Cable relaxation spectrum identification method and system based on forward-backward dynamic modal decomposition

Abstract

The application discloses a cable relaxation spectrum identification method and system based on forward and backward dynamic modal decomposition, relates to the field of high voltage and insulation technology, and aims to solve the technical problems that the branch number of the extended Debye model cannot be automatically determined in the prior art, and the relaxation parameter identification is easily disturbed by false modes under low signal-to-noise ratio, thereby causing the identification result to not conform to the physical attenuation characteristic. The cable relaxation spectrum identification method based on forward and backward dynamic modal decomposition comprises the following steps: acquiring a discrete sequence of a cable insulation depolarization current, constructing a Hankel matrix, and generating a first snapshot matrix and a second snapshot matrix; constructing an augmented matrix, and determining an effective rank order; constructing a projection matrix, and acquiring a first projection snapshot matrix and a second projection snapshot matrix; estimating a discrete attenuation eigenvalue set; acquiring effective relaxation components and corresponding relaxation time constants; solving current amplitudes corresponding to the relaxation components, and outputting a relaxation spectrum containing the relaxation time constants and the current amplitudes.

Description

Technical Field

[0001] This invention relates to the field of high voltage and insulation technology, and more specifically, to a method and system for identifying cable relaxation spectra based on forward and backward dynamic mode decomposition. Background Technology

[0002] Currently, common methods for on-site cable insulation condition testing include insulation resistance testing, partial discharge detection, and dielectric loss tangent testing. These methods are easily affected by environmental factors and have poor data repeatability. Dielectric response methods have attracted attention due to their efficiency and non-destructive nature. Among them, the polarization and depolarization current method (PDC) has advantages such as portable equipment and short testing time, making it suitable for cable maintenance in industrial fields. To evaluate insulation condition based on the polarization and depolarization current method, it is necessary to perform equivalent modeling of the measured current according to the physical relaxation characteristics. The most widely used model is currently the extended Debye model (EDM). However, a single or few branches in the EDM cannot cover relaxation processes with different time constants. While increasing the number of branches can improve simulation accuracy, too many branches can lead to difficulties in parameter solving.

[0003] Researchers have proposed using intelligent algorithms such as cuckoo search and improved genetic algorithms to identify EDM parameters. However, these methods only improve fitting accuracy from a mathematical perspective and fail to solve the problem of automatically determining the number of EDM branches. Existing technologies involve pre-setting multiple sets of branch numbers and selecting the optimal branch based on goodness of fit. However, this approach does not consider the physical meaning of the EDM and relies on manual traversal of preset values, resulting in low efficiency. Intelligent algorithms also require pre-specifying the number of branches and cannot adaptively determine the number of branches based on measurement data. Furthermore, the measured signal-to-noise ratio at the tail of the depolarization current is extremely low, and conventional Dynamic Mode Decomposition (DMD) easily generates pseudo-slow modes (eigenvalues ​​close to 1) and oscillating pseudo-modes, leading to distorted relaxation spectrum estimation. Therefore, how to adaptively suppress noise interference, accurately identify the relaxation time constant and current amplitude of the EDM without manually pre-setting the number of branches, and ensure that the identification results conform to the physical attenuation characteristics of the depolarization current, has become a pressing technical problem in the field of cable insulation condition assessment. Summary of the Invention

[0004] The purpose of this invention is to provide a method and system for identifying cable relaxation spectra based on forward and backward dynamic mode decomposition, to solve the technical problems in the prior art where the number of branches in the extended Debye model cannot be automatically determined, and where the identification of relaxation parameters at low signal-to-noise ratios is easily affected by pseudo-mode interference, resulting in identification results that do not conform to physical attenuation characteristics. In view of this, this invention achieves this through the following scheme.

[0005] In a first aspect, the present invention provides a cable relaxation spectrum identification method based on forward and backward dynamic mode decomposition, comprising: A discrete sequence of cable insulation depolarization current is obtained, a Hankel matrix is ​​constructed by Hankel delay embedding, and a first snapshot matrix and a second snapshot matrix are generated based on the Hankel matrix. Construct an augmented matrix containing the first snapshot matrix and the second snapshot matrix, perform singular value decomposition on the augmented matrix, and adaptively determine the effective rank order using the singular value hard thresholding method; Extract the first few columns of the right singular vector matrix based on the effective rank order, construct a projection matrix, and use the projection matrix to pre-project the first snapshot matrix and the second snapshot matrix respectively to obtain the first projected snapshot matrix and the second projected snapshot matrix. The forward and backward dynamic mode decomposition method is used to estimate the set of discrete attenuation eigenvalues ​​based on the first and second projection snapshot matrices; The discrete attenuation eigenvalue set is mapped to a continuous-time relaxation time constant, and the modes are screened according to the physical prior criterion of cable depolarization current to obtain the effective relaxation components and the corresponding relaxation time constants. The Vandermonde matrix is ​​constructed based on the selected discrete decay eigenvalues. The current amplitude corresponding to each relaxation component is solved by the least squares method, and the relaxation spectrum containing the relaxation time constant and the current amplitude is output.

[0006] Furthermore, in the cable relaxation spectrum identification method based on forward and backward dynamic mode decomposition of the present invention, the discrete sequence of depolarization current is set as follows: The embedding dimension is The constructed Hankel matrix Size is , The first snapshot matrix From the first column to the second column of the Hankel matrix The second snapshot matrix is ​​composed of columns. From the 2nd column to the 3rd column of the Hankel matrix The sequence consists of columns; the discrete sequence has N sampling points. Indicates the first The current amplitude at each sampling time, where m represents the column number.

[0007] Furthermore, in the cable relaxation spectrum identification method based on forward and backward dynamic mode decomposition of the present invention, the adaptive determination of the effective rank order by the singular value hard thresholding method includes: The augmented matrix is ​​decomposed to obtain a sequence of singular values, and a threshold is adaptively obtained. The subspaces corresponding to all singular values ​​greater than the threshold are retained, and their number is the effective rank order.

[0008] Furthermore, in the cable relaxation spectrum identification method based on forward and backward dynamic mode decomposition of the present invention, the right singular vector matrix obtained after performing singular value decomposition on the augmented matrix is: Take the first r columns to form a matrix The projection matrix is ​​constructed as follows: The pre-projection calculation is as follows: , ;in, Represents the first projected snapshot matrix. This represents the first snapshot matrix. Represents the second projection snapshot matrix. This represents the second snapshot matrix.

[0009] Furthermore, in the cable relaxation spectrum identification method based on forward and backward dynamic mode decomposition of the present invention, the step of estimating the discrete attenuation feature set according to the first projection snapshot matrix and the second projection snapshot matrix includes: constructing a forward operator and a backward operator according to the first projection snapshot matrix and the second projection snapshot matrix, constructing an equivalent operator through symmetric combination, and performing feature decomposition on the equivalent operator to obtain the discrete attenuation feature set.

[0010] Furthermore, in the cable relaxation spectrum identification method based on forward and backward dynamic mode decomposition of the present invention, the method of mapping the discrete attenuation feature value set to a continuous-time relaxation time constant is as follows: Obtaining the continuous time growth rate ,in, The sampling period is The discrete decay characteristic value; the relaxation time constant and retain the satisfaction The modality.

[0011] Furthermore, in the cable relaxation spectrum identification method based on forward and backward dynamic mode decomposition of the present invention, the physical prior criteria include attenuation criteria, near-realism criteria, significance criteria, and relaxation time upper limit criteria; wherein: The attenuation criterion is to retain those that satisfy... or The modalities; the near-realism criterion is to retain the modes that satisfy... or Modes with amplitudes less than a preset threshold are excluded to suppress oscillating pseudo-modes; the significance criterion is to construct a current amplitude surrogate quantity and remove modes whose contribution intensity to the original depolarization current is lower than a set threshold; the relaxation time upper limit criterion is to remove modes whose relaxation time constant is significantly greater than the observation time window length.

[0012] Furthermore, in the cable relaxation spectrum identification method based on forward and backward dynamic mode decomposition of the present invention, after the output includes the relaxation spectrum containing the relaxation time constant and the current amplitude, it further includes: The depolarized current sequence is reconstructed using the obtained relaxation time constant and current amplitude, and the goodness of fit or normalized root mean square error between the reconstructed sequence and the original sequence is obtained.

[0013] Furthermore, in the cable relaxation spectrum identification method based on forward and backward dynamic mode decomposition of the present invention, the embedding dimension is based on the total number of sampling points. Set as to The values ​​are between; the sampling period is determined based on the sampling rate of the cable polarization-depolarization test equipment.

[0014] Compared with existing technologies, the cable relaxation spectrum identification method based on forward and backward dynamic mode decomposition of this invention addresses the industry pain points of the need for manual preset of the number of branches in the Extended Debye Model (EDM) and the contamination of the relaxation spectrum by pseudo-modes under low signal-to-noise ratio. It transforms the single-channel depolarized current sequence into snapshot matrix pairs through Hankel embedding, then constructs an augmented matrix and performs adaptive order determination using Singular Value Hard Thresholding (SVHT). This process requires no prior knowledge and automatically outputs the effective rank directly from the singular value distribution of the measurement data, fundamentally solving the problem of the inability to automatically determine the number of branches. However, if the snapshot matrix truncated by SVHT is directly fed into forward and backward dynamic mode decomposition (fbDMD), operator ill-conditioning and a surge in pseudo-modes will occur due to inconsistencies in the column space. Therefore, this invention designs a connection step: taking the first r columns of the right singular vector matrix to form matrix Vr, and constructing a projection matrix... And simultaneously apply this projection to the first snapshot matrix X and the second snapshot matrix Y to obtain a low-rank filtered snapshot with consistent column space. This projection operation effectively suppresses the high-amplitude noise at the tail of the depolarization current while preserving the exponentially decaying main structure, ensuring... and Sharing the same low-dimensional subspace significantly improves the condition number of the forward and backward operator inversions in subsequent fbDMD, resulting in a substantial increase in numerical stability. It also filters out pseudo-slow modes and oscillating pseudo-modes with eigenvalues ​​close to 1 from the source. Subsequently, fbDMD obtains discrete decay eigenvalues ​​through symmetric combination, mapping them to continuous-time relaxation time constants, and introduces a quadruple physical prior screening method specifically for cable depolarization current: a decay criterion to eliminate divergent modes, a near-real criterion (… or The eigenvalues ​​are selected based on several criteria: first, oscillating components that should not exist are eliminated; second, a significance criterion (current amplitude surrogate) is used to discard redundant weak branches; and third, an upper limit criterion for relaxation time is used to prevent extremely slow pseudo-modes from aliasing with baseline drift. The selected eigenvalues ​​have clear physical meanings. Finally, the current amplitude of each relaxation branch is solved with high precision using Vandermonde least squares, outputting a complete relaxation spectrum. Simulation and experimental results show that under 1%~5% noise, this invention can automatically identify all true branches (…). The normalized RMSE is less than 0.03%, and the cable samples at different thermal aging stages show a clear relaxation spectrum evolution law. Through the above-mentioned technical solution of the present invention, the technical problems of the inability to automatically determine the number of branches of the extended Debye model and the easy susceptibility of the identification of relaxation parameters under low signal-to-noise ratio to pseudo-mode interference are solved, resulting in the identification results not conforming to the physical attenuation characteristics.

[0015] Secondly, the present invention provides a cable relaxation spectrum identification system based on forward and backward dynamic mode decomposition, comprising: The data acquisition and embedding module is used to acquire the discrete sequence of cable insulation depolarization current, construct a Hankel matrix through Hankel delayed embedding, and generate a first snapshot matrix and a second snapshot matrix based on the Hankel matrix; The augmentation and order determination module is used to construct an augmented matrix containing the first snapshot matrix and the second snapshot matrix, perform singular value decomposition on the augmented matrix, and adaptively determine the effective rank order by using the singular value hard thresholding method. The pre-projection module is used to extract the first few columns of the right singular vector matrix according to the effective rank order, construct the projection matrix, and use the projection matrix to pre-project the first snapshot matrix and the second snapshot matrix respectively to obtain the first projected snapshot matrix and the second projected snapshot matrix. The eigenvalue estimation module is used to estimate the set of discrete decay eigenvalues ​​based on the first and second projected snapshot matrices using the forward and backward dynamic mode decomposition method. The mapping and filtering module is used to map the discrete attenuation feature set to a continuous-time relaxation time constant, and to filter the modes according to the physical prior criterion of cable depolarization current to obtain the effective relaxation components and the corresponding relaxation time constants. The amplitude calculation and output module is used to construct the Vandermonde matrix based on the selected discrete decay eigenvalues, solve the current amplitude corresponding to each relaxation component by the least squares method, and output the relaxation spectrum containing the relaxation time constant and the current amplitude. Attached Figure Description

[0016] The accompanying drawings, which are included to provide a further understanding of the invention and form part of this invention, illustrate exemplary embodiments of the invention and are used to explain the invention, but do not constitute an undue limitation of the invention. In the drawings: Figure 1 This is a flowchart illustrating a cable relaxation spectrum identification method based on forward and backward dynamic mode decomposition according to the present invention. Figure 2 This is a schematic diagram of an experimental apparatus for cable measurement according to the present invention.

[0017] Figure label: In Figure 2 In the middle, 1. Shielding ring; 2. Copper shielding layer; 3. Polarization current; 4. Picoammeter; 5. Communication connection. Detailed Implementation

[0018] To make the technical problems to be solved, the technical solutions, and the beneficial effects of the present invention clearer, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the present invention and are not intended to limit the present invention.

[0019] It should be noted that when a component is referred to as being "fixed to" or "set on" another component, it can be directly on or indirectly on that other component. When a component is referred to as being "connected to" another component, it can be directly connected to or indirectly connected to that other component.

[0020] Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of indicated technical features. Thus, a feature defined as "first" or "second" may explicitly or implicitly include one or more of that feature. In the description of this invention, "a plurality of" means two or more, unless otherwise explicitly specified. "Several" means one or more, unless otherwise explicitly specified.

[0021] Existing technologies utilize intelligent algorithms such as cuckoo search and improved genetic algorithms to identify EDM parameters. However, these methods only improve fitting accuracy from a mathematical perspective and fail to address the problem of automatically determining the number of EDM branches. Current technologies pre-determine multiple branch numbers and select the optimal branch based on goodness of fit, but this approach does not consider the physical meaning of the EDM and relies on manual iteration, resulting in low efficiency. Similarly, intelligent algorithms require pre-specifying the number of branches and cannot adaptively determine the branch number based on measurement data. Furthermore, the measured signal-to-noise ratio at the tail of the depolarization current is extremely low, and conventional Dynamic Mode Decomposition (DMD) easily generates pseudo-slow modes (eigenvalues ​​close to 1) and oscillating pseudo-modes, leading to distorted relaxation spectrum estimation. Therefore, how to adaptively suppress noise interference, accurately identify the relaxation time constant and current amplitude of the EDM without manually pre-determining the number of branches, and ensure that the identification results conform to the physical attenuation characteristics of the depolarization current, has become a pressing technical problem in the field of cable insulation condition assessment.

[0022] To address the aforementioned technical problems, this invention provides a cable relaxation spectrum identification method based on forward and backward dynamic mode decomposition, comprising: A discrete sequence of cable insulation depolarization current is obtained, a Hankel matrix is ​​constructed by Hankel delay embedding, and a first snapshot matrix and a second snapshot matrix are generated based on the Hankel matrix. Construct an augmented matrix containing the first snapshot matrix and the second snapshot matrix, perform singular value decomposition on the augmented matrix, and adaptively determine the effective rank order using the singular value hard thresholding method; Extract the first few columns of the right singular vector matrix based on the effective rank order, construct a projection matrix, and use the projection matrix to pre-project the first snapshot matrix and the second snapshot matrix respectively to obtain the first projected snapshot matrix and the second projected snapshot matrix. The forward and backward dynamic mode decomposition method is used to estimate the set of discrete attenuation eigenvalues ​​based on the first and second projection snapshot matrices; The discrete attenuation eigenvalue set is mapped to a continuous-time relaxation time constant, and the modes are screened according to the physical prior criterion of cable depolarization current to obtain the effective relaxation components and the corresponding relaxation time constants. The Vandermonde matrix is ​​constructed based on the selected discrete decay eigenvalues. The current amplitude corresponding to each relaxation component is solved by the least squares method, and the relaxation spectrum containing the relaxation time constant and the current amplitude is output.

[0023] In the above-mentioned technical solution, the cable relaxation spectrum identification method based on forward and backward dynamic mode decomposition of the present invention addresses the industry pain points of the need for manual preset of the number of branches in the Extended Debye Model (EDM) and the contamination of the relaxation spectrum by pseudo-modes under low signal-to-noise ratio. It transforms the single-channel depolarized current sequence into snapshot matrix pairs through Hankel embedding, then constructs an augmented matrix and performs adaptive order determination using Singular Value Hard Thresholding (SVHT). This process requires no prior knowledge and automatically outputs the effective rank directly from the singular value distribution of the measurement data, fundamentally solving the problem of the inability to automatically determine the number of branches. However, if the snapshot matrix truncated by SVHT is directly fed into forward and backward dynamic mode decomposition (fbDMD), operator ill-conditioning and a surge in pseudo-modes will occur due to inconsistencies in the column space. Therefore, the present invention designs a connection step: taking the first r columns of the right singular vector matrix to form matrix Vr, and constructing a projection matrix. And simultaneously apply this projection to the first snapshot matrix X and the second snapshot matrix Y to obtain a low-rank filtered snapshot with consistent column space. This projection operation effectively suppresses the high-amplitude noise at the tail of the depolarization current while preserving the exponentially decaying main structure, ensuring... and Sharing the same low-dimensional subspace significantly improves the condition number of the forward and backward operator inversions in subsequent fbDMD, resulting in a substantial increase in numerical stability. It also filters out pseudo-slow modes and oscillating pseudo-modes with eigenvalues ​​close to 1 from the source. Subsequently, fbDMD obtains discrete decay eigenvalues ​​through symmetric combination, mapping them to continuous-time relaxation time constants, and introduces a quadruple physical prior screening method specifically for cable depolarization current: a decay criterion to eliminate divergent modes, a near-real criterion (… or The eigenvalues ​​are selected based on several criteria: first, oscillating components that should not exist are eliminated; second, a significance criterion (current amplitude surrogate) is used to discard redundant weak branches; and third, an upper limit criterion for relaxation time is used to prevent extremely slow pseudo-modes from aliasing with baseline drift. The selected eigenvalues ​​have clear physical meanings. Finally, the current amplitude of each relaxation branch is solved with high precision using Vandermonde least squares, outputting a complete relaxation spectrum. Simulation and experimental results show that under 1%~5% noise, this invention can automatically identify all true branches (…). The normalized RMSE is less than 0.03%, and the cable samples at different thermal aging stages show a clear relaxation spectrum evolution law. Through the above-mentioned technical solution of the present invention, the technical problems of the inability to automatically determine the number of branches of the extended Debye model and the easy susceptibility of the identification of relaxation parameters under low signal-to-noise ratio to pseudo-mode interference are solved, resulting in the identification results not conforming to the physical attenuation characteristics.

[0024] To better understand the present invention, the following specific embodiments further illustrate the content of the present invention, but the content of the present invention is not limited to the following embodiments.

[0025] Example 1 This embodiment provides a cable relaxation spectrum identification method based on forward and backward dynamic mode decomposition, including: Step 1: Obtain the discrete sequence of cable insulation depolarization current, construct the Hankel matrix through Hankel delay embedding, and generate the first snapshot matrix and the second snapshot matrix based on the Hankel matrix; Step 2: Construct an augmented matrix containing the first snapshot matrix and the second snapshot matrix, perform singular value decomposition on the augmented matrix, and adaptively determine the effective rank order using the singular value hard thresholding method. Step 3: Extract the first few columns of the right singular vector matrix according to the effective rank order, construct the projection matrix, and use the projection matrix to pre-project the first snapshot matrix and the second snapshot matrix respectively to obtain the first projected snapshot matrix and the second projected snapshot matrix. Step 4: Using the forward and backward dynamic mode decomposition method, estimate the set of discrete attenuation eigenvalues ​​based on the first and second projected snapshot matrices; Step 5: Map the discrete attenuation eigenvalue set to a continuous-time relaxation time constant, and filter the modes according to the physical prior criterion of cable depolarization current to obtain the effective relaxation components and the corresponding relaxation time constants. Step 6: Construct the Vandermonde matrix based on the selected discrete decay eigenvalues, solve for the current amplitude corresponding to each relaxation component using the least squares method, and output the relaxation spectrum containing the relaxation time constant and the current amplitude.

[0026] Example 2 This embodiment provides a cable relaxation spectrum identification method based on forward and backward dynamic mode decomposition, including: Step 1: Obtain the discrete sequence of cable insulation depolarization current, construct the Hankel matrix through Hankel delay embedding, and generate the first snapshot matrix and the second snapshot matrix based on the Hankel matrix; Furthermore, the discrete sequence of depolarization current is defined as follows: The embedding dimension is The constructed Hankel matrix Size is , First snapshot matrix From the first column to the second column of the Hankel matrix The second snapshot matrix is ​​composed of columns. From the 2nd column to the 3rd column of the Hankel matrix The sequence consists of columns; the discrete sequence has N sampling points. Indicates the first The current amplitude at each sampling time, where m represents the column number; embedding dimension. Based on the total number of sampling points Set as to The values ​​between.

[0027] Step 2: Construct an augmented matrix containing the first snapshot matrix and the second snapshot matrix, perform singular value decomposition on the augmented matrix, and adaptively determine the effective rank order using the singular value hard thresholding method. Furthermore, the effective rank order is adaptively determined using the singular value hard thresholding method, including: The augmented matrix is ​​decomposed to obtain a sequence of singular values, and the threshold is adaptively obtained. Retain the subspaces corresponding to all singular values ​​greater than the threshold, the number of which is the effective rank.

[0028] Step 3: Extract the first few columns of the right singular vector matrix according to the effective rank order, construct the projection matrix, and use the projection matrix to pre-project the first snapshot matrix and the second snapshot matrix respectively to obtain the first projected snapshot matrix and the second projected snapshot matrix. Furthermore, the right singular vector matrix obtained after performing singular value decomposition on the augmented matrix is: Take the first r columns to form a matrix The projection matrix is ​​constructed as follows: The pre-projection calculation is as follows: , ;in, Represents the first projected snapshot matrix. This represents the first snapshot matrix. Represents the second projection snapshot matrix. This represents the second snapshot matrix.

[0029] Step 4: Using the forward and backward dynamic mode decomposition method, forward and backward operators are constructed based on the first and second projection snapshot matrices. Equivalent operators are constructed through symmetric combination, and the equivalent operators are decomposed to obtain a set of discrete decay eigenvalues.

[0030] Step 5: Map the discrete attenuation eigenvalue set to a continuous-time relaxation time constant, and filter the modes according to the physical prior criterion of cable depolarization current to obtain the effective relaxation components and the corresponding relaxation time constants. Furthermore, the method for mapping the discrete decay eigenvalue set to the continuous-time relaxation time constant is as follows: obtain the continuous-time growth rate. ,in, The sampling period is The discrete decay eigenvalues; relaxation time constant. and retain the satisfaction The mode; the sampling period is determined based on the sampling rate of the cable polarization-depolarization test equipment.

[0031] Physical prior criteria include the decay criterion, near-realism criterion, significance criterion, and relaxation time upper bound criterion; among them, the decay criterion preserves the condition that satisfies the given conditions. or The modality; the near-realism criterion is to retain the mode that satisfies or Modes with amplitudes less than a preset threshold are excluded to suppress oscillating pseudo-modes; the significance criterion is to construct a current amplitude surrogate and remove modes whose contribution intensity to the original depolarization current is less than a set threshold; the relaxation time upper limit criterion is to remove modes whose relaxation time constant is significantly greater than the observation time window length.

[0032] Step 6: Construct the Vandermonde matrix based on the selected discrete decay eigenvalues, solve for the current amplitude corresponding to each relaxation component using the least squares method, and output the relaxation spectrum containing the relaxation time constant and the current amplitude.

[0033] Step 7: Reconstruct the depolarized current sequence using the solved relaxation time constant and current amplitude, and obtain the goodness of fit or normalized root mean square error between the reconstructed sequence and the original sequence.

[0034] Example 3 Firstly, please refer to Figure 1 and Figure 2 This embodiment provides a cable relaxation spectrum identification method based on forward and backward dynamic mode decomposition, including: Step 1: Obtain the discrete sequence of cable insulation depolarization current, perform Hankel delayed embedding on the discrete sequence, and construct a snapshot matrix. Specifically, first convert the discrete sequence into a snapshot matrix of Hankel embedding, and then rearrange the one-dimensional current sequence into a Hankel matrix of delayed snapshot.

[0035] Step 2: Perform SVHT singular value hard thresholding for order determination (adaptive noise reduction and effective order determination). Specifically, since real PDC data contains noise (especially in the low-current tail region), the enhanced snapshot data is processed using the singular value hard thresholding method. The SVHT algorithm provides an adaptive rule that can preserve the dominant singular orientation while removing the subspace portion mainly related to noise. Since the number of branches cannot be predetermined, the effective order r is determined based on the singular value distribution rather than manually set; let σ i Let represent the singular values ​​of the enhanced snapshot matrix, then the retained rank is: ; Where, σ th A hard threshold is applied to singular values. This process is a key step in achieving data-driven branch count identification, retaining only the primary subspace supported by the measurement data for subsequent spectral estimation.

[0036] Step 3, Forward-Backward Dynamic Mode Decomposition (fbDMD) spectral estimation; specifically, after low-rank screening, the retained dynamic features are analyzed through forward-backward dynamic mode decomposition. In standard dynamic mode decomposition, sensor noise may affect the estimation results of eigenvalues. The forward-backward decomposition method can effectively reduce this bias and obtain more robust spectral estimation results when data is noisy or of limited length. Subsequently, the obtained discrete decay modes are mapped to relaxation time constants. After fbDMD generates a set of discrete-time eigenvalues, these spectral quantities must be converted into physically meaningful relaxation parameters. This conversion process is achieved as follows: ; ; where μ j It is related to the discrete decay eigenvalue λ j The associated continuous-time decay rate, Indicates the sampling period. μ j The real part of the equation is retained; only the attenuating and almost non-oscillating modes are preserved, thus ensuring that the identified parameters are consistent with the cable depolarization behavior.

[0037] Step 4: Perform physical consistency mode screening; specifically, only retain modes that are physically consistent with the cable depolarization behavior, mainly referring to attenuating, nearly non-oscillatory modes, and whose contribution to the measured current is not negligible.

[0038] Step 5: Generate the relaxation spectrum (relaxation time constant τ, current amplitude A) and reconstruct the depolarization current; specifically, the amplitude is finally obtained using the least squares method and the preserved exponential basis function. After preserving the allowed decay modes, the corresponding amplitude is recovered using the least squares method, thereby reconstructing the depolarization current sequence, expressed as: ; ;in, This represents the current magnitude vector obtained through least squares estimation. This represents the value that minimizes the objective function. value, 2 represents the Euclidean norm of a vector. This represents the original depolarization current sequence vector. It is an exponential base composed of preserved patterns. This represents the set of discrete decay eigenvalues ​​that are ultimately retained after physical screening. This represents the magnitude vector variable to be solved. The first depolarization current sequence representing the reconstructed depolarization current sequence One value, Indicates the first Estimated current amplitude of the relaxation branch. No. Each retained feature value Power of 1 This represents the final effective number of relaxation branches. Therefore, three outputs can be generated: the number of effective branches, the relaxation spectrum parameters, and the reconstructed depolarization current sequence. Compared with direct fixed-order fitting methods, this process is more measurement-oriented. It first determines what structure the measurement data actually supports, then solves for the corresponding parameters, and finally evaluates the cable insulation status.

[0039] Furthermore, in this embodiment, the single-channel sequence is first boosted into a snapshot matrix using Hankel delayed embedding. Then, SVHT adaptive order determination is used, combined with TDMD pre-projection to suppress noise pseudo-modes. Subsequently, fbDMD robustly estimates discrete decay eigenvalues ​​and maps them to obtain the relaxation time constant. Finally, the amplitude parameters are solved using Vandermonde least squares to form a (τ,A) relaxation spectrum representation. Specifically, the EDM expression of the depolarization current and its discretized form are given first, clarifying the mapping relationship between the object to be identified and the parameters, providing a unified mathematical benchmark for subsequent spectrum estimation and amplitude solution. Under ideal conditions, the sampling period is set to T. s The discrete sequence of cable insulation depolarization current is denoted as y. k =i(t k (k=0,1,…,N−1),i(t) k () represents the continuous-time depolarization current at the sampling time. The value of , where N represents the total number of sampling points in the discrete sequence, and the th . Each sampling time: t k =kT s ,but: ; Here, the discrete attenuation characteristic value is defined as follows: ; Therefore, the mapping relationship of the relaxation time constant is obtained as follows: ; in, The first discrete sequence of depolarization current represents the... Each sample value, This represents the total number of effective relaxor branches. Indicates the first The current amplitude of the relaxation branch, Represents an exponential function. Indicates the first The relaxation time constant of a relaxation branch. Indicates the first Discrete attenuation characteristic values ​​corresponding to each branch.

[0040] Furthermore, addressing the characteristics of "depolarization current being a single-channel sequence, significant noise in the field data, and low signal-to-noise ratio at the tail end," this embodiment introduces Hankel delayed embedding to expose the low-rank structure, and employs SVHT adaptive order determination and TDMD pre-projection in the augmented space to suppress the pseudo-slow modes and pseudo-spectral points that are prone to occur in DMD from the source. Specifically, the embedding dimension d is selected, and the depolarization current discrete sequence is... Constructing a Hankel matrix: ; in, d represents the total number of sampling points, and d represents the embedding dimension of the Hankel delayed embedding.

[0041] Further define adjacent column snapshot pairs as: First snapshot matrix Second snapshot matrix ;in, This represents the constructed Hankel matrix, with a size of . , (:) means to retrieve all rows of the matrix, (,) means to separate row index and column index, 1:end-1 means all columns from the first column to the second to last column, 2:end means all columns from the second column to the last column.

[0042] For an ideal multi-exponential sequence, the column space of X and Y is spanned by a finite number of exponential "basis functions", exhibiting an approximately low rank. Noise can raise the singular values ​​at the tail and induce pseudo-modes, so order determination and projection are required first.

[0043] Furthermore, construct the augmented matrix: ; This represents the first snapshot matrix. This represents the second snapshot matrix. express It is a real matrix with rows. The number of columns is d is the embedding dimension, and N is the total number of sampling points in the discrete sequence of depolarized current.

[0044] Perform economic singular value decomposition on the augmented matrix: , ;in, Represents an augmented matrix. Describes a left singular vector matrix. Represents a singular value diagonal matrix. Describes a right singular vector matrix. express The transpose of the matrix, This represents a diagonal matrix whose elements on the main diagonal are singular values. Under the condition of unknown noise variance, the effective order r is determined using the SVHT criterion: let the threshold be σ. th ,but This mechanism preserves the singular subspace contributed by the dominant exponential decay structure and eliminates the tail subspace mainly driven by noise, which can significantly reduce the probability of pseudo-slow modes with λ≈1, and is particularly critical for low signal-to-noise ratio conditions in the tail segment.

[0045] Furthermore, take the first r columns V of the right singular vector matrix. r Construct the projection matrix: ; and project the snapshot onto the effective subspace: first projected snapshot matrix Second projection snapshot matrix , This represents the first snapshot matrix. This represents the second snapshot matrix; the projection is equivalent to implementing adaptive low-rank filtering in the snapshot column space: it preserves the exponentially decaying principal structure, significantly reduces the impact of noise and local outliers on subsequent operator estimation, and improves the condition number of pseudo-inverse operations.

[0046] Furthermore, fbDMD spectral extraction, continuous-time mapping, and physical consistency screening are performed. After completing the low-rank preprojection, fbDMD is used to estimate the discrete attenuation eigenvalue set {λ}. j}. Then the discrete decay eigenvalue λ j Mapped to continuous-time relaxation constant τ j Furthermore, the physical prior of "non-oscillatory, monotonically decaying" depolarization current is used for screening to enhance the interpretability and stability of spectral estimation. Specifically, based on projection snapshots... First, construct the forward and backward operators: , ,in, Indicates the forward operator, Indicates the backward operator, express Moore-Penrose pseudo-reverse, express Moore-Penrose pseudo-inverse; ideally, under noise-free conditions, it should satisfy fbDMD reduces noise bias through symmetric combination and constructs an equivalent operator A. fb (can be considered as a response to A) f A b −1 (The "symmetry" process is performed), and eigenvalues ​​are decomposed to obtain a set of candidate discrete decay eigenvalues ​​{λ}. j Compared to unidirectional DMD, fbDMD typically provides a more "contracted" spectral estimate under conditions of noise and limited data length, which better matches the decay dynamics of the depolarization current.

[0047] Furthermore, based on the discrete decay eigenvalue λ j Continuous time growth rate: And calculate the relaxation time constant: ; when Re(μ j When τ < 0, it corresponds to the decay process, and τ j >0, consistent with the ideal model. Considering that the cable insulation depolarization current exhibits non-oscillatory decay under common test conditions, to suppress numerical pseudo-modes and enhance interpretability, the following screening criteria are adopted for candidate modes: decay (necessary condition) is used to retain modes that satisfy |λ. j |<1 or Re(μ) j Modes with λ < 0; near-real (suppressing oscillatory components) retain Im(λ) j ) or Im(μ j Sufficiently small modes (threshold set by experimental noise level); significance (eliminating weak energy modes) is achieved by constructing a "current amplitude surrogate" to measure the mode's contribution to the original current, eliminating modes with extremely weak contributions; relaxation time upper limit (suppressing pseudo-slow modes under finite observation windows) is achieved by constraining the observation window length τ. j The observation window length should not be significantly larger than the observation window length (controlled by empirical factors), to avoid τ≫T. obs The set of eigenvalues ​​retained after filtering, which includes pseudospectral points where time and baseline / drift aliasing cause λ≈1, is denoted as . , where z is the final effective relaxation component number.

[0048] Furthermore, the amplitude A is... j Solving for and reconstructing the depolarization current, specifically, obtaining the filtered discrete decay eigenvalues. Then, the amplitude parameter A is calculated using the Vandermonde structure. j The estimation is transformed into a linear least squares problem, and a reconstructed expression for the depolarization current is given. For the retained {λ... j Construct the Vandermonde matrix: ; Let A = [A1, A2, ..., A z ] T The original sequence y=[y0,y1,…,y N−1 ] T Then the amplitude estimate can be expressed as: ; Based on this, the depolarization current reconstruction sequence is obtained: , ; in, This represents the total number of sampling points in the discrete sequence of depolarization current. This represents the number of effective relaxation branches that are ultimately retained after physical screening. Indicates the first Each retained discrete decay eigenvalue Represents the Vandermonde matrix The Middle line, number The elements of the column are equal to Let A represent the magnitude vector to be solved. The magnitude vector estimate obtained through least squares estimation Represents the Vandermonde matrix Moore-Penrose pseudo-reverse, This represents the reconstructed depolarization current sequence vector. Indicates the first branch road to the first The contribution of each sample point to the reconstructed value.

[0049] The final output is the set of relaxation spectrum parameters. And the fit consistency index between ŷ and y (such as R) can be used. 2 The verification is performed using normalized RMSE. This embodiment provides a structured summary of the entire identification process, clearly defining the input / output and key hyperparameter locations, facilitating engineering reproduction and subsequent simulation and experimental applications. The input is a discrete sequence of depolarization current. Sampling period T s Embedding dimension d (other orders are adaptively given by SVHT); output is the number of effective modes z and relaxation spectrum parameters. , and the reconstructed sequence ŷ.

[0050] Furthermore, the method of this embodiment was simulated and verified, and the simulated current parameter values ​​under different noise levels were set, as shown in Table 1.

[0051] Table 1 Simulation current parameter values 1 10 800 2 50 300 3 300 20 Furthermore, the SVHT-fbDMD was used to identify the simulated current under different noise conditions, and the calculation results under 1% noise and 5% noise were obtained, as shown in Table 2 and Table 3, respectively.

[0052] Table 2 Calculation results under 1% noise 1 9.9998 799.99 2 49.998 300.01 3 299.25 19.962 Table 3 Calculation results under 5% noise 1 9.995 798.82 2 49.595 299.54 3 285.82 21.888 Furthermore, under both 1% and 5% noise conditions, SVHT-fbDMD can identify z=3 true branches and provide high-precision (τ,A) regression; simultaneously, the overall reconstruction accuracy R2 =0.999998, normalized RMSE=0.0213%, proving that the process maintains stable consistency between spectral estimation and current reconstruction under noisy conditions. The method of this embodiment is applied in practice: to verify the feasibility and accuracy of the SVHT-fbDMD identification method, this method is used to perform parameter identification tests on the depolarization current of the same cable at different thermal aging stages. An unused domestic commercial 8.7 / 10kV XLPE cable is selected for the experiment, and a 50cm section is taken as the sample; thermal aging is carried out at 135℃, and PDC tests are performed in three stages: unaged S1, early thermal aging (20d) S2, and late thermal aging (40d) S3. The test voltage is 1kV, and the wiring diagram is attached. Figure 2 As shown, the test results are presented in Tables 4 to 6; furthermore, in Figure 2 In the middle, shielding ring 1 is grounded, copper shielding layer 2 is connected to the input terminal of picoammeter 4, polarization current 3 enters the ground through picoammeter 4, and picoammeter 4 uploads data through communication connection 5.

[0053] Table 4 S1 Identification Results 1 1.2 3.6284 2 13.022 0.27994 3 47.011 0.14886 4 302.99 0.12597 Table 5 S2 Identification Results 1 4.0003 3.4292 2 27.026 1.4747 3 87.654 0.18142 4 474.14 0.097723 Table 6 S3 Identification Results 1 6.4 4.228 2 35 2.163 3 102 0.256 4 273 0.173 5 641 0.074 The results show that the goodness of fit R for the three groups of samples is... 2 The results were all approximately 99%, and the normalized RMSEs were 0.0032%, 0.0065%, and 0.0090%, respectively, indicating that the method can stably reconstruct the depolarization current and output a reliable parameter spectrum under real data conditions.

[0054] Secondly, the present invention provides a cable relaxation spectrum identification system based on forward and backward dynamic mode decomposition, comprising: The data acquisition and embedding module is used to acquire the discrete sequence of cable insulation depolarization current, construct a Hankel matrix through Hankel delayed embedding, and generate a first snapshot matrix and a second snapshot matrix based on the Hankel matrix; The augmentation and order determination module is used to construct an augmented matrix containing the first snapshot matrix and the second snapshot matrix, perform singular value decomposition on the augmented matrix, and adaptively determine the effective rank order by using the singular value hard thresholding method. The pre-projection module is used to extract the first few columns of the right singular vector matrix according to the effective rank order, construct the projection matrix, and use the projection matrix to pre-project the first snapshot matrix and the second snapshot matrix respectively to obtain the first projected snapshot matrix and the second projected snapshot matrix. The eigenvalue estimation module is used to estimate the set of discrete decay eigenvalues ​​based on the first and second projected snapshot matrices using the forward and backward dynamic mode decomposition method. The mapping and filtering module is used to map the discrete attenuation feature set to a continuous-time relaxation time constant, and to filter the modes according to the physical prior criterion of cable depolarization current to obtain the effective relaxation components and the corresponding relaxation time constants. The amplitude calculation and output module is used to construct the Vandermonde matrix based on the selected discrete decay eigenvalues, solve the current amplitude corresponding to each relaxation component by the least squares method, and output the relaxation spectrum containing the relaxation time constant and the current amplitude.

[0055] In the description of the above embodiments, specific features, structures, materials, or characteristics may be combined in any suitable manner in one or more embodiments or examples.

[0056] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A method for identifying cable relaxation spectra based on forward and backward dynamic mode decomposition, characterized in that, include: A discrete sequence of cable insulation depolarization current is obtained, a Hankel matrix is ​​constructed by Hankel delay embedding, and a first snapshot matrix and a second snapshot matrix are generated based on the Hankel matrix. Construct an augmented matrix containing the first snapshot matrix and the second snapshot matrix, perform singular value decomposition on the augmented matrix, and adaptively determine the effective rank order using the singular value hard thresholding method; Based on the effective rank order, extract the first few columns of the right singular vector matrix, construct a projection matrix, and use the projection matrix to pre-project the first snapshot matrix and the second snapshot matrix respectively to obtain the first projected snapshot matrix and the second projected snapshot matrix. The forward and backward dynamic mode decomposition method is used to estimate the set of discrete attenuation eigenvalues ​​based on the first and second projection snapshot matrices; The discrete attenuation eigenvalue set is mapped to a continuous-time relaxation time constant, and the modes are screened according to the physical prior criterion of cable depolarization current to obtain the effective relaxation components and the corresponding relaxation time constants. The Vandermonde matrix is ​​constructed based on the selected discrete decay eigenvalues. The current amplitude corresponding to each relaxation component is solved by the least squares method, and the relaxation spectrum containing the relaxation time constant and the current amplitude is output.

2. The cable relaxation spectrum identification method based on forward and backward dynamic mode decomposition according to claim 1, characterized in that, The discrete sequence of depolarization current is set as follows The embedding dimension is The constructed Hankel matrix Size is , The first snapshot matrix From the first column to the second column of the Hankel matrix The second snapshot matrix is ​​composed of columns. From the 2nd column to the 3rd column of the Hankel matrix The sequence consists of columns; the discrete sequence has N sampling points. Indicates the first The current amplitude at each sampling time, where m represents the column number.

3. The cable relaxation spectrum identification method based on forward and backward dynamic mode decomposition according to claim 1, characterized in that, The adaptive determination of the effective rank order using the singular value hard thresholding method includes: The augmented matrix is ​​decomposed to obtain a sequence of singular values, and a threshold is adaptively obtained. The subspaces corresponding to all singular values ​​greater than the threshold are retained, and their number is the effective rank order.

4. The cable relaxation spectrum identification method based on forward and backward dynamic mode decomposition according to claim 1, characterized in that, The right singular vector matrix obtained by performing singular value decomposition on the augmented matrix is: Take the first r columns to form a matrix The projection matrix is ​​constructed as follows: The pre-projection calculation is as follows: , ;in, Represents the first projected snapshot matrix. This represents the first snapshot matrix. Represents the second projection snapshot matrix. This represents the second snapshot matrix.

5. The cable relaxation spectrum identification method based on forward and backward dynamic mode decomposition according to claim 1, characterized in that, The step of estimating the set of discrete attenuation eigenvalues ​​based on the first and second projection snapshot matrices includes: Based on the first and second projection snapshot matrices, forward and backward operators are constructed, and equivalent operators are constructed through symmetric combination. The equivalent operators are then subjected to eigenvalue decomposition to obtain the discrete decay eigenvalue set.

6. The cable relaxation spectrum identification method based on forward and backward dynamic mode decomposition according to claim 1, characterized in that, The method for mapping the discrete decay feature value set to a continuous-time relaxation time constant is as follows: Obtaining the continuous time growth rate ,in, The sampling period is The discrete decay characteristic value; the relaxation time constant and retain the satisfaction The modality.

7. The cable relaxation spectrum identification method based on forward and backward dynamic mode decomposition according to any one of claims 1 to 6, characterized in that, The physical prior criteria include the decay criterion, the near-real criterion, the significance criterion, and the upper limit criterion for relaxation time; wherein: The attenuation criterion is to retain those that satisfy... or The modalities; the near-realism criterion is to retain the modes that satisfy... or Modes with amplitudes less than a preset threshold are used to suppress oscillating pseudo-modes; the significance criterion is to construct a current amplitude surrogate quantity and remove modes whose contribution intensity to the original depolarization current is lower than a set threshold; the relaxation time upper limit criterion is to remove modes whose relaxation time constant is significantly greater than the observation time window length.

8. The cable relaxation spectrum identification method based on forward and backward dynamic mode decomposition according to claim 7, characterized in that, The output, after including the relaxation spectrum of the relaxation time constant and the current amplitude, also includes: The depolarized current sequence is reconstructed using the obtained relaxation time constant and current amplitude, and the goodness of fit or normalized root mean square error between the reconstructed sequence and the original sequence is obtained.

9. The cable relaxation spectrum identification method based on forward and backward dynamic mode decomposition according to claim 8, characterized in that, The embedding dimension is based on the total number of sampling points. Set as to The values ​​are between; the sampling period is determined based on the sampling rate of the cable polarization-depolarization test equipment.

10. A cable relaxation spectrum identification system based on forward and backward dynamic mode decomposition, characterized in that, include: The data acquisition and embedding module is used to acquire the discrete sequence of cable insulation depolarization current, construct a Hankel matrix through Hankel delayed embedding, and generate a first snapshot matrix and a second snapshot matrix based on the Hankel matrix; The augmentation and order determination module is used to construct an augmented matrix containing the first snapshot matrix and the second snapshot matrix, perform singular value decomposition on the augmented matrix, and adaptively determine the effective rank order by using the singular value hard thresholding method. The pre-projection module is used to extract the first few columns of the right singular vector matrix according to the effective rank order, construct the projection matrix, and use the projection matrix to pre-project the first snapshot matrix and the second snapshot matrix respectively to obtain the first projected snapshot matrix and the second projected snapshot matrix. The eigenvalue estimation module is used to estimate the set of discrete decay eigenvalues ​​based on the first and second projected snapshot matrices using the forward and backward dynamic mode decomposition method. The mapping and filtering module is used to map the discrete attenuation feature set to a continuous-time relaxation time constant, and to filter the modes according to the physical prior criterion of cable depolarization current to obtain the effective relaxation components and the corresponding relaxation time constants. The amplitude calculation and output module is used to construct the Vandermonde matrix based on the selected discrete decay eigenvalues, solve the current amplitude corresponding to each relaxation component by the least squares method, and output the relaxation spectrum containing the relaxation time constant and the current amplitude.

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