CN115541995A - Power system harmonic frequency estimation method based on mutual prime sampling root finding MUSIC - Google Patents

Power system harmonic frequency estimation method based on mutual prime sampling root finding MUSIC Download PDF

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CN115541995A
CN115541995A CN202211166354.4A CN202211166354A CN115541995A CN 115541995 A CN115541995 A CN 115541995A CN 202211166354 A CN202211166354 A CN 202211166354A CN 115541995 A CN115541995 A CN 115541995A
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岳衡
张小飞
石莎
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Nanjing University of Aeronautics and Astronautics
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Abstract

The invention discloses a power system harmonic frequency estimation method based on mutual-prime sampling root-finding MUSIC, which comprises the steps of carrying out time domain sampling on a signal by using a sampler for expanding a mutual-prime array to obtain sample data; constructing a corresponding estimation covariance matrix; performing characteristic decomposition on the obtained covariance matrix to obtain a noise subspace; defining a polynomial of the sought frequency; solving the polynomial to obtain the estimated value of the frequency. The method combines the idea of sparse sampling with the problem of harmonic and inter-harmonic estimation of a power system, fully utilizes the large array aperture characteristic of a co-prime array and a modern spectrum estimation algorithm, reduces the computational complexity and simultaneously realizes high-precision frequency estimation.

Description

Power system harmonic frequency estimation method based on mutual prime sampling root finding MUSIC
Technical Field
The invention relates to the field of harmonic measurement of a power system, in particular to harmonic and inter-harmonic frequency estimation based on mutual prime sampling root finding MUSIC.
Background
With the wide application of nonlinear devices such as power electronics and the like in a power system, a large number of nonlinear loads are connected into a power grid, so that harmonic wave and inter-harmonic wave pollution in the power grid is increasingly serious, various faults and accidents caused by the harmonic wave and the inter-harmonic wave are continuously generated, and the safety and the power quality of the power grid are seriously influenced. Therefore, it is necessary to treat the harmonics and inter-harmonics, and accurate and effective harmonic and inter-harmonic parameter estimation is a precondition and an important guarantee for harmonic and inter-harmonic treatment.
The classical power system harmonic and inter-harmonic estimation method is mainly based on Fourier transform, and the algorithm has the advantages of high operation speed, easiness in hardware implementation and the like, but the algorithm has 2 defects: firstly, the frequency resolution is limited, only integer harmonic parameter estimation can be realized, and inter-harmonic estimation cannot be realized; and secondly, the algorithm requires synchronous sampling, but interharmonics often exist in an actual power grid, so that the synchronous sampling is difficult to realize, and when the sampling is not synchronous, the frequency spectrums of each subharmonic and the interharmonics are interfered with each other, so that serious frequency leakage and barrier effect are caused, and the estimation of harmonic parameters is invalid.
To address the inherent deficiencies of the classical fourier transform, modern spectral estimation theory is applied to power system harmonic and inter-harmonic parameter estimation. The existing estimation method generally adopts a uniform sampling method to receive and model signals, and is limited by a Nyquist sampling rate. Since the estimation accuracy is proportional to the array aperture, in order to improve the estimation accuracy, the conventional method needs to extend the array aperture by increasing the sampling times, which results in an increase in the computational complexity and hardware complexity of the entire system. Therefore, the existing estimation method has a certain trade-off problem between precision performance and computational complexity.
At present, attention is paid to a mutual-prime sampling scheme, and the mutual-prime sampling breaks through the limitation of the traditional sampling frequency and has a plurality of excellent characteristics. The array aperture larger than that of the traditional uniform sampling can be obtained, on the basis of improving the precision, a more accurate parameter estimation result can be obtained by less sampling number, and the real-time estimation of the harmonic frequency is favorably realized.
The improper use of sparse arrays can obscure the estimation results, and the accuracy of the commonly used spatial smoothing method decreases with the addition of the smoothing process in the case of less signal components estimated by the harmonic estimation problem of the power system. The invention directly uses the root finding MUSIC to estimate the relatively prime sampling data, overcomes the decorrelation process of the spatial smoothing method, has no requirement of continuous uniform arrays in the processing process, and can realize the effective recovery of the undersampled signals.
Disclosure of Invention
The invention aims to solve the technical problem that a power system harmonic and inter-harmonic frequency estimation method based on mutual-prime sampling root MUSIC combines a mutual-prime sampling technology and a power grid signal frequency estimation problem, and has higher estimation precision.
In order to solve the technical problem, the method for estimating the harmonic and inter-harmonic frequencies of the power system based on the mutual-prime sampling root-finding MUSIC provided by the invention comprises the following steps:
(1) Performing time domain sampling on the signal by using a sampler which expands a co-prime array to obtain sample data;
(2) Constructing a corresponding estimation covariance matrix according to the sample data;
(3) Performing characteristic decomposition on the obtained covariance matrix to obtain a noise subspace;
(4) Defining a root polynomial of a polynomial frequency;
(5) Solving the polynomial to obtain the estimated value of the frequency.
Preferably, (1) the implementation process is as follows: and (2) constructing according to an unfolded relatively-prime array structure by adopting the M + N-1 samplers, wherein M and N are relatively prime numbers, sampling the power grid signal, and obtaining sample data when the sampling frequency of the received signal is L.
Preferably, step (2) is implemented as follows, the sampling signal of the first time of a single sampler is as follows,
Figure BDA0003861538050000021
Figure BDA0003861538050000022
wherein M and N represent the serial number of sampling, M is more than or equal to 0 and less than or equal to M-1, N is more than or equal to 1 and less than or equal to N, and 1/T represents the Nyquist sampling frequency;
constructing a vector of sampled signals for two sub-arrays using the samples
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
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
The samples of the two sub-arrays are connected in series, and the signal vector of the whole sampling signal is expressed as
Figure BDA0003861538050000031
Wherein A = [ a (ω) 1 ),a(ω 2 ),...,a(ω D )]Is a frequency matrix, a (ω) d ) Is a frequency vector containing single frequency information expressed as
Figure BDA0003861538050000032
Figure BDA0003861538050000033
White gaussian noise with zero mean; constructing a corresponding estimated covariance matrix
Figure BDA0003861538050000034
Wherein
Figure BDA0003861538050000035
Is S.
Preferably, the step (3) is specifically:
the covariance matrix is subjected to eigenvalue decomposition and is expressed as
Figure BDA0003861538050000036
Wherein Λ s Representing a diagonal matrix of D, D is the number of sine components including fundamental wave, harmonic wave and inter-harmonic wave, and the first D diagonal elements are sorted from large to small
Figure BDA0003861538050000037
Characteristic value composition of n Represents S-D pieces after sorting from big to small
Figure BDA0003861538050000038
A diagonal matrix of eigenvalues of, E s Is D pieces in the front from big to small
Figure BDA0003861538050000039
A matrix formed by eigenvectors corresponding to the eigenvalues of (E) n Then S-D are sequenced from big to small
Figure BDA00038615380500000310
A matrix of eigenvectors corresponding to the eigenvalues of, E s Referred to as signal subspace, E n Referred to as the noise subspace.
Preferably, the step (4) is specifically:
defining a root polynomial
Figure BDA00038615380500000311
Wherein p (z) = [1, z ] M ,...,z (N-1)M ,...,z (N-1)M+(M-1)N ]Z is a parameter to be determined containing a frequency component when
Figure BDA00038615380500000312
When p (z) belongs to the signal subspace, derived from the orthogonality of the signal subspace to the noise subspace, F (z) =0, i.e. the root of the polynomial on the unit circle corresponds to the frequency of the sinusoidal signal.
Preferably, the step (5) is specifically: taking D roots of the polynomial F (z) closest to the unit circle, and respectively taking z 1 、z 2 、…、z D The corresponding conjugate root is (z) * 1 、z * 2 、…、z * D ) Then the analog angular frequency of the complex sinusoidal signal is obtained
Figure BDA00038615380500000313
Has the advantages that:
(1) The proposed method uses a sparse sampling method with a lower sampling rate than the conventional uniform sampling.
(2) The proposed method uses root-MUSIC for analysis, which is less computationally complex than the MUSIC method.
(3) The method is suitable for harmonic and inter-harmonic signals in a power system, is easy to realize in practice, and can realize high-precision frequency estimation
Drawings
FIG. 1 is a flow chart of the present invention.
FIG. 2 is a schematic of the sampling times of the present invention;
FIG. 3 is a distribution diagram of frequency estimation results according to the present invention.
FIG. 4 is a comparison of the variation trend of the signal-to-noise ratio of the performance of uniform sampling and relatively prime sampling under the same sampling number;
fig. 5 is a comparison of the performance of uniform sampling and relatively prime sampling with the variation trend of the sampling times under the condition of 20dB signal-to-noise ratio.
Detailed Description
In order that those skilled in the art will better understand the technical solutions of the present invention, the present invention will be further described in detail with reference to the following detailed description.
The invention provides a power system harmonic and inter-harmonic frequency estimation method based on mutual prime sampling root-finding MUSIC, and the used array structure is composed of a mutual prime array with mutual prime numbers of M and N, wherein M and N are a pair of mutual prime numbers. And respectively sampling the signals to be detected by using two groups of Nyquist samplers, wherein the sampling time intervals are MT and NT respectively, and 1/T represents the Nyquist sampling frequency.
1. Data model
The frequency signal (voltage or current) of the power system with noise, power frequency, harmonic and inter-harmonic components at the receiving end can be expressed as
Figure BDA0003861538050000041
In the formula: d is the number of sinusoidal components including fundamental wave, harmonic wave, inter-harmonic wave and the like; alpha (alpha) ("alpha") d Is the amplitude of the d-th sinusoidal component; omega d Is the angular frequency of the d-th sinusoidal component;
Figure BDA0003861538050000042
is the phase of the d-th sinusoidal component; e (t) is a noise signal.
Can be converted into
Figure BDA0003861538050000051
In the formula: a. The d Is a complex constant for the amplitude of the harmonic signal; omega d Is the normalized frequency to be estimated;
Figure BDA0003861538050000058
the initial phase is normalized and is uniformly distributed in the (-pi, pi) interval; u is uncorrelated complex gaussian white noise with 0 mean.
The signal sampled by the single sampler at the ith time is as follows,
Figure BDA0003861538050000052
Figure BDA0003861538050000053
wherein M and N represent the serial numbers of the samples, M is more than or equal to 0 and less than or equal to M-1, and N is more than or equal to 1 and less than or equal to N.
The samples can be used to construct a vector of sampled signals for two sub-arrays
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
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
By concatenating the samples of the two sub-arrays, the signal vector of the entire sampled signal can be represented as
Figure BDA0003861538050000054
Wherein A = [ a (ω) 1 ),a(ω 2 ),...,a(ω D )]Is a frequency matrix, a (ω) d ) Is a frequency vector containing single frequency information expressed as
Figure BDA0003861538050000055
Figure BDA0003861538050000056
u (l) is zero-mean white Gaussian noise.
2. Frequency estimation method
1. Sampling is carried out according to an expanded co-prime array structure, M and N are co-prime numbers, the power grid signals are sampled, the sampling frequency of the received signals is L, and an array received signal matrix is obtained
Figure BDA0003861538050000057
2. Constructing a corresponding covariance matrix from the received signals
Figure BDA0003861538050000061
3. Performing eigenvalue decomposition on the obtained covariance matrix to obtain a noise subspace
Figure BDA0003861538050000062
Wherein Λ s Representing a diagonal matrix of D by D, the diagonal elements consisting of larger D eigenvalues, Λ n Representing a diagonal matrix of S-D smaller eigenvalues, E s Is a matrix formed by eigenvectors corresponding to D larger eigenvalues, E n Then it is a matrix of eigenvectors corresponding to the other smaller S-D eigenvalues. E s Referred to as signal subspace, E n Referred to as the noise subspace.
4. Defining polynomial
Figure BDA0003861538050000063
In the formula u k Is a covariance matrix
Figure BDA0003861538050000064
Z is a solution parameter, and p (z) = [1, z = M ,...,z (N-1)M ,...,z (N-1)M+(M-1)N ]In order to extract information from all feature vectors simultaneously, it is desirable to denominate the MUSIC spectral function
Figure BDA0003861538050000065
Zero point of (c). The polynomial at this time is not a polynomial of z, and since there is a conjugate term of z, since only the z value on the unit circle is of interest, P can be used T (z -1 ) In place of p H (z)。
5. The above polynomial is changed to a polynomial of z, expressed as
Figure BDA0003861538050000066
When in use
Figure BDA0003861538050000067
When p (z) belongs to the signal subspace, derived from the orthogonality of the signal subspace to the noise subspace, F (z) =0, i.e. the root of the polynomial on the unit circle corresponds to the frequency of the sinusoidal signal. Taking D roots of the polynomial F (z) closest to the unit circle, and respectively taking z 1 、z 2 、…、z D The corresponding conjugate root is (z) * 1 、z * 2 、…、z * D ) Then the analog angular frequency of the complex sinusoidal signal is obtained as
Figure BDA0003861538050000068
The effect of the present invention will be further described with reference to the simulation example.
Suppose that the sensor receives a signal containing harmonics of
x(t)＝0.2cos(2π·25t)+cos(2π·50t)+0.2cos(2π·150t)+e(t)
The signal contains 3 frequency components of power frequency 50Hz, inter-harmonic 25Hz and 3-time harmonic 150Hz, and e (t) is Gaussian white noise.
In the simulation, for a fair comparison, a uniform line array was simulated using classical MUSIC, where M + N-1=9 sensor elements, with the fixed step size of MUSIC set to 0.010. We use Root Mean Square Error (RMSE) of the signal frequency estimation to evaluate the parameter estimation performance of the proposed algorithm, defined as
Figure BDA0003861538050000071
Wherein,
Figure BDA0003861538050000072
for ω in the ith Monte Carlo simulation m An estimate of (d). I is the total number of simulations, in the following simulations we take I =200.
Simulation 1: FIG. 3 is a diagram of the frequency estimation result distribution of the proposed method at a SNR of 20 dB. The parameters defining the coprime array used M =4,n =5. It can be seen from the figure that the algorithm can still effectively identify the fixed frequency at a lower signal-to-noise ratio.
Simulation 2: fig. 4 is a comparison between the proposed method and the signal-to-noise ratio variation trend of the performance of uniform sampling and relatively-prime sampling under the same sampling number, for fair comparison, the same sampling sample is kept, the array used for uniform sampling is set to be a uniform linear array with the number of samplers M + N-1=9, and the sampling number is 300. It can be seen from the figure that the proposed method has better frequency estimation performance than the commonly used uniform linear array.
Simulation 3: fig. 5 is a comparison graph of the performance of frequency estimation as a function of the number of samples for the proposed method at a signal-to-noise ratio of 20 dB. It can be seen from the figure that the proposed method frequency estimation performance is better than the commonly used uniform linear array.

Claims (6)

1. A power system harmonic frequency estimation method based on mutual prime sampling and root finding MUSIC is characterized by comprising the following steps:
(1) Performing time domain sampling on the signal by using a sampler which expands a co-prime array to obtain sample data;
(2) Constructing a corresponding estimation covariance matrix according to the sample data;
(3) Performing characteristic decomposition on the obtained covariance matrix to obtain a noise subspace;
(4) Defining a root polynomial of the polynomial frequency;
(5) Solving the polynomial to obtain the estimated value of the frequency.
2. The method for estimating harmonic frequency of power system based on mutual-prime sampling root-finding MUSIC according to claim 1, wherein the step (1) is implemented as follows:
and constructing according to an unfolded mutual prime array structure by adopting M + N-1 samplers, wherein M and N are mutual prime numbers, sampling the power grid signal, and obtaining sample data by taking the sampling frequency of the received signal as L.
3. The method according to claim 2, wherein the step (2) is implemented as follows, the first sampled signal of the single sampler is as follows,
Figure FDA0003861538040000011
Figure FDA0003861538040000012
wherein M and N represent the serial number of sampling, M is more than or equal to 0 and less than or equal to M-1, N is more than or equal to 1 and less than or equal to N, and 1/T represents the Nyquist sampling frequency;
construction of a vector of sampled signals for two sub-arrays using the samples
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
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
The samples of the two sub-arrays are connected in series, and the signal vector of the whole sampling signal is expressed as
Figure FDA0003861538040000013
Wherein A = [ a (ω) 1 ),a(ω 2 ),...,a(ω D )]Is a frequency matrix, a (ω) d ) Is a frequency vector containing single frequency information expressed as
Figure FDA0003861538040000021
Figure FDA0003861538040000022
u (l) is zero mean Gaussian white noise; constructing a corresponding estimated covariance matrix
Figure FDA0003861538040000023
Wherein
Figure FDA0003861538040000024
Is S.
4. The method according to claim 3, wherein the step (3) is specifically as follows:
the covariance matrix is subjected to eigenvalue decomposition and is expressed as
Figure FDA0003861538040000025
Wherein Λ s Representing a diagonal matrix of D, D is the number of sine components including fundamental wave, harmonic wave and inter-harmonic wave, and the first D diagonal elements are sorted from large to small
Figure FDA0003861538040000026
Characteristic value composition of n Represents S-D pieces after sorting from big to small
Figure FDA0003861538040000027
A diagonal matrix of eigenvalues of, E s The first D pieces are sorted from big to small
Figure FDA0003861538040000028
A matrix formed by eigenvectors corresponding to the eigenvalues of (E) n Then S-D are sequenced from big to small
Figure FDA0003861538040000029
A matrix of eigenvectors corresponding to the eigenvalues of, E s Referred to as signal subspace, E n Referred to as the noise subspace.
5. The method according to claim 4, wherein the step (4) comprises:
defining a root polynomial
Figure FDA00038615380400000210
Wherein
Figure FDA00038615380400000211
z is a parameter to be determined containing a frequency component when
Figure FDA00038615380400000212
When p (z) belongs to the signal subspace, as determined by the orthogonality of the signal subspace and the noise subspace, F (z) =0, i.e. the polynomial is in unityThe root on the circle corresponds to the frequency of the sinusoidal signal.
6. The method according to claim 5, wherein the step (5) is specifically as follows:
taking D roots of the polynomial F (z) closest to the unit circle, and respectively taking z 1 、z 2 、…、z D The corresponding conjugate root is (z) * 1 、z * 2 、…、z * D ) Then the analog angular frequency of the complex sinusoidal signal is obtained as
Figure FDA00038615380400000213
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