CN104459315A

CN104459315A - Inter-harmonic detection method based on non-base 2FFT transformation

Info

Publication number: CN104459315A
Application number: CN201410620585.7A
Authority: CN
Inventors: 程玉华; 陈凯; 刘葱茜; 张旭霞; 张�杰
Original assignee: University of Electronic Science and Technology of China
Current assignee: University of Electronic Science and Technology of China
Priority date: 2014-11-06
Filing date: 2014-11-06
Publication date: 2015-03-25

Abstract

The invention discloses an inter-harmonic detection method based on non-base 2FFT transformation. When the number N of FFT points is equal to 2<M>*N', N1 is made to be equal to 2<M>, N2 is made to be equal to N', the FFT operation data are divided into N2 groups, each group has N1 point data, the data serial number n extracted from original FFT operation data is equal to N2n1+n2, 2FFT transformation is carried out on the N1 point data in each group, the k is equal to k1+N1k2, and the kth FFT transformation result in the original FFT operation data is obtained according to the calculation formula. The new non-base 2FFT algorithm is provided, FFT transformation of data with the point number N equal to 2<M>*N' is achieved, no interpolation zero-padding is needed, inter-harmonic detection precision can be improved, the frequency resolution meets national standard requirements, a transformation result needing to be calculated can be selected according to actual requirements, and operation time is saved.

Description

Inter-harmonic detection method based on non-basis 2FFT (fast Fourier transform)

Technical Field

The invention belongs to the technical field of inter-harmonic detection, and particularly relates to an inter-harmonic detection method based on non-basis 2FFT (fast Fourier transform).

Background

With the rapid development of scientific technology and the wide application of a large number of impact loads, current conversion and other electric equipment, the harmonic pollution of a power grid is increasingly serious, the safety of an electric power system is greatly influenced, and particularly with the continuous popularization of various complex and precise electric equipment sensitive to the quality of electric energy, fractional harmonics (inter-harmonics) become objects of continuous attention in the engineering and research fields increasingly. Inter-harmonics can cause voltage flicker, television image roll and noise of radio or other audio equipment, and can also cause strong noise and vibration of a motor to cause waveform distortion of a power grid, so that the power factor of a load is reduced, various energy losses are increased, and accurate detection of inter-harmonics is required to be enhanced to ensure the quality and safety of the power grid so as to be convenient for the treatment of the power quality of the power grid.

At present, the most widely applied harmonic measurement method is Fourier transform, amplitude and phase of harmonic can be obtained simultaneously through Fourier transform, functions are multiple, and hardware is easy to implement. The Fourier transform method of the inter-harmonics comprises the following steps: extending the sample sequence to adjacent 2 by interpolation zero-filling^MPoint, then proceed to 2 using the base 2 algorithm^MPoint FFT (Fast Fourier Transform). The frequency domain decimated radix-2 FFT algorithm is described below.

Defined by DFT (Discrete Fourier Transform):

where N represents the number of points, f [ N ]]Which represents a discrete signal that is,representing the twiddle factor.

Dividing fn into two halves, i.e. fn and fn + N/2 (N is more than or equal to 0 and less than or equal to N/2-1), substituting into formula (1) to obtain

Due to the fact that

W_{N}^{k (n + N / 2)} = W_{N}^{kn} W_{N}^{kN / 2} = {(- 1)}^{k} W_{N}^{kn},

(2) Two terms in the formula can be combined into:

when k is an even number, 2r, note (-1)^k＝1，

(3) The formula becomes:

when k is an odd number 2r +1,

(3) the formula becomes:

thus, an N point sequence (f [ N ]) is formed]) N-point DFT (F [ k ]) of (1)]) Calculated as two N/2 point sequences (g [ N ]]And p [ n ]]) N/2 point DFT (G [ r ]) of]And P [ r ]]) And (4) calculating. From f [ n ]]Is divided into g [ n ]]And p [ n ]]Is calculated as N additions (i.e., f N]+f[n+N/2]And f [ n ]]-f[n+N/2]0. ltoreq. n.ltoreq.N/2-1) and N/2 multiplications (i.e., (N ≦ N/2-1)0≤n≤N/2-1)。

Since the N/2-point DFT calculated by g [ N ] is half of the N-point DFT (Fk) of F [ N ] in which k is an even number and half of k is an even number calculated by p [ N ], it is necessary to extract and put the even-numbered K F [ k ] together as the DFT (G (r)) of g [ N ] and extract and put the odd-numbered K F [ k ] together as the DFT (P (r)) of p [ N ] for output. Since k is a frequency domain variable, this algorithm is called the FFT algorithm of frequency domain decimation.

Then, the two N/2 point DFTs can still be divided into two N/4 point DFTs (simultaneously, corresponding frequency domain extraction is also carried out) by N/4 times N/2 additions by the method, so that the DFTs are divided into 4N/2 point DFTs in total, and the total division calculation amount is still N additions and N/2 times. Such partitioning can be done step by step, and it can be seen that the total partition computation amount of each step is N sums and N/2 multiples.

When N is 2^MAfter M-1 step division, N/2 DFTs are calculated

Fig. 1 is a schematic diagram of an 8-point FFT computation for frequency domain decimation. According to fig. 1, an 8-point FFT is divided into 4 two-point DFTs through 2 steps. (6) The amount of computation required for the equation is 2 summations and 1 multiplication, so the total amount of computation to complete N/2 two-point DFTs is still N summations and N/2 multiplications. Thereby completing the total calculation amount M × N ═ Nlog of all N-point DFTs₂N sums MxN/2 ═ N/2 (log)₂N multiplications, the ratio being calculated by definition directly for the required N²The number of multiplications and additions is much less. For example, N ═ 2¹⁰When it is 1024, M is 10, the number of multiplications required for the calculation by the radix-2 FFT algorithm is mxn/2, 5 × 1024, and the number of multiplications required for the calculation by definition is N²1024 × 1024, the difference between the two is 1024/5 ≈ 200 times, and the larger N is, the more significant the efficiency improvement of FFT is.

In general, since M-1 parity extraction is performed, F [ k ] calculated by the last N/2 two-point DFTs of the algorithm is not extracted sequentially. The change in order can be illustrated by a binary code: the parity divided by the first extraction is distinguished by that the 1 st bit of the binary code is 1 or 0, when the bit is 0, the bit is even, when the bit is 1, the bit is odd, the parity extracted by the second extraction is … … distinguished by that the 2 nd bit of the binary code is 1 or 0, each extraction places the even number item on the front (left) side and the odd number item on the back (right) side, so that the binary codes for extracting the subsequent numbers are arranged in sequence from left to right according to the binary bits, and the rule of arranging the common binary numbers in sequence from right to left is just opposite, so the extraction is called the inverted order. The order of inversion is changed to order after F [ k ] is calculated.

In inter-harmonic detection, since the amplitude of the inter-harmonics is to be calculated, it is necessary to process data for several cycles, fromTo reach 2 based on the radix-2 FFT algorithm^MIn general, the number of sampling points in each signal period is a power of 2, but the number of data processing periods is not necessarily a power of 2, so that some values of inter-harmonics may be obtained by interpolation, which causes inaccurate measurement precision.

Disclosure of Invention

The invention aims to overcome the defects of the prior art and provides a non-base 2 FFT-based inter-harmonic detection method, which realizes non-base 2FFT conversion, does not need zero padding, improves the accuracy of inter-harmonic detection, ensures that a detection result can completely accord with the IEC standard, and ensures that the frequency resolution can obtain an accurate value at 5Hz under the conditions that the fundamental wave is 50Hz and the width of a sampling window is 10 cycles.

In order to achieve the above object, the present invention provides a non-basis 2FFT transform-based inter-harmonic detection method, comprising the steps of:

s1: sampling the signal, and setting the number of sampling points N to 2^MX N', let N₁＝2^M，N₂Dividing the sampled data into N ═ N₂Groups of N₁Dot data, n-th₂N in the group₁The point data is denoted as f [ n ]₁,n₂]The data number N extracted from the sample data is N₂n₁+n₂Wherein n is₁Has a value range of n not less than 0₁≤N₁-1，n₂Has a value range of n not less than 0₂≤N₂-1；

S2: respectively for N in each group₁Performing base 2FFT on the point data to obtain the n-th data₂Kth in group₁The transformation result is denoted as F' [ k ]₁,n₂]，k₁K is not less than 0₁≤N₁-1；

S3: let k equal to k₁+N₁k₂，k₂K is not less than 0₂≤N₂-1, computing the kth Fourier transform junction of the sampled dataFruit F [ k ]₁,k₂]And outputting as a detection result, wherein the calculation formula is as follows:

wherein,

the invention discloses a non-basis 2FFT (fast Fourier transform) -based inter-harmonic detection method, wherein when the number N of FFT points is 2^MX N', let N₁＝2^M，N₂Dividing FFT calculation data into N ═ N₂Groups of N₁Dot data, where N is the data number extracted from the original FFT-calculated data₂n₁+n₂For N in each group respectively₁The point data is subjected to a radix-2 FFT transform, and then k is k₁+N₁k₂And obtaining the kth FFT transformation result in the original FFT operation data according to a calculation formula. The invention provides a new non-radix 2FFT algorithm, which realizes the point number N-2^MThe FFT conversion of the data of x N' does not need interpolation zero filling, can improve the detection precision of inter-harmonic, enable the frequency resolution to meet the national standard requirement, and can also select the conversion result that needs to be calculated according to the actual need, save the operating time.

Drawings

FIG. 1 is a schematic diagram of an 8-point FFT calculation for frequency domain decimation;

FIG. 2 is a flow chart of an embodiment of the method for detecting inter-harmonics based on non-basis 2 FFT;

fig. 3 is a diagram illustrating the result of inter-harmonic detection using the present invention.

Detailed Description

The following description of the embodiments of the present invention is provided in order to better understand the present invention for those skilled in the art with reference to the accompanying drawings. It is to be expressly noted that in the following description, a detailed description of known functions and designs will be omitted when it may obscure the subject matter of the present invention.

Fig. 2 is a flow chart of an embodiment of the non-basis 2FFT transform-based inter-harmonic detection method of the present invention. As shown in fig. 2, the method for detecting inter-harmonic based on non-basis 2FFT of the present invention includes the following steps:

s201: sampling data packet:

sampling the signal, and setting the number of sampling points N to 2^MX N', let N₁＝2^M，N₂Dividing FFT calculation data into N ═ N₂Groups of N₁Dot data, n-th₂N in the group₁The point data is denoted as f [ n ]₁,n₂]The data number N extracted from the sample data is N₂n₁+n₂Wherein n is₁Has a value range of n not less than 0₁≤N₁-1，n₂Has a value range of n not less than 0₂≤N₂-1。

S202: and performing radix-2 FFT on each group of data:

for N in each group₁Performing base 2FFT on the point data to obtain the n-th data₂Kth in group₁The transformation result is denoted as F' [ k ]₁,n₂]，k₁K is not less than 0₁≤N₁-1。

S203: calculating a Fourier transform result:

let k equal to k₁+N₁k₂，k₂K is not less than 0₂≤N₂-1, computing the kth Fourier transform result F [ k ] of the sampled data₁,k₂]And outputting a detection result, wherein the calculation formula is as follows:

wherein,

the principles and derivation of the present invention are explained below.

In the present invention, when performing an N-point FFT, N is 2^MX N', let N₁＝2^M，N₂N', then by f [ N]DFT of (2):

by mapping:

it is possible to obtain:

W_{N}^{nk} = W_{N}^{(N_{2} n_{1} + n_{2}) (k_{1} + N_{1} k_{2})} = W_{N}^{(N_{2} n_{1} k_{1} + N_{1} N_{2} n_{1} k_{2} + n_{2} k_{1} + N_{1} n_{2} k_{2})},

and N is equal to N₁N₂，

W_{N}^{N_{1}} = W_{N_{2}},

W_{N}^{N_{2}} = W_{N_{1}},

Can be simplified as follows:

W_{N}^{nk} = W_{N_{1}}^{n_{1} k_{1}} W_{N}^{n_{2} k_{1}} W_{N_{2}}^{n_{2} k_{2}} - - - (10)

thereby converting formula (8) to:

wherein k is not less than 0₁≤N₁-1,0≤k₂≤N₂-1。

Combining the first two twiddle factors in the formula (11) according to W_NAvailability of reducibility of

W_{N_{2}}^{n_{2} k_{2}} W_{N}^{n_{2} k_{1}} = W_{N_{1} N_{2}}^{N_{1} n_{2} k_{2}} W_{N}^{n_{2} k_{1}} = W_{N}^{N_{1} n_{2} k_{2} + n_{2} k_{1}} = W_{N}^{n_{2} (N_{1} k_{2} + k_{1})} - - - (12)

Substituting the formula (9) into the formula (12) to obtain

Equation (11) can thus be:

it can be seen that according to n₂Is selected from the group consisting of]Is divided into N₂Group data f [ n ]₁,n₂]Each group of data has N₁Point of formula (14)Indicates that will₂K-th obtained after group data conversion₁The transformation result is denoted as F' [ k ]₁,n₂]. Due to N₁＝2^MThus N is₁The FFT transformation of the point data can be found by the radix-2 FFT algorithm. This pattern (14) can be:

equation (15) is analogous to the DFT calculation, whereNot a rotation factor in the strict sense but

Examples

The present invention was simulated to illustrate the technical effects of the present invention. The signal expression is y-0.8 sin (10 π t) +0.6sin (80 π t) + sin (100 π t). According to the IEC standard, the fundamental frequency of the signal is 50 Hz. The intermediate harmonic frequencies in this embodiment are 5Hz and 40 Hz. The window width is 10 cycles. The signal is sampled 1024 points per cycle for a total of 10240 points. Fig. 3 is a diagram illustrating the result of inter-harmonic detection using the present invention. As shown in fig. 3, the inter-harmonic detection result obtained by the present invention is consistent with the signal.

In practical application, according to signal characteristics and IEC standard requirements, it is not necessary to change all results, but only to select a part of results, if the traditional FFT method is used, it is necessary to calculate all results and then select them₁+N₁k₂The mapping of the transformation result is performed, so that corresponding data can be selected and calculated according to the sequence number of the required result, taking this embodiment as an example, the data is divided into 5 groups, the FFT transformation result of 2048 data in 5 groups is calculated first, and assuming that the sequence number of the currently required transformation result is 511, then k is equal to k according to the formula₁+N₁k₂Then k can be obtained₁And k₂And then selecting 5 groups of data corresponding to the FFT transformation results for calculation. In this way, further savings in computation time may be achieved. The time required to complete 10240 points using the present invention was 0.062962 seconds as verified by MATLAB simulation. According to the IEC standard requirements for measuring frequency ranges, only a limited number of points can be calculated, for example, 2000 points, which requires 0.021317 seconds. The invention has high operation speed and can meet the measurement speed requirement of inter-harmonic detection.

According to the current national standard, the fundamental frequency f of a signal is 50Hz, the width D of a sampling window is 10 periods, and the frequency resolution is required to be 5 Hz. Counting the number of sampling points in each period as X, then samplingSample frequency f_sIf Xf, the total number of samples N is DX, then the frequency resolution f₀Xf DX f D. If the radix-2 FFT algorithm is adopted, interpolation zero filling is needed, and the sampling period is prolonged to reach the nearest 2^M′Point, namely D 'is 16, so the frequency resolution f D' is 3.125Hz, and the detection precision can not be ensured when the frequency value of 5Hz multiple is the estimated value obtained by interpolation; by adopting the inter-harmonic detection method based on the non-basis 2FFT algorithm, the frequency resolution f D is 5Hz, so that the inter-harmonic detection precision can be met, and the national standard requirement is met.

The invention aims at the point number N is 2^M×N′＝N₁×N₂Due to N₁＝2^MIs the integer power number of 2, and can call the FFT library function of the DSP to calculate N₂Group N₁Point-based 2FFT conversion with fast operation speed, and then calculating N₁Group N₂The point DFT is similar to the operation, and the Fourier transform of the non-base 2 is completed. Compared with the traditional method, the method for detecting the inter-harmonic wave has the advantages that the frequency resolution can reach the national standard, the detection precision is higher, and the operation speed can also meet the requirements of the existing detection equipment. In addition, the present invention can calculate N-2^MThe number of multiplied N' points, and the application range is wider.

Although illustrative embodiments of the present invention have been described above to facilitate the understanding of the present invention by those skilled in the art, it should be understood that the present invention is not limited to the scope of the embodiments, and various changes may be made apparent to those skilled in the art as long as they are within the spirit and scope of the present invention as defined and defined by the appended claims, and all matters of the invention which utilize the inventive concepts are protected.

Claims

1. A non-basis 2FFT transform-based inter-harmonic detection method is characterized by comprising the following steps:

s1: sampling the signal, and setting the number of sampling points N to 2^MX N', let N₁＝2^M，N₂Dividing the sampled data into N ═ N₂Groups of N₁Dot data, n-th₂N in the group₁The point data is denoted as f [ n ]₁,n₂]The data number N extracted from the sample data is N₂n₁+n₂Wherein n is₁The value range of (A) is not more than 0n₁≤N₁-1，n₂Has a value range of n not less than 0₂≤N₂-1；

S2: for N in each group₁Performing non-base 2FFT on the point data to obtain the n-th data₂Kth in group₁The transformation result is denoted as F' [ k ]₁,n₂]，k₁K is not less than 0₁≤N₁-1；

S3: let k equal to k₁+N₁k₂，k₂K is not less than 0₂≤N₂-1, computing the kth Fourier transform result F [ k ] of the sampled data₁,k₂]And outputting as a detection result, wherein the calculation formula is as follows:

wherein,