting and receiving ends is:

[0055]

[0056] After the above matrix transformation processing, the sum of the αβ modulus components of the sending-end current and the sum of the αβ modulus components of the receiving-end current can be obtained; among them, Δg1 and Δg2 represent the sum of the αβ modulus components of the sending-end current and the sum of the αβ modulus components of the receiving-end current, respectively.

[0057] S13, taking the square root of the difference between the sum of the αβ modulus components of the sending-end current and the sum of the αβ modulus components of the receiving-end current and dividing the square root by 2 to obtain the fault initiation criterion Δg;

[0058] Among them, the calculation formula of the fault start criterion Δg is:

[0059]

[0060] In S1, the fault start criterion is compared with a preset fault start threshold to determine whether a fault occurs in the half-wavelength transmission line, specifically:

[0061] The fault start criterion Δg is compared with the preset fault start threshold Δg set (Δg set 0.5) can be taken for comparison. If the fault start criterion Δg is greater than the preset fault start threshold Δg set , it is determined that the half-wavelength transmission line has a fault, and the current time t is recorded as the fault start time t p If the fault start criterion Δg is not greater than the preset fault start threshold Δg set , it is determined that the half-wavelength transmission line has no fault.

[0062] In some other embodiments, the first preset sampling frequency f l Set to 500kHz, the first preset time window length t s If it is set to 2μs, then under this condition, the number of sampling points n+1 in the first preset time window is 1001; under this condition, the waveform formed by collecting 1001 sampling points is as follows Figure 3 As shown; then the fault start criterion Δg is calculated to be 1.2×10 3 >Δg set , and record the current time t as the fault start time t P , it is determined that the half-wavelength transmission line is in an abnormal working state, and then the subsequent steps are carried out; wherein, the schematic diagram of the starting judgment criterion is as follows Figure 4 shown.

[0063] In S2, within a second preset time window before and after the fault initiation moment, the three-phase voltage and three-phase current at the receiving end are sampled at a second preset sampling frequency to obtain a second three-phase voltage and current matrix of the receiving end, specifically:

[0064] The fault start time t P As a benchmark, the second preset time window is intercepted as [t p -t q , t p +t q -1 / f h ], where t q is the second preset time window half length; uses the second preset sampling frequency f h The three-phase voltages and currents of the sending end and the receiving end within the second preset time window are sampled; a second three-phase voltage and current matrix M1 of the sending end and a second three-phase voltage and current matrix M2 of the receiving end are formed using the sampling points within the second preset time window; the specific representations of the second three-phase voltage and current matrix M1 of the sending end and the second three-phase voltage and current matrix M2 of the receiving end are:

[0065]

[0066]

[0067] Among them, u′ A 、u′ B 、u′ C and i A ′、i′ B , i′ C Respectively represent the second preset sampling frequency f within the second preset time window h The three-phase voltage sequence and three-phase current sequence collected at the sending end M; u′ AA 、u′ BB 、u′ CC and i A ' A , i′ BB , i′ CC Respectively represent the second preset sampling frequency f within the second preset time window h Three-phase voltage sequence and three-phase current sequence collected at the receiving end N;

[0068] and Respectively expressed in t p -k / 2f h The three-phase voltage and three-phase current collected at the sending end M at all times; and Respectively expressed in t p -k / 2f hThe three-phase voltage and three-phase current collected at the receiving end N at all times;

[0069] k represents the second preset sampling frequency f h The number of sampling points in the second preset time window, and k=2t q f h In this embodiment, the second preset sampling frequency f can be set h =1MHz (sampling interval is 1μs), set the second preset time window half length t q The second preset time window is 5 ms, so the number of sampling points in the second preset time window is 10,000.

[0070] The above is a comprehensive explanation of "sampling the three-phase voltage and three-phase current at the sending end and the receiving end at a second preset sampling frequency within a second preset time window before and after the fault start moment to obtain a second three-phase voltage and current matrix at the sending end and the receiving end." In actual application, it is only necessary to sample the three-phase voltage and three-phase current at the receiving end at a second preset sampling frequency within the second preset time window before and after the fault start moment to obtain the second three-phase voltage and current matrix at the receiving end.

[0071] In the step S2, a sending-end current line mode reverse traveling wave is synthesized according to the receiving-end second three-phase voltage and current matrix, which specifically includes the following steps S21 to S24:

[0072] S21, performing phase mode transformation processing on the second three-phase voltage and current matrix at the receiving end to obtain a second three-phase voltage and current phase mode transformation matrix at the receiving end.

[0073] Specifically, according to formula (17), the second three-phase voltage and current matrix M1 at the sending end and the second three-phase voltage and current matrix M2 at the receiving end are subjected to phase mode transformation processing, and the second three-phase voltage and current phase mode transformation matrix M1′ at the sending end and the second three-phase voltage and current phase mode transformation matrix M2′ at the receiving end are obtained accordingly:

[0074] M1′=AM1, M2′=AM2; (17)

[0075] Wherein, A represents the phase mode transformation matrix, which is consistent with formula (4).

[0076] The specific expressions of the second three-phase voltage and current phase model transformation matrix M1′ at the sending end and the second three-phase voltage and current phase model transformation matrix M2′ at the receiving end are:

[0077]

[0078] Wherein, u′1, u′2, u′0, i′1, i′2, and i′0 represent the sending-end voltage α modulus component, the sending-end voltage β modulus component, the sending-end voltage 0 modulus component, the sending-end current α modulus component, the sending-end current β modulus component, and the sending-end current 0 modulus component within the second preset time window, respectively; u′ 11 、u' 22 、u' 00 , i′ 11 、i' 22 、i' 00 Respectively represent the α-mode component of the receiving-end voltage, the β-mode component of the receiving-end voltage, the 0-mode component of the receiving-end voltage, the α-mode component of the receiving-end current, the β-mode component of the receiving-end current, and the 0-mode component of the receiving-end current within the second preset time window;

[0079] Respectively represent t p -k / 2f h The sending end α mode voltage, sending end β mode voltage, sending end 0 mode voltage, sending end α mode current, sending end β mode current, sending end 0 mode current at the time t p -k / 2f h The three-phase voltage and current collected at the sending end M at the moment Phase mode transformation results; Respectively represent t p -k / 2f h The receiving end α mode voltage, receiving end β mode voltage, receiving end 0 mode voltage, receiving end α mode current, receiving end β mode current, receiving end 0 mode current at the moment, that is, at t p -k / 2f h The three-phase voltage and current collected at the receiving end N at the moment The phase transformation result of .

[0080] S22, respectively subtract the receiving end voltage α mode component, receiving end voltage β mode component, receiving end current α mode component and receiving end current β mode component in the second half time window of the receiving end second three-phase voltage and current phase mode transformation matrix from the receiving end voltage α mode component, receiving end voltage β mode component, receiving end current α mode component and receiving end current β mode component in the first half time window, and obtain the receiving end voltage α mode fault component, receiving end voltage β mode fault component, receiving end current α mode fault component and receiving end current β mode fault component respectively.

[0081] Specifically, extract the row vectors of the first, second, fourth, and fifth rows of the second three-phase voltage and current phase mode transformation matrix M1′ at the sending end and the second three-phase voltage and current phase mode transformation matrix M2′ at the receiving end, and then subtract the elements of the first half time window from the elements of the second half time window to obtain eight new row vectors: Δu′1, Δu′2, Δu′ 11 , Δu′22 , Δi1′, Δi′2, Δi1′1, Δi′ 22 ,in:

[0082]

[0083] Among them, Δu′1 and Δi1′ represent the α-mode fault component of the sending-end voltage and the α-mode fault component of the sending-end current, Δu′2 and Δi′2 represent the β-mode fault component of the sending-end voltage and the β-mode fault component of the sending-end current, and Δu′ 11 , Δi1′1 represents the α-mode fault component of the receiving voltage and the α-mode fault component of the receiving current, Δu′ 22 , Δi′ 22 Indicates the β-mode fault component of the receiving-end voltage and the β-mode fault component of the receiving-end current.

[0084] S23, correspondingly add each element of the α-mode fault component of the receiving-end voltage and each element of the β-mode fault component of the receiving-end voltage to obtain the line-mode fault component of the receiving-end voltage; correspondingly add each element of the α-mode fault component of the receiving-end current and each element of the β-mode fault component of the receiving-end current to obtain the line-mode fault component of the receiving-end current.

[0085] Specifically, Δu′1 and Δu′2 are combined to obtain the matrix M1″; Δi1′ and Δi′2 are combined to obtain the matrix M2″; Δu′ 11 and Δu′ 22 Combine them to get the matrix M3″; replace Δi1′1 and Δi′ 22 Combining them, we get the matrix M4″, where:

[0086]

[0087] Perform matrix transformation on the four matrices M1″, M2″, M3″, and M4″ according to formula (32) to obtain the sending-end voltage line mode fault component ΔU1, the sending-end current line mode fault component ΔI1, the receiving-end voltage line mode fault component ΔU2, and the receiving-end current line mode fault component ΔI2. Then combine these four row vectors to obtain a new matrix N as shown in formula (33):

[0088]

[0089]

[0090] Wherein, D represents the third transformation matrix, and its specific expression is as follows:

[0091]

[0092] S24 , calculating the sending-end current line-mode reverse traveling wave according to the receiving-end voltage line-mode fault component and the receiving-end current line-mode fault component.

[0093] Specifically, extract the sending-end voltage line mode fault component ΔU1, the sending-end current line mode fault component ΔI1, the receiving-end voltage line mode fault component ΔU2, and the receiving-end current line mode fault component ΔI2 from the matrix N and perform matrix calculation according to formula (35) to obtain the j-th element fu in the sending-end voltage line mode reverse traveling wave TU j And the jth element fi in the sending-end current line mode reverse traveling wave TI j Finally, these elements are combined to obtain the sending-end voltage line mode reverse traveling wave TU and the sending-end current line mode reverse traveling wave TI as shown in Equations (36) and (37):

[0094]

[0095] Among them, TU represents the line mode reverse traveling wave of the sending end voltage, fu j represents the jth element in the line-mode reverse traveling wave TU of the sending-end voltage; TI represents the line-mode reverse traveling wave of the sending-end current, fi j represents the jth element in the line mode reverse traveling wave TI of the sending-end current; ΔU 1,j represents the jth element in the line mode fault component ΔU1 of the sending end voltage, ΔI 1,j represents the jth element in the line mode fault component ΔI1 of the sending-end current; ΔU 2,j represents the jth element in the line mode fault component ΔU2 of the receiving voltage, ΔI 2,j represents the jth element in the line mode fault component ΔI2 of the receiving-end current; Z represents the wave impedance.

[0096] The above S21 to S24 are a comprehensive description of the specific steps for obtaining the sending-end voltage line-mode reverse traveling wave TU and the sending-end current line-mode reverse traveling wave TI; in actual applications, it is only necessary to obtain the sending-end current line-mode reverse traveling wave TI, so it is only necessary to collect and process the data required to obtain the sending-end current line-mode reverse traveling wave TI.

[0097] In S3, a short-time Fourier transform is performed on the line-mode reverse traveling wave of the sending-end current to obtain a time-frequency distribution of the line-mode reverse traveling wave of the sending-end current, specifically:

[0098] Extract the sending-end current line mode reverse traveling wave TI, and use the sending-end current line mode reverse traveling wave TI as the target of short-time Fourier transform (STFT); the waveform of the sending-end current line mode reverse traveling wave TI is as follows Figure 5 shown.

[0099] Perform short-time Fourier transform on the sending-end current line-mode reverse traveling wave TI, that is, decompose the sending-end current line-mode reverse traveling wave TI in two dimensions of time and frequency, and obtain the time-frequency distribution tfr(t,ω) of the sending-end current line-mode reverse traveling wave TI, which is expressed as:

[0100]

[0101] Where TI(t) is the time domain expression of the line mode reverse traveling wave TI of the sending-end current. ω is the angular frequency, ω(t-τ) is the sliding window function, and e -i(t-τ) is a complex exponential function, which converts TI(t) from the time domain to the frequency domain. The time-frequency distribution of TI(t) after transformation is as follows: Figure 6 shown.

[0102] The time-frequency distribution is subjected to a second-order transient extraction transformation process to obtain a first time-frequency spectrum, specifically:

[0103] The second-order transient-extracting transform (STET) is performed on the time-frequency distribution tfr(t,ω). The second-order transient-extracting transform (STET) is mainly completed in two steps. First, the frequency expression of the current line mode reverse traveling wave is obtained according to formula (39):

[0104]

[0105] A(ω) and is the amplitude and phase of the sending-end current line mode reverse traveling wave TI in the frequency domain;

[0106] Use Dirac function to get the second-order frequency-varying model Then the second-order frequency-varying model is obtained by short-time Fourier transform. Frequency domain expression G(t,ω):

[0107]

[0108]

[0109] Where σ is the window function in the short-time Fourier transform Elements in .

[0110] The second-order group delay (2D GD) expression is obtained by G(t,ω) right Perform the transformation to obtain a new expression.

[0111]

[0112] Define a new second-order group delay estimation expression For two second-order group delay estimates The partial derivatives are calculated in the frequency and time dimensions, and then the second-order group delay estimate corrected by the second-order model is obtained according to formula (49):

[0113]

[0114] Where Re(·) is the real part function.

[0115] Calculated in Short-time Fourier transform at pass To correct the amplitude in formula (41), we can get the corrected G [2] (t,ω):

[0116]

[0117] combination and G [2] (t,ω) to obtain the second-order transient extraction transform Te [2] (t,ω).

[0118]

[0119] Where δ(t) is the Dirac function. The first time spectrum S1(t,ω) obtained after the STET algorithm transformation is as follows: Figure 7 shown.

[0120] In the step S3 , the preset frequency bands are specifically a low frequency band of 25 to 170 kHz and a high frequency band of 285 to 425 kHz.

[0121] The ridges of the 25-170kHz low-frequency band and the 285-425kHz high-frequency band in the time-spectrum S1(t,ω) are intercepted to obtain the second time-spectrum S2(t,ω). Then, the wave arrival times (WATs) of the second time-spectrum S2(t,ω) are obtained by using the different times corresponding to different frequency bands. In this embodiment, the time-spectrum S2(t,ω) consisting of 30 elements in the 25-170kHz low-frequency band and the 285-425kHz high-frequency band is intercepted for analysis. Among them:

[0122] S1(t,ω)=[tω1 tω2...tω j ...tω k tω k+1 ] 1×(k+1) ; (53)

[0123] S2(t,ω)=[tω′1 tω′2 ... tω′ i ... tω 29 ′ tω 30 ′] 1×30 ; (54)

[0124] In the formula, tω j is the jth arrival time in the first time spectrum S1(t,ω), tω i ′ is the i-th arrival time in the second time-spectrum S2(t,ω).

[0125] In S3, determining whether there is a sudden change in the arrival time of the second time-frequency spectrum specifically includes:

[0126] extracting corresponding elements from the line-mode reverse traveling wave of the sending-end current according to each arrival moment of the second time-frequency spectrum to obtain extracted elements;

[0127] Determining whether all the extracted elements are less than or equal to a preset value; if so, determining that there is no sudden change in the arrival time of the second time spectrum; if not, determining that there is a sudden change in the arrival time of the second time spectrum;

[0128] The arrival time corresponding to the extracted element in the second time-frequency spectrum that is greater than the preset value is the sudden arrival time.

[0129] Specifically, this embodiment determines whether there is a sudden change in the arrival time of the second time-frequency spectrum by determining whether there is a sudden change in the line-mode reverse traveling wave TI of the sending-end current. The formula for determining whether there is a sudden change in the line-mode reverse traveling wave TI of the sending-end current is:

[0130]

[0131] Where b i is the i-th element in the sending-end current line mode reverse traveling wave TI, which corresponds to the arrival time of the second time-frequency spectrum; b ref Indicates a preset value. In this embodiment, b can be set ref = 50. When equation (55) is satisfied, it indicates that there is no mutation in the line-mode reverse traveling wave TI of the sending-end current, and thus it can be determined that there is no mutation in the arrival time of the second time-frequency spectrum; otherwise, a mutation occurs.

[0132] In S3, the Lagrange interpolation method is used to fill in the arrival time that is eliminated in the second time-frequency spectrum, specifically:

[0133] Extract the time spectrum S2(t,ω), find the part where the mutation occurs according to formula (55), delete this part first, and then calculate the Lagrange interpolation polynomial. The Lagrange interpolation polynomial is expressed as:

[0134]

[0135] Where o is L i The independent variable in (o), o i 、o j is the horizontal coordinate of the data point, and represents the corresponding time in the reverse traveling wave TI of the sending-end current line mode; j≠i (oo j ) is (o-o0)(o-o1)…(oo k ), that is, except (oo i All except (oo j ), ∏ j≠i (o i -o j ) is (o i -o0)(o i -o1)…(o i -o k ); At the same time, in this embodiment, j≠i (o i -o j ) is set to a constant to ensure that L i (x i )=1 and L i (x j )=0(i≠j).

[0136] The missing part of the Lagrange interpolation polynomial L i (o) and c i Multiply and sum to get the Lagrange interpolation polynomial P of the missing part k (o).

[0137]

[0138] Where c i is the frequency corresponding to the line mode reverse traveling wave TI of the sending-end current.

[0139] The deleted elements are filled in by interpolation through equation (57) to obtain the third time-frequency spectrum S3(t,ω):

[0140] S3(t,ω)=[tω1″tω2″...tω e ″tω (e+1) ″...tω 29 ″tω 30 ″] 1×30 ; (58)

[0141] In the formula, tω e ″、tω (e+1) ″ is the element padded after Lagrange interpolation.

[0142] In S3, median filtering is performed on the second time-frequency spectrum or the third time-frequency spectrum, specifically:

[0143] Set a filter window of size M0, select an element tω(m) in the time spectrum S2(t,ω) or S3(t,ω), and then expand (M0-1) / 2 elements to the left and right with this element as the center to form a filter window U containing M0 elements as shown in formula (59) M0 :

[0144]

[0145] Where, is an element of the extension, and the other elements can be obtained similarly.

[0146] Take the filter window U M0 The median of the elements in the filter window is obtained by arranging these M0 elements in ascending order to obtain a new filter window U. M1 :

[0147]

[0148] Median is for U M0 The median of the M0 elements in is taken; in this embodiment, the value of M0 is 30.

[0149] According to formula (60), the fourth time-frequency spectrum S4(t,ω) is obtained, and S4(t,ω) is used to obtain a more realistic arrival time.

[0150] S4(t,ω)=[tω1(m) tω2(m)…tω i (m)…tω 29 (m) tω 30 (m)] 1×30 ; (61)

[0151] In the formula, tω i (m) is one of the results after median filtering. The schematic diagram of the fourth time spectrum S4(t,ω) after median filtering of low frequency band and high frequency band is as follows: Figure 8 、 Figure 9 shown.

[0152] In the step S4, multiple frequency components are extracted from the fourth time-frequency spectrum, and the time components corresponding to the respective frequency components are found, which specifically includes the following steps S401 to S405:

[0153] S401 : extracting a plurality of initial frequency components from the fourth time-frequency spectrum S4 (t, ω).

[0154] In this embodiment, 30 initial frequency components are extracted from the fourth time-frequency spectrum S4 (t, ω).

[0155] S402: sharpen each initial frequency component to obtain a corresponding frequency component.

[0156] Specifically, the corresponding frequency components are obtained according to formula (62);

[0157] f HLx =f hlx -f m ; (62)

[0158] Among them, f HLx represents the xth frequency component, f hlx represents the xth initial frequency component, f m represents the preset fundamental frequency, x=1, 2, ..., a, a represents the total number of initial frequency components extracted from the fourth time-frequency spectrum S4(t, ω), and also represents the total number of frequency components; in this embodiment, a=30, f m =20kHz. Each frequency component corresponds to a small frequency band.

[0159] Combine all frequency components to form a frequency component matrix M HL , the frequency component matrix M HL Expressed as:

[0160] M HL =[f HL1 f HL2 ...f HLx ...f HL(a-1) f HLa ] 1×a (63)

[0161] S403 , extracting the frequency value of each frequency component respectively to obtain a corresponding small frequency band frequency sequence.

[0162] Specifically, the small frequency band sequence corresponding to the xth frequency component is expressed as:

[0163]

[0164] in, Represents the rth frequency value in the xth frequency component, r = 1, 2, ..., e, and e represents the total number of frequency values in a small frequency band.

[0165] Combine the small frequency band frequency sequences corresponding to all frequency components to form a small frequency band frequency matrix M A , the small frequency band frequency matrix M A Expressed as:

[0166]

[0167] Among them, the small frequency band frequency matrix M A The row vector in is the corresponding small frequency band frequency sequence.

[0168] S404 , respectively obtaining the small-band time sequence corresponding to each small-band frequency sequence.

[0169] Specifically, each small frequency band has a frequency value and its corresponding time value. Formula (65) is the frequency value corresponding to each small frequency band, so the time value corresponding to each small frequency band can be expressed as:

[0170]

[0171] Among them, M a represents the small frequency band time matrix; t HLx Represents the x-th small frequency band time series, t HLx With f HLx Corresponding to; t HLxr Represents the rth time value in the xth small frequency band time series; and correspond.

[0172] S405 , extracting the maximum time value in each small frequency band time series respectively to obtain the time component corresponding to each frequency component.

[0173] Specifically, the time component corresponding to each frequency component is expressed as:

[0174]

[0175] in, is the maximum time value in the x-th small frequency band time series, indicating the time component corresponding to the x-th frequency component. That is, the maximum time value in each small frequency band time series is the time component corresponding to the corresponding frequency component.

[0176] Combine the time components corresponding to all frequency components to form a time matrix The time matrix Expressed as:

[0177]

[0178] In this embodiment, 15 frequencies are selected in the low frequency band and 15 frequencies are selected in the high frequency band. The time when each frequency reaches the measurement point (ie, the arrival time) is shown in Table 1.

[0179] Table 1 The time it takes for each frequency of low-frequency and high-frequency components to reach the measurement point

[0180]

[0181] In S4, multiple frequency components and their corresponding time components are freely combined to obtain multiple pairs of different frequency components. That is, multiple frequency components are arbitrarily combined in pairs, and their corresponding time components are also combined along with the frequency component combination. The free combination expression is as follows:

[0182]

[0183] Wherein, s represents the total logarithm of different frequency components, C(·) represents the free combination function, and a represents the total number of frequency components. In this embodiment, a=30.

[0184] In the above S4, an overdetermined equation is written based on the wave velocities of multiple pairs of different frequency components, specifically including S406 to S408:

[0185] S406: Calculate the time of each pair of hetero-frequency components according to the time components corresponding to the two frequency components in each pair of hetero-frequency components, and combine the time of all hetero-frequency components to obtain the overdetermined equation time matrix.

[0186] Specifically, any pair of heterofrequency components is composed of any two frequency components extracted above. The higher frequency component in the heterofrequency components is defined as the high-frequency component, and the lower frequency component in the heterofrequency components is defined as the low-frequency component. The time matrix T of the overdetermined equation is expressed as:

[0187] T=[t1 t2 ... t s′ ... t s-1 t s ] 1×s ; (70)

[0188] Among them, t s′ is the s′th element in the time matrix T of the overdetermined equation, which represents the time of the s′th pair of heterofrequency components. The time value corresponding to the s′th pair of heterofrequency components is the difference between the time components corresponding to the two frequency components that make up the s′th pair of heterofrequency components, that is, t s′ =t ls′ -t hs′ , t hs′ Indicates the time component corresponding to the high-frequency component in the s′th pair of heterofrequency components, t ls′ It represents the time component corresponding to the low-frequency component in the s′th pair of hetero-frequency components, t hs′ and t ls′ is an element in formula (68); s′=1,2,…,s.

[0189] S407 , calculating the wave velocity of each pair of hetero-frequency components according to the wave velocities of the two frequency components in each pair of hetero-frequency components, and combining the wave velocities of all hetero-frequency components to obtain the overdetermined equation wave velocity matrix.

[0190] Specifically, the wave velocity of the frequency component is calculated according to formula (71):

[0191]

[0192] Among them, v x Indicates the wave velocity of the xth frequency component, L x is the inductance of the half-wavelength transmission line at the xth frequency component, C x is the capacitance value of the half-wavelength transmission line at the xth frequency component.

[0193] In each pair of heterofrequency components, the wave velocity corresponding to the high-frequency component is the high-frequency wave velocity, and the wave velocity corresponding to the low-frequency component is the low-frequency wave velocity. The calculation formula for the wave velocity of the heterofrequency component is:

[0194]

[0195] Specific, vs. ′ represents the wave velocity of the s′th pair of heterofrequency components, v hs′ It represents the wave velocity of the high-frequency component in the s′th pair of heterofrequency components, that is, the high-frequency wave velocity, v ls′ It represents the wave velocity of the low-frequency component in the s′th pair of heterofrequency components, that is, the low-frequency wave velocity.

[0196] The wave velocities of all different frequency components are combined to obtain the overdetermined equation wave velocity matrix; the overdetermined equation wave velocity matrix is expressed as:

[0197] v a =[v1 v2 ... v s′ ... v s-1 v s ] 1×s ; (73)

[0198] Among them, v a represents the wave velocity matrix of the overdetermined equation.

[0199] S408: Construct an overdetermined equation for the fault distance matrix based on the overdetermined equation time matrix and the overdetermined equation wave velocity matrix. Specifically, the overdetermined equation is expressed as:

[0200] v a d=T; (74)

[0201] d=[d1 d2...d s′ ...d s-1 d s ] 1×s ; (75)

[0202] d represents the fault distance matrix to be determined, d s′represents the s′th element in the fault distance matrix.

[0203] In S4, the gradient descent method is used to find the optimal solution of the overdetermined equation, which specifically includes the following S409 to S412:

[0204] S409: Obtain a fault distance function according to the overdetermined equation; the fault distance function is expressed as:

[0205] f(d)=v a dT. (76)

[0206] S410, setting an initial point d0, performing gradient transformation on the fault distance function, and obtaining a gradient matrix of the initial point d0; wherein the formula for performing gradient transformation on the fault distance function is:

[0207]

[0208] Specifically, is a negative gradient; the gradient matrix of the initial point d0 is expressed as:

[0209] D′=[d′1 d'2 ... d′ s′ ... d′ s-1 d′ s ] 1×s ; (78)

[0210] D′ represents the gradient matrix of the initial point d0, d s ' ′ Represents the s′th element in the gradient matrix of the initial point d0.

[0211] S411, iterating the fault distance function starting from the initial point d0, and in each iteration, the fault distance function moves a preset step size in the direction of the fastest descent speed, and recording the two iteration values before and after the iteration; wherein the relationship between the two iteration values before and after the iteration is:

[0212]

[0213] Specifically, d s′ and d″ s′-1 represents the two iteration values before and after the iteration, γ represents the preset step size, and in this embodiment, γ=0.0001.

[0214] S412: Set an iteration threshold ε. When the absolute value of the difference between the two iteration values before and after the iteration is less than the iteration threshold ε, the iteration is stopped. The point corresponding to the last iteration is the fault distance of the half-wavelength transmission line. The absolute value of the difference between the two iteration values before and after the iteration is less than the iteration threshold ε is expressed as follows:

[0215] |d″ s′ -d″ s′-1 |<ε. (80)

[0216] In this embodiment, a=30, and 435 pairs of different frequency components are obtained according to formula (69). The overdetermined equation is written according to formula (74), and the final fault distance is 1500.428 km, which is 428 m different from the fault distance set in this embodiment. The fault distance of the half-wavelength AC transmission line is calculated very accurately.

[0217] The present invention discloses a high-resolution fault ranging method for half-wavelength transmission lines, which can accurately locate the fault position without requiring dual-end synchronous timing. The method also utilizes a high-time-frequency resolution second-order transient extraction transformation algorithm to process the time-frequency distribution of a short-time Fourier transform to obtain an energy-concentrated time-frequency spectrum, enabling more accurate acquisition of the independent arrival times of each frequency component of the traveling wave and reducing ranging errors caused by the "uncertainty" principle. Furthermore, the method utilizes an overdetermined equation and a gradient descent method to estimate the fault position, further reducing random errors. Overall, the ranging error can be controlled to below 1 km. Furthermore, the ranging performance of the present invention is not affected by transition resistance.

[0218] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A high-resolution fault location method for half-wavelength transmission lines, characterized in that: include: S1, within a first preset time window before the current moment, sampling the three-phase voltages and three-phase currents at a sending end and a receiving end of a half-wavelength transmission line at a first preset sampling frequency to obtain a first three-phase voltage and current matrix at the sending end and a first three-phase voltage and current matrix at the receiving end; and calculating a fault initiation criterion based on the first three-phase voltage and current matrix at the sending end and the first three-phase voltage and current matrix at the receiving end; Comparing the fault start criterion with a preset fault start threshold to determine whether a fault occurs in the half-wavelength transmission line; if not, repeatedly executing S1; if yes, recording the current moment as the fault start moment, and executing S2 to S4; S2, sampling the three-phase voltage and three-phase current at the receiving end at a second preset sampling frequency within a second preset time window before and after the fault initiation moment to obtain a second three-phase voltage and current matrix at the receiving end; synthesizing a sending-end current line mode reverse traveling wave according to the receiving-end second three-phase voltage and current matrix; S3, performing a short-time Fourier transform on the line-mode reverse traveling wave of the sending-end current to obtain a time-frequency distribution of the line-mode reverse traveling wave of the sending-end current; performing a second-order transient extraction transformation on the time-frequency distribution to obtain a first time-frequency spectrum; intercepting a time-frequency spectrum of a preset frequency band from the first time-frequency spectrum to obtain a second time-frequency spectrum; judging whether there is a sudden change in the arrival time of the second time-frequency spectrum; if so, removing the sudden change in the arrival time of the second time-frequency spectrum, and using a Lagrange interpolation method to fill the removed arrival time in the second time-frequency spectrum to obtain a third time-frequency spectrum, and performing a median filter on the third time-frequency spectrum to obtain a fourth time-frequency spectrum; if not, performing a median filter on the second time-frequency spectrum to obtain a fourth time-frequency spectrum; S4, extracting multiple frequency components from the fourth time-frequency spectrum, and finding the time component corresponding to each frequency component; freely combining multiple frequency components and their corresponding time components to obtain multiple pairs of heterofrequency components; writing overdetermined equations according to the wave velocity and time of the multiple pairs of heterofrequency components, and using the gradient descent method to find the optimal solution of the overdetermined equation to obtain the fault distance of the half-wavelength transmission line.

2. The high-resolution fault location method for half-wavelength transmission lines according to claim 1, characterized in that: In S1, a fault start criterion is calculated based on the first three-phase voltage and current matrix of the sending end and the first three-phase voltage and current matrix of the receiving end, specifically including: Performing phase mode transformation processing on the first three-phase voltage and current matrix of the sending end and the first three-phase voltage and current matrix of the receiving end respectively, to obtain the first three-phase voltage and current phase mode transformation matrix of the sending end and the first three-phase voltage and current phase mode transformation matrix of the receiving end; Accumulate all elements of the α modulus component of the sending-end current and all elements of the β modulus component of the sending-end current in the first three-phase voltage and current phase modulus transformation matrix of the sending-end to obtain the sum of the αβ modulus components of the sending-end current; accumulate all elements of the α modulus component of the receiving-end current and all elements of the β modulus component of the receiving-end current in the first three-phase voltage and current phase modulus transformation matrix of the receiving-end to obtain the sum of the αβ modulus components of the receiving-end current; The fault starting criterion is obtained by square rooting the difference between the sum of the αβ modulus components of the sending-end current and the sum of the αβ modulus components of the receiving-end current and dividing the square by 2.

3. The high-resolution fault location method for half-wavelength transmission lines according to claim 1, characterized in that: In S1, the fault start criterion is compared with a preset fault start threshold to determine whether a fault occurs in the half-wavelength transmission line, specifically: The fault start criterion is compared with a preset fault start threshold. If the fault start criterion is greater than the preset fault start threshold, it is determined that a fault has occurred in the half-wavelength transmission line. If the fault start criterion is not greater than the preset fault start threshold, it is determined that no fault has occurred in the half-wavelength transmission line.

4. The high-resolution fault location method for half-wavelength transmission lines according to claim 1, characterized in that: In the step S2, synthesizing a sending-end current line mode reverse traveling wave according to the receiving-end second three-phase voltage and current matrix specifically includes: Performing phase mode transformation processing on the second three-phase voltage and current matrix of the receiving end to obtain a second three-phase voltage and current phase mode transformation matrix of the receiving end; subtracting the receiving end voltage α mode component, receiving end voltage β mode component, receiving end current α mode component, and receiving end current β mode component in the second half time window from the receiving end voltage α mode component, receiving end voltage β mode component, receiving end current α mode component, and receiving end current β mode component in the second three-phase voltage and current phase mode transformation matrix of the receiving end from the receiving end voltage α mode component, receiving end voltage β mode component, receiving end current α mode component, and receiving end current β mode component in the first half time window, respectively, to obtain the receiving end voltage α mode fault component, receiving end voltage β mode fault component, receiving end current α mode fault component, and receiving end current β mode fault component; Adding correspondingly each element of the receiving-end voltage α-mode fault component and each element of the receiving-end voltage β-mode fault component to obtain the receiving-end voltage line mode fault component; adding correspondingly each element of the receiving-end current α-mode fault component and each element of the receiving-end current β-mode fault component to obtain the receiving-end current line mode fault component; The sending-end current line-mode reverse traveling wave is calculated according to the receiving-end voltage line-mode fault component and the receiving-end current line-mode fault component.

5. The high-resolution fault location method for half-wavelength transmission lines according to claim 4, characterized in that: The formula for calculating the line mode reverse traveling wave of the sending-end current is: Wherein, TI represents the line mode reverse traveling wave of the sending end current, fi j represents the jth element in the line mode reverse traveling wave of the sending-end current, k represents the number of sampling points in the second preset time window, ΔU 2,j represents the jth element in the line mode fault component of the receiving voltage, ΔI 2,j represents the jth element in the line mode fault component of the receiving-end current, and Z represents the wave impedance.

6. The high-resolution fault location method for half-wavelength transmission lines according to claim 1, characterized in that: In the step S3 , the preset frequency bands are specifically a low frequency band of 25 to 170 kHz and a high frequency band of 285 to 425 kHz.

7. The high-resolution fault location method for half-wavelength transmission lines according to claim 1, characterized in that: In S3, determining whether there is a sudden change in the arrival time of the second time-frequency spectrum specifically includes: extracting corresponding elements from the line-mode reverse traveling wave of the sending-end current according to each arrival moment of the second time-frequency spectrum to obtain extracted elements; Determining whether all the extracted elements are less than or equal to a preset value; if so, determining that there is no sudden change in the arrival time of the second time spectrum; if not, determining that there is a sudden change in the arrival time of the second time spectrum; The arrival time corresponding to the extracted element in the second time-frequency spectrum that is greater than the preset value is the sudden arrival time.

8. The high-resolution fault location method for half-wavelength transmission lines according to claim 1, characterized in that: In S4, extracting multiple frequency components from the fourth time-frequency spectrum and finding the time component corresponding to each frequency component specifically includes: extracting a plurality of initial frequency components from the fourth time-frequency spectrum; Sharpen each initial frequency component separately to obtain the corresponding frequency component; Extract the frequency value of each frequency component respectively to obtain the corresponding small frequency band frequency sequence; Obtain the small frequency band time series corresponding to each small frequency band frequency sequence respectively; The maximum time value in each small frequency band time series is extracted respectively to obtain the time component corresponding to each frequency component.

9. The high-resolution fault location method for half-wavelength transmission lines according to claim 1, characterized in that: In S4, an overdetermined equation is written based on the wave velocities of multiple pairs of different frequency components, specifically including: According to the time components corresponding to the two frequency components in each pair of heterofrequency components, the time of each pair of heterofrequency components is calculated, and the time of all heterofrequency components is combined to obtain the time matrix of the overdetermined equation; According to the wave velocities of the two frequency components in each pair of heterofrequency components, the wave velocity of each pair of heterofrequency components is calculated, and the wave velocities of all heterofrequency components are combined to obtain the wave velocity matrix of the overdetermined equation; An overdetermined equation about the fault distance matrix is constructed according to the overdetermined equation time matrix and the overdetermined equation wave velocity matrix; the overdetermined equation is expressed as: v a d=T; Among them, v a represents the overdetermined equation wave velocity matrix, T represents the overdetermined equation time matrix, and d represents the fault distance matrix to be determined; specifically, v a =[v1 v2 ... v s′ ... in s-1 v s ] 1×s , v s′ represents the wave velocity of the s′th pair of heterofrequency components, v hs′ represents the wave velocity of the high-frequency component in the s′th pair of heterofrequency components, v ls′ represents the wave velocity of the low-frequency component in the s′th pair of heterofrequency components, and s represents the total number of pairs of heterofrequency components; T=[t1 t2 ... t s′ ... t s-1 t s ] 1×s ,t s′ =t ls′ -t hs′ ; t s′ Indicates the time of the s′th pair of heterofrequency components, t hs′ Indicates the time component corresponding to the high-frequency component in the s′th pair of heterofrequency components, t ls′ represents the time component corresponding to the low-frequency component in the s′th pair of hetero-frequency components; d=[d1 d2...d s′ ...d s-1 d s ] 1×s ; d s′ represents the s′th element in the fault distance matrix.

10. The high-resolution fault location method for half-wavelength transmission lines according to claim 9, characterized in that: In S4, the gradient descent method is used to find the optimal solution of the overdetermined equation, which specifically includes: The fault distance function is obtained according to the overdetermined equation; wherein the fault distance function is expressed as: f(d)=v a d-T; An initial point is set, and a gradient transformation is performed on the fault distance function to obtain a gradient matrix of the initial point. The formula for gradient transformation of the fault distance function and the gradient matrix of the initial point are respectively expressed as: D=[d1' d'2 ... d′ s′ ... d s ' -1 d s '] 1×s ; Specifically, is a negative gradient, D′ represents the gradient matrix of the initial point, d′ s′ The s′th element in the gradient matrix representing the initial point; The fault distance function is iterated starting from the initial point, and in each iteration, the fault distance function moves a preset step size in the direction of the fastest descent speed, and the two iteration values before and after the iteration are recorded; wherein the relationship between the two iteration values before and after the iteration is: Specifically, d s′ and d″ s′-1 represents the two iteration values before and after the iteration, and γ represents the preset step size; An iteration threshold is set, and the iteration is stopped when the absolute value of the difference between the two iteration values before and after the iteration is less than the iteration threshold. The point corresponding to the last iteration is the fault distance of the half-wavelength transmission line.

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