A method for identifying parameters of an extended debye model based on dynamic modal decomposition

Abstract

The present application relates to the technical field of identification of dielectric response characteristics and aging state evaluation of capacitive power equipment, and particularly relates to a dynamic modal decomposition-based extended Debye model parameter identification method, which comprises the following steps: obtaining depolarization current data changing with time, constructing the data into a Hankel matrix, taking the first N columns and the last N columns to form a matrix A and a matrix B; performing economic SVD decomposition on the matrix A, setting a singular value threshold, discarding singular values less than the singular value threshold, and reconstructing the SVD decomposition matrix; solving X=A ‑1 B, then solving the eigenvalues and eigenvectors of X; solving the modes of X and sorting them in the order of decreasing energy; screening the modes, restoring the depolarization current by using the screened modes, judging the effect of dynamic modal decomposition, and judging whether to adjust the singular value threshold according to the restoration effect; calculating the branch number, time constant and relaxation intensity coefficient of the insulation extended Debye model. The parameter identification result obtained by the method has clear physical meaning.

Description

Technical Field

[0001] This invention relates to the field of insulation dielectric response characteristic identification and aging condition assessment technology for capacitive power equipment, and in particular to a method for parameter identification of extended Debye model based on dynamic mode decomposition. Background Technology

[0002] Serious accidents involving large capacitive power equipment (such as generators and transformers) not only damage the equipment itself but also disrupt power supply, causing enormous economic losses to society. With increasing service life and more complex operating conditions, the insulation systems of large capacitive power equipment face the combined effects of electrical, thermal, mechanical, and environmental stresses, leading to increasingly serious insulation aging problems and a greater likelihood of shutdown failures, posing a significant threat to the stable operation of the power system.

[0003] Traditional insulation condition assessment methods (such as insulation resistance testing, power frequency dielectric loss testing, partial discharge testing, and breakdown voltage testing) provide limited information on the dielectric response and are easily affected by field noise, potentially even damaging electrical equipment. In contrast, the time-domain dielectric response method—Polarization and Depolarization Current (PDC)—is gaining popularity due to its non-destructive nature and high accuracy. Current research on PDC for characterizing insulation condition primarily involves performing PDC tests on electrical equipment to obtain polarization and depolarization currents. These current data are then fitted to an extended Debye model to obtain circuit branch parameters and characteristic parameters, thereby achieving condition assessment of the electrical equipment. Therefore, the key to successfully and accurately obtaining the circuit parameters of the extended Debye model is the ability to apply the PDC method to power equipment in the field.

[0004] The Extended Debye Model (EDM) is a classic dielectric response model that represents the insulation of power equipment as equivalent to insulation resistance branches, geometric capacitance branches, and multiple RC series branches. By fitting PDC test data to the EDM, the insulation condition of power equipment can be diagnosed using the model parameters. Current research on branch identification methods using the EDM based on the PDC method mainly falls into the following categories: ① Pre-setting the number of branches, substituting the polarization / depolarization currents into the parameter matrix of each branch, and using optimization algorithms to obtain the optimal solution of the equations, thus achieving branch identification and parameter extraction. However, this method is prone to getting trapped in local optima, the polarization / depolarization current data needs to be filtered beforehand, and different filtering algorithms have a significant impact on the branch identification results. ② Using dominant mode parameter identification methods for branch identification. Among common methods, the Prony algorithm is too sensitive to noise in the input data, and interference is unavoidable in the PDC test data, resulting in less than ideal model recognition performance; the TLS-ESPRIT algorithm has some robustness to noise in the input data, but this algorithm usually assumes that the signal satisfies rotation invariance, but complex extended Debye models may not satisfy this assumption; the CMIF algorithm is also quite sensitive to noise in the input data, and in complex scenarios, the Debye model will exhibit dense modes, causing the algorithm to fail to perform effective recognition.

[0005] For example, in the prior art, Chinese invention patent document with publication number CN113779914A and publication date of December 10, 2021, discloses the following technical solution: A method for identifying parameters of extended Debye equivalent circuit of transformer oil-paper insulation based on TLS-ESPRIT. This method, based on the physical characteristic that the depolarization current is a superposition of exponential functions in the extended Debye equivalent model, performs singular value decomposition on the Hankel matrix constructed from sample data, determines the number of polarization branches based on the rate of change of singular values, and then uses the total least squares-rotational vector invariant technique (TLS-ESPRIT) algorithm to obtain the relaxation coefficient and time constant of each polarization branch. These are then substituted into the Debye equivalent circuit parameter identification formula to identify the resistance and capacitance parameters of each polarization branch.

[0006] In the above technical solution, the singular value change rate is solved, and the singular value change rate of 0 is used as the truncation criterion. The number of singular values ​​after truncation is the number of branches in the Debye model, but the physical meaning is not clear enough. Summary of the Invention

[0007] To address the aforementioned technical problems, this invention proposes an extended Debye model parameter identification method based on dynamic mode decomposition. After solving for the eigenvalues ​​of X, DMD modes representing different frequency components are constructed. From all modes, modes that conform to exponential decay are selected, and their physical meaning matches the response characteristics of each branch of the Debye model. Therefore, the parameter identification results obtained by this method have clear physical meaning.

[0008] This invention is achieved by adopting the following technical solution:

[0009] A parameter identification method for an extended Debye model based on dynamic mode decomposition includes the following steps:

[0010] Step S1. Obtain the data on the change of depolarization current over time, construct it into a Hankel matrix, and take the first N columns and the last N columns to form matrix A and matrix B;

[0011] Step S2. Perform economical SVD decomposition on matrix A, set a singular value threshold, discard singular values ​​smaller than the singular value threshold, and reconstruct the SVD decomposition matrix;

[0012] Step S3. Solve for X = A -1 B, then solve for the eigenvalues ​​and eigenvectors of X;

[0013] Step S4. Solve for the modes of X and sort them in order of decreasing energy;

[0014] Step S5. Screen the modes, select the screened modes to restore the depolarization current, judge the effect of dynamic mode decomposition, and judge whether to adjust the singular value threshold based on the restoration effect.

[0015] Step S6. Calculate the number of branches, time constant, and relaxation coefficient of the insulation extension Debye model.

[0016] Step S1 specifically includes the following steps:

[0017] Step S 11 Obtain two vectors: depolarization current current(i) (i = 1, 2, 3, ..., n) and time t(i) (i = 1, 2, 3, ..., n);

[0018] Step S 12 Step S 11 The length of the depolarization current vector becomes an even number n. If the vector length is odd, the last element of the depolarization current vector is discarded.

[0019] Step S 13 Using step S 12 The depolarization current data in the data is used to construct the Hankel matrix as shown below;

[0020]

[0021] Step S 14 Extract the first N columns and the last N columns from the Hankel matrix to form matrices A and B.

[0022] The value of N is n / 2.

[0023] Step S2 specifically includes the following steps:

[0024] Step S 21 Perform an economical SVD decomposition on matrix A to extract all singular values;

[0025] Step S 22 Set a singular value threshold and discard all singular values ​​that are less than the threshold.

[0026] Step S 23 Reconstruct the SVD decomposition matrix based on the retained singular values.

[0027] Step S3 specifically includes the following steps:

[0028] Step S 31 Using the reconstructed SVD decomposition matrix and the B matrix, solve for X = A. -1 B;

[0029] Step S 32 Find the eigenvalues ​​and eigenvectors of matrix X.

[0030] Step S4 specifically includes the following steps:

[0031] Step S 41 The modes are solved using the eigenvalues ​​of X, the SVD decomposition results, and the B matrix;

[0032] Step S 42 Solve for the energy of each mode and sort the modes in descending order of energy.

[0033] The mode screening in step S5 specifically refers to: based on the extended Debye equivalent circuit model of stator bar insulation, the modes are screened. When the mode slope is non-negative and the initial value is positive, the exponential decay requirement is met, the modes that meet the exponential decay requirement are retained, and the modes with negative slopes or non-positive initial values ​​are removed.

[0034] Step S6 specifically includes the following steps: Assume the eigenvalues ​​of X are λ1, λ2, ..., λ m Then the amplitude, time constant, and relaxation strength coefficient of each branch of the extended Debye model are:

[0035]

[0036] A i =|a i |,

[0037]

[0038] Where Ts is the sampling time interval.

[0039] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0040] 1. This method is highly robust to noise in field tests. It does not require high-order filtering of PDC test data and has low computational complexity. It only requires setting the decomposition threshold according to the magnitude of PDC test data to achieve modal decomposition of PDC test data, accurately identify the branch parameters of the extended Debye model, and thus diagnose the insulation status.

[0041] This invention differs from the conventional method for determining the number of branches in the Debye model.

[0042] Conventional methods for determining the number of branches in a Debye model include solving for the singular value rate of change, using a singular value rate of change of 0 as the truncation criterion, and determining the number of branches in the Debye model by the number of singular values ​​after truncation; or solving for the sparse augmentation magnitude vector and determining the number of branches in the extended Debye model based on the number of non-zero elements. However, these methods are not intuitive enough and their physical meaning is not clear enough.

[0043] In contrast, in this invention, after solving for the eigenvalues ​​of X, DMD modes representing different frequency components are constructed, and modes that conform to exponential decay are selected from all modes. Their physical meaning is consistent with the response characteristics of each branch of the Debye model. Therefore, the parameter identification results obtained by this method have clear physical meaning.

[0044] Furthermore, compared with traditional optimization fitting methods, this invention does not require presetting the number of branches of the extended Debye model, thus avoiding the problem of getting trapped in local optima. Moreover, the input data does not require high-order filtering, making it convenient to quickly identify the branch parameters of the extended Debye model on-site and thereby diagnose the insulation status.

[0045] 2. In existing technologies, conventional designs only remove smaller singular values, thus eliminating only minor noise. However, generator maintenance and testing sites typically experience significant interference, with noise levels potentially comparable to the depolarization current. Removing smaller singular values ​​is insufficient to effectively eliminate larger noise levels, potentially causing substantial interference with the solution results. Existing technologies also employ sparsity penalty functions to eliminate spurious modes introduced by noise interference, but this process requires multiple iterations to achieve noise immunity.

[0046] Conversely, this invention prevents computational divergence by setting a singularity threshold for deleting smaller singular values. This step removes smaller singular values ​​representing smaller noise, improving the algorithm's noise resistance. Then, by filtering modes at each order, modes exhibiting exponential decay are retained, and oscillating modes caused by noise are eliminated, further enhancing the algorithm's noise resistance. Through the combined effect of these two steps, without iteration, excellent noise resistance can be achieved with less computation, making it more suitable for on-site maintenance and testing.

[0047] Furthermore, the singular value threshold is not a completely fixed value; it can be updated during the identification process to make the identification results more accurate.

[0048] Furthermore, since this invention removes modes that do not conform to exponential decay, mode screening can be performed without other redundant calculations. This not only avoids excessive computational load and long processing time, but also ensures the parameter identification effect, has low equipment and instrument requirements, and is more suitable for on-site maintenance and testing.

[0049] 3. This invention aims to extract the dielectric response information contained in the time series obtained from PDC testing. Based on the principle of Dynamic Mode Decomposition (DMD), the depolarization current data is constructed into a Hankel matrix, with the first N columns designated as matrix A and the last N columns as matrix B. Matrix A and matrix B have a lead-lag relationship on the time scale, and thus the following mapping relationship exists: B = AX, where X is the mapping matrix containing the dielectric response information of the test system. By solving for the eigenvalues ​​of matrix X, the eigenvalues ​​representing the dielectric response information are separated from the eigenvalues ​​representing noise, allowing for the further solution of the Debye model. This step fully considers the lead-lag state information contained in the PDC test data. Attached Figure Description

[0050] The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments, wherein:

[0051] Figure 1 This is a schematic diagram of the process of the present invention;

[0052] Figure 2 This is a diagram illustrating the meaning of each branch of the extended Debye model in this invention;

[0053] Figure 3 This is a schematic diagram of PDC testing of generator stator bars in this invention;

[0054] Figure 4 This is a schematic diagram of the unfiltered depolarization current data input in this invention;

[0055] Figure 5 This is a schematic diagram showing the distribution of the characteristic roots obtained from the decomposition in this invention;

[0056] Figure 6 This is a diagram showing the effect of using the screened mode reduction depolarization current in this invention. Detailed Implementation

[0057] Example 1

[0058] As a basic embodiment of the present invention, the present invention includes a method for parameter identification of an extended Debye model based on dynamic mode decomposition, comprising the following steps:

[0059] Step S1. Obtain the data on the change of depolarization current over time, construct it into a Hankel matrix, and take the first N columns and the last N columns to form matrix A and matrix B.

[0060] Step S2. Perform economical SVD decomposition on matrix A, set a singular value threshold, discard singular values ​​smaller than the singular value threshold, and reconstruct the SVD decomposition matrix.

[0061] Step S3. Solve for X = A -1 B, then solve for the eigenvalues ​​and eigenvectors of X.

[0062] Step S4. Solve for the modes of X and sort them in order of decreasing energy.

[0063] Step S5. Filter the modes, select the filtered modes to restore the depolarization current, judge the effect of dynamic mode decomposition, and determine whether to adjust the singular value threshold based on the restoration effect.

[0064] Step S6. Calculate the number of branches, time constant, and relaxation coefficient of the insulation extension Debye model.

[0065] Example 2

[0066] As a preferred embodiment of the present invention, the present invention includes a method for parameter identification of an extended Debye model based on dynamic mode decomposition, comprising the following steps:

[0067] Step S1. Obtain the time-varying depolarization current data, construct it into a Hankel matrix, and take the first N columns and the last N columns to form matrices A and B. Specifically, this includes the following steps:

[0068] Step S 11 Obtain two vectors: depolarization current current(i) (i = 1, 2, 3, ..., n) and time t(i) (i = 1, 2, 3, ..., n).

[0069] Step S 12 Step S 11 The length of the depolarization current vector becomes an even number n. If the vector length is odd, the last element of the polarization current vector is discarded.

[0070] Step S 13 Using step S 12 The depolarization current data in the data are used to construct the Hankel matrix.

[0071] Step S 14Extract the first N columns and the last N columns from the Hankel matrix to form matrices A and B.

[0072] Step S2. Perform economical SVD decomposition on matrix A, set a singular value threshold, discard singular values ​​smaller than the singular value threshold, and reconstruct the SVD decomposition matrix.

[0073] Step S3. Solve for X = A -1 B, then solve for the eigenvalues ​​and eigenvectors of X.

[0074] Step S4. Solve for the modes of X and sort them in order of decreasing energy.

[0075] Step S5. Filter the modes, select the filtered modes to restore the depolarization current, judge the effect of dynamic mode decomposition, and determine whether to adjust the singular value threshold based on the restoration effect. Specifically, mode filtering refers to: based on the extended Debye equivalent circuit model of stator bar insulation, filtering the modes; when the mode slope is non-negative and the initial value is positive, it meets the exponential decay requirement, retaining the modes that meet the exponential decay requirement, and removing modes with negative slopes or non-positive initial values.

[0076] Step S6. Calculate the number of branches, time constant, and relaxation coefficient of the insulation extension Debye model.

[0077] Example 3

[0078] In another preferred embodiment of the present invention, the present invention includes a method for parameter identification of an extended Debye model based on dynamic mode decomposition, comprising the following steps:

[0079] Step S1. Obtain the data on the change of depolarization current over time, construct it into a Hankel matrix, and take the first N columns and the last N columns to form matrix A and matrix B.

[0080] Step S2. Perform economical SVD decomposition on matrix A, set a singular value threshold, discard singular values ​​smaller than the threshold, and reconstruct the SVD decomposition matrix. Specifically, this includes the following steps:

[0081] Step S 21 Perform an economical SVD decomposition on matrix A to extract all singular values;

[0082] Step S 22 Set a singular value threshold and discard all singular values ​​that are less than the threshold.

[0083] Step S 23 Reconstruct the SVD decomposition matrix based on the retained singular values.

[0084] Step S3. Solve for X = A -1B, then solve for the eigenvalues ​​and eigenvectors of X. Specifically, this includes the following steps:

[0085] Step S 31 Using the reconstructed SVD decomposition matrix and the B matrix, solve for X = A. -1 B;

[0086] Step S 32 Find the eigenvalues ​​and eigenvectors of matrix X.

[0087] Step S4. Solve for the modes of X and sort them in order of decreasing energy.

[0088] Step S5. Filter the modes, select the filtered modes to restore the depolarization current, judge the effect of dynamic mode decomposition, and determine whether to adjust the singular value threshold based on the restoration effect.

[0089] Step S6. Calculate the number of branches, time constant, and relaxation coefficient of the insulation extension Debye model.

[0090] Example 4

[0091] As another preferred embodiment of the present invention, the present invention includes a method for parameter identification of an extended Debye model based on dynamic mode decomposition, as described in the appendix to the specification. Figure 1 This includes the following steps:

[0092] Step S1. Obtain the time-varying depolarization current data, construct it into a Hankel matrix, and take the first N columns and the last N columns to form matrices A and B. Specifically, this includes the following steps:

[0093] Step S 11 The PDC testing device is used to test the object under test and obtain two vectors: depolarization current current(i) (i = 1, 2, 3, ..., n) and time t(i) (i = 1, 2, 3, ..., n).

[0094] Step S 12 Step S 11 The length of the depolarization current vector becomes an even number n. If the vector length is odd, the last element of the depolarization current vector is discarded.

[0095] Step S 13 Using step S 12 The depolarization current data in the data is used to construct the Hankel matrix as shown below;

[0096]

[0097] Step S 14Extract the first N columns and the last N columns from the Hankel matrix to form matrices A and B. The value of N can be n / 2.

[0098] Step S2. Perform economical SVD decomposition on matrix A, set a singular value threshold, discard singular values ​​smaller than the threshold, and reconstruct the SVD decomposition matrix. Specifically, this includes the following steps:

[0099] Step S 21 Perform an economical SVD decomposition on matrix A to extract all singular values.

[0100] Step S 22 Set a singular value threshold and discard all singular values ​​that are less than the threshold.

[0101] Step S 23 Reconstruct the SVD decomposition matrix based on the retained singular values.

[0102] Step S3. Solve for X = A -1 B, then solve for the eigenvalues ​​and eigenvectors of X. Specifically, this includes the following steps:

[0103] Step S 31 Using the reconstructed SVD decomposition matrix and the B matrix, solve for X = A. -1 B;

[0104] Step S 32 Find the eigenvalues ​​and eigenvectors of matrix X.

[0105] Step S4. Solve for the modes of X and sort them in order of decreasing energy. This specifically includes the following steps:

[0106] Step S 41 The modes are solved using the eigenvalues ​​of X, the SVD decomposition results, and the B matrix.

[0107] Step S 42 Solve for the energy of each mode and sort the modes in descending order of energy.

[0108] Step S5. Screen the modes, select the screened modes to restore the depolarization current, evaluate the effect of dynamic mode decomposition, and determine whether to adjust the singular value threshold based on the restoration effect. Specifically, this includes the following steps:

[0109] Step S 51 Based on the extended Debye equivalent circuit model of stator bar insulation, modes are screened. When the mode slope is non-negative and the initial value is positive, the exponential decay requirement is met. Modes that meet the exponential decay requirement are retained, while modes with negative slopes or non-positive initial values ​​are removed.

[0110] Step S 52The selected modes are used to reconstruct the signal, that is, the reconstructed data is obtained by superimposing the selected modes, and the difference between the reconstructed data and the original data is judged.

[0111] Step S 53 If the difference between the reconstructed data and the original data is small, proceed to step S6; if the difference between the reconstructed data and the original data is large, return to step S7. 22 Adjust the singularity threshold and re-decompose the depolarization current data.

[0112] This method has been tested by introducing a quantization accuracy coefficient, and the restoration rate has reached 99%.

[0113] Step S6. Calculate the number of branches, time constant, and relaxation coefficients of the extended insulation Debye model. Specifically, under conditions of high fidelity, the number of branches, time constant, and relaxation coefficients of the extended insulation Debye model are solved using, for example, the first 5 selected modes. Assume the eigenvalues ​​of X are λ1, λ2, ..., λ m Then the amplitude, time constant, and relaxation strength coefficient of each branch of the extended Debye model are:

[0114]

[0115] A i =|a i |,

[0116]

[0117] Where Ts is the sampling time interval.

[0118] Example 5

[0119] As another specific embodiment of the present invention, the appendix to the specification... Figure 2 To expand the meaning of the branch diagrams in the Debye model, the instruction manual includes... Figure 3 This invention utilizes a PDC testing device to test the object under test. Specifically, this invention includes a method for parameter identification of an extended Debye model based on dynamic mode decomposition, comprising the following steps:

[0120] 1. According to step S1 in any of the embodiments 1 to 4, refer to the appendix of the specification. Figure 3 A PDC testing device was used to perform a PDC test on a stator bar of a hydro-generator that had been in operation for 17 years. The depolarization voltage was 1kV, the depolarization time was 90s, and the sampling interval was 0.07s. The data on the change of depolarization current over time were obtained as shown in the attached instruction manual. Figure 4 As shown.

[0121] 2. Solve sequentially according to steps S2 and S3 in any of the embodiments 1 to 4 to obtain the eigenvalue distribution as shown in the appendix to the specification. Figure 5 As shown.

[0122] 3. Solve sequentially according to steps S4, S5, and S6 in any of the embodiments 1 to 4, and use the selected modes to restore the depolarization current signal. The restoration effect is shown in the appendix of the instruction manual. Figure 6 As shown.

[0123] In summary, any other corresponding modifications made by those skilled in the art after reading this invention document, without requiring creative mental effort, based on the technical solutions and concepts of this invention, are all within the scope of protection of this invention.

Claims

1. A method for parameter identification of an extended Debye model based on dynamic mode decomposition, characterized in that: Includes the following steps: Step S1. Obtain the data on the change of depolarization current over time, construct it into a Hankel matrix, and take the first N columns and the last N columns to form matrix A and matrix B; Step S2. Perform economical SVD decomposition on matrix A, set a singular value threshold, discard singular values ​​smaller than the singular value threshold, and reconstruct the SVD decomposition matrix; Step S3. Solve Then, the eigenvalues ​​and eigenvectors of X are solved; Step S4. Solve for the modes of X and sort them in order of decreasing energy; Step S5. Screen the modes, select the screened modes to restore the depolarization current, judge the effect of dynamic mode decomposition, and determine whether to adjust the singular value threshold based on the restoration effect. Step S6. Calculate the number of branches, time constant, and relaxation coefficient of the insulation extension Debye model.

2. The method for parameter identification of the extended Debye model based on dynamic mode decomposition according to claim 1, characterized in that: Step S1 specifically includes the following steps: Step S 11 Obtain the depolarization current. ,time Two vectors, ; Step S 12 Step S 11 The length of the depolarization current vector becomes an even number n. If the vector length is odd, the last element of the depolarization current vector is discarded. Step S 13 Using step S 12 The depolarization current data in the data is used to construct the Hankel matrix as shown below; Step S 14 Extract the first N columns and the last N columns from the Hankel matrix to form matrices A and B.

3. The method for parameter identification of the extended Debye model based on dynamic mode decomposition according to claim 2, characterized in that: The value of N is n / 2.

4. The method for parameter identification of the extended Debye model based on dynamic mode decomposition according to claim 1, characterized in that: Step S2 specifically includes the following steps: Step S 21 Perform an economical SVD decomposition on matrix A to extract all singular values; Step S 22 Set a singular value threshold and discard all singular values ​​that are less than the threshold. Step S 23 Reconstruct the SVD decomposition matrix based on the retained singular values.

5. The method for parameter identification of the extended Debye model based on dynamic mode decomposition according to claim 1, characterized in that: Step S3 specifically includes the following steps: Step S 31 Using the reconstructed SVD decomposition matrix and B matrix, solve for... ; Step S 32 Find the eigenvalues ​​and eigenvectors of matrix X.

6. The method for parameter identification of the extended Debye model based on dynamic mode decomposition according to claim 1, characterized in that: Step S4 specifically includes the following steps: Step S 41 The modes are solved using the eigenvalues ​​of X, the SVD decomposition results, and the B matrix; Step S 42 Solve for the energy of each mode and sort the modes in descending order of energy.

7. The method for parameter identification of the extended Debye model based on dynamic mode decomposition according to claim 1, characterized in that: The mode screening in step S5 specifically refers to: based on the extended Debye equivalent circuit model of stator bar insulation, the modes are screened. When the mode slope is non-negative and the initial value is positive, the exponential decay requirement is met, the modes that meet the exponential decay requirement are retained, and the modes with negative slopes or non-positive initial values ​​are removed.

8. The method for parameter identification of the extended Debye model based on dynamic mode decomposition according to claim 7, characterized in that: Step S6 specifically includes the following steps: Assume the eigenvalues ​​of X are... Then the amplitude, time constant, and relaxation strength coefficient of each branch of the extended Debye model are: ， ， ， Where Ts is the sampling time interval.

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