A Power System Harmonic Frequency Estimation Method Based on Coprime Sampling Root-Finding MUSIC

Abstract

This invention discloses a power system harmonic frequency estimation method based on coprime sampling root-finding MUSIC. The method uses a sampler with an expanded coprime array to sample the signal in the time domain, obtaining sample data; constructs the corresponding estimated covariance matrix; performs eigenvalue decomposition on the obtained covariance matrix to obtain the noise subspace; defines a polynomial for the desired frequency; and solves the polynomial to obtain the frequency estimate. This invention combines the idea of ​​sparse sampling with the problem of power system harmonic and interharmonic estimation, fully utilizing the large array aperture characteristics of coprime arrays and modern spectral estimation algorithms to achieve high-precision frequency estimation while reducing computational complexity.

Description

Technical Field

[0001] This invention relates to the field of power system harmonic measurement, and particularly to harmonic and interharmonic frequency estimation based on coprime sampling root-finding MUSIC. Background Technology

[0002] With the widespread application of nonlinear devices such as power electronics in power systems, a large number of nonlinear loads are connected to the power grid, leading to increasingly serious harmonic and interharmonic pollution. Various faults and accidents caused by harmonics and interharmonics are occurring continuously, severely impacting power grid safety and power quality. Therefore, it is necessary to control harmonics and interharmonics, and accurate and effective estimation of harmonic and interharmonic parameters is a prerequisite and important guarantee for harmonic and interharmonic control.

[0003] Classical methods for estimating harmonics and interharmonics in power systems are mainly based on Fourier transform. These algorithms have advantages such as fast computation speed and ease of hardware implementation. However, they have two drawbacks: First, the frequency resolution is limited, and they can only estimate integer harmonic parameters, not interharmonic parameters. Second, the algorithm requires synchronous sampling, but interharmonics are often present in actual power grids, making synchronous sampling difficult. During asynchronous sampling, the spectra of different harmonics and interharmonics interfere with each other, causing severe frequency leakage and picket fence effects, leading to the failure of harmonic parameter estimation.

[0004] To address the inherent limitations of the classical Fourier transform, modern spectral estimation theory has been applied to the estimation of harmonic and interharmonic parameters in power systems. Existing estimation methods typically employ uniform sampling for signal reception and modeling, which is limited by the Nyquist sampling rate. Since the estimation accuracy is proportional to the array aperture, traditional methods require increasing the number of samples to expand the array aperture in order to improve accuracy, leading to an increase in both computational and hardware complexity. Therefore, existing estimation methods present a trade-off between accuracy and computational complexity.

[0005] Currently, coprime sampling schemes have attracted attention. Coprime sampling breaks through the limitations of traditional sampling frequencies and has many excellent characteristics. It can obtain a larger array aperture than traditional uniform sampling, and while improving accuracy, it can obtain relatively accurate parameter estimation results with fewer samples, which is beneficial for realizing real-time estimation of harmonic frequencies.

[0006] Improper use of sparse arrays can lead to ambiguity in estimation results, and the accuracy of commonly used spatial smoothing methods decreases with the addition of smoothing processes when the signal components estimated in power system harmonic estimation problems are few. This invention directly uses root-finding MUSIC to estimate coprime sample data, overcoming the decorrelation process of spatial smoothing methods. Furthermore, it eliminates the need for a continuous uniform array during processing, enabling effective recovery of undersampled signals. Summary of the Invention

[0007] The technical problem to be solved by this invention is a method for estimating the harmonic and interharmonic frequencies of a power system based on coprime sampling root-finding MUSIC, which combines coprime sampling technology with the problem of power grid signal frequency estimation and has high estimation accuracy.

[0008] To address the aforementioned technical problems, this invention provides a method for estimating the harmonic and interharmonic frequencies of a power system based on coprime sampling root-finding MUSIC, comprising the following steps:

[0009] (1) Use a sampler with an expanded coprime array to sample the signal in the time domain to obtain sample data;

[0010] (2) Construct the corresponding estimated covariance matrix based on the sample data;

[0011] (3) Perform eigenvalue decomposition on the obtained covariance matrix to obtain the noise subspace;

[0012] (4) Define the root polynomial for the frequency of a polynomial;

[0013] (5) Solve the polynomial to obtain the estimated frequency.

[0014] Preferably, (1) the implementation process is as follows: M+N-1 samplers are used, and the array is constructed according to the expanded coprime array structure, where M and N are coprime numbers. The power grid signal is sampled, and the number of times the received signal is sampled is L, so as to obtain sample data.

[0015] Preferably, step (2) is implemented as follows, and the sampling signal of a single sampler for the lth time is as follows:

[0016]

[0017]

[0018] Where m and n represent the sampling sequence number, 0≤m≤M-1, 1≤n≤N, and 1 / T represents the Nyquist sampling frequency;

[0019] The above samples are used to construct the sampled signal vectors of two subarrays.

[0020] Y M (l)=[x M ((M+N-1)l+0),x M ((M+N-1)l+1),...,x M ((M+N-1)l+N-1)] T

[0021] Y N (l)=[xN ((M+N-1)l+N+1),x N ((M+N-1)l+N+2),...,x N ((M+N-1)l+N+M-1)] T

[0022] By concatenating the samples from the two subarrays, the signal vector of the entire sampled signal is represented as:

[0023]

[0024] Where A=[a(ω1),a(ω2),...,a(ω D )] is the frequency matrix, a(ω) d () is a frequency vector containing information about a single frequency, represented as

[0025] Use zero-mean Gaussian white noise; construct the corresponding estimated covariance matrix. in The rank of is S.

[0026] Preferably, step (3) specifically includes:

[0027] The covariance matrix is ​​decomposed into eigenvalues, which is represented as follows: Among them Λ s This represents a D*D diagonal matrix, where D is the number of sinusoidal components including the fundamental, harmonics, and interharmonics. The diagonal elements are sorted in descending order, with the first D elements being the most significant. The eigenvalues ​​of Λ n This represents the SD numbers after sorting from largest to smallest. E is a diagonal matrix composed of the eigenvalues ​​of . s The top D items are sorted from largest to smallest. The matrix formed by the eigenvectors corresponding to the eigenvalues ​​of E, n Then it is sorted from largest to smallest by SD units. The matrix formed by the eigenvectors corresponding to the eigenvalues ​​of E is... s This is called the signal subspace, E n This is called the noise subspace.

[0028] Preferably, step (4) specifically includes:

[0029] Define the root-finding polynomial

[0030] Where p(z) = [1, z] M ,...,z (N-1)M ,...,z (N-1)M+(M-1)N ], z is the parameter to be determined containing frequency components, when When p(z) belongs to the signal subspace, due to the orthogonality between the signal subspace and the noise subspace, F(z) = 0, that is, the root of the polynomial on the unit circle corresponds to the frequency of the sinusoidal signal.

[0031] Preferably, step (5) specifically involves: taking the D roots of the polynomial F(z) that are closest to the unit circle, namely z1, z2, ..., z3. D The corresponding conjugate root is (z * 1, z * 2、…、z * D Then, the analog angular frequency of the complex sinusoidal signal can be obtained as follows:

[0032] Beneficial effects:

[0033] (1) The proposed method uses a sparse sampling method, which has a lower sampling rate than the traditional uniform sampling.

[0034] (2) The proposed method uses root-MUSIC for analysis, and its computational complexity is lower than that of the MUSIC method.

[0035] (3) This method is applicable to harmonic and interharmonic signals in power systems, is easy to implement in practice, and can achieve high-precision frequency estimation. Attached Figure Description

[0036] Figure 1 This is a flowchart of the present invention.

[0037] Figure 2 This is a schematic diagram of the sampling time of the present invention;

[0038] Figure 3 This is a distribution diagram of the frequency estimation results of the present invention.

[0039] Figure 4 This invention compares the performance trends of uniform sampling and coprime sampling with the signal-to-noise ratio under the same number of samples.

[0040] Figure 5 This invention compares the performance trends of uniform sampling and coprime sampling with the number of samplings under a signal-to-noise ratio of 20dB. Detailed Implementation

[0041] To enable those skilled in the art to better understand the technical solution of the present invention, the present invention will be further described in detail below with reference to specific embodiments.

[0042] This invention provides a method for estimating the harmonic and interharmonic frequencies of a power system based on coprime sampling root-finding MUSIC. The array structure used consists of coprime arrays with coprime numbers M and N, where M and N are a pair of coprime numbers. Two sets of Nyquist samplers are used to sample the signal under test, with sampling time intervals of MT and NT, respectively, where 1 / T represents the Nyquist sampling frequency.

[0043] I. Data Model

[0044] A power system frequency signal (voltage or current) containing noise, power frequency, harmonics, and interharmonics at the receiving end can be represented as:

[0045]

[0046] In the formula: D is the number of sinusoidal components including the fundamental wave, harmonics, and interharmonics; α d ω is the amplitude of the d-th sine component; d Let be the angular frequency of the d-th sine component; t represents the phase of the d-th sinusoidal component; e(t) represents the noise signal.

[0047] It can be transformed using Euler's formula.

[0048]

[0049] In the formula: A d The amplitude of the harmonic signal is a complex constant; ω d The normalized frequency to be estimated; The normalized initial phase is uniformly distributed in the interval (-π, π); u is uncorrelated complex Gaussian white noise with zero mean.

[0050] The sampled signal of a single sampler at the lth time is as follows:

[0051]

[0052]

[0053] Where m and n represent the sampling sequence numbers, 0≤m≤M-1, 1≤n≤N.

[0054] The sampled signal vectors of the two subarrays can be constructed using the above samples.

[0055] Y M (l)=[x M ((M+N-1)l+0),x M ((M+N-1)l+1),...,x M ((M+N-1)l+N-1)] T

[0056] Y N (l)=[x N ((M+N-1)l+N+1),x N ((M+N-1)l+N+2),...,x N ((M+N-1)l+N+M-1)] T

[0057] By concatenating the samples from the two subarrays, the signal vector of the entire sampled signal can be represented as:

[0058]

[0059] Where A=[a(ω1),a(ω2),...,a(ω D )] is the frequency matrix, a(ω) d () is a frequency vector containing information about a single frequency, represented as u(l) is zero-mean Gaussian white noise.

[0060] II. Frequency Estimation Methods

[0061] 1. Sampling is performed according to the unfolded coprime array structure, where M and N are coprime numbers. The power grid signal is sampled, and the number of samplings of the received signal is L, to obtain the array received signal matrix.

[0062] 2. Construct the corresponding covariance matrix based on the received signal.

[0063]

[0064] 3. Perform eigenvalue decomposition on the obtained covariance matrix to obtain the noise subspace.

[0065]

[0066] Among them Λ s Let Λ represent a D*D diagonal matrix, where the diagonal elements consist of the D largest eigenvalues. n E represents a diagonal matrix consisting of SD smaller eigenvalues. s E is a matrix composed of the eigenvectors corresponding to D larger eigenvalues. n It is a matrix composed of the eigenvectors corresponding to the other SD smaller eigenvalues. E s This is called the signal subspace, E n This is called the noise subspace.

[0067] 4. Define a polynomial In the formula, u k It is the covariance matrix The k-th eigenvector, z is the parameter to be calculated, and p(z) = [1, z]. M ,...,z (N-1)M ,...,z (N-1)M+(M-1)N To extract information from all eigenvectors simultaneously, we want to calculate the denominator of the MUSIC spectral function. The zero point. At this point, the polynomial is not yet a polynomial of z, because there exists a conjugate term for z. Since we are only interested in the z-value on the unit circle, we can use P. T (z -1 ) replace p H (z).

[0068] 5. Rewrite the polynomial above as a polynomial of degree z, expressed as: when When p(z) belongs to the signal subspace, due to the orthogonality of the signal and noise subspaces, F(z) = 0, meaning the roots of the polynomial on the unit circle correspond to the frequencies of the sinusoidal signal. We take the D roots of the polynomial F(z) that are closest to the unit circle, denoted as z1, z2, ..., z... D The corresponding conjugate root is (z * 1, z * 2、…、z * D Then, the analog angular frequency of the complex sinusoidal signal can be obtained as follows:

[0069]

[0070] The effects of the present invention will be further described below with reference to simulation examples.

[0071] Assume the harmonic signal received by the sensor is

[0072] x(t)=0.2cos(2π·25t)+cos(2π·50t)+0.2cos(2π·150t)+e(t)

[0073] This signal contains three frequency components: 50Hz power frequency, 25Hz interharmonic, and 150Hz third harmonic. e(t) is Gaussian white noise.

[0074] In the simulation, to ensure fair comparison, a uniform linear array was simulated using a classic MUSIC model, with M+N-1 = 9 sensor elements, and the fixed step size of the MUSIC model was set to 0.010. We use the root-mean-square error (RMSE) of the signal frequency estimation to evaluate the parameter estimation performance of the proposed algorithm, defined as...

[0075]

[0076] in, For the i-th Monte Carlo simulation, ω m The estimated value is I. I represents the total number of simulations; in the simulations below, we take I = 200.

[0077] Simulation 1: Figure 3 The graph shows the frequency estimation results of the proposed method at a signal-to-noise ratio of 20 dB. The parameters of the coprime array used are defined as M = 4 and N = 5. The graph shows that the algorithm can still effectively identify fixed frequencies even at low signal-to-noise ratios.

[0078] Simulation 2: Figure 4 To compare the performance of the proposed method with that of uniform sampling and coprime sampling under the same number of samples, and to ensure a fair comparison, the same sample size was maintained. The array used for uniform sampling was set to a uniform linear array with M+N-1=9 samplers and 300 sampling times. The figure shows that the frequency estimation performance of the proposed method is superior to that of commonly used uniform linear arrays.

[0079] Simulation 3: Figure 5 The graph shows a comparison of the frequency estimation performance of the proposed method with the number of samplings at a signal-to-noise ratio of 20 dB. It can be seen from the graph that the proposed method outperforms commonly used uniform linear arrays in frequency estimation performance.

Claims

1. A method for estimating harmonic frequencies in power systems based on coprime sampling root-finding MUSIC, characterized in that, Includes the following steps: (1) Use a sampler with an expanded coprime array to sample the signal in the time domain to obtain sample data; (2) Construct the corresponding estimated covariance matrix based on the sample data; (3) Perform eigenvalue decomposition on the obtained covariance matrix to obtain the noise subspace; (4) Define the root-finding polynomial for the frequency of a polynomial; (5) Solve the polynomial to obtain an estimate of the frequency; The implementation process of step (1) is as follows: Using M+N-1 samplers, structured according to an expanded coprime array, where M and N are coprime numbers, the power grid signal is sampled, and the number of times the received signal is sampled is L, to obtain sample data; The implementation process of step (2) is as follows. The sampling signal of a single sampler for the lth time is as follows: ； ； Where m and n represent the sampling sequence numbers, Where 1 / T represents the Nyquist sampling frequency; The above samples are used to construct the sampled signal vectors of two subarrays. ； ； By concatenating the samples from the two subarrays, the signal vector of the entire sampled signal is represented as: ； in For frequency matrix, A frequency vector, containing information about a single frequency, is represented as... , u(l) is zero-mean Gaussian white noise; construct the corresponding estimated covariance matrix. ,in The rank of is S.

2. The power system harmonic frequency estimation method based on coprime sampling root-finding MUSIC as described in claim 1, characterized in that, Step (3) specifically involves: The covariance matrix is ​​decomposed into eigenvalues, which is represented as follows: ;in This represents a D*D diagonal matrix, where D is the number of sinusoidal components including the fundamental, harmonics, and interharmonics. The diagonal elements are sorted in descending order, with the first D elements being the most significant. The eigenvalues ​​are composed of, This represents the SD numbers after sorting from largest to smallest. A diagonal matrix composed of the eigenvalues ​​of . The top D items are sorted from largest to smallest. The matrix formed by the eigenvectors corresponding to the eigenvalues ​​of . Then it is sorted from largest to smallest by SD units. The matrix formed by the eigenvectors corresponding to the eigenvalues ​​of . This is called the signal subspace. This is called the noise subspace.

3. The power system harmonic frequency estimation method based on coprime sampling root-finding MUSIC as described in claim 2, characterized in that, Step (4) is as follows: Define the root-finding polynomial ; in Let z be the parameter to be determined containing frequency components, when hour, Belonging to the signal subspace, and due to the orthogonality between the signal subspace and the noise subspace, F(z) = 0, that is, the root of the polynomial on the unit circle corresponds to the frequency of the sinusoidal signal.

4. The power system harmonic frequency estimation method based on coprime sampling root-finding MUSIC as described in claim 3, characterized in that, Step (5) specifically involves: Take the D roots of the polynomial F(z) that are closest to the unit circle, namely z1, z2, ..., z3. D The corresponding conjugate root is (z * 1, z * 2、…、z * D Then the analog angular frequency of the complex sine signal is obtained as follows: 。

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