Traveling wave head extraction method based on multi-scale time-frequency enhancement and low-rank sparse optimization

Abstract

The invention belongs to the field of power distribution network traveling wave head analysis, and particularly relates to a traveling wave head extraction method based on multi-scale time-frequency enhancement and low-rank sparse optimization. According to the method, starting from two aspects of time-frequency transformation precision improvement and feature enhancement denoising, high-resolution time-frequency distribution is obtained by adopting adaptive window width wide-sense S transformation, and the resolvability and robustness of a wave head mutation point are remarkably improved through multi-scale sparse-low rank joint optimization; therefore, stable and accurate traveling wave head detection can still be realized under the condition of low signal-to-noise ratio, and the fault positioning accuracy is finally improved.

Description

Technical Field

[0001] This invention belongs to the field of traveling wave front analysis in power distribution networks, specifically involving a method for extracting traveling wave fronts based on multi-scale time-frequency enhancement and low-rank sparsity optimization. Background Technology

[0002] With the increasing complexity of modern power distribution network structures and the continuous improvement of the proportion of distributed power sources and renewable energy integration, the uncertainty and frequency of power grid operation have increased significantly, and the probability of transient faults such as single-phase grounding, short circuits, and lightning strikes has risen markedly. Under these circumstances, rapid fault location and isolation have become the core link in ensuring the safe, stable, and continuous power supply of the power distribution network.

[0003] The traveling wave method (TWM) is widely considered an effective means of high-precision fault location due to its advantages such as reliance on high-frequency transient information of the initial fault state, minimal impact from system operating conditions, and high positioning accuracy. Accurate detection of the traveling wave front arrival time (ToA) is a crucial step in the TWM, as its accuracy directly determines the spatial resolution of fault location. However, in the actual operating environment of power distribution networks, traveling wave signals propagate over long distances and along complex paths, and are also subject to various interference factors such as switching operations, harmonics, and communication noise. The wavefront characteristics are often submerged in strong noise backgrounds, posing a significant challenge to accurate detection.

[0004] To extract the traveling wavefront position from noisy measurement signals, academia and engineering have proposed various time-frequency analysis methods. The core idea is to map the fault current / voltage signal from the time domain to a two-dimensional time-frequency space to enhance the observability of abrupt changes. Common methods include:

[0005] 1. Short-time Fourier Transform (STFT): Based on the Fourier transform with a fixed window length, it can provide time-frequency joint information, but the fixed window width makes it difficult to balance high-frequency resolution and low-frequency integrity; the abrupt change characteristics of high-frequency traveling waves are often smoothed, limiting the positioning accuracy.

[0006] 2. Wavelet Transform (WT) captures different frequency components through multi-scale decomposition and can better characterize transient features; it is highly sensitive to the selection of wavelet basis functions, and different mother wavelets have significantly different effects on wavefront localization; high-frequency components are easily affected by noise and lack an adaptive optimization mechanism.

[0007] 3. Dynamic Time Warping (DTW) is mainly used for nonlinear matching of signal sequences. It is sensitive to pattern similarity, but the impulse characteristics of transient wavefronts are unstable, resulting in large matching errors. It also has high computational complexity and is not suitable for large-scale real-time processing.

[0008] 4. The traditional S-transform (Stockwell Transform, ST) combines the full-frequency information of the Fourier transform with the localization characteristics of the wavelet transform; the fixed form with the window width inversely proportional to the frequency has insufficient resolution in the high-frequency region, there is redundancy in the spectrum sampling, and the anti-interference performance of low signal-to-noise ratio signals is limited.

[0009] Existing invention patent CN120336830A discloses a traveling wave front extraction method based on Short-Time Fourier Transform (STFT). This method utilizes STFT to simultaneously obtain time and frequency domain information, making the wavefront extraction process intuitive. However, due to the limitation of the fixed window width of STFT, there is a contradiction between insufficient high-frequency resolution and insufficient low-frequency time resolution, resulting in a decrease in the ability to extract weak wavefront signals in noisy environments. Invention patent CN119474839A discloses a traveling wave front extraction method based on Discrete S-Transform. In this method, Discrete S-Transform has an adaptive window width compared to STFT, improving the balance between time and frequency resolution. However, in low signal-to-noise ratio (SNR < 10dB) environments, the problems of false wavefront detection and increased location errors still occur. In 2024, Cheng Chen et al. proposed a "Rapid Fault Location Method for Urban Distribution Networks Based on Traveling Wave Theory." This method utilizes the multi-resolution characteristics of wavelet transform in the time and frequency domains to accurately identify fault traveling wave fronts and combines traveling wave theory to quickly locate faults. Wavelet transform is sensitive to abrupt signals and has high accuracy in wavefront identification. However, in high-noise environments, the modulus maxima are easily disturbed, requiring additional noise suppression steps. Summary of the Invention

[0010] To address the shortcomings of existing technologies, this invention aims to provide a traveling wave front extraction method based on multi-scale time-frequency enhancement and low-rank sparse optimization. This method addresses both the improvement of time-frequency transformation accuracy and feature enhancement and denoising. It employs an adaptive window width generalized S-transform to obtain a high-resolution time-frequency distribution and significantly improves the distinguishability and robustness of wavefront abrupt change points through multi-scale sparse-low-rank joint optimization. Thus, it can still achieve stable and accurate traveling wave front detection under low signal-to-noise ratio conditions, ultimately improving fault location accuracy.

[0011] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0012] A method for extracting traveling wavefronts based on multi-scale time-frequency enhancement and low-rank sparsity optimization is proposed. This method is implemented based on the following steps:

[0013] S1. Signal preprocessing: The original current signal is preprocessed to filter out power frequency and high frequency interference components, and the sampling rate is unified as the input for subsequent processing;

[0014] S2. Time-frequency matrix construction: The preprocessed time-domain signal is transformed by the adaptive window width generalized S-transform to obtain the energy distribution matrix with time and frequency as two-dimensional coordinates, which intuitively reflects the local characteristics of the wavefront propagation process at different frequencies.

[0015] S3. A multi-scale sparse-low-rank joint optimization time-frequency enhancement method is adopted for optimization: In order to enhance the wavefront information in the signal and suppress interference noise, multiple denoising mechanisms are used to jointly optimize the time-frequency energy map.

[0016] S4. Wavehead Recognition: On the optimized energy matrix, energy accumulation and sliding window smoothing are performed along the time dimension. The abrupt inflection point is extracted through differential operation and dynamic threshold criterion, thereby determining the moment of the first significant response of the wavehead.

[0017] Furthermore, the signal preprocessing method in step S1 specifically includes:

[0018] S1-1. Bandpass Filtering: Considering the significant power frequency fundamental component and various broadband high-frequency noises in the transient signals of the power system, a 6th-order linear phase Butterworth bandpass filter is adopted; the low-frequency cutoff point can effectively suppress DC bias and fundamental components, and the high-frequency cutoff point can filter out non-target high-frequency noise introduced by switching action and electromagnetic interference, thereby preserving the main spectral energy of the fault traveling wave; the filtering process adopts zero-phase processing to avoid the influence of phase distortion on wavefront timing identification;

[0019] S1-2. Sampling Rate Renormalization: To address the differences in sampling rate and signal length in different acquisition environments, the preprocessing module resamples the signal to 10 kHz and adjusts the data length to a fixed value through truncation or zero padding; ensuring that the input data is completely consistent in scale and time domain.

[0020] Furthermore, step S2, the time-frequency analysis method, specifically includes:

[0021] To achieve high-precision characterization of the fault traveling wavefront in transient current waveforms, an adaptive window width generalized S-transform (AW-GST) is proposed. This transforms the preprocessed time-domain signal to obtain a two-dimensional high-resolution time-spectrum image, providing accurate input for subsequent feature compression and wavefront localization. AW-GST incorporates targeted improvements in window width control strategy, target bandwidth enhancement, and noise immunity, including:

[0022] S2-1. Frequency-adaptive window width control:

[0023] Based on the dominant frequency distribution of the target wavefront characteristics, a frequency-dependent Gaussian window function is designed. And introduce adjustable parameters This allows for dynamic adjustment of the window width; the window width decreases as the frequency increases, enhancing high-frequency detail resolution while suppressing high-frequency leakage interference with low-frequency components; the mathematical expression is as follows:

[0024] (1);

[0025] (2);

[0026] in, For the signal at time With frequency Adaptive window generalized S-transform coefficients; It is the time-domain signal to be analyzed; For frequency-dependent Gaussian windows (with Centered on the window, the width of the window follows Changes, used for local weighting); The imaginary unit ( ); To analyze frequency (Hz); It is a continuous time variable (s); The bandwidth / standard deviation parameter of the Gaussian window at this frequency is calculated according to... Adaptive settings: Control the slope of the window width as a function of frequency. To control the overall scale, the higher the frequency, the narrower the window; the lower the frequency, the wider the window.

[0027] S2-2. Target band focusing:

[0028] To address the frequency characteristics of traveling wave fronts in distribution networks, the analysis frequency band is limited to 200Hz-3kHz to avoid redundant calculations for irrelevant frequencies and to improve the concentration of target energy.

[0029] S2-3. Adaptive Weighted Noise Reduction:

[0030] In the time-frequency transformation process, high-noise frequency bands are assigned lower weights, thus maintaining clear and separable wavefront features under strong noise. Finally, AW-GST outputs a complex-valued time-frequency matrix, in which:

[0031] Number of rows M: Number of frequency sampling points;

[0032] Number of columns N: Number of time sampling points;

[0033] Each line represents the timing response at the corresponding frequency;

[0034] Each column represents the local spectrum for the corresponding time period;

[0035] This matrix serves as the input to the subsequent multi-scale sparse-low-rank joint optimization module, providing a clear contrast between the energy jumps before and after the wavefront, enabling traveling wave arrival time detection to maintain high accuracy and stability even under low signal-to-noise ratio conditions.

[0036] Furthermore, step S3 specifically includes:

[0037] S3-1. Low-rank feature extraction:

[0038] Singular value decomposition (SVD) is performed on the time-frequency energy matrix to analyze the energy distribution of the characteristic spectrum. Principal components are adaptively retained in the subspace based on the cumulative energy ratio, thereby completing the low-rank approximate reconstruction, preserving the main structural features and suppressing background interference.

[0039] S3-2. Robust Principal Component Analysis (RPCA):

[0040] The time-frequency matrix is ​​modeled as a superposition of low-rank and sparse terms. By minimizing the joint objective of the nuclear norm and the sparse norm, the wavefront component with strong periodicity and well-defined structure and the local sparse noise term are explicitly separated, thereby achieving anomaly detection and edge enhancement.

[0041] S3-3. Wavelet soft thresholding for noise reduction:

[0042] Wavelet transform is applied in the main direction, and a soft threshold function is applied to the signal coefficients to filter out high-frequency noise components and enhance the gradient abrupt change morphology of the wavefront to improve positioning accuracy; finally, the energy map after multi-model processing has more prominent structural sparsity and edge clarity.

[0043] Furthermore, step S3-1, truncating the singular value decomposition (SVD), specifically includes:

[0044] Let the original time-frequency feature matrix be X∈R m×n Then its SVD form is:

[0045] (3);

[0046] Where: U∈R m×m V∈R n×n It is an orthogonal matrix; It is a diagonal matrix whose diagonal elements are singular values. ;

[0047] In practice, the signal is mainly concentrated in the first k principal components, while the noise is distributed in the dimensions corresponding to low singular values; the truncation and reconstruction method is as follows:

[0048] (4);

[0049] Where X is the time-frequency feature matrix ( It is the number of rows / feature dimension. (This is the number of columns / samples or time frames); its singular value decomposition is... ,in , It is an orthogonal matrix. For a diagonal matrix, the elements on the main diagonal are... These are called singular values ​​(according to...) (Sort). The truncated rank is Reconstruction writing :in For the first A singular value, For the corresponding left singular vector ( ), Right singular vector ( );

[0050] This process achieves the reduction and compression of high-dimensional noise features.

[0051] Furthermore, step S3-2, Robust Principal Component Analysis (RPCA), includes the following:

[0052] Robust Principal Component Analysis (RPCA) is introduced to decompose the characteristic matrix into low-rank terms and sparse terms:

[0053] (5);

[0054] in, Represents the nuclear norm, used to measure low-rank property; express Norm, used to measure sparsity; λ is a moderating term that balances low rank and sparsity weights; L represents the low-rank background / principal components of the matrix (continuous, common, and slowly varying structure). This represents sparse anomalies / mutations (pulses, spikes, local variations); that is, in RPCA, the observation matrix is ​​decomposed into... Smooth background Sparse interference and transients such as wavefronts ;

[0055] The model separates stable components from background noise while detecting and eliminating mutations or anomalous noise.

[0056] Further, step S3-3, wavelet soft thresholding, includes:

[0057] To further suppress high-frequency noise components, a soft threshold filter based on multi-scale wavelet transform is introduced; let the signal in the wavelet domain have coefficients d. i The filtering formula is:

[0058] (6);

[0059] Where θ is a threshold selected based on experience or data.

[0060] Further, step S4, wavefront identification and localization, specifically includes:

[0061] S4.1 Energy Projection and One-Dimensional Signal Reconstruction:

[0062] Projecting the two-dimensional time-frequency energy matrix along the frequency dimension yields the total energy sequence in the time domain:

[0063] (7);

[0064] Where Senhanced(f,t) represents the time-frequency energy matrix after optimization by SVD + RPCA + wavelet filtering, and E(t) is the comprehensive energy magnitude at each time point; and These are the lower and upper cutoff frequencies (in Hz) for summing / integrating along the frequencies when performing energy projection. In other words, they represent the enhanced time-frequency matrix within the frequency band. Accumulation along the inner frequency direction yields They are taken from the previously defined target / adaptive analysis band; in discrete implementation, this is equivalent to the pair falling within... The center frequency grid of the interval (index Summation;

[0065] This process essentially maps two-dimensional time-frequency features into a one-dimensional energy curve, providing a foundation for subsequent mutation detection;

[0066] S4.2 Difference and Threshold Judgment:

[0067] To accurately capture the wavefront abrupt change point, a first-order difference operation is performed on E(t) to obtain the instantaneous energy increment sequence. :

[0068] (8);

[0069] in, This represents the energy projection value from the previous moment, i.e., the energy projection value from the time series. The one-time lag term in first-order difference finite element ... ( For sampling index, the actual time is ),therefore It is the difference between the current energy and the energy at the previous sampling moment;

[0070] Subsequently, the local mean μ and standard deviation σ of the differential signal are calculated to construct a dynamic criterion threshold:

[0071] (9);

[0072] Wherein, α is an empirical hyperparameter, with a value between 3 and 5;

[0073] Ultimately, the time point that first meets the following conditions is determined as the wavefront arrival time t0:

[0074] (10).

[0075] The beneficial effects of this invention are as follows:

[0076] 1. This invention proposes a wavefront time-frequency enhancement method based on the generalized S-transform of the frequency adaptive window function and multi-scale sparse-low-rank joint optimization. Its core innovation lies in accurately enhancing the energy transition boundary before and after the wavefront from the time-frequency distribution, thereby achieving high-precision wavefront extraction.

[0077] 2. This invention improves the ability to locate high-frequency details: by using the generalized S-transform (GST) of the frequency adaptive window function, the window width is reduced in the high-frequency region, which significantly improves the temporal positioning accuracy of wavefront abrupt change points.

[0078] 3. This invention suppresses low-frequency leakage and high-frequency interference: By combining selective time-frequency analysis and frequency weighting strategy, the analysis focus is concentrated on the wavefront characteristic region of 200 Hz to 3 kHz, reducing the interference of power frequency main components and broadband noise.

[0079] 4. This invention enhances noise resistance by introducing multi-scale sparse decomposition, robust principal component analysis (RPCA), and wavelet soft thresholding on the time-frequency matrix to achieve time-frequency denoising and feature enhancement under strong noise conditions.

[0080] 5. This invention incorporates information compression and feature enhancement: weighted truncated SVD reconstruction is employed to achieve low-rank approximation that preserves the main energy structure and highlights the energy difference before and after the wavefront, making detection more stable and reliable.

[0081] 6. Since existing methods such as STFT, DTW, WT, and ST all have shortcomings in terms of high-frequency resolution, anti-interference performance, and feature enhancement, this invention effectively solves the core technical problem of "difficulty in accurately extracting wavefronts in complex noise backgrounds" by using an adaptive window width generalized S-transform and multi-scale sparse-low-rank joint optimization technique, which significantly improves the accuracy and robustness of fault location in distribution networks. Attached Figure Description

[0082] Figure 1 This is a flowchart illustrating the overall technical process of the present invention.

[0083] Figure 2 This is a time-domain diagram of the signal in this invention;

[0084] Figure 3 This is the time-frequency domain matrix diagram of the present invention;

[0085] Figure 4 This is a diagram of the power distribution network fault simulation model of the present invention;

[0086] Figure 5 This is the time-frequency diagram of the discrete original S-transform wavefront of the present invention;

[0087] Figure 6 This is the reconstructed wavefront time-frequency diagram of the present invention;

[0088] Figure 7 This is a graph showing the energy curve and wavefront detection results of the present invention;

[0089] Figure 8 This is a diagram showing the wavehead detection results of the present invention. Detailed Implementation

[0090] The principles and features of the present invention are described below with reference to the accompanying drawings. The examples given are for illustrative purposes only and are not intended to limit the scope of the invention.

[0091] This invention proposes a traveling wave front (TWF) identification method based on time-frequency enhancement and multi-scale sparse-low-rank joint optimization, applicable to TWF Fault Location in distribution networks. The method executes sequentially according to a fixed process, comprising four stages: signal preprocessing, time-frequency matrix construction, multi-scale sparse-low-rank joint optimization time-frequency enhancement, and TWF identification. This forms a closed-loop algorithm structure from beginning to end, as illustrated in the flowchart below. Figure 1 As shown. This method is implemented based on the following steps:

[0092] Step S1. Signal preprocessing:

[0093] The raw current signal is preprocessed to filter out power frequency and high-frequency interference components, and a unified sampling rate is used as the input for subsequent processing. Specifically, this includes:

[0094] This invention aims to standardize and denoise the acquired three-phase fault current signal to ensure the accuracy and stability of subsequent time-frequency feature extraction and wavefront detection. The preprocessing mainly includes two parts: bandpass filtering and sampling rate renormalization.

[0095] S1-1. Bandpass Filtering: Considering the significant power frequency fundamental component (50 Hz) and various broadband high-frequency noises in the transient signals of the power system, a 6th-order linear-phase Butterworth bandpass filter is adopted. The passband range is set to [500 Hz, 5000 Hz]. The low-frequency cutoff point can effectively suppress DC bias and fundamental components, while the high-frequency cutoff point can filter out non-target high-frequency noise introduced by switching actions, electromagnetic interference, etc., thereby preserving the main spectral energy of the fault traveling wave. The filtering process adopts zero-phase processing (bidirectional filtering) to avoid the influence of phase distortion on wavefront timing identification.

[0096] S1-2. Sampling Rate Renormalization: To address the differences in sampling rate and signal length across different acquisition environments, the preprocessing module resamples the signal to a uniform 10 kHz and adjusts the data length to a fixed value through truncation or zero-padding. This sampling rate balances the acquisition capabilities of the hardware device with the temporal resolution requirements of the algorithm analysis, ensuring complete consistency of the input data in both scale and time domain.

[0097] After this step, the signal has undergone noise suppression and sampling standardization, providing unified, clean, and high-fidelity input data for subsequent multi-scale time-frequency analysis and wavefront localization.

[0098] Step S2. Time-frequency matrix construction:

[0099] The preprocessed time-domain signal is subjected to time-frequency transformation using the adaptive window width generalized S-transform to obtain an energy distribution matrix with time and frequency as two-dimensional coordinates, which intuitively reflects the local characteristics of the wavefront propagation process at different frequencies. Specific content includes:

[0100] After signal preprocessing, the crucial time-frequency analysis stage begins. To achieve high-precision characterization of the fault traveling wave front in transient current waveforms, this invention proposes an Adaptive-Window Generalized Stockwell Transform (AW-GST) to analyze the preprocessed time-domain signal (such as...). Figure 2 The transformation is performed on the spectrum (as shown) to obtain a two-dimensional high-resolution time spectrum (as shown). Figure 3 As shown in the figure, it provides accurate input for subsequent feature compression and wavefront localization.

[0101] Compared with traditional Discrete S-Transform (DST) and Fixed Window Wide Generalized S-Transform (GST), AW-GST has made targeted improvements in window width control strategy, target bandwidth enhancement, and noise immunity performance:

[0102] S2-1. Frequency-adaptive window width control:

[0103] Based on the dominant frequency distribution of the target wavefront characteristics, a frequency-dependent Gaussian window function is designed. And introduce adjustable parameters This allows for dynamic adjustment of the window width; the window width decreases as the frequency increases, enhancing high-frequency detail resolution while suppressing high-frequency leakage interference with low-frequency components; the mathematical expression is as follows:

[0104] (1);

[0105] (2);

[0106] in, For the signal at time With frequency Adaptive window generalized S-transform coefficients; It is the time-domain signal to be analyzed; For frequency-dependent Gaussian windows (with Centered on the window, the width of the window follows Changes, used for local weighting); The imaginary unit ( ); To analyze frequency (Hz); Let be a continuous time variable (s). In equation (2), The bandwidth / standard deviation parameter of the Gaussian window at this frequency is calculated according to... Adaptive settings: Control the slope of the window width as a function of frequency. To control the overall scale, the higher the frequency, the narrower the window; the lower the frequency, the wider the window.

[0107] S2-2. Target band focusing:

[0108] To address the frequency characteristics of traveling wave fronts in distribution networks, the analysis frequency band is limited to 200Hz-3kHz to avoid redundant calculations for irrelevant frequencies and to improve the concentration of target energy.

[0109] S2-3. Adaptive Weighted Noise Reduction:

[0110] In the time-frequency transformation process, high-noise frequency bands are assigned lower weights, thus maintaining clear and separable wavefront features under strong noise. Finally, AW-GST outputs a complex-valued time-frequency matrix, in which:

[0111] Number of rows M: Number of frequency sampling points;

[0112] Number of columns N: Number of time sampling points;

[0113] Each line represents the timing response at the corresponding frequency;

[0114] Each column represents the local spectrum for the corresponding time period;

[0115] This matrix serves as the input to the subsequent multi-scale sparse-low-rank joint optimization module, providing a clear contrast between the energy jumps before and after the wavefront, enabling traveling wave arrival time detection to maintain high accuracy and stability even under low signal-to-noise ratio conditions.

[0116] Step S3. Optimize using a multi-scale sparse-low-rank joint optimization time-frequency enhancement method:

[0117] To improve the recognizability of wavefront signals in the S-transform time-frequency matrix under strong noise background, this invention proposes a joint optimization mechanism that integrates multiple time-frequency denoising strategies. It sequentially utilizes truncated singular value decomposition (SVD), robust principal component analysis (RPCA), and wavelet soft thresholding to perform multi-level denoising and feature enhancement, ultimately forming a reconstructed feature matrix with clear structure and strong noise suppression capability, providing a stable input for subsequent wavefront recognition.

[0118] Traditional discrete S-transform has poor robustness in locating wavefronts of noisy signals. Its frequency spectrum contains a large number of pseudo-high energy points caused by sudden noise and stationary interference, which leads to an increase in the misjudgment rate of wavefront algorithms based on energy threshold or gradient identification.

[0119] To address this issue, this invention employs a triple optimization approach to process the two-dimensional time-frequency amplitude matrix after the S-transform, reducing background noise disturbances and highlighting the intrinsic structure of the signal. Specifically, this includes:

[0120] S3-1. Truncated Singular Value Decomposition (SVD):

[0121] Singular value decomposition (SVD) is performed on the time-frequency energy matrix to analyze the energy distribution of the characteristic spectrum. Principal components are adaptively preserved in the subset space based on the cumulative energy ratio, thereby completing a low-rank approximate reconstruction that retains the main structural features and suppresses background interference. Specifically, this includes:

[0122] SVD is a classic method for handling low-rank structures. Let the original time-frequency characteristic matrix be X∈R. m×n Then its SVD form is:

[0123] (3);

[0124] Where: U∈R m×m V∈R n×n It is an orthogonal matrix; It is a diagonal matrix whose diagonal elements are singular values. ;

[0125] In practice, the signal is mainly concentrated in the first k principal components, while the noise is distributed in the dimensions corresponding to low singular values; the truncation and reconstruction method is as follows:

[0126] (4);

[0127] Where X is the time-frequency feature matrix ( It is the number of rows / feature dimension. (This is the number of columns / samples or time frames); its singular value decomposition is... ,in , It is an orthogonal matrix. For a diagonal matrix, the elements on the main diagonal are... These are called singular values ​​(according to...) (Sort). The truncated rank is Reconstruction writing :in For the first A singular value, For the corresponding left singular vector ( ), Right singular vector ( ).

[0128] This process achieves the reduction and compression of high-dimensional noise features.

[0129] S3-2. Robust Principal Component Analysis (RPCA):

[0130] The time-frequency matrix is ​​modeled as a superposition of low-rank and sparse terms. By minimizing the joint objective of the nuclear norm and sparse norm, the wavefront components with strong periodicity and well-defined structure are explicitly separated from the locally sparse noise terms, achieving anomaly detection and edge enhancement. The content includes the following:

[0131] To address the issue of traditional SVD's insensitivity to outliers (such as mutation noise), this invention further introduces Robust Principal Component Analysis (RPCA) to decompose the feature matrix into low-rank terms and sparse terms:

[0132] (5);

[0133] in, Represents the nuclear norm, used to measure low-rank property; express Norm, used to measure sparsity; λ is a moderating term that balances low rank and sparsity weights; L represents the low-rank background / principal components of the matrix (continuous, common, and slowly varying structure). This represents sparse anomalies / mutations (pulses, spikes, local variations); that is, in RPCA, the observation matrix is ​​decomposed into... Smooth background Sparse interference and transients such as wavefronts .

[0134] This model can effectively separate stable components (such as wavefront features) from background noise, while detecting and eliminating abrupt or abnormal noise.

[0135] S3-3. Wavelet Soft Thresholding:

[0136] Wavelet transform is applied along the main direction (time or frequency), and a soft thresholding function is applied to the signal coefficients to filter out high-frequency noise components and enhance the gradient abrupt change morphology of the wavefront leading edge, thereby improving positioning accuracy. Ultimately, the energy map after multi-model processing exhibits more prominent structural sparsity and edge sharpness, significantly improving the robustness of wavefront identification. This includes:

[0137] To further suppress high-frequency noise components, a soft threshold filter based on multi-scale wavelet transform is introduced; let the signal in the wavelet domain have coefficients d. i The filtering formula is:

[0138] (6);

[0139] Here, θ is a threshold selected empirically or based on data. This method has the advantages of strong time-frequency locality and excellent anti-interference ability, removing high-frequency background noise while preserving wavefront edge energy.

[0140] Summary of effects and advantages:

[0141] 1. Truncation of SVD removes dimensional redundancy;

[0142] 2. RPCA improves robustness to localized anomalous noise;

[0143] 3. Wavelet soft thresholding enhances the ability to preserve local weak wavefront features.

[0144] The combined use of these three methods makes the wavefront signal more concentrated and significant in the time-frequency plane after the S-transform, ultimately achieving higher robustness and accuracy in wavefront extraction. In low SNR and high noise environments, this method is superior to the original discrete S-transform for wavefront extraction.

[0145] Step S4. Wavehead recognition:

[0146] On the optimized energy matrix, energy accumulation and sliding window smoothing are performed along the time dimension. The abrupt inflection point is extracted by differential operation and dynamic threshold criterion, thereby determining the first significant response time t0 of the wavefront. Compared with traditional methods, this strategy has stronger anti-interference ability and positioning accuracy, and is especially suitable for signal recognition tasks in high noise backgrounds.

[0147] Based on the time-frequency energy map optimized by multi-scale sparse-low-rank joint optimization, accurate wavefront identification and arrival time localization are further required. As the earliest occurrence of energy abrupt changes in traveling wave signals, the accurate extraction of the wavefront is a crucial prerequisite for fault segment localization. Therefore, this invention proposes a wavefront identification method based on time-frequency energy abrupt change detection, which improves the accuracy and noise resistance of wavefront detection through statistical inference and differential operators. Specifically, it includes:

[0148] S4.1 Energy Projection and One-Dimensional Signal Reconstruction:

[0149] Projecting the two-dimensional time-frequency energy matrix along the frequency dimension yields the total energy sequence in the time domain:

[0150] (7);

[0151] Where Senhanced(f,t) represents the time-frequency energy matrix after optimization by SVD + RPCA + wavelet filtering, and E(t) is the comprehensive energy magnitude at each time point; and These are the lower and upper cutoff frequencies (in Hz) for summing / integrating along the frequencies when performing energy projection. In other words, they represent the enhanced time-frequency matrix within the frequency band. Accumulation along the inner frequency direction yields They are taken from the previously defined target / adaptive analysis band; in discrete implementation, this is equivalent to the pair falling within... The center frequency grid of the interval (index Summation.

[0152] This process essentially maps two-dimensional time-frequency features into a one-dimensional energy curve, providing a foundation for subsequent mutation detection;

[0153] S4.2 Difference and Threshold Judgment:

[0154] To accurately capture the wavefront abrupt change point, a first-order difference operation is performed on E(t) to obtain the instantaneous energy increment sequence. :

[0155] (8);

[0156] This represents the energy projection value from the previous moment, i.e., the energy projection value from the time series. The one-time lag term in first-order difference finite element ... ( For sampling index, the actual time is ),therefore It is the difference between the current energy and the energy at the previous sampling moment.

[0157] Subsequently, the local mean μ and standard deviation σ of the differential signal are calculated to construct a dynamic criterion threshold:

[0158] (9);

[0159] Wherein, α is an empirical hyperparameter, with a value between 3 and 5;

[0160] Ultimately, the time point that first meets the following conditions is determined as the wavefront arrival time t0:

[0161] (10).

[0162] Simulation and experimental comparison verification:

[0163] To verify the effectiveness and superiority of the proposed "wavefront time-frequency enhancement method based on multi-scale sparse-low-rank joint optimization" in practical applications, this invention conducted systematic simulation experiments and comparative analyses, the specific process of which is as follows:

[0164] 5-1. Simulation Model Construction and Data Acquisition:

[0165] This invention is based on typical overhead power distribution lines and is constructed on the MATLAB / Simulink platform as follows: Figure 4 The diagram shows a power distribution network fault simulation model. The model includes modules for multiple nodes, branches, transformers, loads, and fault switches, capable of simulating single-phase grounding faults and acquiring current signals. By setting different fault types, fault location, fault impedance, and system noise intensity, a current waveform dataset with varying signal-to-noise ratios is obtained, providing a rich sample base for subsequent processing.

[0166] 5-2. Comparison of wavefront time-frequency diagrams under different methods:

[0167] For the original current signal obtained from the simulation, time-frequency analysis wavefront extraction is performed using both the traditional discrete S-transform method and the low-rank sparse joint optimization reconstruction method proposed in this invention. Figure 5 and Figure 6 The wavefront time-frequency energy distribution maps under the traditional discrete S-transform and reconstruction results are shown respectively. It can be observed that although wavefront energy can be observed in both, there is obvious frequency domain redundancy and noise interference in the traditional discrete S-transform result, and the energy distribution area is more diffused; while the image after low-rank compression reconstruction shows a clearer energy concentration area and more prominent local structural features, laying a good foundation for subsequent wavefront detection.

[0168] 5-3. Wavehead Extraction Comparative Analysis:

[0169] Based on the original and reconstructed time-frequency diagrams, energy accumulation and abrupt change detection are performed using the wavefront recognition algorithm proposed in Section 4. Figure 7 The energy curves and wavefront detection results for typical samples are shown. A comparison of the coordinate values ​​in the figure reveals:

[0170] In the original S-transform energy curve, the wavefront abrupt change point identification time is t = 3.14ms;

[0171] The reconstructed energy curve identification result is t = 3.04ms, with smaller error and steeper abrupt change edges.

[0172] This indicates that the method of the present invention can significantly improve the accuracy and stability of wavefront detection, especially when there is strong interference in the high-frequency band, it can still maintain good detection capability.

[0173] 5-4. Performance comparison under multiple noise levels:

[0174] Furthermore, to examine the robustness of this invention under different noise environments, the wavefront detection results of the S-transform and the method of this invention were compared at multiple signal-to-noise ratio levels (SNR=0, 10, 20, 30). Figure 8 As shown.

[0175] Experimental results show that:

[0176] Both methods can achieve wavefront identification when SNR ≥ 20 dB.

[0177] Below SNR = 10 dB, the recognition performance of traditional S-transform drops sharply, the error increases and false detections occur frequently;

[0178] The method of this invention still maintains a stable wavehead recognition effect and has higher noise resistance and practical adaptability.

[0179] In summary, the simulation experiments demonstrated that the method proposed in this invention outperforms the traditional discrete S-transform scheme in wavefront time-frequency enhancement processing. In particular, it can effectively suppress interference and highlight wavefront characteristics under strong noise conditions, significantly improving the accuracy and reliability of wavefront detection. It has strong engineering practical value and promotion potential.

[0180] This invention, within the traditional S-transform framework, adjusts the matching relationship between the window function and the scale parameter to improve the separation capability of high-frequency transient components and low-frequency background noise. The introduction of a variable window width function enables time-frequency resolution to adapt to signal characteristics.

[0181] The proposed multi-scale sparse-low-rank joint optimization time-frequency enhancement method utilizes matrix factorization (RPCA, SV) to decompose the time-frequency matrix into two parts: a low-rank background and a sparse transient component, effectively suppressing background noise while preserving wavefront features. Wavelet soft thresholding is introduced to further filter the sparse component, achieving a balance between denoising and edge fidelity preservation.

[0182] The robust wavefront extraction algorithm proposed in this invention locates the wavefront based on the energy mutation characteristics of the optimized time-frequency matrix, thereby improving detection accuracy in low signal-to-noise ratio environments. Adaptive thresholding and multi-channel consistency criteria are employed to reduce false positives and false negatives.

[0183] Obviously, the embodiments described above are only some embodiments of this application, not all embodiments. The accompanying drawings show preferred embodiments of this application, but do not limit the patent scope of this application. This application can be implemented in many different forms; rather, the purpose of providing these embodiments is to provide a more thorough and comprehensive understanding of the disclosure of this application. Although this application has been described in detail with reference to the foregoing embodiments, those skilled in the art can still modify the technical solutions described in the foregoing specific embodiments, or make equivalent substitutions for some of the technical features. Any equivalent structures made using the content of this application's specification and drawings, directly or indirectly applied to other related technical fields, are similarly within the scope of patent protection of this application.

Claims

1. A method for extracting the wavefront of a traveling wave based on multi-scale time-frequency enhancement and low-rank sparse optimization, characterized in that: This method is implemented based on the following steps: S1. Signal preprocessing: The original current signal is preprocessed to filter out power frequency and high frequency interference components, and the sampling rate is unified as the input for subsequent processing; S2. Time-frequency matrix construction: The preprocessed time-domain signal is transformed by the adaptive window width generalized S-transform to obtain the energy distribution matrix with time and frequency as two-dimensional coordinates, which intuitively reflects the local characteristics of the wavefront propagation process at different frequencies. S3. A multi-scale sparse-low-rank joint optimization time-frequency enhancement method is adopted for optimization: In order to enhance the wavefront information in the signal and suppress interference noise, multiple denoising mechanisms are used to jointly optimize the time-frequency energy map. S4. Wavehead Recognition: On the optimized energy matrix, energy accumulation and sliding window smoothing are performed along the time dimension. The abrupt inflection point is extracted through differential operation and dynamic threshold criterion, thereby determining the moment of the first significant response of the wavehead.

2. The method for extracting traveling wavefronts according to claim 1, characterized in that: Step S1, the signal preprocessing method, specifically includes: S1-1. Bandpass Filtering: Considering the significant power frequency fundamental component and various broadband high-frequency noises in the transient signals of the power system, a 6th-order linear phase Butterworth bandpass filter is adopted; the low-frequency cutoff point can effectively suppress DC bias and fundamental components, and the high-frequency cutoff point can filter out non-target high-frequency noise introduced by switching action and electromagnetic interference, thereby preserving the main spectral energy of the fault traveling wave; the filtering process adopts zero-phase processing to avoid the influence of phase distortion on wavefront timing identification; S1-2. Sampling Rate Renormalization: To address the differences in sampling rate and signal length in different acquisition environments, the preprocessing module resamples the signal to 10 kHz and adjusts the data length to a fixed value through truncation or zero padding; ensuring that the input data is completely consistent in scale and time domain.

3. The method for extracting traveling wavefronts according to claim 1, characterized in that: Step S2, the time-frequency analysis method, specifically includes: To achieve high-precision characterization of the fault traveling wavefront in transient current waveforms, an adaptive window width generalized S-transform (AW-GST) is proposed. This transforms the preprocessed time-domain signal to obtain a two-dimensional high-resolution time-spectrum image, providing accurate input for subsequent feature compression and wavefront localization. AW-GST incorporates targeted improvements in window width control strategy, target bandwidth enhancement, and noise immunity, including: S2-1. Frequency-adaptive window width control: Based on the dominant frequency distribution of the target wavefront characteristics, a frequency-dependent Gaussian window function is designed. And introduce adjustable parameters This allows for dynamic adjustment of the window width; the window width decreases as the frequency increases, enhancing high-frequency detail resolution while suppressing high-frequency leakage interference with low-frequency components; the mathematical expression is as follows: （1）； （2）； in, For the signal at time With frequency The adaptive window generalized S-transform coefficients on the surface; It is the time-domain signal to be analyzed; A frequency-dependent Gaussian window; The imaginary unit; For frequency analysis; For continuous time variables; The bandwidth / standard deviation parameter of the Gaussian window at this frequency is calculated according to... Adaptive settings: Control the slope of the window width as a function of frequency. To control the overall scale, the higher the frequency, the narrower the window; the lower the frequency, the wider the window. S2-2. Target band focusing: To address the frequency characteristics of traveling wave fronts in distribution networks, the analysis frequency band is limited to 200Hz-3kHz to avoid redundant calculations for irrelevant frequencies and to improve the concentration of target energy. S2-3. Adaptive Weighted Noise Reduction: In the time-frequency transformation process, high-noise frequency bands are assigned lower weights, thus maintaining clear and separable wavefront features under strong noise. Finally, AW-GST outputs a complex-valued time-frequency matrix, in which: Number of rows M: Number of frequency sampling points; Number of columns N: Number of time sampling points; Each line represents the timing response at the corresponding frequency; Each column represents the local spectrum for the corresponding time period; This matrix serves as the input to the subsequent multi-scale sparse-low-rank joint optimization module, providing a clear contrast between the energy jumps before and after the wavefront, enabling traveling wave arrival time detection to maintain high accuracy and stability even under low signal-to-noise ratio conditions.

4. The method for extracting traveling wavefronts according to claim 1, characterized in that: Step S3 specifically includes: S3-1. Low-rank feature extraction: Singular value decomposition (SVD) is performed on the time-frequency energy matrix to analyze the energy distribution of the characteristic spectrum. Principal components are adaptively retained in the subspace based on the cumulative energy ratio, thereby completing the low-rank approximate reconstruction, preserving the main structural features and suppressing background interference. S3-2. Robust Principal Component Analysis (RPCA): The time-frequency matrix is ​​modeled as a superposition of low-rank and sparse terms. By minimizing the joint objective of the nuclear norm and the sparse norm, the wavefront component with strong periodicity and well-defined structure and the local sparse noise term are explicitly separated, thereby achieving anomaly detection and edge enhancement. S3-3. Wavelet soft thresholding for noise reduction: Wavelet transform is applied in the main direction, and a soft threshold function is applied to the signal coefficients to filter out high-frequency noise components and enhance the gradient abrupt change morphology of the wavefront to improve positioning accuracy. Finally, the energy map after multi-model processing has more prominent structural sparsity and edge clarity.

5. The method for extracting the traveling wave head according to claim 4, characterized in that: Step S3-1 involves truncating the singular value decomposition (SVD), specifically including: Let the original time-frequency feature matrix be X∈R m×n Then its SVD form is: (3)； Where: U∈R m×m V∈R n×n It is an orthogonal matrix; It is a diagonal matrix whose diagonal elements are singular values. ; In practice, the signal is mainly concentrated in the front. There are principal components, and the noise is distributed in the dimension corresponding to the low singular values; the truncation and reconstruction method is as follows: （4）； Where X is the time-frequency feature matrix, It is the number of rows / feature dimension. It is the number of columns / samples or time frames; its singular value decomposition is , , It is an orthogonal matrix. For a diagonal matrix, the elements on the main diagonal are... These are called singular values; the truncated rank is Reconstruction writing ,in For the first A singular value, For the corresponding left singular vector ( ), A right singular vector ( ); This process achieves the reduction and compression of high-dimensional noise features.

6. The method for extracting the traveling wave head according to claim 5, characterized in that: Step S3-2, Robust Principal Component Analysis (RPCA), includes the following: Robust Principal Component Analysis (RPCA) is introduced to decompose the characteristic matrix into low-rank terms and sparse terms: （5）； in, Represents the nuclear norm, used to measure low-rank property; express The 1-norm is used to measure sparsity; λ is a moderating term that balances low rank with sparsity weights; L represents the low-rank background / principal component of the matrix. This represents sparse outlier / mutation components; that is, in RPCA, the observation matrix is ​​decomposed into... Smooth background Sparse interference and transients such as wavefronts ; The model separates stable components from background noise while detecting and eliminating mutations or anomalous noise.

7. The method for extracting the traveling wave head according to claim 6, characterized in that: Step S3-3, wavelet soft thresholding, includes: To further suppress high-frequency noise components, a soft threshold filter based on multi-scale wavelet transform is introduced; let the signal in the wavelet domain have coefficients d. i The filtering formula is: (6)； Where θ is a threshold selected based on experience or data.

8. The method for extracting the traveling wave head according to claim 1, characterized in that: Step S4, wavefront identification and localization, specifically includes: S4.1 Energy Projection and One-Dimensional Signal Reconstruction: Projecting the two-dimensional time-frequency energy matrix along the frequency dimension yields the total energy sequence in the time domain: （7）； Where Senhanced(f,t) represents the time-frequency energy matrix after optimization by SVD + RPCA + wavelet filtering, and E(t) is the comprehensive energy magnitude at each time point; and These are the lower and upper cutoff frequencies for summing / integrating along the frequencies when performing energy projection; that is, the enhanced time-frequency matrix in the frequency band. Accumulation along the inner frequency direction yields They are taken from the previously defined target / adaptive analysis band; in discrete implementation, they are equivalent to pairs falling within... The center frequency grid of the interval Summation; This process essentially maps two-dimensional time-frequency features into a one-dimensional energy curve, providing a foundation for subsequent mutation detection; S4.2 Difference and Threshold Judgment: To accurately capture the wavefront abrupt change point, a first-order difference operation is performed on E(t) to obtain the instantaneous energy increment sequence. : （8）； in, This represents the energy projection value from the previous moment, i.e., the energy projection value from the time series. The one-time lag term in first-order difference finite element ... ,therefore It is the difference between the current energy and the energy at the previous sampling time; Subsequently, the local mean μ and standard deviation σ of the differential signal are calculated to construct a dynamic criterion threshold: （9）； Wherein, α is an empirical hyperparameter, with a value between 3 and 5; Ultimately, the time point that first meets the following conditions is determined as the wavefront arrival time t0: (10)。

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