A method for evaluating harmonic resistance of transformer bushing

Abstract

The application relates to the technical field of evaluation, in particular to a transformer bushing anti-harmonic performance evaluation method, which comprises the following steps: acquiring real-time sampling sequences of voltage and current of a transformer bushing, filtering the real-time sampling sequences by adopting a fractional order Kalman filter to obtain noiseless sequences; constructing a data matrix and mapping to a reproducing kernel Hilbert space by utilizing kernel canonical correlation analysis, solving a generalized eigenvalue problem, acquiring a canonical correlation coefficient and identifying a broadband state space model; performing Schur decomposition on a model state transition matrix, constructing a weighted function by taking the reciprocal of the canonical correlation coefficient as an uncertainty penalty index, weighting non-diagonal elements and calculating a Frobenius norm to obtain a modal coupling credibility index; mapping discrete system eigenvalues to a continuous time domain by utilizing logarithmic transformation, extracting self-vibration frequency and a decay factor, and combining the modal coupling credibility index to comprehensively evaluate the anti-harmonic performance of the transformer bushing; the application has high identification precision, and the evaluation result is suitable for complex harmonic field working conditions.

Description

Technical Field

[0001] This application relates to the field of evaluation technology, and in particular to a method for evaluating the harmonic resistance performance of transformer bushings. Background Technology

[0002] Transformers are crucial core equipment in power systems, and transformer bushings, as electrical insulation and mechanical support components between the leads and the transformer tank, are critical to the safety and stability of the entire power grid. When transformer bushings are exposed to harmonic environments for extended periods, the internal insulation medium can experience localized overheating, increased dielectric loss, and accelerated insulation aging due to the combined effects of high-frequency electrical and thermal stresses. In extreme cases, this can even lead to insulation breakdown and equipment explosions. Therefore, real-time monitoring of the operating status of transformer bushings under harmonic environments and scientific evaluation of their harmonic resistance performance have become important research topics for preventing major equipment accidents and ensuring the reliable operation of power systems.

[0003] In actual industrial settings, the voltage and current operating signals acquired are often mixed with background noise and measurement interference, posing a challenge to obtaining high-quality data representing bushing characteristics. The electrical response of transformer bushings under broadband harmonics is extremely complex, exhibiting strong high-dimensional nonlinear characteristics and cross-coupling effects of multiple system modes. Conventional system modeling and state assessment methods often fail to identify a spatial model that can completely represent the operating state of the equipment under strong noise interference and broadband complex operating conditions. Furthermore, in the analysis of system characteristic modes, there is a lack of scientific quantitative indicators that integrate modal coupling strength with parameter identification reliability, making it impossible to systematically judge the stability margin of the equipment under various harmonic impacts. Currently, there is an urgent need to propose a performance evaluation method that can suppress environmental noise, extract broadband characteristic models, and comprehensively quantify the overall stability and harmonic resistance capability of transformer bushings from multiple dimensions. Summary of the Invention

[0004] To improve the accuracy and reliability of transformer bushing condition assessment under harmonic environments, this application provides a method for assessing the harmonic resistance performance of transformer bushings.

[0005] This application provides a method for evaluating the harmonic resistance performance of transformer bushings, including the following steps:

[0006] S1. Obtain the real-time sampling sequence of transformer bushing port voltage and current; filter the real-time sampling sequence to obtain a noiseless voltage and current sequence for model identification; S2. Construct a data matrix, map the data matrix to the regenerative kernel Hilbert space, solve the generalized eigenvalue problem formed by the covariance matrix in the space, obtain typical correlation coefficients and state sequences, and identify the broadband state-space model representing the characteristics of transformer bushings. The model includes: state transition matrix A; S3. Perform Schul decomposition on the state transition matrix A to obtain the upper triangular form of the Schul matrix T and the unitary matrix U. The diagonal elements of the Schul matrix T are the system eigenvalues, representing the system modes. Construct a weighted function matrix using the reciprocal of the canonical correlation coefficient as the uncertainty penalty index of the system modes. Weight the off-diagonal elements of the Schul matrix T, calculate the Frobenius norm of the weighted off-diagonal element matrix, and obtain the modal coupling credibility index that quantifies the modal coupling strength and identification credibility. S4. Based on the eigenvalues ​​of all discrete systems in the state transition matrix A, the logarithmic transformation formula is used to map them to the continuous time domain to obtain the continuous poles of the system in the complex plane. The natural frequencies and attenuation factors corresponding to the continuous poles are extracted to evaluate the anti-harmonic resonance capability of each mode. Combined with the mode coupling credibility index, the anti-harmonic performance of the transformer bushing is comprehensively calculated and evaluated.

[0007] This application can effectively extract and identify a broadband state-space model that fully represents the operating state of equipment under strong noise interference and wide-band complex operating conditions. In addition, it proposes a scientific quantitative index that integrates modal coupling strength and parameter identification reliability, which solves the problem that existing methods cannot systematically judge the stability margin of equipment under various harmonic impacts, thereby realizing a multi-dimensional comprehensive quantification of the overall harmonic resistance capability of transformer bushings.

[0008] Preferably, the real-time sampling sequence of the transformer bushing port voltage and current includes: connecting the voltage transformer and the current transformer to the end test terminal and the grounding lead of the transformer bushing, respectively. High-frequency noise in the output signals of the voltage transformer and current transformer is filtered out using a low-pass anti-aliasing filter; The filtered signal is converted from analog to digital using a synchronous acquisition terminal at a set sampling frequency, and the data points at each moment are extracted to form a real-time sampling sequence of the port voltage and the current flowing through it.

[0009] This application constructs a non-intrusive real-time voltage and current monitoring circuit and inserts a low-pass anti-aliasing filter, which can effectively reduce high-frequency electromagnetic interference noise caused by switching operations, lightning pulses or partial discharges, effectively prevent frequency aliasing caused by high-frequency interference during subsequent sampling, and ensure the authenticity and quality of the benchmark dataset for feature analysis.

[0010] Preferably, obtaining the noiseless voltage and current sequence for model identification includes: setting initial values ​​for the process noise covariance matrix and the measurement noise covariance matrix of the fractional-order Kalman filter, calibrating the fractional-order difference order, and constructing a fractional-order state-space model with voltage and current as state variables. Discretize the state equations of the fractional-order state-space model to obtain the predicted state value and prediction error covariance matrix at the current time step; Calculate the Kalman gain matrix at the current time step, use the real-time sampling sequence as the observation input, and update the state prediction value in combination with the observation matrix to obtain the optimal state estimation sequence; The voltage and current components in the optimal state estimation sequence are output as noiseless voltage and current sequences for model identification.

[0011] This application combines the memory characteristics of the insulating dielectric response under harmonic distortion to construct a fractional-order state-space model. By utilizing the memory-forgetting characteristics of the fractional-order derivative terms, the state is updated more accurately. This allows for deep smoothing and correction of the original signal contaminated by noise, and the extraction of high-quality noise-free voltage and current components, providing a clean time-series data foundation for subsequent high-precision broadband model identification.

[0012] Preferably, the model further includes: an input matrix B, an output matrix C, and an action matrix D.

[0013] Preferably, the identification of the broadband state-space model includes: concatenating historical voltage and current sequences at different times to form a historical data Hankel matrix, and concatenating voltage and current sequences corresponding to future times to form a future data Hankel matrix; selecting a Gaussian radial basis kernel function to calculate the kernel matrix corresponding to the historical and future data Hankel matrices to achieve mapping to the regenerative kernel Hilbert space; constructing a correlation matrix and performing singular value decomposition based on the empirical covariance matrix and empirical cross-covariance matrix in the kernel matrix calculation space, extracting the largest non-zero singular values ​​of the same order as the broadband state-space model as canonical correlation coefficients, transposing the corresponding left singular vectors and combining them to form a state sequence; using the state sequence, solving the state-space evolution process of the system using the orthogonal projection method, and executing the least squares regression algorithm to calculate the state transition matrix A, input matrix B, output matrix C, and action matrix D of the broadband state-space model.

[0014] This application effectively addresses the strong high-dimensional nonlinear characteristics exhibited by transformer bushings under broadband harmonic effects by introducing a Gaussian radial basis kernel function to map the data matrix to a high-dimensional regenerative kernel Hilbert space. By combining a regularization constant to eliminate mean drift and prevent high-dimensional matrix singularity, the application ensures the numerical absolute stability of the generalized eigenvalue problem and the solution process of the state space matrix.

[0015] Preferably, the step of performing Schul decomposition on the state transition matrix A to obtain an upper triangular form Schul matrix T and a unitary matrix U, where the diagonal elements of the Schul matrix T are system eigenvalues ​​representing system modes, includes: reducing the state transition matrix A to a Shanghai Semburg matrix using the Householder transformation algorithm; performing a shifted QR iteration calculation on the Shanghai Semburg matrix, converging iteratively to a quasi-upper triangular matrix in real Schul form; extracting the real elements on the main diagonal of this quasi-upper triangular matrix and the conjugate complex eigenvalues ​​of the 2×2 real diagonal blocks on the secondary diagonal, which together serve as the system eigenvalue set of the state transition matrix A to represent each system mode; or performing a unitary similarity transformation in the complex domain to obtain a pure upper triangular form complex Schul matrix T and extracting its main diagonal complex eigenvalues; multiplying all orthogonal transformation matrices in each iteration to obtain the corresponding orthogonal transformation composite matrix, which is then used as the unitary matrix U.

[0016] This application employs Householder transformation and QR iterative algorithm with implicit shift to perform Schul decomposition on the state transition matrix. This method can stably and accurately converge the matrix and extract complex elements representing system characteristics as system modes, avoiding the numerical instability that may be caused by direct eigenvalue decomposition. This lays a reliable mathematical orthogonal transformation foundation for subsequent quantification of the degree of mode coupling.

[0017] Preferably, constructing the weighted function matrix includes: Extract all canonical correlation coefficients and calculate their reciprocals one by one to form a main diagonal matrix, which is then used as the weighting function matrix of the system.

[0018] Preferably, obtaining the modal coupling credibility index, which quantifies the modal coupling strength and identification credibility, includes: Multiply the weighted function matrix on the left by the Schul matrix T after removing the diagonal elements to form a pure off-diagonal matrix, and then perform matrix multiplication to obtain the weighted off-diagonal matrix. Squaring the absolute value of each element in the weighted off-diagonal matrix. The Frobenius norm is calculated by summing the results of all squaring operations in the matrix and taking the square root of the sum. The obtained Frobenius norm value is used as the modal coupling credibility index.

[0019] Preferably, the evaluation of the harmonic resonance resistance of each mode includes: The continuous poles corresponding to all eigenvalues ​​are obtained by using the conversion formula from discrete to continuous, and the target harmonic poles within the preset harmonic frequency band are selected. Calculate the ratio of the absolute value of the real part of each target harmonic pole to the complex modulus of the pole to obtain the damping ratio. Select the minimum damping ratio among all target harmonic poles and use this minimum damping ratio as the harmonic resonance suppression margin. By setting a fixed performance weighting coefficient, multiplying the harmonic resonance suppression margin by the performance weighting coefficient, and subtracting the modal coupling confidence index, the value of the harmonic suppression performance evaluation is obtained.

[0020] Preferably, the comprehensive calculation and evaluation of the harmonic resistance performance of transformer bushings includes: The value of the anti-harmonic performance assessment is compared with a pre-set safety judgment threshold. When the value of the anti-harmonic performance assessment is greater than the safety judgment threshold, the transformer bushing is determined to have anti-harmonic performance. When the value of the anti-harmonic performance assessment is not greater than the safety judgment threshold, the transformer bushing is determined to have lost its anti-harmonic performance.

[0021] The technical solution of this application has the following beneficial technical effects: This application uses a fractional-order Kalman filter to process the sampling sequence under harmonic conditions, effectively filtering out complex background noise and providing high-quality noise-free data for subsequent model identification. By using kernelized canonical correlation analysis to map the data to the regenerative kernel Hilbert space, the nonlinear characteristics of the data can be deeply explored, and the state-space model characterizing the broadband characteristics of transformer bushings can be accurately identified.

[0022] Meanwhile, this application innovatively performs Schul decomposition on the state transition matrix and constructs a weighting function using the reciprocal of the canonical correlation coefficient as an uncertainty penalty index. The off-diagonal elements of the Schul matrix are weighted and the Frobenius norm is calculated. The modal coupling credibility index constructed in this way not only quantifies the coupling strength between system modes, but also fully reflects the reliability of the identification results.

[0023] Furthermore, this application uses logarithmic transformation to accurately map discrete eigenvalues ​​to the continuous-time complex plane, extracts natural frequencies and attenuation factors to intuitively reflect the anti-harmonic resonance capability of each mode, and uses the modal coupling credibility index for comprehensive evaluation, avoiding the one-sidedness of relying solely on eigenvalue evaluation, and greatly improving the accuracy and engineering practical value of transformer bushing performance evaluation in complex harmonic environments. Attached Figure Description

[0024] Figure 1 A flowchart of a method for evaluating the harmonic resistance performance of transformer bushings; Figure 2 This is a schematic diagram of the current time-domain waveform; Figure 3 The diagram shows the performance of the three schemes. Detailed Implementation

[0025] The technical solutions in the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, and not all embodiments.

[0026] Figure 1 This is a flowchart of a method for evaluating the harmonic resistance performance of transformer bushings according to an embodiment of this application. Figure 1 As shown, the method for evaluating the harmonic resistance performance of transformer bushings includes steps S1 to S4, which are described in detail below.

[0027] S1. Acquire the real-time sampling sequence and perform fractional-order Kalman filtering.

[0028] Specifically, the real-time sampling sequence of the port voltage and current flowing through the transformer bushing under harmonic environment is obtained; a fractional-order Kalman filter is used to filter the real-time sampling sequence to obtain a noiseless voltage and current sequence for model identification.

[0029] Furthermore, analog electrical signals are acquired through wideband voltage and current transformers deployed on the transformer bushing side. These signals are then discretized using a high-precision analog-to-digital converter chip, ADS1256. The sampling frequency is set to 10,000 times per second. A first-in-first-out circular queue is constructed in the hardware industrial control computer memory based on the C++ standard template library to cache the data and obtain the real-time sampling sequence. The specific process of using a fractional-order Kalman filter for filtering involves establishing a discrete-time fractional-order state-space model based on Grunwald-Letnikov fractional-order calculus theory, setting the system fractional-order constant and the initial prior error covariance matrix. In the prediction update step, the optimal state estimate of the previous time step is used to predict the prior state at the current time step, and the prior error covariance matrix is ​​updated by combining the memory-forgetting characteristics of the fractional-order derivative term. In the measurement update step, the inverse function is used to calculate the Kalman gain matrix, and the prior state estimate is corrected by combining the currently acquired real-time sampling sequence to obtain the optimal posterior state estimate. The posterior error covariance matrix is ​​updated synchronously. After iterative convergence for a preset maximum number of loops, the filtered and smoothed state vector time series is extracted, and the sequence is used as the noiseless voltage sequence and noiseless current sequence for subsequent model identification.

[0030] In an optional embodiment, acquiring the real-time sampling sequence of the transformer bushing's port voltage and flowing current under harmonic conditions includes: Connect the voltage transformer and the current transformer to the end test terminal and the grounding lead of the transformer bushing, respectively. High-frequency noise in the output signals of the voltage transformer and current transformer is filtered out using a low-pass anti-aliasing filter; The filtered signal is converted from analog to digital using a synchronous acquisition terminal at a set sampling frequency, and the data points at each moment are extracted to form a real-time sampling sequence of the port voltage and the current flowing through it.

[0031] It should be noted that, in order to avoid the failure of high-frequency aliasing and severe waveform distortion caused by conventional direct sampling of transformer bushings under strong electromagnetic harmonic environment and high-frequency partial discharge interference, this application uses a non-intrusive circuit to connect a low-pass anti-aliasing filter in series and simultaneously performs high-frequency analog-to-digital conversion to forcibly cut off the interference in the over-limit frequency band.

[0032] In actual transformer bushing operation scenarios, wideband voltage transformers and wideband current transformers with an accuracy class of not less than 0.2 are selected and electrically connected to the end flange test terminals and the grounding lead of the end screen of the transformer bushing, respectively, to construct a non-intrusive real-time voltage and current monitoring loop. To prevent frequency aliasing caused by high-frequency interference during subsequent sampling, an 8th-order Butterworth low-pass anti-aliasing filter is inserted in series in both sensor output links, and the cutoff frequency of the filter is set to 10kHz to reduce high-frequency electromagnetic interference noise caused by switching operations, lightning pulses, or partial discharges.

[0033] After filtering, a data terminal supporting multi-channel synchronous acquisition is used. The ADC analog-to-digital conversion bit width is set to 16 bits, and the sampling frequency is configured to 25.6kHz, that is, 512 synchronous data points are acquired per cycle at the power frequency of 50Hz. After the terminal starts, the analog-to-digital conversion process is continuously executed. For example, a continuous observation time window with a duration of 0.2 seconds is captured to extract waveform data points at each discrete moment. Two one-dimensional real number vectors with a dimension of 5120×1 are constructed from 5120 discrete voltage sampling values ​​and 5120 discrete current sampling values, which serve as the real-time sampling sequence of the port voltage and the current flowing through it. The data is then imported into the database as the benchmark dataset for feature analysis.

[0034] In an optional embodiment, the step of employing a fractional-order Kalman filter to filter the real-time sampling sequence to obtain a noiseless voltage and current sequence for model identification includes: Initial values ​​for the process noise covariance matrix and measurement noise covariance matrix of the fractional-order Kalman filter are set, the fractional-order difference order is calibrated, and a fractional-order state-space model with voltage and current as state variables is constructed. Discretize the state equations of the fractional-order state-space model to obtain the predicted state value and prediction error covariance matrix at the current time step; Calculate the Kalman gain matrix at the current time step, use the real-time sampling sequence as the observation input, and update the state prediction value in combination with the observation matrix to obtain the optimal state estimation sequence; The voltage and current components in the optimal state estimation sequence are output as noiseless voltage and current sequences for model identification.

[0035] In the initialization phase of the filtering algorithm, the initial value of the main diagonal of the 2×2 dimensional process noise covariance matrix Q is set to... The initial value of the main diagonal of the 2×2 dimensional measurement noise covariance matrix R is set to 0.01. Based on the memory characteristics of the insulating dielectric response under harmonic distortion, a fractional-order constant is set. The value is 0.85; the two-dimensional state variable vector of the system is constructed as a state vector. A continuous domain fractional state-space model with a 0.85 fractional derivative operator is constructed by combining the state transition matrix A.

[0036] Furthermore, the system-level calibration method for setting the initial value of the main diagonal of the measurement noise covariance matrix R specifically includes: when the transformer bushing is in a physical dormant state with zero voltage and zero current, continuously capturing the background high-frequency environmental white noise time series data of the underlying line using a synchronous acquisition terminal; calculating the static signal variance of the environmental white noise time series data within a preset observation time window, and using it as a physical benchmark to anchor the initial value of the main diagonal of the measurement noise covariance matrix R, for example, calibrating it to 0.01.

[0037] Furthermore, the method for calibrating the initial value of the main diagonal of the process noise covariance matrix Q specifically includes: pre-obtaining the maximum time-series drift rate of the leakage current of the transformer bushing insulation medium under the allowable temperature rise operating limit, using the square of the maximum time-series drift rate as the upper limit characterizing the uncertainty of the system state evolution, and calibrating the initial value of the main diagonal of the process noise covariance matrix Q, for example, calibrating it as follows: .

[0038] Furthermore, the physical testing method for calibrating the fractional-order differential order specifically includes: applying a broadband dielectric frequency response scanning test to the insulating dielectric samples of the same batch of transformer bushings to obtain the broadband time-series evolution curve of the complex dielectric constant during polarization relaxation; extracting the characteristic logarithmic slope at the inflection point where the high-frequency asymptote and the low-frequency asymptote intersect in the low-frequency dispersion region of the time-series evolution curve; and physically mapping the absolute value of the characteristic logarithmic slope to the fractional-order differential order of the state-space model, which is preferably calibrated to 0.85 in this embodiment.

[0039] During the filtering iteration, the Grünwaldretnikov calculus formula with a truncated memory length L of 30 is used to discretize and expand the differential operator of the model's state equation. The optimal state from the past 30 historical time steps and the binomial attenuation coefficient are extracted and superimposed to obtain the state prediction value for the current time step and the updated prior error covariance matrix. The current Kalman gain matrix is ​​calculated, using the real-time voltage and current sampling sequence as the observation input. The observation matrix H is set to a standard second-order identity matrix and substituted into the filtering correction equation to correct the state prediction value. After traversing all 5120 sampling times, the voltage and current data components in the truncated smooth sequence are separated, resulting in noise-free voltage and current sequences. (Refer to...) Figure 2 The waveform is taken from the original sampled current in the 0.2-second observation window, superimposed with operating condition harmonics and electromagnetic interference glitches. It can verify the integrity of the output data of the current transformer, anti-aliasing filter and synchronous ADC acquisition link, and serve as the original timing reference material before fractional-order Kalman filter noise reduction.

[0040] It should be added that the calibration procedure for the above-mentioned truncated memory length L includes: based on the physical latency of memory addressing and the computing power overhead of a single floating-point operation of the underlying data acquisition industrial control computer, the maximum historical state vector stacking throughput that the microprocessor can tolerate within a single discrete time step is calculated, and under the physical boundary constraint that the comprehensive computation time of a single filtering prediction correction is strictly less than the microsecond-level sampling interval under the power frequency cycle, the throughput is rounded down to obtain the truncated memory length L.

[0041] The procedure for obtaining the preset maximum number of iterations includes: under the physical reference condition of no load and no harmonic injection, extracting the standard voltage timing sequence of the transformer bushing, artificially injecting Gaussian white noise with known variance through a signal generator and then importing it into the Kalman filter algorithm iterative model, monitoring the convergence difference of the Frobenius norm of the state prediction error covariance matrix between two adjacent iterations in the underlying register in real time, and locking the total number of iterations corresponding to the first decay and stabilization of this difference to the quantization noise floor physical threshold of the underlying analog-to-digital converter chip as the preset maximum number of iterations.

[0042] In this way, by combining the high-frequency anti-aliasing hardware constraints of the front-end physical monitoring link with the fractional-order state-space prediction software algorithm with wideband historical memory forgetting characteristics, a pure electrical sequence that can faithfully reflect the intrinsic polarization evolution law of the bushing insulation medium can be accurately extracted from the composite stray signal heavily contaminated by electromagnetic shock and background noise. This blocks the error transmission of high-frequency random disturbances to the subsequent high-dimensional matrix mapping process from the physical source, providing a high-precision data base without phase shift and amplitude distortion for the system's wideband identification model.

[0043] S2. Construct a data matrix and identify a broadband state-space model of transformer bushing characteristics.

[0044] Specifically, a data matrix is ​​constructed with the noiseless voltage sequence as input and the noiseless current sequence as output. Kernelized canonical correlation analysis is used to map the data matrix to the regenerative kernel Hilbert space. The generalized eigenvalue problem formed by the covariance matrix in the space is solved to obtain the canonical correlation coefficient and state sequence. A broadband state-space model representing the characteristics of the transformer bushing is identified. The model includes a state transition matrix A, an input matrix B, an output matrix C, and an action matrix D.

[0045] The noiseless voltage and current sequences are arranged in time-lag order to construct input-output Hankel matrices representing past and future time-domain information, respectively, as initial data matrices. The inner product of the data is calculated using a Gaussian radial basis function, and the Hankel matrices are nonlinearly mapped to a high-dimensional regenerative kernel Hilbert space. Within this space, the autocovariance matrices of the past and future data matrices, as well as their mutual covariance matrices, are calculated, and a generalized eigenvalue decomposition equation for the canonical correlation variables is constructed. The linalgeigh function from the Scipy library is used to solve the generalized eigenvalue problem using the Choliski decomposition method. The generalized eigenvalues ​​are obtained, and the corresponding eigenvectors are projected back into the original low-dimensional space through the kernel matrix to obtain the system state sequence. The arithmetic square root of the generalized eigenvalues ​​is extracted as the canonical correlation coefficient.

[0046] The current state sequence and the noiseless voltage input sequence are combined to construct the independent variable matrix, and the next state sequence and the noiseless current output sequence are combined to construct the dependent variable matrix. Singular value decomposition is used to perform least squares regression calculation. The state transition matrix A, input matrix B, output matrix C and action matrix D in the broadband state space model are obtained by matrix pseudo-inverse operation in one step.

[0047] It should be noted that the broadband state-space model is a model used to quantitatively describe the mapping relationship between the dynamic response characteristics of the internal insulating medium of a transformer bushing and the electrical quantities at the external ports under broadband excitation including fundamental and higher harmonics. Since transformer bushings exhibit complex frequency-varying characteristics and high-frequency distributed parameter effects under harmonic environments, this application uses a broadband state-space model to characterize their global dynamic behavior.

[0048] Discrete-time broadband state-space model is based on state equations and output equation Composition, in which The input variable is the noiseless voltage sequence. The output variable is the noiseless current sequence. The hidden variables, or state sequences, represent the transient evolution process within the system.

[0049] The four core matrices A, B, C, and D in the model define the broadband transmission characteristics of the transformer bushing, specifically: State transition matrix A: Represents the change of the internal state of the transformer bushing over time. It is the core of the system's inherent properties, containing all the system's eigenvalues, and determines the self-resonant frequency and damping attenuation characteristics of the transformer bushing in a wide frequency band, i.e., the attenuation factor source.

[0050] Input matrix B: represents the degree of coupling influence of external input excitation on various state variables inside the system, that is, how the external voltage signal drives or excites the polarization state change inside the bushing.

[0051] Output matrix C: Represents how the linear combination of the internal state variables of the system is transformed into an externally observable output response, reflecting the mapping relationship between the energy release and charge displacement inside the bushing to the port current.

[0052] Action matrix D: Represents the instantaneous effect of the input voltage on the output current at the current moment; For capacitive devices such as transformer bushings, under high-frequency harmonic conditions, high-frequency abrupt changes in the input voltage will generate a direct displacement current. Matrix D is used to fit the high-frequency direct coupling effect that bypasses internal state changes and feeds forward from voltage to current.

[0053] In an optional embodiment, the construction of a data matrix with the noiseless voltage sequence as input and the noiseless current sequence as output is performed. Kernelized canonical correlation analysis is used to map the data matrix to a regenerative kernel Hilbert space. The generalized eigenvalue problem formed by the covariance matrix in the space is solved to obtain canonical correlation coefficients and state sequences. A broadband state-space model representing the characteristics of the transformer bushing is identified. The model includes a state transition matrix A, an input matrix B, an output matrix C, and an action matrix D, comprising: Historical voltage and current sequences at different times are concatenated to form a historical data Hankel matrix, and voltage and current sequences corresponding to future times are concatenated to form a future data Hankel matrix. The kernel matrix corresponding to the historical data Hankel matrix and the future data Hankel matrix is ​​calculated by selecting the Gaussian radial basis kernel function, so as to realize the mapping to the regenerating kernel Hilbert space; Calculate the empirical covariance matrix and empirical cross-covariance matrix in the space based on the kernel matrix, construct the correlation matrix and perform singular value decomposition, extract the largest non-zero singular values ​​with the same number of orders as the broadband state space model as the canonical correlation coefficient, and combine the corresponding left singular vectors after transposing them to form a state sequence. The state space evolution process of the system is solved by orthogonal projection using the state sequence, and the least squares regression algorithm is executed to calculate the state transition matrix A, input matrix B, output matrix C and action matrix D of the broadband state space model.

[0054] Optionally, Z-score normalization is performed on the extracted noiseless voltage and current sequences, respectively. For the noiseless voltage and current sequences at 5120 time steps, the number of rows / blocks in the model fitting is specified as 20. Using a time sliding window, the dataset containing inputs and outputs is divided into past and future windows, reconstructing a 40×5000 historical data Hankel matrix and a future data Hankel matrix of the same dimension. A Gaussian radial basis kernel function is selected for mapping calculation, and the hyperparameter kernel width is set to 2.5, generating two kernel matrices of size 5000×5000, which are then mapped to the regenerating kernel Hilbert space.

[0055] Furthermore, the specific calibration method for the hyperparameter kernel width of the Gaussian radial basis kernel function is as follows: extract the noiseless voltage and noiseless current of the transformer bushing under steady-state operating conditions, and reconstruct them into a reference time-series characteristic distribution matrix; traverse and calculate the sum of squared Euclidean distances between all pairwise state column vectors in the reference time-series characteristic distribution matrix, and obtain the mean of the global spatial distance; use the reciprocal of the arithmetic square root of the mean of the global spatial distance as the physical constraint scale to calibrate the hyperparameter kernel width, for example, calibrating it to 2.5, so that the mapping boundary of the regenerative kernel Hilbert space can perfectly enclose the broadband dynamic response domain of the transformer bushing.

[0056] Input within this space The regularization constant is used to calculate the centered historical experience autocovariance matrix, the future experience autocovariance matrix, and the empirical cross-covariance matrix associated with the two. Specifically, a centered matrix is ​​constructed based on the dimension of the kernel matrix by subtracting the mean matrix from the identity matrix. This matrix is ​​then multiplied from the left and right sides into the historical data kernel matrix and the future data kernel matrix, respectively, to eliminate the mean drift caused by the nonlinear mapping of the data, thus obtaining the centered historical kernel matrix and the future kernel matrix.

[0057] To prevent singularities in high-dimensional matrices and ensure numerical stability in subsequent solutions, the centered historical kernel matrix is ​​multiplied by itself, and the regularization constant is added to the diagonal of the resulting matrix to obtain a centered historical empirical autocovariance matrix. The same self-multiplication plus regularization term operation is then applied to the future kernel matrix to obtain a centered future empirical autocovariance matrix. Matrix multiplication is then performed between the centered historical kernel matrix and the centered future kernel matrix to quantify the cross-correlation characteristics of the two time series in the high-dimensional reproducing kernel Hilbert space, thereby obtaining the empirical cross-covariance matrix of their association.

[0058] A correlation matrix is ​​constructed and singular value decomposition is performed. Specifically, the modal order of the target system is set to 6, and the top 6 largest non-zero singular values ​​in descending order, such as 0.982 and 0.914, are extracted as canonical correlation coefficients. The corresponding left singular vectors are then transposed to generate a 6-dimensional state sequence. Based on the orthogonal projection method, the input and output data are combined as observation vectors. Using QR orthogonal decomposition and least squares regression algorithms, the state transition matrix A (6×6), input matrix B (6×1), output matrix C (1×6), and action matrix D (1×1) are calculated.

[0059] S3. Perform the Schul decomposition of the state transition matrix A and calculate the modal coupling confidence index.

[0060] Specifically, the state transition matrix A is decomposed using Schuler to obtain an upper triangular Schuler matrix T and a unitary matrix U. The diagonal elements of the Schuler matrix T are system eigenvalues, representing system modes. The reciprocal of the canonical correlation coefficient is used as the uncertainty penalty index of the system modes to construct a weighted function matrix. The off-diagonal elements of the Schuler matrix T are weighted, and the Frobenius norm of the weighted off-diagonal element matrix is ​​calculated to obtain the modal coupling credibility index, which quantifies the modal coupling strength and identification credibility.

[0061] In an optional embodiment, the state transition matrix A is subjected to a similarity transformation using a balanced transformation matrix constructed based on the canonical correlation coefficient of CCA as a non-singular transformation matrix, and then projected onto the internal balanced realization coordinate system, so that the transformed state space coordinates are arranged strictly in descending order according to the size of the canonical correlation coefficient, thereby obtaining the balanced state transition matrix.

[0062] The linalg Schul function from the SciPy library is used to perform an orthogonal similarity transformation on the equilibrium state transition matrix using a QR iterative algorithm with implicit displacement. This decomposes the matrix and outputs a unitary matrix U and an upper triangular Schul matrix T. The complex elements on the main diagonal of the Schul matrix T are extracted as system eigenvalues ​​representing specific resonant frequencies and damping, i.e., system modes. The canonical correlation coefficients obtained in the preceding steps are normalized to their minimum and maximum values, linearly mapping them to the range of 0 to 1. The reciprocal of the normalized canonical correlation coefficient is used as the penalty amplification weight for the corresponding system mode identification results. Initialize a zero-weighted function matrix with the same row and column dimensions as the Shure matrix T. Calculate the arithmetic mean of the penalty amplification weights of the i-th and j-th system modes and assign it to the element in the i-th row and j-th column of the weighted function matrix to form the complete weighted function matrix. Using the triu function, with the diagonal offset parameter k=1, extract the strictly upper triangular part of the Shure matrix T after removing the main diagonal elements as a pure off-diagonal element matrix. Perform a Hadamard dot product operation on the weighted function matrix and this off-diagonal element matrix to obtain the weighted off-diagonal element matrix. Calculate the arithmetic square root of the sum of the squares of the absolute values ​​of all elements in the weighted off-diagonal element matrix to obtain the Frobenius norm of the weighted off-diagonal element matrix. Use this calculated single scalar value as the modal coupling confidence index.

[0063] In an optional embodiment, the state transition matrix A is decomposed using Schuler to obtain an upper triangular Schuler matrix T and a unitary matrix U. The diagonal elements of the Schuler matrix T are system eigenvalues, representing system modes, including: The state transition matrix A is reduced to a Shanghai Semper matrix using the Householder transformation algorithm; The Shanghai Senberg matrix is ​​subjected to a QR iterative calculation with displacement, which converges to a quasi-upper triangular matrix in real Schul form after cyclic iteration. The real elements on the main diagonal and the conjugate complex eigenvalues ​​of the 2×2 real diagonal blocks on the secondary diagonal of this quasi-upper triangular matrix are extracted and used as the system eigenvalue set of the state transition matrix A to represent each system mode. Alternatively, a unitary similarity transformation is performed in the complex domain to obtain a complex Schul matrix T in pure upper triangular form and its main diagonal complex eigenvalues ​​are extracted. Multiply all the orthogonal transformation matrices in each iteration to obtain the corresponding orthogonal transformation composite matrix, and use the composite matrix as the unitary matrix U.

[0064] For the 6×6 state transition matrix A obtained by identification, a series of Householder orthogonal transformation matrices are constructed. The matrices are multiplied before and after the original matrix A, and the values ​​below the main and secondary diagonals are set to zero to simplify the original matrix. The original matrix is ​​reduced to a 6×6-dimensional Heisenberg matrix that retains the non-zero elements of the main diagonal and the first secondary diagonal.

[0065] A QR iterative calculation with displacement is performed on the Hessenberg matrix (upper Hessenberg matrix). In each iteration, the eigenvalues ​​of the 2×2 sub-block in the lower right corner of the matrix are extracted as displacements for QR decomposition and recombination calculations. After iterative iteration, the elements on the lower diagonal are continuously monitored. When the residual is lower than the convergence tolerance... The calculation is terminated at the specified time. This converged quasi-upper triangular matrix is ​​taken as the real Schul matrix T. The real elements on the main diagonal of this matrix, as well as the conjugate complex eigenvalues ​​calculated by the 2×2 real diagonal blocks on the secondary diagonal, such as a pair of conjugate poles containing 0.85±0.12j, together constitute 6 system poles to represent each system mode. All single-step orthogonal transformation matrices used from the initial Householder transformation to each QR iteration are multiplied to generate the corresponding 6×6 orthogonal transformation composite matrix, which is taken as the unitary matrix U.

[0066] In an optional embodiment, the step of constructing a weighted function matrix using the reciprocal of the canonical correlation coefficient as an uncertainty penalty index for the system modes, weighting the off-diagonal elements of the Shure matrix T, calculating the Frobenius norm of the weighted off-diagonal element matrix, and obtaining the modal coupling credibility index that quantifies the modal coupling strength and identification credibility includes: Extract all the canonical correlation coefficients and calculate their reciprocals one by one to form a main diagonal matrix, and use the main diagonal matrix as the weighting function matrix of the system; Multiply the weighted function matrix on the left by the pure off-diagonal matrix formed by removing the diagonal elements of the Schul matrix T, and perform matrix multiplication to obtain the weighted off-diagonal matrix. Squaring the absolute value of each element in the weighted off-diagonal matrix. The Frobenius norm is calculated by summing the results of all squaring operations in the matrix and taking the square root of the sum. The obtained Frobenius norm value is used as the modal coupling credibility index.

[0067] Extract the six typical correlation coefficient values ​​from the aforementioned singular value decomposition stage, such as 0.982 to 0.45, calculate their reciprocals one by one, and arrange the calculated reciprocal results sequentially to generate a 6×6 main diagonal matrix. Use the main diagonal matrix as the weighting function matrix of the system. Obtain the 6×6 upper triangular Schul matrix T obtained from the aforementioned decomposition, set all eigenvalues ​​on the main diagonal to zero, and form a pure off-diagonal element matrix. Multiply the weighting function matrix on the left by the pure off-diagonal element matrix, and calculate the 6×6-dimensional weighted off-diagonal element matrix through matrix multiplication.

[0068] Multiply the weighted function matrix on the left by the pure off-diagonal element matrix to obtain a 6×6 weighted off-diagonal element matrix through matrix multiplication. Calculate the absolute value of each complex component in this matrix and perform a square operation on each of them. Sum the results of all the square operations in the matrix and perform a square root operation on the sum to calculate the Frobenius norm of the matrix. Extract the norm value obtained from this calculation, for example, 0.345, and use this norm value as the modal coupling confidence index.

[0069] S4. Extract the natural frequency and attenuation factor and comprehensively evaluate the harmonic resistance performance of the transformer bushing.

[0070] Specifically, based on the eigenvalues ​​of all discrete systems in the state transition matrix A, the logarithmic transformation formula is used to map them to the continuous time domain to obtain the continuous poles of the system in the complex plane. The natural frequencies and attenuation factors corresponding to the continuous poles are extracted to evaluate the harmonic resonance resistance of each mode. Combined with the mode coupling credibility index, the harmonic resistance performance of the transformer bushing is comprehensively calculated and evaluated.

[0071] The complex modulus of all system eigenvalues ​​in the state transition matrix A is calculated. Based on the Julius stability criterion for discrete-time systems, it is determined whether the modulus of all eigenvalues ​​is less than 1. If an eigenvalue with a modulus greater than or equal to 1 is detected during memory traversal and comparison, an alarm program is triggered, and the assessment conclusion that the transformer bushing's harmonic suppression performance has completely failed is output on the terminal interface. All continuous poles after transformation are traversed, and target harmonic poles with natural frequencies within a preset harmonic frequency band, such as 150Hz to 2500Hz, are selected. The ratio of the absolute value of the real part of each target harmonic pole to the complex modulus of the pole is calculated to obtain the harmonic damping ratio of each mode. The minimum value among all calculated damping ratios is extracted as the harmonic resonance suppression margin of the system.

[0072] It should be noted that the absolute value of the real part of each target harmonic pole is used as the corresponding attenuation factor, and the complex modulus of each target harmonic pole is used as the corresponding natural frequency.

[0073] The initial performance benchmark coefficient is set to 1. A multi-source parameter linear fusion evaluation mathematical model is constructed. The product of the harmonic resonance suppression margin and the preset first proportional coefficient is added to the benchmark full score as a positive reward gain, and the product of the modal coupling credibility index and the preset second proportional coefficient is subtracted as a negative penalty deduction to obtain the anti-harmonic performance evaluation value. According to the interval mapping threshold table stored in the local database, the anti-harmonic performance evaluation value is classified into four health level labels: excellent, good, medium and poor, and the results are output to complete the comprehensive quantitative evaluation of the transformer bushing's anti-harmonic performance.

[0074] In an optional embodiment, the eigenvalues ​​of the entire discrete system based on the state transition matrix A are mapped to the continuous time domain using a logarithmic transformation formula to obtain the continuous poles of the system in the complex plane. The natural frequencies and attenuation factors corresponding to these continuous poles are extracted to evaluate the harmonic resistance capability of each mode. Combined with the modal coupling confidence index, the harmonic resistance performance of the transformer bushing is comprehensively calculated and evaluated, including: The continuous poles corresponding to all the eigenvalues ​​are obtained by using the discrete-to-continuous conversion formula, and the target harmonic poles within the preset harmonic frequency band are selected. Calculate the ratio of the absolute value of the real part of each target harmonic pole to the complex modulus of the pole to obtain the damping ratio. Select the minimum damping ratio among all target harmonic poles and use this minimum damping ratio as the harmonic resonance suppression margin. Set a fixed performance weighting coefficient, multiply the harmonic resonance suppression margin by the performance weighting coefficient, and subtract the modal coupling confidence index to obtain the harmonic resistance performance evaluation value. The anti-harmonic performance evaluation value is compared with a preset safety judgment threshold. When the anti-harmonic performance evaluation value is greater than the safety judgment threshold, the transformer bushing is determined to have anti-harmonic performance. When the anti-harmonic performance evaluation value is not greater than the safety judgment threshold, the transformer bushing is determined to have lost its anti-harmonic performance.

[0075] Six discrete system eigenvalues ​​are extracted from the main diagonal of the Shure matrix T. Combined with the set sampling frequency of 25.6 kHz, they are uniformly mapped to complex poles in the continuous time domain using the complex logarithmic function. From the obtained set of continuous poles, the target harmonic poles with the resonant frequency corresponding to the imaginary part in the harmonic band are selected, and their damping ratio parameters are calculated. The minimum value of the calculated damping ratio, such as 0.142, is selected as the harmonic resonance suppression margin.

[0076] A fixed performance weighting coefficient with a value ranging from 5 to 20 is set; in this embodiment, a value of 10 is preferred. The harmonic resonance suppression margin of 0.142 calculated above is multiplied by this coefficient of 10 to obtain 1.42. The modal coupling confidence index of 0.345 measured in the previous step is subtracted to generate a value of 1.075 as the harmonic resistance performance evaluation value. A pre-set safety judgment threshold is called; in this embodiment, it is preferably set to 0.8. The value of 1.075 is compared with the safety judgment threshold. When 1.075 is greater than the threshold of 0.8, the transformer bushing is determined to have harmonic resistance performance; otherwise, if the calculated evaluation value is not greater than the threshold of 0.8, the transformer bushing is determined to have lost harmonic resistance performance.

[0077] Furthermore, the underlying physical calibration method for the pre-set safety judgment threshold specifically includes: performing a high-frequency thermal stress accelerated aging limit test on the same type of transformer bushing sample in a laboratory environment, and using a high-frequency current sensor to monitor and record the temporal evolution characteristics of the partial discharge quantity of the sample in real time; locking the physical critical failure time point when the partial discharge quantity suddenly increases and exceeds the International Electrotechnical Commission's transformer bushing safety standard; retrieving the anti-harmonic performance evaluation values ​​within several preceding observation time windows before the physical critical failure time point, and obtaining its lower limit envelope boundary value as the safety judgment threshold, for example, calibrated to 0.8, thereby ensuring that a hard correspondence is established between the evaluation value of the pure algorithm dimension and the actual attenuation of physical insulation performance.

[0078] Under the same transformer bushing harmonic operating environment, test signals containing high-frequency noise and multiple harmonics were injected into the system. The sampling frequency was set to 25.6 kHz, and 5120 continuous data points were extracted from each group to form a dataset. Three control experiments were set up: integer-order Kalman filtering combined with linear canonical correlation analysis was set as the baseline group; fractional-order Kalman filtering combined with linear canonical correlation analysis was set as the transition group; and the fractional-order Kalman filtering combined with kernelized canonical correlation analysis proposed in this paper was set as the complete group. The number of model fitting rows in each group was uniformly set to 20, the modal order was set to 6, the safety judgment threshold was set to 0.8, and the performance weight coefficient was set to 10. Figure 3 The three sets of control experiments were configured with the same sampling parameters and model order. The bar charts correspond to the anti-harmonic performance evaluation results of each group. The dashed line marks the 0.8 safety judgment threshold, which can be used to compare the deviation of the evaluation values ​​from the safety threshold under the three algorithm combinations, indicating the difference in the identification effect of different schemes.

[0079] It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of this application, and these all fall within the scope of protection of this application. Therefore, the scope of protection of this patent application shall be determined by the appended claims.

Claims

1. A method for evaluating the harmonic resistance performance of transformer bushings, characterized in that, Includes the following steps: S1. Obtain the real-time sampling sequence of transformer bushing port voltage and current; filter the real-time sampling sequence to obtain a noiseless voltage and current sequence for model identification; S2. Construct a data matrix, map the data matrix to the regenerative kernel Hilbert space, solve the generalized eigenvalue problem formed by the covariance matrix in the space, obtain the typical correlation coefficients and state sequences, and identify the broadband state-space model representing the characteristics of the transformer bushing; the model includes: state transition matrix A; S3. Perform Schul decomposition on the state transition matrix A to obtain the upper triangular form of the Schul matrix T and the unitary matrix U. The diagonal elements of the Schul matrix T are the system eigenvalues, representing the system modes. Construct a weighted function matrix using the reciprocal of the canonical correlation coefficient as the uncertainty penalty index of the system modes. Weight the off-diagonal elements of the Schul matrix T, calculate the Frobenius norm of the weighted off-diagonal element matrix, and obtain the modal coupling credibility index that quantifies the modal coupling strength and identification credibility. S4. Based on the eigenvalues ​​of all discrete systems in the state transition matrix A, the logarithmic transformation formula is used to map them to the continuous time domain to obtain the continuous poles of the system in the complex plane. The natural frequencies and attenuation factors corresponding to the continuous poles are extracted to evaluate the anti-harmonic resonance capability of each mode. Combined with the mode coupling credibility index, the anti-harmonic performance of the transformer bushing is comprehensively calculated and evaluated.

2. The method according to claim 1, characterized in that, The real-time sampling sequence for obtaining the transformer bushing port voltage and current includes: connecting the voltage transformer and the current transformer to the end test terminal and the grounding lead of the transformer bushing, respectively. High-frequency noise in the output signals of the voltage transformer and current transformer is filtered out using a low-pass anti-aliasing filter; The filtered signal is converted from analog to digital using a synchronous acquisition terminal at a set sampling frequency, and the data points at each moment are extracted to form a real-time sampling sequence of the port voltage and the current flowing through it.

3. The method according to claim 1, characterized in that, The process of obtaining noiseless voltage and current sequences for model identification includes: setting initial values ​​for the process noise covariance matrix and measurement noise covariance matrix of the fractional-order Kalman filter, calibrating the fractional-order difference order, and constructing a fractional-order state-space model with voltage and current as state variables. Discretize the state equations of the fractional-order state-space model to obtain the predicted state value and prediction error covariance matrix at the current time step; Calculate the Kalman gain matrix at the current time step, use the real-time sampling sequence as the observation input, and update the state prediction value in combination with the observation matrix to obtain the optimal state estimation sequence; The voltage and current components in the optimal state estimation sequence are output as noiseless voltage and current sequences for model identification.

4. The method according to claim 1, characterized in that, The model also includes: input matrix B, output matrix C, and action matrix D.

5. The method according to claim 4, characterized in that, Identifying the broadband state-space model includes: concatenating historical voltage and current sequences at different times to form a historical data Hankel matrix, and concatenating voltage and current sequences corresponding to future times to form a future data Hankel matrix; selecting a Gaussian radial basis kernel function to calculate the kernel matrix corresponding to the historical and future data Hankel matrices to achieve mapping to the regenerative kernel Hilbert space; constructing a correlation matrix and performing singular value decomposition based on the empirical covariance and empirical cross-covariance matrices in the kernel matrix calculation space, extracting the largest non-zero singular values ​​of the same order as the broadband state-space model as canonical correlation coefficients, and transposing the corresponding left singular vectors to form a state sequence; using the state sequence, solving the system's state-space evolution process using the orthogonal projection method, and executing the least squares regression algorithm to calculate the state transition matrix A, input matrix B, output matrix C, and action matrix D of the broadband state-space model.

6. The method according to claim 1, characterized in that, The process of performing Schul decomposition on the state transition matrix A to obtain an upper triangular Schul matrix T and a unitary matrix U, where the diagonal elements of the Schul matrix T are system eigenvalues ​​representing system modes, includes: reducing the state transition matrix A to a Shanghai Semburg matrix using the Householder transformation algorithm; performing QR iteration calculations with displacement on the Shanghai Semburg matrix, converging iteratively to a quasi-upper triangular matrix in real Schul form; extracting the real elements on the main diagonal and the conjugate complex eigenvalues ​​of the 2×2 real diagonal blocks on the secondary diagonal of this quasi-upper triangular matrix, which together serve as the set of system eigenvalues ​​of the state transition matrix A to represent each system mode; or performing a unitary similarity transformation in the complex domain to obtain a pure upper triangular complex Schul matrix T and extracting its main diagonal complex eigenvalues; multiplying all orthogonal transformation matrices in each iteration to obtain the corresponding orthogonal transformation composite matrix, which is then used as the unitary matrix U.

7. The method according to claim 1, characterized in that, Construct the weighted function matrix, including: Extract all canonical correlation coefficients and calculate their reciprocals one by one to form a main diagonal matrix, which is then used as the weighting function matrix of the system.

8. The method according to claim 7, characterized in that, The modal coupling credibility index, which quantifies the modal coupling strength and identification credibility, includes: Multiply the weighted function matrix on the left by the Schul matrix T after removing the diagonal elements to form a pure off-diagonal matrix, and then perform matrix multiplication to obtain the weighted off-diagonal matrix. Squaring the absolute value of each element in the weighted off-diagonal matrix. The Frobenius norm is calculated by summing the results of all squaring operations in the matrix and taking the square root of the sum. The obtained Frobenius norm value is used as the modal coupling credibility index.

9. The method according to claim 1, characterized in that, The evaluation of the harmonic resonance resistance of each mode includes: The continuous poles corresponding to all eigenvalues ​​are obtained by using the conversion formula from discrete to continuous, and the target harmonic poles within the preset harmonic frequency band are selected. Calculate the ratio of the absolute value of the real part of each target harmonic pole to the complex modulus of the pole to obtain the damping ratio. Select the minimum damping ratio among all target harmonic poles and use this minimum damping ratio as the harmonic resonance suppression margin. By setting a fixed performance weighting coefficient, multiplying the harmonic resonance suppression margin by the performance weighting coefficient, and subtracting the modal coupling confidence index, the value of the harmonic suppression performance evaluation is obtained.

10. The method according to claim 9, characterized in that, The comprehensive calculation and evaluation of the transformer bushing's harmonic resistance performance includes: The value of the anti-harmonic performance assessment is compared with a pre-set safety judgment threshold. When the value of the anti-harmonic performance assessment is greater than the safety judgment threshold, the transformer bushing is determined to have anti-harmonic performance. When the value of the anti-harmonic performance assessment is not greater than the safety judgment threshold, the transformer bushing is determined to have lost its anti-harmonic performance.

Images