CN109490625B - Harmonic signal analysis method based on sliding window and semi-definite programming - Google Patents

Harmonic signal analysis method based on sliding window and semi-definite programming Download PDF

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CN109490625B
CN109490625B CN201811357725.0A CN201811357725A CN109490625B CN 109490625 B CN109490625 B CN 109490625B CN 201811357725 A CN201811357725 A CN 201811357725A CN 109490625 B CN109490625 B CN 109490625B
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semi
harmonic
subset
definite programming
sliding window
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CN109490625A (en
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沈兴来
姜华
白金花
胡凯
李磊
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State Grid Corp of China SGCC
Xuzhou Power Supply Co of State Grid Jiangsu Electric Power Co Ltd
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Xuzhou Power Supply Co of State Grid Jiangsu Electric Power Co Ltd
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    • G01—MEASURING; TESTING
    • G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
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Abstract

The invention relates to a harmonic signal analysis method based on a sliding window and semi-definite programming, which is different from the traditional frequency domain solving method. The method has the advantages that the method is processed in the time domain, the influence of frequency domain and spectrum leakage and asynchronous sampling on the calculation precision is overcome, and the method has higher frequency resolution and analysis precision.

Description

Harmonic signal analysis method based on sliding window and semi-definite programming
Technical Field
The invention relates to the field of power quality analysis and control, in particular to a harmonic signal analysis method based on a sliding window and semi-definite programming.
Background
In recent years, with the increase of interference loads and the increase of sensitivity of equipment to harmonics, the influence of the harmonics on a power system is more serious, and harmonic pollution becomes a problem to be paid much attention. It is generally considered that the frequency of the integer-order harmonics is an integer multiple of the fundamental frequency. In addition to this, the grid has a rich non-integer number of harmonics, i.e. inter-harmonics, whose frequencies are not integer multiples of the fundamental frequency, and whose frequency spectrum may be in discrete or continuous form.
There are many harmonic analysis methods at present, but most of them are based on FFT, which is also recommended by IEC standard 61000-4-7 as the most basic method for harmonic measurement. Indeed, the methods are simple and have low calculation cost, and can realize high-precision analysis of harmonics under synchronous sampling conditions, however, the FFT-based methods are sensitive to frequency offset, are harsh on sampling conditions, and have problems of spectrum leakage, barrier effect and the like, and meanwhile, the FFT-based parameter estimation methods are all limited by the fourier resolution of the uncertain principle, in particular, two sine wave signals with a frequency difference of Δ ω require a data length longer than 2 pi/Δ ω, and these defects are more obvious in the analysis of inter-harmonics. Therefore, it is of great value to find a method that can circumvent the FFT resolution limit.
The invention realizes the harmonic/inter-harmonic signal analysis of the power system by utilizing the sliding window and the semi-definite programming, and overcomes the spectrum leakage and the barrier effect of FFT.
Disclosure of Invention
According to the defects of the prior art, the invention provides a harmonic signal analysis method based on a sliding window and semi-definite programming, which realizes the harmonic/inter-harmonic signal analysis of a power system by utilizing the sliding window and the semi-definite programming and overcomes the spectrum leakage and the barrier effect of FFT.
The invention is realized according to the following technical scheme:
a harmonic signal analysis method based on a sliding window and semi-definite programming comprises the following steps:
(1) for signal x (n), starting from the first sample point, selecting M successive points to form a subset
Figure GDA0001890908760000011
Then starting from the second point, selecting M points which are continuous to form a subset
Figure GDA0001890908760000012
And the like, in the form of sliding windows, to form subsets
Figure GDA0001890908760000021
Here, the subscript M ∈ [1, N-M +1 ]]Denotes that the subset starts with x (m), the range of the subset being
Figure GDA0001890908760000022
(2) For the
Figure GDA0001890908760000023
Solving the following semi-definite programming problem by using ADMM method
Figure GDA0001890908760000024
Solving the optimal solution (x, t, u) in the formula;
(3) the subsets can be obtained by triangulating T (u)
Figure GDA0001890908760000025
Frequency of power system harmonics and interharmonics
Figure GDA0001890908760000026
Sum amplitude
Figure GDA0001890908760000027
(4) Averaging the estimation results of N-M +1 different windows with the size of M into the frequency and amplitude of the harmonic wave and the interharmonic wave, namely:
Figure GDA0001890908760000028
the invention has the beneficial effects that:
the method has the advantages that the method is processed in the time domain, the influence of frequency domain and spectrum leakage and asynchronous sampling on the calculation precision is overcome, and the method has higher frequency resolution and analysis precision.
Detailed Description
In order to make the implementation objects, technical solutions and advantages of the present invention clearer, the technical solutions in the embodiments of the present invention are described in more detail below. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention. The following provides a detailed description of embodiments of the invention.
The traditional Fourier transform is a complete orthogonal decomposition method, a frequency spectrum leakage phenomenon exists, the accuracy of harmonic analysis is reduced, for this reason, the invention hopes to find the least atoms (not the most complete atoms in Fourier change) to represent harmonic information through a continuous parameter space, which is actually an underdetermined problem, but the harmonic analysis of the power systemHas the characteristics that only a small amount of harmonic and inter-harmonic components exist, and high-frequency information has no application value, so the patent considers that the atom l is adopted1The norm solves the problem of harmonic analysis of the power system.
Let it be an atomic set, if its convex hull conv () is a centrosymmetric tight set with respect to the origin and contains the origin as an interior point, this means that any element in it, υ e, cannot be located in the convex hull conv (\ υ) formed by other elements except for υ, i.e., the elements in it are all extreme points of conv (), and υ e is if and only if- υ e. The norm defined by the scale function of the convex hull conv () then becomes the atomic norm, as | · | | survivalThis means that there are:
Figure GDA0001890908760000031
atomic norm | · | non-conducting phosphorIn effect, a sparse constraint is added to the collection, which treats the collection as an infinite dictionary describing continuously varying parameters.
Setting steady-state signals composed of k steady-state power harmonics and inter-harmonics as follows:
Figure GDA0001890908760000032
wherein { f }i}i＝1,2,...kFor harmonic/inter-harmonic frequencies, { phii}i＝1,2,...kFor the harmonic/inter-harmonic phase, let J ═ {0,1, 2., N-1}, define atoms
Figure GDA0001890908760000033
Thus a set of atoms can be written as { a (f, φ): f > 0, [ φ ∈ [0,2 π) }, with the atomic norm of the signal x (n) being defined according to the atomic norm of formula:
Figure GDA0001890908760000034
setting the sampling point of the harmonic signal as {0, 1., N-1}, the sampling period as Δ T, and sequentially intercepting the harmonic signal subset T e J by using a sliding window with the size of M from the first sampling point, as shown in the following table:
Figure GDA0001890908760000035
it is apparent that for a harmonic signal with a total number of samples N, there are N-M +1 such subsets of signals.
According to the sparse nature of the harmonic signals, the missing harmonic signals can be recovered using the following atomic norm minimization problem:
Figure GDA0001890908760000041
here | | s | non-conducting phosphorAThe atomic norm minimization problem can be converted into the following semi-definite programming problem by adopting a linear semi-definite programming theory method to solve the atomic norm in polynomial time according to the Caratheodory theorem and carrying out Vandermonde decomposition on any semi-definite Toeplitz matrix:
Figure GDA0001890908760000042
Figure GDA0001890908760000043
where T (u) is a Toeplitz matrix, and the first behavior u ═ u of T (u)1,u2,...,uN]∈CNNamely:
Figure GDA0001890908760000044
when the semi-definite programming problem is solved, if the number of compressed measurements is enough and the intervals among a plurality of harmonic frequencies are out of a certain range, missing signal sampling points can be accurately recovered and each harmonic frequency can be determined through the semi-definite programming.
In order to solve the semi-definite programming, the ADMM method is adopted for solving, and the principle is as follows:
Figure GDA0001890908760000045
Figure GDA0001890908760000046
Z≥0
the lagrange function of this problem is:
Figure GDA0001890908760000047
in the formula, | · the luminance | |FIs Frobenius norm.
Therefore, the ADMM solving steps are as follows:
Figure GDA0001890908760000051
Figure GDA0001890908760000052
Figure GDA0001890908760000053
where k is the number of iterations.
And solving the optimal solution (x, t, u) in the formula, and performing triangular decomposition on T (u) to obtain the frequency f and amplitude | s | of the harmonic wave and the interharmonic wave of the power system.
T(u)＝A(f)diag(|s|)AH(f)。
With the sliding of the window, the estimation results of N-M +1 different windows with the size of M are estimated, and then the frequency and the amplitude of the harmonic wave and the inter-harmonic wave can be obtained by averaging the estimation results.
In summary, the method comprises the following specific steps:
1. for signal x (n), starting from the first sample point, selecting M successive points to form a subset
Figure GDA0001890908760000054
Then starting from the second point, selecting M points which are continuous to form a subset
Figure GDA0001890908760000055
And the like, in the form of sliding windows, to form subsets
Figure GDA0001890908760000056
Here, the subscript M ∈ [1, N-M +1 ]]Denotes that the subset starts with x (m), the range of the subset being
Figure GDA0001890908760000057
2. For the
Figure GDA0001890908760000058
Solving the following semi-definite programming problem by using ADMM method
Figure GDA0001890908760000059
And solving to obtain the optimal solution (x, t, u) in the formula.
3. The subsets can be obtained by triangulating T (u)
Figure GDA0001890908760000061
Frequency of power system harmonics and interharmonics
Figure GDA0001890908760000062
Sum amplitude
Figure GDA0001890908760000063
4. Averaging the estimation results of N-M +1 different windows with the size of M into the frequency and amplitude of the harmonic wave and the interharmonic wave, namely:
Figure GDA0001890908760000064
the above-mentioned embodiments only express one embodiment of the present invention, and the description thereof is more specific and detailed, but not construed as limiting the scope of the present invention. It should be noted that various changes and modifications can be made by those skilled in the art without departing from the spirit and principle of the present invention, and these changes and modifications are within the scope of the present invention. Therefore, the protection scope of the present patent shall be subject to the appended claims.

Claims (1)

1. A harmonic signal analysis method based on a sliding window and semi-definite programming is characterized by comprising the following steps:
(1) for signal x (n), starting from the first sample point, selecting M successive points to form a subset
Figure FDA0001865283960000011
Then starting from the second point, selecting M points which are continuous to form a subset
Figure FDA0001865283960000012
And the like, in the form of sliding windows, to form subsets
Figure FDA0001865283960000013
Here, the subscript M ∈ [1, N-M +1 ]]Denotes that the subset starts with x (m), the range of the subset being
Figure FDA0001865283960000014
(2) For the
Figure FDA0001865283960000015
Solving the following semi-definite programming problem by using ADMM method
Figure FDA0001865283960000016
Solving the optimal solution (x, t, u) in the formula;
(3) the subsets can be obtained by triangulating T (u)
Figure FDA0001865283960000017
Frequency of power system harmonics and interharmonics
Figure FDA0001865283960000018
Sum amplitude
Figure FDA0001865283960000019
(4) Averaging the estimation results of N-M +1 different windows with the size of M into the frequency and amplitude of the harmonic wave and the interharmonic wave, namely:
Figure FDA00018652839600000110
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