CN102243272A - High-precision harmonic analysis method of sampling data synchronization - Google Patents

Abstract

The invention discloses a high-precision harmonic analysis method of sampling data synchronization, which is suitable for the field of power grid harmonic analysis and can realize accuracy computation of frequency, amplitude and phase of a power grid signal. The technical scheme of the invention is as follows: 1, a quadratic polynomial reverse interpolation method for accurately determining a fundamental wave period ensures that the computing precision of the fundamental wave period is improved by 2 orders of magnitude (reaching 1.51*10<-7> percent); and 2, a radial primary function interpolation method used for synchronizing nonsynchronously-sampled data ensures that synchronization data errors subjected to radial primary function interpolation are improved by 3-4 orders of magnitude (reaching 1.1*10<-9> percent). The invention has the advantage that the computing precision of harmonic parameters (amplitude and phase) on the basis of synchronization data is improved by 4-5 orders of magnitude.

Description

A kind of harmonic analysis method of sampling with high precision data sync

Technical field

The present invention relates to a kind of high precision harmonic analysis method, comprise the quadratic polynomial inverse interpolation method that is used for accurately measuring the primitive period, the radial basis function interpolation method that is used for the processing of non-synchronous sampling data sync at steady periodic signal.The invention belongs to the frequency analysis field of power network signal.

Background technology

In the electric energy metrical field, need the each harmonic amplitude and the phase relation of voltage, electric current accurately when calculating meritorious and capacity of idle power based on the frequency domain Power Theory; In the electrical network electric energy quality monitoring, also need the Harmonic Distribution situation of voltage, electric current; Therefore, frequency analysis is the important content in the power network signal research.Present frequency analysis is mainly by what realize power network signal sampling and digitized processing, considers that mains frequency fluctuation etc. has caused non-synchronous sampling, directly carries out the spectrum analysis meeting and has a strong impact on computational accuracy because of the leakage of frequency spectrum.

By interpolation or adaptive algorithm, make the data sequence after the processing approach idealized sample sequence, and then to carry out the computing of parameter and analyze be a class important method that improves spectrum analysis to sample sequence.The synchronized algorithm of existing realization mainly contains: 1. differential interpolation algorithm, mainly utilize the periodicity and the Taylor expansion of signal, and by actual samples and differential approximate representation, and differential can be derived according to the periodicity of signal with the ideal synchronisation sample sequence; 2. interpolation of data algorithm specifically comprises insert method before linearity, quadratic interpolation, the quadravalence again, with the ideal synchronisation sample sequence by near actual samples data by linear, second order parabola or quadravalence before the interlude method obtain.

The deficiency that above-mentioned method for synchronizing exists is: 1. in the differential interpolation, and calculation of parameter complexity such as fundamental frequency, each harmonic amplitude, and must call data in the last signal period during synchronization process, thereby the time-delay of the minimum of calculation of parameter is greater than the primitive period of signal; 2. in the interpolation of data, interpolation curve slickness and precision are limited; And it is restricted that interpolation is counted, and promptly requires interpolation point and sampling number approaching substantially, and it adjusts difficulty, also needs the complicated derivation of equation and compensation to calculate in addition; The precision of said method is lower, and error can not be applicable in the high-precision frequency analysis about 10-2%～101% magnitude.

Summary of the invention

The present invention is directed to the deficiencies in the prior art, provide a kind of high-precision sampled data synchronized harmonic analysis method, realize the accurate calculating of frequency, amplitude and the phase place of power network signal; The quadratic polynomial inverse interpolation method and the radial basis function interpolation method that is used for the processing of non-synchronous sampling data sync, the precision height of frequency analysis of primitive period have been proposed to be used for accurately measuring.

In order to realize the foregoing invention purpose, technical scheme of the present invention is to carry out according to the following steps:

(1), a kind of quadratic polynomial inverse interpolation method that is used for accurately measuring the primitive period has been proposed:

At first, the primitive period is to determine by the difference of adjacent zero passage moment point; Thereby the accuracy of detection of zero crossing is crucial, and it is not only the prerequisite of primitive period calculating, also is the basis of phase compensation in the spectrum analysis; The calculating primitive period is divided into the zero crossing detection and the primitive period is calculated two steps:

1) detection of zero crossing: find out x (t in the sample sequence ₀)＜0, x (t ₁)＞0, x (t ₂Three neighbouring sample points of)＞0 can determine that zero crossing is positioned at t ₀And t ₁Between; Adopted quadratic polynomial inverse interpolation method to determine zero crossing, its principle is: get quadratic interpolation polynomial expression p ₂(t)=x (t ₀)+x[t ₀, t ₁] (t-t ₀)+x[t ₀, t ₁, t ₂] (t-t ₀) (t-t ₁), the p of its zero crossing place ₂(t _Zero)=c=0 determines zero crossing t _ZeroStep be:

1. adopt successive approximation method, get earlier

Make:

$x\left(t_0\right)+x[{\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="HSA00000464996000021.png,HSA00000464996000022.png"\; state="\{\{state\}\}">t</figure-callout>}}_0,t_1](t_{\mathrm{zero}}^{\left(0\right)}-t_0)=c\&DoubleRightArrow;t_{\mathrm{zero}}^{\left(0\right)}={\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="HSA00000464996000021.png,HSA00000464996000022.png"\; state="\{\{state\}\}">t</figure-callout>}}_0+\frac{c-x\left(t_0\right)}{x[{\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="HSA00000464996000021.png,HSA00000464996000022.png"\; state="\{\{state\}\}">t</figure-callout>}}_0,t_1]}$

The 1. x[t in the step ₀, t ₁] represent that x (t) is at t ₀The first order difference at place;

2. ask again Make and satisfy:

$x\left(t_0\right)+x[{\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="HSA00000464996000021.png,HSA00000464996000022.png"\; state="\{\{state\}\}">t</figure-callout>}}_0,t_1](t_{\mathrm{zero}}^{\left(1\right)}-t_0)+x[{\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="HSA00000464996000021.png,HSA00000464996000022.png"\; state="\{\{state\}\}">t</figure-callout>}}_0,{\mathrm{<figure-callout\; id="1"\; label="t"\; filenames="HSA00000464996000021.png,HSA00000464996000022.png"\; state="\{\{state\}\}">t</figure-callout>}}_1,t_2](t_{\mathrm{zero}}^{\left(0\right)}-t_0)(t_{\mathrm{zero}}^{\left(0\right)}-t_1)=c$

$\&DoubleRightArrow;t_{\mathrm{zero}}^{\left(1\right)}={\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="HSA00000464996000021.png,HSA00000464996000022.png"\; state="\{\{state\}\}">t</figure-callout>}}_0+\frac{c-x\left(t_0\right)}{x[{\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="HSA00000464996000021.png,HSA00000464996000022.png"\; state="\{\{state\}\}">t</figure-callout>}}_0,t_1]}-\frac{x[{\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="HSA00000464996000021.png,HSA00000464996000022.png"\; state="\{\{state\}\}">t</figure-callout>}}_0,{\mathrm{<figure-callout\; id="1"\; label="t"\; filenames="HSA00000464996000021.png,HSA00000464996000022.png"\; state="\{\{state\}\}">t</figure-callout>}}_1,t_2]}{x[{\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="HSA00000464996000021.png,HSA00000464996000022.png"\; state="\{\{state\}\}">t</figure-callout>}}_0,t_1]}(t_{\mathrm{zero}}^{\left(0\right)}-t_0)(t_{\mathrm{zero}}^{\left(0\right)}-t_1)$

The 2. x[t in the step ₀, t ₁, t ₂] be illustrated in t ₀The second order difference at place;

3. utilize iterative formula then:

$t_{\mathrm{zero}}^{\left(k\right)}={\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="HSA00000464996000021.png,HSA00000464996000022.png"\; state="\{\{state\}\}">t</figure-callout>}}_0+\frac{c-x\left(t_0\right)}{x[{\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="HSA00000464996000021.png,HSA00000464996000022.png"\; state="\{\{state\}\}">t</figure-callout>}}_0,t_1]}-\frac{x[{\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="HSA00000464996000021.png,HSA00000464996000022.png"\; state="\{\{state\}\}">t</figure-callout>}}_0,{\mathrm{<figure-callout\; id="1"\; label="t"\; filenames="HSA00000464996000021.png,HSA00000464996000022.png"\; state="\{\{state\}\}">t</figure-callout>}}_1,t_2]}{x[{\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="HSA00000464996000021.png,HSA00000464996000022.png"\; state="\{\{state\}\}">t</figure-callout>}}_0,t_1]}(t_{\mathrm{zero}}^{(k-1)}-t_0)(t_{\mathrm{zero}}^{(k-1)}-t_1)$

At equidistant node (is t _i=t ₀+ ih) situation, available difference interpolation formula is got t=t ₀+ λ h, Xiang Yingyou:

${\mathrm{\&lambda;}}^{\left(k\right)}=t^{\left(0\right)}+\frac{{\mathrm{\&Delta;}}^2{x}_0}{2\mathrm{\&Delta;}{x}_0}{\mathrm{\&lambda;}}^{(k-1)}({\mathrm{\&lambda;}}^{(k-1)}-1),$ $t^{\left(0\right)}=\frac{c-x_0}{\mathrm{\&Delta;}{x}_0}$

Its zero crossing position is: t _Zero=t ₀+ λ ^(k)H, wherein Δ x ₀=x (t ₀+ h)-x (t ₀), Δ ²x ₀=Δ x (t ₀+ h)-Δ x (t ₀)=x (t ₀+ 2h)-2x (t ₀+ h)+x (t ₀);

In addition, for improving the precision that zero crossing detects, on quadratic polynomial inverse interpolation basis, adopt the average treatment method, two groups of data points of promptly getting near zero-crossing point: x (t ₀)＜0, x (t ₁)＞0, x (t ₂)＞0 and x (t ₀)＜0, x (t ₁)＜0, x (t ₂)＞0, calculate respectively and get its on average approximate as zero crossing behind the zero passage point value:

$\left\{\begin{array}{cc}{t}_{\mathrm{zero}\_1}={\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="HSA00000464996000021.png,HSA00000464996000022.png"\; state="\{\{state\}\}">t</figure-callout>}}_0+{\mathrm{\&lambda;}}^{\left(k\right)}h& (x\left(t_0\right)<0,x\left(t_1\right)>0,x\left(t_2\right)>0)\\ t_{\mathrm{zero}\_2}={\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="HSA00000464996000021.png,HSA00000464996000022.png"\; state="\{\{state\}\}">t</figure-callout>}}_0+{\mathrm{\&lambda;}}^{\left(k\right)}h& (x\left(t_0\right)<0,x\left(t_1\right)<0,x\left(t_2\right)>0)\end{array}\right.\&DoubleRightArrow;t_{\mathrm{zero}}=\frac{(t_{\mathrm{zero}\_1}+t_{\mathrm{zero}\_2})}{2};$

2) calculating of primitive period: the 1st) step has realized determining of single zero crossing, can provide the primitive period T of signal then by the mistiming between adjacent zero crossing ₀, promptly provide adjacent two zero crossings and be designated as t according to aforementioned zero crossing detection method _StartAnd t _End, its primitive period should be T mutually ₀=t _Start-t _End

(2), the synchronized radial basis function interpolation method of a kind of sampled data has been proposed:

The signal of analyzing is x (t), and its actual sample sequence is x (n), and the ideal synchronisation sample sequence is x ₀(n), the data synchronization process nature is mapping f:|x (the n) → x that seeks sampling from the actual samples to the ideal ₀(n); All relation between actual samples point and desirable sampled point realizes existing method by directly setting up, thereby precision and computing formula are complicated; The present invention proposes a kind of synchronization thinking, the steps include:

1) according to sampled value x (n), by function of interpolation method structure

Approximate signal x (t), promptly

The interpolating function that its interpolation process is selected for use is a radial basis function

Wherein c is the center of radial basis function, and a is a form parameter.Radial basis function interpolation principle is: the non-synchronous sampling sequence x (n) of known x (t) is the linear combination of radial basis function with x (t) approximate representation:

$x\left(t\right)\&ap;\hat{x}\left(t\right)={\mathrm{\&lambda;}}_1{\mathrm{\&phi;}}_1\left(t\right)+{\mathrm{\&lambda;}}_2{\mathrm{\&phi;}}_2\left(t\right)+...+{\mathrm{\&lambda;}}_N{\mathrm{\&phi;}}_N\left(t\right)=\mathrm{\&Sigma;}{\mathrm{\&lambda;}}_j{\mathrm{\&phi;}}_j\left(t\right)=\left[\mathrm{\&Phi;}\left(t\right)\right]\left[\mathrm{\&lambda;}\right]$

Utilize the interpolation condition at discrete point place:

$x\left(n\right)=\mathrm{\&Sigma;}{\mathrm{\&lambda;}}_j\mathrm{\&phi;}\left(\right||t_n-t_m|\left|\right)\&DoubleRightArrow;\left[\mathrm{\&Phi;}\left(t\right)\right]\left[\mathrm{\&lambda;}\right]=\left[x\right]$

Wherein, [x]=(x (1), x (2) ..., (N)) ^T, [λ]=(λ ₁, λ ₂..., λ _N) ^TBe undetermined coefficient, form is:

$\left[\begin{array}{cccc}\mathrm{\&phi;}\left(\right||t_1-{\mathrm{<figure-callout\; id="1"\; label="t"\; filenames="HSA00000464996000021.png,HSA00000464996000022.png"\; state="\{\{state\}\}">t</figure-callout>}}_1|\left|\right)& \mathrm{\&phi;}\left(\right||t_1-{\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="HSA00000464996000021.png"\; state="\{\{state\}\}">t</figure-callout>}}_2|\left|\right)& ...& \mathrm{\&phi;}\left(\right||t_1-t_N|\left|\right)\\ \mathrm{\&phi;}\left(\right||t_2-{\mathrm{<figure-callout\; id="1"\; label="t"\; filenames="HSA00000464996000021.png,HSA00000464996000022.png"\; state="\{\{state\}\}">t</figure-callout>}}_1|\left|\right)& \mathrm{\&phi;}\left(\right||t_2-{\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="HSA00000464996000021.png"\; state="\{\{state\}\}">t</figure-callout>}}_2|\left|\right)& ..& \mathrm{\&phi;}\left(\right||t_2-t_N|\left|\right)\\ .& .& .& .\\ .& .& .& .\\ .& .& .& .\\ \mathrm{\&phi;}\left(\right||t_N-{\mathrm{<figure-callout\; id="1"\; label="t"\; filenames="HSA00000464996000021.png,HSA00000464996000022.png"\; state="\{\{state\}\}">t</figure-callout>}}_1|\left|\right)& \mathrm{\&phi;}\left(\right||t_N-{\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="HSA00000464996000021.png"\; state="\{\{state\}\}">t</figure-callout>}}_2|\left|\right)& ...& \mathrm{\&phi;}\left(\right||t_N-t_N|\left|\right)\end{array}\right]\left[\begin{array}{c}{\mathrm{\&lambda;}}_1\\ {\mathrm{\&lambda;}}_2\\ .\\ .\\ .\\ {\mathrm{\&lambda;}}_N\end{array}\right]\left[\begin{array}{c}x\left(1\right)\\ x\left(2\right)\\ .\\ .\\ .\\ x\left(N\right)\end{array}\right]$

And then can get coefficient:

[λ]＝[Φ] ^-1[x]

Therefore, the approximate value of the x (t) at any t place

For:

$\hat{x}\left(t\right)=\left[\mathrm{\&Phi;}\left(t\right)\right]\left[\mathrm{\&lambda;}\right]=\left[\mathrm{\&Phi;}\left(t\right)\right]{\left[\mathrm{\&Phi;}\right]}^{-1}\left[x\right]$

In the above-mentioned interpolation structure approximate function, considered the influence of sampled data, thereby interpolation curve is Paint Gloss, approximation accuracy is higher;

2) get function

Synchronized sampling approximate representation ideal synchronisation sampling, promptly

The pairing approximation function

Get synchronously sampled data

Only need during process the substitution of synchronized sampling moment point coordinate

In.

The present invention compared with prior art, its technique effect is:

(1) adopt quadratic polynomial inverse interpolation method to determine zero crossing, (in 10kHz sampling and fundamental frequency is under the 49.5Hz to its computational accuracy, and precision has improved 2 orders of magnitude (from 1.31 * 10 than the linear interpolation height ^-2% brings up to 1.29 * 10 ^-4%); Corresponding primitive period computational accuracy has significantly raising (from 9.16 * 10 ^-5% brings up to 1.51 * 10 ^-7% has improved 2 orders of magnitude).Thereby, fundamental frequency computational accuracy height.

(2) adopt the radial basis function interpolation method to realize the synchronous of sampled data, sample-synchronous is shone upon f:|x (n) → x from setting up ₀(n) form is improved to the interpolation realization

Two processes; Solved restriction that interpolation is counted (be interpolation point and synchronized sampling are counted approaching); In addition, existing interpolating method adopts local interpolation form, the limited (linear interpolation about 1.1 * 10 of interpolation curve slickness and precision ^-4%, quadratic interpolation about 4.1 * 10 ^-6%), approach mode and the radial basis function interpolation is the overall situation, precision significantly improves and (reaches 1.1 * 10 ^-9% has improved 3～4 orders of magnitude).

(3) the harmonic parameters analysis precision has significantly and to improve, and adopts the frequency spectrum computational accuracy of radial basis function interpolation synchronization data, with respect to linear and quadratic interpolation method for synchronizing:

1. amplitude precision, first-harmonic brings up to 5 * 10 from existing 0.1% ^-6%; Third harmonic brings up to 2 * 10 from existing 0.02%～0.08% ^-4%; Quintuple harmonics brings up to 5 * 10 from existing 1～2.2% ^-4%; Nine subharmonic bring up to 7 * 10 from existing 2～3% ^-4%;

2. phase accuracy, first-harmonic brings up to 8 * 10 from existing 2% ^-6%, three times, five times, nine subharmonic bring up to 1 * 10 from existing 5% ^-4%;

Thereby amplitude, phase calculation precision have improved 4～5 orders of magnitude.

Description of drawings

Fig. 1 is sampling with high precision data syncization and harmonic analysis method process flow diagram;

Fig. 2 is that zero crossing is determined the scheme diagram; (1) figure expression among Fig. 2 is chosen and is satisfied x (t ₀)＜0, x (t ₁)＞0, x (t ₂Three neighbouring sample points of)＞0 condition are to determine zero crossing t _{Zero_1}, t wherein ₀The last sampling instant of expression zero crossing, t ₁A sampling instant behind the expression zero crossing, t ₂Second sampling instant behind the expression zero crossing; (2) figure expression among Fig. 2 is chosen and is satisfied x (t ₀)＜0, x (t ₁)＜0, x (t ₂Three neighbouring sample points of)＞0 condition are to determine zero crossing t _{Zero_2}, t wherein ₀Preceding two sampling instants of expression zero crossing, t ₁The last sampling instant of expression zero crossing, t ₂Sampling instant behind the expression zero crossing;

Fig. 3 is non-synchronous sampling and the synchronization in the single cycle;

Fig. 4 is non-synchronous sampling and synchronization data comparison diagram thereof.

Embodiment

The present invention will be described in further detail in conjunction with the accompanying drawings.

As shown in Figure 1, the process flow diagram of the present invention's realization carries out according to the following steps:

(1) input non-synchronous sampling data x (n) and sampling period T _s, each sampling instant point is n*T as can be known _s

(2) determine zero crossing position process, comprising:

1) scanning sample data x (n) satisfies x (t with the neighbouring sample point ₀)＜0, x (t ₁)＞0 condition be defined as the first zero crossing position;

2) for realizing contrary quadratic interpolation, choose x (t again ₂)＞0 sampled point is by x (t ₀), x (t ₁), x (t ₂) 3 can determine zero crossing position t _{Zero_1}, see (1) figure among Fig. 2;

3) for improving the zero crossing accuracy of detection, get x (t again ₀)＜0, x (t ₁)＜0, x (t ₂)＞0 three point is used contrary quadratic interpolation and is determined zero crossing position t _{Zero_2}, see (2) figure among Fig. 2;

4) the zero crossing weighted mean that two groups of data are determined is got

As being similar to of actual zero crossing, and the first zero crossing position is designated as t _Start

(3) determine the primitive period, adopt with step (2) similar procedure and determine next adjacent zero crossing position and be designated as t _End, primitive period T then ₀=t _Start-t _End, fundamental frequency

(4) radial basis function interpolation synchronization process comprises:

1) time of getting is positioned at [t _Start, t _End] interior sample sequence value, the non-synchronous sampling data number of establishing wherein is M, then its time coordinate corresponds to: { t _Start(m ₁+ 1) T _s..., (m ₁+ M) T _s, t _End; Corresponding signal value is: { x (t _Start), x[(m ₁+ 1) T _s] ..., x[(m ₁+ M) T _s], x (t _End), promptly 0, x[(m ₁+ 1) T _s] ..., x[(m ₁+ M) T _s], 0}, wherein m ₁Sequence number for zero crossing position sample sequence;

2) use the radial basis function interpolation method, with time coordinate { t _Start, (m ₁+ 1) T _s..., (m ₁+ M) T _s, t _EndAs the center of radial basis function with join a position, get form parameter a=5T _s Corresponding signal value 0, x[(m ₁+ 1) T _s] ..., x[(m ₁+ M) T _s], 0} can get the approximate value of any t x of place (t) as joining a condition

See Fig. 3 and Fig. 4;

3) at [t _Start, t _End] choosing N synchronized sampling moment point in the interval, its synchronous points number N can choose arbitrarily according to the frequency analysis needs, is not subjected to the restriction of the asynchronous M of counting; Distance is during synchronous points

The moment of corresponding synchronous points is { t _Start, t _Start+ Δ T, t _Start+ 2 Δ T ..., t _Start+ (N-1) Δ T, t _End; Again synchronization point is put the substitution approximate value

In can obtain the approximate of synchronously sampled data

4) to data through synchronization process

Carrying out Fast Fourier Transform (FFT) can get

The amplitude of signal corresponds to

Be phase place; Because when FFT calculated, synchronization data was t in the starting point of time shaft _Start, therefore need phase parameter is carried out Mode compensates.

Simulation example

Below further specify embodiment of the present invention, establish the power network signal model and be:

$x\left(t\right)=\cos ({\mathrm{\&omega;}}_{0}t+\frac{\mathrm{\&pi;}}{4})+0.5\cos (3{\mathrm{\&omega;}}_{0}t+\frac{\mathrm{\&pi;}}{3})+0.3\cos (5{\mathrm{\&omega;}}_{0}t+\frac{\mathrm{\&pi;}}{6})+0.1\cos (9{\mathrm{\&omega;}}_{0}t+\frac{\mathrm{\&pi;}}{3}),$ Get sample frequency F _S=10kHz, it is about 200 to be positioned at two data between zero crossing, gets 200 synchronously sampled datas after the radial basis function interpolation again and carries out frequency spectrum and calculate; System frequency changes at 49.50～50.50Hz, and the computational accuracy of analytical algorithm is as follows:

Table 1 frequency computation part precision and synchrodata errors table

Annotate: go up in the table, the error define method is as follows:

Frequency error (%) is defined as:

Interpolated data error (%) is defined as:

As seen from Table 1: during (1) was determined in the primitive period, quadratic polynomial inverse interpolation method had than the linear interpolation precision and has significantly improved 2 orders of magnitude; (2) in the synchronization data error, linear interpolation about 1.1 * 10 ^-4%, quadratic interpolation about 4.1 * 10 ^-6% approaches mode and the radial basis function interpolation is the overall situation, and precision significantly improves, and reaches 1.1 * 10 ^-9% has improved 3～4 orders of magnitude;

The frequency spectrum result of calculation of table 2RBF interpolation synchronization data

Table 3 linear interpolation synchronization data frequency spectrum result of calculation

Table 4 quadratic interpolation synchronization data frequency spectrum result of calculation

From table 2,3 and 4 contrast as seen: after adopting radial basis function interpolation synchronization data to handle, humorous wave amplitude and phase accuracy effect significantly improve, and reach 1.0 * 10 ^-4The % magnitude, with respect to linearity and quadratic interpolation, its amplitude, phase calculation precision have improved 4～5 orders of magnitude.

Technical scheme of the present invention can be applicable to electric harmonic analysis, electric energy metrical and electric energy quality monitoring.

Claims (1)

1. the harmonic analysis method of a sampling with high precision data syncization is characterized in that this method carries out according to the following steps:

(1), a kind of quadratic polynomial inverse interpolation method that is used for accurately measuring the primitive period has been proposed:

At first, the primitive period is to determine by the difference of adjacent zero passage moment point; Thereby the accuracy of detection of zero crossing is crucial, and it is not only the prerequisite of primitive period calculating, also is the basis of phase compensation in the spectrum analysis; The calculating primitive period is divided into the zero crossing detection and the primitive period is calculated two steps:

1) detection of zero crossing: find out x (t in the sample sequence ₀)＜0, x (t ₁)＞0, x (t ₂Three neighbouring sample points of)＞0 can determine that zero crossing is positioned at t ₀And t ₁Between; Adopted quadratic polynomial inverse interpolation method to determine zero crossing, its principle is: get quadratic interpolation polynomial expression p ₂(t)=x (t ₀)+x[t ₀, t ₁] (t-t ₀)+x[t ₀, t ₁, t ₂] (t-t ₀) (t-t ₁), the p of its zero crossing place ₂(t _Zero)=c=0 determines zero crossing t _ZeroStep be:

1. adopt successive approximation method, get earlier

Make:

$x\left(t_0\right)+x[t_0,t_1](t_{\mathrm{zero}}^{\left(0\right)}-t_0)=c\&DoubleRightArrow;t_{\mathrm{zero}}^{\left(0\right)}=t_0+\frac{c-x\left(t_0\right)}{x[t_0,t_1]}$

The 1. x[t in the step ₀, t ₁] represent that x (t) is at t ₀The first order difference at place;

2. ask again

Make and satisfy:

$x\left(t_0\right)+x[t_0,t_1](t_{\mathrm{zero}}^{\left(1\right)}-t_0)+x[t_0,t_1,t_2](t_{\mathrm{zero}}^{\left(0\right)}-t_0)(t_{\mathrm{zero}}^{\left(0\right)}-t_1)=c$

$\&DoubleRightArrow;t_{\mathrm{zero}}^{\left(1\right)}=t_0+\frac{c-x\left(t_0\right)}{x[t_0,t_1]}-\frac{x[t_0,t_1,t_2]}{x[t_0,t_1]}(t_{\mathrm{zero}}^{\left(0\right)}-t_0)(t_{\mathrm{zero}}^{\left(0\right)}-t_1)$

The 2. x[t in the step ₀, t ₁, t ₂] be illustrated in t ₀The second order difference at place;

3. utilize iterative formula then:

$t_{\mathrm{zero}}^{\left(k\right)}=t_0+\frac{c-x\left(t_0\right)}{x[t_0,t_1]}-\frac{x[t_0,t_1,t_2]}{x[t_0,t_1]}(t_{\mathrm{zero}}^{(k-1)}-t_0)(t_{\mathrm{zero}}^{(k-1)}-t_1)$

At equidistant node (is t ₁=t ₀+ ih) situation, available difference interpolation formula is got t=t ₀+ λ h, Xiang Yingyou:

${\mathrm{\&lambda;}}^{\left(k\right)}=t^{\left(0\right)}+\frac{{\mathrm{\&Delta;}}^2{x}_0}{2\mathrm{\&Delta;}{x}_0}{\mathrm{\&lambda;}}^{(k-1)}({\mathrm{\&lambda;}}^{(k-1)}-1),$ $t^{\left(0\right)}=\frac{c-x_0}{\mathrm{\&Delta;}{x}_0}$

Its zero crossing position is: t _Zero=t ₀+ λ ^(k)H, wherein Δ x ₀=x (t ₀+ h)-x (t ₀), Δ ²x ₀=Δ x (t ₀+ h)-Δ x (t ₀)=x (t ₀+ 2h)-2x (t ₀+ h)+x (t ₀);

In addition, for improving the precision that zero crossing detects, on quadratic polynomial inverse interpolation basis, adopt the average treatment method, two groups of data points of promptly getting near zero-crossing point: x (t ₀)＜0, x (t ₁)＞0, x (t ₂)＞0 and x (t ₀)＜0, x (t ₁)＜0, x (t ₂)＞0, calculate respectively and get its on average approximate as zero crossing behind the zero passage point value:

$\left\{\begin{array}{cc}{t}_{\mathrm{zero}\_1}=t_0+{\mathrm{\&lambda;}}^{\left(k\right)}h& (x\left(t_0\right)<0,x\left(t_1\right)>0,x\left(t_2\right)>0)\\ t_{\mathrm{zero}\_2}=t_0+{\mathrm{\&lambda;}}^{\left(k\right)}h& (x\left(t_0\right)<0,x\left(t_1\right)<0,x\left(t_2\right)>0)\end{array}\right.\&DoubleRightArrow;t_{\mathrm{zero}}=\frac{(t_{\mathrm{zero}\_1}+t_{\mathrm{zero}\_2})}{2};$

2) calculating of primitive period: the 1st) step has realized determining of single zero crossing, can provide the primitive period T of signal then by the mistiming between adjacent zero crossing ₀, promptly provide adjacent two zero crossings and be designated as t according to aforementioned zero crossing detection method _StartAnd t _End, its primitive period should be T mutually ₀=t _Start-t _End

(2), the synchronized radial basis function interpolation method of a kind of sampled data has been proposed:

The signal of analyzing is x (t), and its actual sample sequence is x (n), and the ideal synchronisation sample sequence is x ₀(n), the data synchronization process nature is mapping f:|x (the n) → x that seeks sampling from the actual samples to the ideal ₀(n); All relation between actual samples point and desirable sampled point realizes existing method by directly setting up, thereby precision and computing formula are complicated; The present invention proposes a kind of synchronization thinking, the steps include:

1) according to sampled value x (n), by function of interpolation method structure

Approximate signal x (t), promptly

The interpolating function that its interpolation process is selected for use is a radial basis function

Wherein c is the center of radial basis function, and a is a form parameter.Radial basis function interpolation principle is: the non-synchronous sampling sequence x (n) of known x (t) is the linear combination of radial basis function with x (t) approximate representation:

$x\left(t\right)\&ap;\hat{x}\left(t\right)={\mathrm{\&lambda;}}_1{\mathrm{\&phi;}}_1\left(t\right)+{\mathrm{\&lambda;}}_2{\mathrm{\&phi;}}_2\left(t\right)+...+{\mathrm{\&lambda;}}_N{\mathrm{\&phi;}}_N\left(t\right)=\mathrm{\&Sigma;}{\mathrm{\&lambda;}}_j{\mathrm{\&phi;}}_j\left(t\right)=\left[\mathrm{\&Phi;}\left(t\right)\right]\left[\mathrm{\&lambda;}\right]$

Utilize the interpolation condition at discrete point place:

$x\left(n\right)=\mathrm{\&Sigma;}{\mathrm{\&lambda;}}_j\mathrm{\&phi;}\left(\right||t_n-t_m|\left|\right)\&DoubleRightArrow;\left[\mathrm{\&Phi;}\left(t\right)\right]\left[\mathrm{\&lambda;}\right]=\left[x\right]$

Wherein, [x]=(x (1), x (2) ..., x (N)) ^T, [λ]=(λ ₁, λ ₂..., λ _N) ^TBe undetermined coefficient, form is:

$\left[\begin{array}{cccc}\mathrm{\&phi;}\left(\right||t_1-t_1|\left|\right)& \mathrm{\&phi;}\left(\right||t_1-t_2|\left|\right)& ...& \mathrm{\&phi;}\left(\right||t_1-t_N|\left|\right)\\ \mathrm{\&phi;}\left(\right||t_2-t_1|\left|\right)& \mathrm{\&phi;}\left(\right||t_2-t_2|\left|\right)& ..& \mathrm{\&phi;}\left(\right||t_2-t_N|\left|\right)\\ .& .& .& .\\ .& .& .& .\\ .& .& .& .\\ \mathrm{\&phi;}\left(\right||t_N-t_1|\left|\right)& \mathrm{\&phi;}\left(\right||t_N-t_2|\left|\right)& ...& \mathrm{\&phi;}\left(\right||t_N-t_N|\left|\right)\end{array}\right]\left[\begin{array}{c}{\mathrm{\&lambda;}}_1\\ {\mathrm{\&lambda;}}_2\\ .\\ .\\ .\\ {\mathrm{\&lambda;}}_N\end{array}\right]\left[\begin{array}{c}x\left(1\right)\\ x\left(2\right)\\ .\\ .\\ .\\ x\left(N\right)\end{array}\right]$

And then can get coefficient:

[λ]＝[Φ] ^-1[x]

Therefore, the approximate value of the x (t) at any t place

For:

$\hat{x}\left(t\right)=\left[\mathrm{\&Phi;}\left(t\right)\right]\left[\mathrm{\&lambda;}\right]=\left[\mathrm{\&Phi;}\left(t\right)\right]{\left[\mathrm{\&Phi;}\right]}^{-1}\left[x\right]$

In the above-mentioned interpolation structure approximate function, considered the influence of sampled data, thereby interpolation curve is Paint Gloss, approximation accuracy is higher;

2) get function Synchronized sampling approximate representation ideal synchronisation sampling, promptly The pairing approximation function

Get synchronously sampled data

During process only need with

The substitution of synchronized sampling moment point coordinate

In.

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