A high-precision measurement method for feeder calibration reference voltage based on PIA-bidirectional FIR

Abstract

The application belongs to the technical field of power system signal processing, and discloses a high-precision measurement method for feeder checking reference voltage based on PIA-bidirectional FIR, which comprises the following steps: collecting a bus voltage signal of a power distribution network; reconstructing the waveform of the collected voltage signal by using PIA cubic spline Hermite interpolation; performing bidirectional FIR filtering on the reconstructed voltage signal to obtain a filtered voltage signal, wherein the bidirectional FIR filtering comprises forward filtering, secondary filtering after time reverse and sequence recovery; performing adaptive decomposition on the filtered voltage signal by using improved variational mode decomposition to obtain a fundamental signal; and extracting the amplitude, phase and frequency parameters of the fundamental signal by using a TLS-ESPRIT algorithm. The application is aimed at the fact that the voltage signal in the operation of the measurement loop of the feeder of the power distribution network is easily affected by load fluctuation, harmonic interference and non-uniform sampling distortion, and through the reconstruction of the pure voltage waveform, the accurate measurement of the key parameters of the measurement loop of the power distribution network is realized, and the online checking of the polarity, phase sequence, transformer accuracy and protection device performance is supported.

Description

Technical Field

[0001] This invention relates to the field of power system signal processing technology, specifically to a high-precision measurement method for feeder verification reference voltage based on PIA-bidirectional FIR. Background Technology

[0002] With the advancement of new power system construction, distributed generation and nonlinear loads are widely integrated, making voltage signals in feeder measurement circuits susceptible to load fluctuations, harmonic interference, and non-uniform sampling distortion. This leads to voltage waveform distortion, severely impacting the accuracy of online verification. The distribution network bus voltage serves as the reference benchmark for online verification of feeder measurement circuits, and the accurate extraction of its amplitude, phase, and frequency parameters is a prerequisite for ensuring verification reliability. Traditional offline verification methods rely on bus power outages, which are ill-suited to the uninterrupted operation and maintenance requirements of modern distribution networks.

[0003] In voltage signal processing, interpolation reconstruction, filtering and denoising, mode decomposition, and parameter identification are key steps, but existing technologies still have shortcomings. Existing signal reconstruction methods have limited ability to repair non-uniform sampling distortion and lack collaborative design with phase-distortion-free filtering; conventional filtering methods easily introduce phase delay, affecting signal phase characteristics. Existing VMD methods suffer from problems such as parameter dependence on empirical settings, low efficiency of automatic optimization, insufficient optimization objectives, and inadequate decomposition stability, making it difficult to guarantee ideal mode separation results; while TLS-ESPRIT has high resolution, it is sensitive to the quality of the input signal, and if distorted voltage signals are processed directly, the parameter estimation results are easily affected by noise and harmonic interference; traditional parameter extraction methods such as Kalman filtering and discrete Fourier transform also have limited accuracy under dynamic conditions. In the online verification system of distribution network feeders, the bus voltage mainly serves as a vector reference for generating synchronous standard currents. The verification device needs to generate a standard current with the same frequency and phase as the bus voltage based on its amplitude, phase, and frequency, and inject it into the measured circuit for closed-loop comparison with the sampled values ​​of the protection device. Since polarity verification, phase sequence verification, and protection device performance verification all rely on phase criteria, the phase accuracy of the reference voltage directly determines the reliability of these verification criteria. In this scenario, the requirements for the reference voltage signal are not only the accurate acquisition of amplitude and frequency parameters, but also that the phase characteristics be as unaffected as possible by additional phase shifts during signal processing.

[0004] However, existing signal processing methods are prone to residual waveform distortion when interpolation reconstruction accuracy is insufficient, while conventional filtering methods may introduce phase delay, thus affecting the reliability of subsequent parameter extraction and online verification results. Therefore, there is an urgent need for a feeder verification reference voltage measurement method that balances non-uniform sampling distortion repair, broadband interference suppression, and phase fidelity. Summary of the Invention

[0005] The purpose of this invention is to provide a high-precision measurement method for feeder calibration reference voltage based on PIA-bidirectional FIR, so as to solve the problems mentioned in the background art.

[0006] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0007] A high-precision measurement method for feeder verification reference voltage based on PIA-bidirectional FIR, characterized by the following steps:

[0008] Step S1: Acquire the voltage signal of the distribution network bus;

[0009] Step S2: The acquired voltage signal is reconstructed using PIA cubic spline Hermite interpolation to obtain the reconstructed voltage signal;

[0010] Step S3: Perform bidirectional FIR filtering on the reconstructed voltage signal to obtain the filtered voltage signal. Bidirectional FIR filtering includes forward filtering, time reversal followed by secondary filtering, and sequence recovery.

[0011] Step S4: Adaptively decompose the filtered voltage signal using improved variational mode decomposition to obtain the fundamental signal;

[0012] Step S5: Use the TLS-ESPRIT algorithm to extract the amplitude, phase, and frequency parameters of the fundamental signal.

[0013] Furthermore, in step S2, PIA cubic spline Hermite interpolation is used to reconstruct the waveform of the acquired voltage signal to obtain the reconstructed voltage signal, specifically as follows:

[0014] Step S2-1: Construct a piecewise cubic polynomial between adjacent sampling points so that the interpolation curve satisfies both position constraints and derivative constraints at each sampling point.

[0015] Step S2-2: Generate the augmented control vertex matrix using an asymptotic iterative approximation algorithm, and iterate over the augmented control vertices;

[0016] Step S2-3: Perform convergence judgment on the augmented control vertex matrix, calculate the Euclidean displacement of each augmented control vertex between two adjacent iterations, and determine that the algorithm has reached convergence when the Euclidean displacement of all augmented control vertices is less than the preset tolerance in N consecutive iteration cycles, and terminate the iteration.

[0017] Further, in step S3, the reconstructed voltage signal is subjected to bidirectional FIR filtering to obtain the filtered voltage signal. The bidirectional FIR filtering includes forward filtering, time reversal followed by secondary filtering, and sequence recovery, specifically as follows:

[0018] Step S3-1: Perform forward FIR filtering on the reconstructed voltage signal to obtain a forward filtered signal;

[0019] Step S3-2: Reverse the time sequence of the forward filtered signal and perform the same FIR filter to obtain the reverse filtered signal;

[0020] Step S3-3: Restore the original time sequence of the reverse filtered signal and output the filtered voltage signal.

[0021] Furthermore, the FIR filter described in step S3 is a 255th-order Kaiser window FIR filter with a cutoff frequency of 2.75kHz. The filter's unit impulse response satisfies the even symmetry condition, and the stopband attenuation is 80dB. The output of the FIR filter is calculated using the linear convolution formula:

[0022] ;

[0023] in, This is the output signal sequence of the FIR filter. The filter's unit impulse response. Let the filter order be . The input is the reconstructed voltage signal to be filtered; The index variable for the convolution summation. The sampling point number of the discrete-time signal. This is a linear convolution summation operation.

[0024] Furthermore, in step S4, an improved variational mode decomposition is used to adaptively decompose the filtered voltage signal to obtain the fundamental signal, specifically as follows:

[0025] Step S4-1: Start the iteration with an initial number of modes K=1, introduce the total entropy of residual loss, and iterate successively to increase the number of decomposed modes K and calculate the total entropy of residual loss. Convergence occurs when the total entropy of residual loss is less than a set threshold. Adaptively determine the optimal number of modes K to avoid under-decomposition or over-decomposition of the signal; the expression for the total entropy of residual loss is:

[0026] ;

[0027] In the formula, The total entropy is the residual loss. These are the indexes of the intrinsic mode components obtained from VMD decomposition. To decompose the mode number; This represents the number of iterations. For the nth iteration The second norm of each intrinsic modal component; For the (n-1)th iteration The second norm of each intrinsic modal component; The second norm of the filtered voltage signal in the frequency domain;

[0028] Set a threshold for total entropy loss During the iteration process, if Greater than the set threshold Then let Continue decomposing until... Less than or equal to Determine the optimal number of modes ;

[0029] Step S4-2: Construct a comprehensive evaluation index by combining the modal center frequency, Chebyshev distance, and improved edge permutation entropy. With the goal of maximizing Chebyshev distance and minimizing improved edge permutation entropy, traverse the search to optimize and determine the optimal penalty factor. This improves mode separation and the purity of the fundamental component;

[0030] Step S4-3: Based on the optimal number of modes and optimal penalty factor The filtered voltage signal is decomposed using VMD, and the fundamental signal is obtained by screening the intrinsic mode components.

[0031] Furthermore, in step S4-2, the penalty factor is optimized by combining the modal center frequency Chebyshev distance with the improved edge arrangement entropy. Specifically:

[0032] Chebyshev distance ;

[0033] in, The maximum separation of modal center frequencies is the Chebyshev distance. These are the indexes of the intrinsic mode components obtained from VMD decomposition. ,and K is the total number of modes in the VMD decomposition; For the first The center frequencies of the intrinsic mode components; For the first The center frequencies of the intrinsic mode components; It represents the absolute difference between the center frequencies of two different intrinsic mode components; To traverse all intrinsic mode component indices , ,and Take the maximum value of the absolute differences in frequency between all different mode pairs;

[0034] Improved edge permutation entropy ;

[0035] in, To improve edge arrangement entropy; The embedding dimension for reconstructing the phase space determines the number of permutation patterns; The time delay for phase space reconstruction; This is the sequence number of the arrangement pattern; To determine the first by combining the weights of the short sequence The probability of a certain permutation pattern occurring; This is a term used for Shannon entropy calculation; For the normalized denominator, is the theoretical maximum value of Shannon entropy;

[0036] A comprehensive evaluation index is derived by combining the modal center frequency Chebyshev distance with the improved edge arrangement entropy, in order to optimize the penalty factor. The expression is: ;

[0037] in, The subscript is a comprehensive evaluation index for adaptive optimization of VMD parameters. Corresponding to the VMD parameters to be optimized, Chebyshev distance; To improve edge arrangement entropy.

[0038] Further, in step S4-3, the fundamental signal is obtained by screening the intrinsic mode components, specifically as follows:

[0039] The fundamental eigenmode component with the highest energy is selected and defined as the fundamental principal component, which is then used as the fundamental signal.

[0040] Furthermore, in another implementation, the fundamental signal is obtained by filtering the intrinsic mode components in step S4-3, specifically as follows:

[0041] The fundamental wave eigenmode component with the highest energy is defined as the fundamental wave principal component.

[0042] Calculate the Spearman correlation coefficient between each intrinsic mode component other than the fundamental main component and the filtered voltage signal, and select the intrinsic mode components with Spearman correlation coefficients greater than a preset threshold as residual fundamental components.

[0043] The filtered residual fundamental component is reconstructed with the fundamental principal component, and the reconstructed signal is used as the fundamental signal.

[0044] Furthermore, in step S5, the TLS-ESPRIT algorithm is used to extract the amplitude, phase, and frequency parameters of the fundamental signal, specifically as follows:

[0045] Step S5-1: Resample the fundamental signal and construct a dimension-reduced Hankel matrix;

[0046] Step S5-2: Perform singular value decomposition on the Hankel matrix to divide it into signal subspace and noise subspace;

[0047] Step S5-3: Solve for the rotation property matrix and calculate the eigenvalues;

[0048] Step S5-4: Solve for the relaxation coefficient, and combine the eigenvalues ​​and relaxation coefficients to calculate the voltage amplitude, frequency, and initial phase;

[0049] Step S5-5: Perform complex plane quadrant correction on the initial phase and output the final phase.

[0050] Compared with the prior art, the present invention has the following improvements and advantages:

[0051] 1. This invention employs progressive iterative approximation cubic spline Hermite interpolation (PIA cubic spline Hermite interpolation) to reconstruct waveforms of non-uniformly sampled signals. While ensuring that the interpolation curve simultaneously satisfies position constraints and derivative constraints at each sampling point, it effectively repairs waveform distortion caused by sampling clock jitter or network delay by iteratively correcting the control vertex, thereby improving the reconstruction fidelity of distorted voltage signals and providing high-quality input signals for subsequent processing.

[0052] 2. This invention employs a bidirectional FIR filter structure. By performing forward FIR filtering on the reconstructed voltage signal, then reversing the time sequence, and finally restoring the original time sequence of the reverse-filtered signal, this symmetrical filter structure utilizes the phase delays of the two filters to cancel each other out, achieving zero phase delay characteristics. This effectively suppresses broadband interference while preserving the phase characteristics of the fundamental signal, which is beneficial for improving phase measurement accuracy.

[0053] 3. This invention achieves blind source separation of the fundamental correlation component by employing improved variational mode decomposition (VMD). During the VMD process, the total entropy of the residual loss is introduced to adaptively determine the number of decomposed modes. The penalty factor is optimized by combining the modal center frequency, Chebyshev distance, and improved edge arrangement entropy. It can reduce the risk of under-decomposition or over-decomposition caused by manual experience in parameter setting, and improve the separation effect of fundamental component and interference component.

[0054] 4. This invention utilizes the Total Least Squares-Rotation Invariant Technique (TLS-ESPRIT) algorithm to estimate the parameters of the separated fundamental signal. By constructing a Hankel matrix, dividing the signal subspace and noise subspace, and combining eigenvalues ​​and relaxation coefficients to obtain voltage amplitude, frequency, and phase parameters, and performing complex plane quadrant correction on the initial phase, the resolution and stability of parameter estimation can be improved.

[0055] 5. This invention addresses the application requirements of non-uniform sampling distortion, broadband interference, and phase sensitivity in feeder calibration reference voltage measurement. It constructs a collaborative processing chain with sequential dependencies. Step S2 is used to repair sampling distortion and restore waveform continuity. Step S3 suppresses interference while maintaining phase fidelity. Step S4 stabilizes and separates the fundamental component based on the previous purified signal. Step S5 extracts high-precision parameters from the separated fundamental signal. Each step is interconnected and interdependent. The preceding step provides the necessary input conditions for the following step, and the following step depends on the output quality of the preceding step, thereby improving the stability and accuracy of feeder calibration reference voltage measurement, especially beneficial for the accurate acquisition of phase parameters. Attached Figure Description

[0056] The present invention will be further described below with reference to the accompanying drawings and embodiments:

[0057] Figure 1 This is a flowchart of the present invention;

[0058] Figure 2 This is a schematic diagram of the online verification architecture for the measurement circuit of the distribution network feeder.

[0059] Figure 3 The image shows the effect of Hermite interpolation of cubic splines in PIA.

[0060] Figure 4 This is a diagram showing the interpolation effect of four PIA spectral lines.

[0061] Figure 5 This is a diagram showing the effect of a one-way filter.

[0062] Figure 6 This is a diagram showing the effect of bidirectional filtering.

[0063] Figure 7 This is a schematic diagram illustrating the signal processing effect of the joint algorithm at a frequency of 49.5Hz.

[0064] Figure 8 This is a schematic diagram illustrating the signal processing effect of the joint algorithm at a frequency of 49.8Hz.

[0065] Figure 9 This is a schematic diagram illustrating the signal processing effect of the joint algorithm at a frequency of 50Hz.

[0066] Figure 10 This is a schematic diagram illustrating the signal processing effect of the joint algorithm at a frequency of 50.3Hz.

[0067] Figure 11 This is a schematic diagram of the signal processing effect of the joint algorithm at a frequency of 50.5Hz. Detailed Implementation

[0068] To make the content of this invention easier to understand, the invention will be further described in detail below with reference to specific embodiments and accompanying drawings.

[0069] This invention is applied to the online verification architecture of the feeder measurement loop in the distribution network, such as... Figure 2 As shown in the figure, this architecture uses the distribution network bus voltage as an online reference benchmark. It obtains amplitude, phase, and frequency parameters through high-precision measurement of the reference voltage, generates a synchronous standard current that is injected into the circuit under test, and forms a closed-loop comparison with the sampling feedback of the protection device. This enables online verification of the accuracy, polarity, phase sequence of the instrument transformer, and the performance of the protection device under uninterrupted power supply conditions.

[0070] like Figure 1 As shown, this invention provides a high-precision measurement method for feeder calibration reference voltage based on PIA-bidirectional FIR. Figure 1 This is a flowchart illustrating the overall process of the method of this invention, sequentially showing the complete processing chain of signal acquisition, PIA cubic spline Hermite interpolation reconstruction, bidirectional FIR filtering, improved VMD decomposition, and TLS-ESPRIT parameter extraction, demonstrating the dependencies between each step. The method includes the following steps:

[0071] Step S1: Acquire the voltage signal of the distribution network bus;

[0072] Step S2: The acquired voltage signal is reconstructed using PIA cubic spline Hermite interpolation to obtain the reconstructed voltage signal;

[0073] The specific implementation of the PIA cubic spline Hermite interpolation in step S2 is as follows:

[0074] Step S2-1: Construct a piecewise cubic polynomial between adjacent sampling points so that the interpolation curve simultaneously satisfies position constraints and derivative constraints at each sampling point. The position constraint means that the function value of the interpolation curve at each sampling point is strictly equal to the actual sampled voltage value at that point. The derivative constraint means that the first-order derivative value of the interpolation curve at each sampling point is equal to a preset derivative value for that point. The preset derivative value is the initial first-order derivative estimate obtained before the iteration starts, based on the voltage values ​​and time intervals of the original sampling points, calculated using the central difference formula or forward / backward difference, ensuring first-order continuity of adjacent curve segments at connection points, i.e., a smooth slope transition. By simultaneously applying these dual constraints, the interpolation curve not only accurately passes through all original sampling points, but also maintains first-order continuity globally, avoiding waveform sharp angles or oscillations caused by abrupt slope changes. For a non-uniform node system defined on the interval [a,b], data reconstruction is performed by constructing a PIA cubic spline Hermite function, satisfying:

[0075] Each short curve segment between two adjacent points is described by a cubic equation;

[0076] All small curve segments maintain a smooth transition at the connection points;

[0077] The curve height and tilt angle at each connection point precisely correspond to the preset data values.

[0078] Step S2-2: Iteratively refine the control vertices using an asymptotic iterative approximation algorithm, generating the j-th augmented control vertex vector based on the following formula. All local interpolation windows (j=1,2,…,n) are combined column-wise to form a complete augmented control vertex matrix. Its mathematical characteristics satisfy the geometric constraint equations.

[0079] ;

[0080] in, This is the index of the original control vertex. This refers to the group number of the local interpolation window. No. The augmented control vertex vector is a 4-dimensional column vector containing... Four augmented control vertices, which simultaneously carry position information and first-order derivative information, are used to construct the PIA cubic spline Hermite interpolation curve; To augment the group number of the control vertex, the index of the corresponding local interpolation segment is used; The original control vertex is taken from the actual sampled value of the voltage signal from the non-uniform sampling of the distribution network, and is used as the initial input for PIA iteration. After the iteration converges, the reconstructed voltage signal is obtained by restoring the converged augmented control vertex through the basis function transformation matrix. Given the original control vertex index, construct an interpolation segment using four consecutive local control vertices; This is the basis function transformation matrix for PIA cubic spline Hermite interpolation, used to map the original control vertices to Hermite interpolation basis coefficients; The matrix inverse of the basis function transformation matrix is ​​used to convert the weighted coefficients into augmented control vertices; the matrix coefficients in the formula (1, , , ) represents the inverse matrix element of the transformation matrix of the Hermite interpolation basis function for PIA cubic splines. Its specific values ​​are calculated by substituting the numerical values ​​of the Hermite interpolation basis function at the nodes. Each group of augmented control vertex vectors It consists of its four corresponding consecutive original control vertices It is obtained by calculating the basis function transformation matrix.

[0081] The specific process of iterative correction is as follows: Using the original sampled data points as the initial values ​​of the augmented control vertices, i.e., when the iteration number k=0, each element in the augmented control vertex matrix takes the actual sampled voltage value and its first derivative at the corresponding sampled point, forming the initial iteration vector. In each iteration, an interpolation curve is constructed using the current augmented control vertex, and the deviation between the curve within each local interpolation window and the original sampled point is calculated. Based on this, a correction amount is generated, and the augmented control vertex position is updated. For the beginning and end of the augmented control vertex sequence, since only a single-sided adjacent window exists, the calculation of the updated values ​​for these two vertices only utilizes the information of their single-sided local window for interpolation and correction, without the need to introduce additional virtual vertices or periodic extension conditions. After the above iterations, the augmented control vertex matrix updated in each iteration is obtained.

[0082] Step S2-3: For the augmented control vertex matrix output in step S2-2, calculate the Euclidean displacement of each vertex between two adjacent iterations, and determine whether the iteration terminates according to the following convergence criterion. The iteration termination criterion adopts a dynamic convergence monitoring mechanism. When the Euclidean displacement of all augmented control vertices is less than the preset tolerance within N consecutive iteration cycles, the algorithm is considered to have reached convergence. Here, N consecutive iteration cycles refer to N consecutive iterations starting from the k-th iteration in the iteration process where the criterion condition is met. N is a preset positive integer representing the number of iteration cycles in which the convergence criterion is met consecutively. This is used to avoid premature termination of the iteration due to a single accidental fulfillment of the condition. It can be an integer between 2 and 5, flexibly determined according to the reliability of the convergence judgment and the computational efficiency requirements. The preset tolerance is used to determine whether the Euclidean displacement of the control vertex between two adjacent iterations has converged. Its value should match the magnitude of the voltage signal amplitude and the final interpolation accuracy, and is usually taken as 10 times the maximum amplitude of the signal. -8 The interpolation error is adjusted to approximately [value missing] times to ensure that the interpolation error meets the requirements of subsequent processing and to avoid the iteration failing to terminate. The convergence criterion is shown in the following formula:

[0083] ;

[0084]

[0085] in, For the first During the nth iteration, the 1st Updated values ​​for each augmented control vertex; For the first In the next iteration, the original value of the corresponding augmented control vertex; ( ) is the first The PIA cubic spline Hermite interpolation basis function coefficient matrix elements of each local interpolation window are obtained by mapping the augmented control vertex of the current iteration in step S2-2 through the basis function transformation matrix; For the first Position constraint vectors for local windows; No. The first-order derivative constraint vector of each local window ensures the first-order continuity of the interpolation curve. The position constraint vector... With first derivative constraint vector The interpolation curve defined in step S2-1 must simultaneously satisfy the position constraint and derivative constraint at each sampling point; The product of the coefficients of the local interpolation polynomial and the time step is used for time axis correction of non-uniform sampling. Its value is determined by the interval between adjacent sampling points of the original non-uniform sampling. This represents the number of iterations in the PIA algorithm. To augment the control vertex index, This is the local interpolation window number. It is the vector norm 2, used to calculate the iterative displacement of the augmented control vertex; This is a preset convergence tolerance threshold used to control interpolation accuracy;

[0086] Step S3: Perform bidirectional FIR filtering on the reconstructed voltage signal to obtain the filtered voltage signal. Bidirectional FIR filtering includes forward filtering, time reversal followed by secondary filtering, and sequence recovery.

[0087] In step S3, the bidirectional FIR filter employs a 255th-order Kaiser window FIR filter with a cutoff frequency of 2.75kHz. The filter's unit impulse response satisfies the even symmetry condition, and the stopband attenuation is 80dB. The unit impulse response of the FIR filter... Given an ordered sequence of length M+1, the output signal is... With input signal The linear convolution relationship is shown in the following formula:

[0088] ;

[0089] in, This is the output signal sequence of the FIR filter. The filter's unit impulse response. Let the filter order be . The input is the reconstructed voltage signal to be filtered; The index variable for the convolution summation. The sampling point number of the discrete-time signal. This is a linear convolution summation operation.

[0090] when When the even symmetry condition is met, the output can be simplified by the following formula:

[0091] ;

[0092] in, The output signal of the bidirectional FIR filter is the filtered distribution network voltage sequence, which has no phase distortion. The input signal to the filter, This is the unit impulse response (tap coefficient) of the FIR filter. Use the convolution summation index variable to iterate through all filter taps; The order of the FIR filter; This is the sampling point number (time index) of the discrete voltage signal. For linear convolution summation, the forward and backward filtering steps are combined into a single calculation to achieve zero-phase filtering.

[0093] The specific process is as follows:

[0094] Step S3-1: Perform forward FIR filtering on the reconstructed voltage signal to obtain a forward filtered signal;

[0095] Step S3-2: Reverse the time sequence of the forward filtered signal and pass it through the same FIR filter again to obtain the reverse filtered signal;

[0096] Step S3-3: Restore the original time sequence of the reverse filtered signal and output the filtered voltage signal.

[0097] In this embodiment, the 255th-order Kaiser window FIR filter with a cutoff frequency of 2.75kHz described in step S3 is applicable to the reconstructed voltage signal obtained after reconstruction in step S2. The original sampling rate of this signal is 2kHz, and the signal density is effectively increased after reconstruction by PIA cubic spline Hermite interpolation in step S2. The cutoff frequency of 2.75kHz is selected because after PIA cubic spline Hermite interpolation in step S2, the original non-uniform sampled voltage signal is reconstructed into a reconstructed voltage sequence with equal time intervals. In this embodiment, the equivalent sampling frequency of the reconstructed voltage sequence is 8kHz, the Nyquist frequency is 4kHz, and the 2.75kHz cutoff frequency, while fully preserving the fundamental 50Hz and necessary harmonic components, can effectively suppress broadband interference such as high-frequency quantization noise and power electronic switching noise introduced during the interpolation process in step S2, and reserves approximately 30% frequency safety margin to avoid signal truncation due to device tolerances or frequency fluctuations. The selection of filter order 255 and Kaiser window is to achieve a better engineering balance between 80dB stopband attenuation and zero-phase filtering performance under this signal condition. As shown in Table 6, the average delay of a single complete processing flow measured under this set of parameters is 403μs, which meets the real-time requirements of online feeder verification.

[0098] In the method of this application, step S2 provides an input waveform without abrupt distortion for step S3, and step S3 provides zero-phase interference suppression for the reconstruction result of step S2. After the two steps are superimposed, the output signal is purified to a level suitable for precise back-end processing while maintaining time domain integrity and phase fidelity. By first reconstructing and repairing the sampling distortion, and then applying bidirectional FIR filtering to the reconstructed voltage signal, it is beneficial to avoid the adverse effects of sampling gaps and local distortions on the filtering effect and the accuracy of subsequent parameter extraction.

[0099] Step S4: Adaptively decompose the filtered voltage signal using improved variational mode decomposition to obtain the fundamental signal;

[0100] Step S4 is used to separate the fundamental frequency of the voltage signal after reconstruction in step S2 and filtering in step S3. Since the previous steps have processed the non-uniform sampling distortion and broadband interference, the input signal of step S4 has the characteristics of fundamental frequency dominance, weak residual interference but still local energy leakage. Therefore, in the variational mode decomposition process, this invention has made targeted settings for determining the number of modes, optimizing the penalty factor, and screening the fundamental frequency component to improve the stability and completeness of fundamental frequency extraction.

[0101] The improved variational mode decomposition algorithm is implemented as follows:

[0102] Step S4-1: Introduce the total residual loss entropy to adaptively determine the number of VMD decomposition modes K. The total residual loss entropy is defined as shown in the following formula:

[0103] ;

[0104] in, The total entropy is the residual loss. This refers to the index of the intrinsic mode components (IMFs) obtained from VMD decomposition. To decompose the mode number; This represents the number of iterations. For the nth iteration The second norm of each intrinsic modal component; For the (n-1)th iteration The second norm of each intrinsic modal component; Let L2 be the frequency norm of the filtered voltage signal. Set a threshold for the total entropy loss. This embodiment The value is set to 0.005. The rationale is that when the energy contribution of the newly added mode is less than 0.5% of the total energy of the original signal, further increasing the number of modes will no longer separate the substantial signal components and may even introduce spurious modes. Therefore, during the iteration process, if... Greater than the set threshold This indicates that the signal has not been completely decomposed, so let Continue decomposing until... convergence( Less than or equal to the set threshold (i.e., considered convergent), determine the optimal number of modes. .

[0105] Step S4-2: Optimize the penalty factor by combining the modal center frequency, Chebyshev distance, and improved edge arrangement entropy. The Chebyshev distance is shown in the following formula:

[0106] It is used to measure the independence of the center frequencies of each intrinsic mode component;

[0107] in, The maximum separation of modal center frequencies is the Chebyshev distance. These are the indexes of the intrinsic mode components obtained from VMD decomposition. ,and K is the total number of modes in the VMD decomposition; For the first The center frequencies of the intrinsic mode components; For the first The center frequencies of the intrinsic mode components; It represents the absolute difference between the center frequencies of two different intrinsic mode components; To traverse all intrinsic mode component indices , ,and The maximum value of the absolute frequency difference of all different mode pairs is taken to characterize the maximum frequency interval of each mode.

[0108] The improved edge permutation entropy is shown in the following formula:

[0109] This is used to evaluate the noise content of the component.

[0110] in, To improve edge arrangement entropy; The embedding dimension for reconstructing the phase space determines the number of permutation patterns; The time delay for phase space reconstruction is adapted to the time scale of non-uniformly sampled signals; This is the sequence number of the arrangement pattern; For the first The probability of a certain permutation pattern occurring; This is a Shannon entropy calculation term used to measure the degree of disorder in the signal arrangement pattern; The normalized denominator term is the theoretical maximum value of Shannon entropy, used to normalize the entropy value to the [0,1] interval, enabling quantitative comparison under different parameters.

[0111] Traditional permutation entropy assumes equal intervals between sequences and sufficient samples when calculating the probability of occurrence of each pattern. However, the time axis of the signal reconstructed in step S2 has non-equal intervals, and the sample size is limited under short window conditions. Direct application can lead to deviations in entropy value estimation. This invention assigns extra weight to short sequences with the same sorting result when calculating improved marginal permutation entropy, so that the entropy value can stably reflect the true noise content and complexity level of the signal even under short data windows, thus providing a more reliable basis for the selection of penalty factors.

[0112] Specifically, the "short sequence" refers to a subsequence obtained by reconstructing the phase space of the filtered voltage signal. (Setting the embedding dimension) and time delay Extract all sequences of length from the signal sequence. subsequence of Each subsequence is a short sequence. For each short sequence... The elements are sorted in ascending order based on their numerical values, and a permutation pattern is obtained based on the index of each element's position in the original sequence. Traditional permutation entropy directly calculates the frequency of each permutation pattern as a probability, but this probability estimation has a large bias under short data windows. This invention assigns weights to each short sequence. This weight reflects the degree of fluctuation between adjacent points within a short sequence. The weighted probability of a pattern is obtained by summing the weights of all short sequences within the same permutation pattern and dividing by the total weight of all short sequences. Let there be a total of... Let there be a short sequence, and let the permutation pattern number be . The set of indices for all short sequences is Then the formula for the weighted probability is: This weighting method allows the entropy value to stably reflect the true noise content and complexity level of the signal even under short data windows and non-uniform sampling conditions, thus providing a more reliable basis for the selection of the penalty factor.

[0113] The comprehensive evaluation indicators are shown in the following formula:

[0114] In the formula, The subscript is a comprehensive evaluation index for adaptive optimization of VMD parameters. Corresponding to the VMD parameters to be optimized, this is used to quantify the mode decomposition quality under different parameters. The larger the size, the better the decomposition quality; Chebyshev distance is used to measure the degree of frequency separation of the intrinsic mode components after VMD decomposition. The larger the value, the less modal aliasing occurs and the more thorough the decomposition. To improve edge alignment entropy, which is used to measure the signal purity and noise level of intrinsic mode components, The smaller the value, the more ordered the signal and the lower the noise content. This index assigns weights to short sequences based on traditional permutation entropy, making it suitable for short-window data from non-uniformly sampled distribution networks. The penalty factor search range is set to... With a step size of 100, the optimal value is determined with the goal of maximizing the comprehensive evaluation index. Because the residual interference in the distribution network feeder voltage signal after preprocessing in steps S2 and S3 is mainly narrowband noise with few modal components, the mode aliasing is severe when the penalty factor is below 500, and the fundamental frequency extraction error exceeds 0.1%; while when it is above 2500, the waveform distortion is significant, the amplitude error exceeds 0.15%, and the calculation time increases by 30%. Therefore, the search interval corresponding to the optimal penalty factor is relatively concentrated. The step size is set to 100 because the search is performed with appropriate granularity within this concentrated interval, avoiding the step size being too large and crossing the optimal solution, or the step size being too small and increasing ineffective computing power.

[0115] Step S4-3: Combine the optimal parameters Substituting the augmented Lagrangian expression of VMD, the mode decomposition is completed by iteratively updating the expression using the alternating direction multiplier method until convergence. The convergence criterion is:

[0116] ;

[0117] in, For the summation operation, iterate through all... Each intrinsic mode component The component number is the component sequence number. This represents the number of decomposition modes in the VMD algorithm. For the first After the nth iteration A signal sequence of eigenmode components. This represents the number of VMD iterations. For the first After the nth iteration The signal sequence of each intrinsic mode component is the iterative update value; The square of the vector's second norm is used to calculate the total energy of the signal; For the first The energy change of each component in adjacent iterations is used to measure the convergence stability of the component. For the first The total energy of each component is used for normalization, converting absolute error into relative error. The preset convergence tolerance threshold is a manually set, extremely small positive number used to determine whether the iteration has converged.

[0118] After completing the modal decomposition, several intrinsic mode components are obtained. The fundamental intrinsic mode component with the highest energy is selected and defined as the fundamental principal component, which is used as the input for subsequent parameter extraction. This step uses energy characteristics as the core selection criterion.

[0119] After preprocessing steps S2 and S3, although the fundamental energy is absolutely dominant, a small amount of fundamental energy may leak to adjacent modes when the grid frequency deviates slightly. If only a single intrinsic mode component is selected as the fundamental wave based on its energy level, the leaked energy will be lost, leading to subsequent TLS-ESPRIT phase identification errors. Therefore, in a preferred approach, to further ensure the integrity of the fundamental wave components, a Spearman correlation coefficient threshold is used for auxiliary screening: first, the intrinsic mode component with the highest energy is selected as the principal component of the fundamental wave; then, the Spearman correlation coefficient between each intrinsic mode component and the filtered voltage signal is calculated, and the correlation coefficient threshold is set to 0.15. The residual fundamental wave components with correlation coefficients greater than the threshold are selected to reconstruct the fundamental wave signal together with the principal component, resulting in a more complete fundamental wave signal. The Spearman correlation coefficient threshold of 0.15 is primarily used to distinguish between the true fundamental frequency leakage component and the noise component. After preprocessing, the Spearman correlation coefficient between the noise component and the filtered voltage signal is typically below 0.1, while the fundamental frequency leakage component, due to its shared origin with the fundamental frequency principal component, exhibits a significantly higher correlation coefficient. The threshold of 0.15 falls within the safe boundary region between the two, effectively distinguishing between the two types of components while balancing fundamental frequency integrity and noise removal. This mechanism effectively improves the fundamental frequency energy recovery rate and significantly reduces phase identification error compared to single energy screening, while effectively eliminating noise component interference, all without increasing the number of decomposition layers.

[0120] Step S5: Using the TLS-ESPRIT algorithm, extract the amplitude, phase, and frequency parameters of the fundamental signal, specifically:

[0121] Step S5-1: Resample the fundamental signal obtained in step S4 with a step size m=5 to construct a dimension-reduced Hankel matrix. The selection of m needs to ensure that the Hankel matrix after dimension reduction can still completely preserve the fundamental signal characteristics, while effectively reducing the computational cost of subsequent SVD decomposition. It is usually a positive integer between 5 and 20. In this embodiment, the original sampling rate is 2kHz and the fundamental frequency is 50Hz. Choosing m=5 can ensure a reasonable ratio of the number of rows and columns in the matrix, so that the amount of data after dimension reduction is about 1 / 5 of the original data, significantly reducing the computational cost without losing the fundamental periodic characteristics. The Hankel matrix is ​​shown in the following formula:

[0122] ;

[0123] in, The Hankel matrix is ​​used to reduce the dimension of the distribution network voltage sequence and expand it into a multi-dimensional phase space, providing a data foundation for the singular value decomposition and subspace partitioning of the subsequent TLS-ESPRIT algorithm. This is a time delay (delay step), used to increase the distance between phase points in phase space; The embedding dimension (phase space dimension) corresponds to the number of columns in the matrix. This represents the total number of vectors in the phase space and the corresponding number of rows in the matrix.

[0124] Step S5-2: Perform singular value decomposition on the reduced-dimensional Hankel matrix, as shown in the following formula:

[0125] In the formula, The reduced-dimensional Hankel matrix is ​​constructed from a one-dimensional distribution network voltage signal sequence using the delay coordinate method and is used to characterize the dynamic evolution trajectory of the signal. Let be a left singular vector matrix, and be a reduced-dimensional Hankel matrix. Obtained through singular value decomposition (SVD), it is used to extract the phase space characteristic directions of the signal; It is a singular value diagonal matrix, where the diagonal elements are singular values, representing the energy weights of the corresponding feature patterns, and are used to quantify the importance of the feature patterns; is the right singular vector matrix, obtained by SVD decomposition, used to characterize the time delay evolution structure of the signal in phase space.

[0126] Through differential spectral sequence The dimension of the signal subspace is determined, and the signal subspace and noise subspace are divided. Specifically, the singular values ​​obtained from SVD decomposition are arranged in descending order, and the difference spectrum sequence of adjacent singular values ​​is calculated. The singular values ​​of the signal subspace are much larger than those of the noise subspace, with a significant numerical difference between them. The difference spectrum sequence exhibits a peak at this boundary. By searching for the peak coordinates of the difference spectrum sequence, the dimension of the signal subspace can be determined. The right singular vectors corresponding to the first few large singular values ​​are assigned to the signal subspace, and the right singular vectors corresponding to the remaining singular values ​​are assigned to the noise subspace. In the formula, For the first The relative rate of change of each singular value, a dimensionless characteristic index, is used to adaptively distinguish the singular value threshold from effective signals and noise. For the first The singular values ​​are derived from the dimension-reduced Hankel matrix. The SVD decomposition, arranged in descending order from largest to smallest, represents the energy / information content of the corresponding feature components. Large singular values ​​correspond to effective signals, and small singular values ​​correspond to noise. For the first The singular value is the th singular value after sorting. The next component of a singular value corresponds to a feature with lower energy.

[0127] This step determines the dimension of the signal subspace using the differential spectrum sequence, divides the signal subspace into a noise subspace, and retains only the signal subspace portion corresponding to large singular values, thereby effectively suppressing noise components and achieving subspace denoising. This provides a clean signal data foundation for subsequent parameter solving.

[0128] Step S5-3: Divide the right singular vector of the signal subspace into two interleaved submatrices, and solve for the rotation characteristic matrix as shown in the equation:

[0129] As shown, calculate its characteristic sequence. This characteristic value directly contains the frequency information of the voltage signal. In the formula, This is the rotation characteristic matrix of the TLS-ESPRIT algorithm, used to characterize the rotation characteristics of the signal subspace, and is the core intermediate matrix for solving the frequency information of the voltage signal; For the dimension-reduced Hankel matrix The right singular vector matrix obtained by singular value decomposition (SVD) is: Orthogonal matrix. It is the cross-coupling submatrix of the right singular vector matrix, corresponding to the coupling relationship between the effective signal component and the noise component, reflecting the degree of interference of noise on the phase space characteristics of the effective signal; This is a submatrix in the right singular vector matrix corresponding to the noise component. It is its inverse matrix, used to decouple noise from the effective signal and characterize the phase space structure of the noise component; the negative sign is the direction correction factor, used to unify the projection direction of the correction coefficient.

[0130] Step S5-4: Utilize the eigenvalue sequence Construct a system of linear equations and solve for the relaxation coefficients using the least squares method. As shown in the following formula:

[0131] ;

[0132] in, The amplitude values ​​of discrete sampling points of the distribution network voltage signal after PIA cubic spline Hermite interpolation, FIR filtering, phase space reconstruction, and SVD subspace noise reduction are given. For phase space time delay step, Embedding dimensions into phase space; The Vandermonde matrix is ​​the core structure for TLS-ESPRIT parameter identification. The number of fundamental signals to be identified; For the first Discrete-time complex exponential factor of each fundamental correlation component , For the first The frequencies of the fundamental correlated components The sampling period; For the first Complex amplitude coefficients of each fundamental correlated component, , For amplitude, Let be the phase, and be the core parameter to be determined. In this step, the fundamental signal output by S4 is the complete fundamental component, so the subscript j in the above parameters takes the value of only 1, that is, only the complex amplitude coefficients of the fundamental signal are solved.

[0133] Combining the relaxation coefficient and eigenvalue, the voltage amplitude, frequency, and initial phase are calculated using the following formula.

[0134] ;

[0135] in, For the first The voltage amplitude of each fundamental frequency correlated component; The solution to the Vandermonde matrix equations yields the first... The complex amplitude coefficient of each component, Its modulus length; For the first The frequency of each component; For the first Discrete-time complex exponential factor of each component These are the real part and the imaginary part, respectively. The time step corresponds to the signal sampling interval; For the first The initial phase of each component is the core parameter for PMU synchronous phasor measurement and fault location.

[0136] In this invention, It is consistent with the time scale of the sampling period. Where is the sampling frequency, and m is the delay step size (number of sampling points) for phase space reconstruction. To determine the actual time step corresponding to the delay, the three follow the general conversion relationship of the TLS-ESPRIT algorithm. In this invention, m=5, that is, every 5 original sampling points, 1 sample point is taken to construct a dimension-reduced Hankel matrix, so as to reduce the matrix dimension and reduce the computational cost of subsequent SVD decomposition.

[0137] Step S5-5: Perform complex plane quadrant correction on the initial phase: If the phase is located in the 1st or 4th quadrant of the complex plane, take the opposite number to obtain the final phase.

[0138] Phase through formula During calculation, the inherent output range of the arctan function is only... Unable to cover the whole The phase interval leads to a calculation error in the phase sign of the first and fourth quadrants of the complex plane. Therefore, quadrant correction is used to restore the true phase. This step is based on... The sign correction of the quadrant distribution is a calibration of the accuracy of the phase calculation results in S5-4. The specific rule for the correction is: in step S5-4, the initial phase... Through formula Calculate. The output range of the arctan function is... This only covers the first and fourth quadrants of the complex plane, while the true phase may lie in the second and third quadrants. To reconstruct the true phase, this step relies on the complex amplitude coefficients. The sign of the real and imaginary parts determines the quadrant in which it is located: when Located in quadrants 1 and 4 (i.e. )hour, The calculated value is the opposite of the calculated value; when Located in the 2nd and 3rd quadrants (i.e. )hour, The calculated value remains unchanged. This correction rule is based on the complex amplitude coefficients in the TLS-ESPRIT algorithm. The definition is directly exported. The sign of the real part determines the half-plane in which the initial phase lies, and the true phase can be uniquely determined by its correspondence with the arctan range.

[0139] Example 1

[0140] This embodiment uses the IEEE 1159 standard voltage signal as the test object, with a fundamental frequency of 50Hz, including the third harmonic (content of 5%) and high-frequency noise (signal-to-noise ratio of 40dB), a sampling rate of 2kHz, and a total duration of 100ms. A 20% frame drop rate was artificially set to simulate non-uniform sampling distortion, and the measurement method of this invention was used for experimental verification.

[0141] First, the non-uniformly sampled signal was reconstructed using PIA cubic spline Hermite interpolation, and the interpolation error was tested under two smoothing conditions, G1 and G2. Condition G1 requires the interpolation curve to satisfy positional constraints and first-order derivative constraints at the sampling points, while condition G2 further requires continuity of the second-order derivative, placing more stringent demands on interpolation accuracy. The interpolation errors under the two conditions are shown in Tables 1 and 2, respectively. As can be seen from the tables, after 10 iterations, the error under condition G1 decreased to 1.16E-10, and the error under condition G2 decreased to 2.37E-10, both reducing the error by eight orders of magnitude compared to the traditional PIA four-line interpolation method. This indicates that the method of this invention can quickly converge to high accuracy under different smoothing levels, effectively correcting the distortion of non-uniform sampling and providing a high-fidelity data foundation for subsequent signal processing.

[0142] Table 1: Error of G1 point data in curve interpolation

[0143]

[0144] Table 2: Error of G2 point data in curve interpolation

[0145]

[0146] Interpolation effect, for example Figure 3 , Figure 4 As shown. Figure 3 In the diagram, the red scatter points represent the original non-uniform sampled signal, and the blue curve represents the waveform reconstructed by PIA cubic spline Hermite interpolation. The curve is smooth and passes precisely through the sampling points, indicating that this method can quickly and accurately repair waveform distortion. Figure 4 The image shows the interpolation results of the PIA four-line method. The discrete points represent the original sampled signal, with the ideal signal at the top and the interpolation curves for 20, 10, 5, and 1 iterations below. It can be seen that the curves gradually approach the ideal signal with increasing iteration count, but noticeable deviations still exist after 20 iterations. In comparison, the method of this invention converges faster and has higher approximation accuracy.

[0147] Using bidirectional FIR filtering to process the interpolated signal can eliminate the phase delay of the filtered voltage signal, and the filtering effect is comparable to... Figure 5 , Figure 6 As shown. Figure 5 The result is from a one-way filter, which shows that the one-way filter introduces a significant phase lag. Figure 6 As a bidirectional filtering result, after the "forward-reverse-inverted-recovery" operation, the output waveform completely coincides with the input waveform in phase, and the phase delay is eliminated. This demonstrates that the bidirectional FIR filter structure of this invention can maintain the phase characteristics of the signal while suppressing interference.

[0148] An improved variational mode decomposition method was used to decompose the filtered voltage signal. The Spearman correlation coefficients between each intrinsic mode component and the filtered voltage signal after decomposition were 0.9893, 0.0743, 0.0758, and 0.5824, respectively. A correlation coefficient threshold of 0.15 was set, and the first and fourth intrinsic mode components were selected to reconstruct the fundamental signal.

[0149] The amplitude, phase, and frequency of the fundamental signal were extracted using the TLS-ESPRIT algorithm. Ten repeated measurements were first performed at the 50Hz power frequency. Comparative tests were then conducted at 49.5Hz, 49.8Hz, 50.3Hz, and 50.5Hz. The results are shown in Tables 3 and 4.

[0150] Table 3: Comparison of voltage phase errors at different frequencies (′′)

[0151]

[0152] Table 4: Comparison of voltage amplitude error at different frequencies (%)

[0153]

[0154] As shown in Tables 3 and 4, the maximum phase error at the 50Hz power frequency is 0.29″ and the minimum is 0.10″; the maximum amplitude error is 0.042% and the minimum is 0.017%, both of which are better than the preset accuracy requirements.

[0155] The signal processing effects of the joint algorithm of this invention at different frequencies are as follows: Figure 7 , Figure 8 , Figure 9 , Figure 10 , Figure 11 As shown, the corresponding operating conditions are 49.5Hz, 49.8Hz, 50Hz, 50.3Hz, and 50.5Hz, respectively. The blue curve represents the original signal without processing, the green curve represents the waveform processed by the method of this invention, and the red curve represents the measurement standard signal. It can be seen that under different frequency conditions, the green curve highly overlaps with the red curve, indicating that this method can effectively recover the signal waveform within a frequency deviation range of ±0.5Hz, demonstrating good frequency adaptability and phase tracking accuracy.

[0156] Example 2:

[0157] In this embodiment, keeping other conditions unchanged, the fundamental frequency was adjusted to 49.5Hz, 49.8Hz, 50.3Hz, and 50.5Hz respectively to verify the measurement accuracy under frequency deviation conditions. The frequency calculation error is shown in Table 5.

[0158] Table 5: Frequency Calculation Error (Hz)

[0159]

[0160] As shown in Table 5, within the frequency deviation range of ±0.5Hz, the absolute value of the frequency error is less than 0.001Hz, which is far better than the preset ±0.02Hz.

[0161] Example 3:

[0162] This embodiment tests the real-time performance of the method of the present invention. Using the test signal of Embodiment 1, the complete measurement process (including interpolation, filtering, mode decomposition, and parameter identification) was run 10 times consecutively, and the processing delay of each measurement was statistically analyzed. The results are shown in Table 6.

[0163] Table 6: Statistics of Time for Measuring Voltage Parameters

[0164]

[0165] As shown in Table 6, the average delay is 403μs, which meets the real-time requirements of the project.

[0166] A comparative experiment was conducted between the method of this invention and existing traditional signal processing methods (PIA four-line interpolation, conventional one-way FIR filtering, traditional artificial parameter VMD, and Kalman filtering). The results show that the method of this invention requires the fewest iterations to reach the convergence state and has the best final voltage parameter extraction accuracy (amplitude, phase, and frequency error). This indicates that the overall processing performance of this invention is superior to various existing signal processing algorithms and is more suitable for the high-precision and real-time requirements of distribution network feeder verification.

[0167] The above description is merely an embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principle of the present invention should be included within the scope of the claims of the present invention.

Claims

1. A high-precision measurement method for feeder verification reference voltage based on PIA-bidirectional FIR, characterized in that, Includes the following steps: Step S1: Acquire the voltage signal of the distribution network bus; Step S2: The acquired voltage signal is reconstructed using PIA cubic spline Hermite interpolation to obtain the reconstructed voltage signal; Step S3: Perform bidirectional FIR filtering on the reconstructed voltage signal to obtain the filtered voltage signal. Bidirectional FIR filtering includes forward filtering, time reversal followed by secondary filtering, and sequence recovery. Step S4: Adaptively decompose the filtered voltage signal using improved variational mode decomposition to obtain the fundamental signal; Step S5: Use the TLS-ESPRIT algorithm to extract the amplitude, phase, and frequency parameters of the fundamental signal.

2. The high-precision measurement method for feeder verification reference voltage based on PIA-bidirectional FIR according to claim 1, characterized in that, In step S2, PIA cubic spline Hermite interpolation is used to reconstruct the waveform of the acquired voltage signal to obtain the reconstructed voltage signal, specifically as follows: Step S2-1: Construct a piecewise cubic polynomial between adjacent sampling points so that the interpolation curve satisfies both position constraints and derivative constraints at each sampling point. The position constraints are determined by the voltage values ​​of each sampling point, and the derivative constraints are determined by the first derivative estimates calculated from the sampling time intervals corresponding to each sampling point, in order to reconstruct the non-uniform sampling voltage signal. Step S2-2: Generate the augmented control vertex matrix using an asymptotic iterative approximation algorithm, and iterate over the augmented control vertices; Step S2-3: Perform convergence judgment on the augmented control vertex matrix, calculate the Euclidean displacement of each augmented control vertex between two adjacent iterations, and determine that the algorithm has reached convergence when the Euclidean displacement of all augmented control vertices is less than the preset tolerance in N consecutive iteration cycles, and terminate the iteration.

3. The high-precision measurement method for feeder verification reference voltage based on PIA-bidirectional FIR according to claim 1, characterized in that, In step S3, the reconstructed voltage signal is subjected to bidirectional FIR filtering to obtain the filtered voltage signal. The bidirectional FIR filtering includes forward filtering, time reversal followed by secondary filtering, and sequence recovery, specifically: Step S3-1: Perform forward FIR filtering on the reconstructed voltage signal to obtain a forward filtered signal; Step S3-2: Reverse the time sequence of the forward filtered signal and perform the same FIR filter to obtain the reverse filtered signal; Step S3-3: Restore the original time sequence of the reverse filtered signal and output the filtered voltage signal.

4. The high-precision measurement method for feeder verification reference voltage based on PIA-bidirectional FIR according to claim 3, characterized in that, In step S3, the FIR filter is a 255th-order Kaiser window FIR filter with a cutoff frequency of 2.75kHz. The filter's unit impulse response satisfies the even symmetry condition, and the stopband attenuation is 80dB. The output of the FIR filter is calculated using the linear convolution formula. ； in, This is the output signal sequence of the FIR filter. The filter's unit impulse response. Let the filter order be . The input is the reconstructed voltage signal to be filtered; The index variable for the convolution summation. The sampling point number of the discrete-time signal. This is a linear convolution summation operation.

5. The high-precision measurement method for feeder verification reference voltage based on PIA-bidirectional FIR according to claim 1, characterized in that, In step S4, improved variational mode decomposition is used to adaptively decompose the filtered voltage signal to obtain the fundamental signal, specifically as follows: Step S4-1: Start the iteration with an initial number of modes K=1, introduce the total entropy of residual loss, and iterate successively to increase the number of decomposed modes K and calculate the total entropy of residual loss. Convergence occurs when the total entropy of residual loss is less than a set threshold. Adaptively determine the optimal number of modes K to avoid under-decomposition or over-decomposition of the signal; the expression for the total entropy of residual loss is: ； In the formula, The total entropy is the residual loss. These are the indexes of the intrinsic mode components obtained from VMD decomposition. To decompose the mode number; This represents the number of iterations. For the nth iteration The second norm of each intrinsic modal component; For the (n-1)th iteration The second norm of each intrinsic modal component; The second norm of the filtered voltage signal in the frequency domain; Set a threshold for total entropy loss During the iteration process, if Greater than the set threshold Then let Continue decomposing until... Less than or equal to Determine the optimal number of modes ; Step S4-2: Construct a comprehensive evaluation index by combining the modal center frequency, Chebyshev distance, and improved edge permutation entropy. With the goal of maximizing Chebyshev distance and minimizing improved edge permutation entropy, traverse the search to optimize and determine the optimal penalty factor. This improves mode separation and the purity of the fundamental component; Step S4-3: Based on the optimal number of modes and optimal penalty factor The filtered voltage signal is decomposed using VMD, and the fundamental signal is obtained by screening the intrinsic mode components.

6. The high-precision measurement method for feeder verification reference voltage based on PIA-bidirectional FIR according to claim 5, characterized in that, In step S4-2, the penalty factor is optimized by combining the modal center frequency Chebyshev distance and the improved edge arrangement entropy. Specifically: Chebyshev distance ; in, The maximum separation of modal center frequencies is the Chebyshev distance. These are the indexes of the intrinsic mode components obtained from VMD decomposition. ,and K is the total number of modes in the VMD decomposition; For the first The center frequencies of the intrinsic mode components; For the first The center frequencies of the intrinsic mode components; It represents the absolute difference between the center frequencies of two different intrinsic mode components; To traverse all intrinsic mode component indices , ,and Take the maximum value of the absolute differences in frequency between all different mode pairs; Improved edge permutation entropy ; in, To improve edge arrangement entropy; The embedding dimension for reconstructing the phase space determines the number of permutation patterns; The time delay for phase space reconstruction; This is the sequence number of the arrangement pattern; To determine the first by combining the weights of the short sequence The probability of a certain permutation pattern occurring; This is a term used for Shannon entropy calculation; For the normalized denominator, is the theoretical maximum value of Shannon entropy; A comprehensive evaluation index is derived by combining the modal center frequency Chebyshev distance with the improved edge arrangement entropy, in order to optimize the penalty factor. The expression is: ; in, The subscript is a comprehensive evaluation index for adaptive optimization of VMD parameters. Corresponding to the VMD parameters to be optimized, Chebyshev distance; To improve edge arrangement entropy.

7. The high-precision measurement method for feeder verification reference voltage based on PIA-bidirectional FIR according to claim 5, characterized in that, The process of filtering intrinsic mode components to obtain the fundamental signal in step S4-3 is as follows: The fundamental eigenmode component with the highest energy is selected and defined as the fundamental principal component, which is then used as the fundamental signal.

8. The high-precision measurement method for feeder verification reference voltage based on PIA-bidirectional FIR according to claim 5, characterized in that, In step S4-3, the fundamental signal is obtained by filtering the intrinsic mode components. As another method, the specific steps are as follows: The fundamental wave eigenmode component with the highest energy is defined as the fundamental wave principal component. Calculate the Spearman correlation coefficient between each intrinsic mode component other than the fundamental main component and the filtered voltage signal, and select the intrinsic mode components with Spearman correlation coefficients greater than a preset threshold as residual fundamental components. The filtered residual fundamental component is reconstructed with the fundamental principal component, and the reconstructed signal is used as the fundamental signal.

9. The high-precision measurement method for feeder verification reference voltage based on PIA-bidirectional FIR according to claim 1, characterized in that, In step S5, the TLS-ESPRIT algorithm is used to extract the amplitude, phase, and frequency parameters of the fundamental signal, specifically as follows: Step S5-1: Resample the fundamental signal and construct a dimension-reduced Hankel matrix; Step S5-2: Perform singular value decomposition on the Hankel matrix to divide it into signal subspace and noise subspace; Step S5-3: Solve for the rotation property matrix and calculate the eigenvalues; Step S5-4: Solve for the relaxation coefficient, and combine the eigenvalues ​​and relaxation coefficients to calculate the voltage amplitude, frequency, and initial phase; Step S5-5: Perform complex plane quadrant correction on the initial phase and output the final phase.

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