Summary of the invention
The object of this invention is to provide a kind of ultralow frequency harmonic detection method of considering negative frequency impact, it is characterized in that, comprise the following steps:
Step 1: under operation of power networks normal condition, set up the unifrequency discrete signal model of power network signal:
(1) wherein
for the amplitude of power network signal,
for the frequency of power network signal and
for the initial phase of power network signal,
,
for the total sampling number of discrete signal intercepting,
for power network signal sample frequency;
Step 2: right
the normalization discrete Fourier transform (DFT) (Discrete Fourier Transform, DFT) that adds rectangular window obtains
(2), wherein k=0,1 ... N-1;
Step 3: the frequency resolution of establishing spectral line is
,
, and will in formula (1) substitution formula (2), obtain
X（k）
（3），
If normalized frequency is
,
, formula (3) is further reduced to:
(4), in formula
;
Step 3: get adjacent three spectral lines
,
with
spectrum value
,
with
, here
for integer and to meet its corresponding spectral line be maximum spectral line, here
for integer and to meet its corresponding spectral line be maximum spectral line, obtain through calculating abbreviation
(5)
Note
,
, have according to formula (4):
(6), wherein
=1,0,1, three unknown numbers in formula are
,
with
, when
=1,0,1, can become a ternary linear function group by simultaneous, therefrom can obtain
,
with
, wherein solve frequency normalization to be asked
expression formula be
(7)
Step 4: obtain the highest spectral line by formula (4)
equation:
(8),
In formula
,
,
, formula (9) is got to conjugation and obtains:
(9) obtain
after, simultaneous formula (8) and formula (9) composition system of equations, this system of equations the inside only has
with
be unknown quantity, all the other are known quantity, therefore can solve:
(10)
Step 5: formula (7) and formula (10) are considers humorous wave frequency, amplitude and phase estimation formula between negative frequency and interpolation ultralow.
Detection in full frequency band time harmonic frequency comprises the following steps:
1) signal electrical network being collected, intercepts one section with rectangular window and makes FFT(DFT) spectrum analysis;
2) for harmonic component between more than 50Hz, the regulation detecting according to harmonic wave between IEC61000-4-7 standard pair, with the form of a harmonic wave group or subgroup, measure frequency range between two harmonic waves and between first-harmonic and second harmonic between harmonic parameters, and frequency using the mean value of two harmonic wave sequence numbers as corresponding harmonic wave group or subgroup;
3) must consider the impact of negative frequency for harmonic wave between 0 ~ 50Hz frequency range; Between the ultralow frequency of 0 ~ 10Hz frequency range, harmonic wave calculates harmonic parameters amplitude between ultralow frequency according to ultralow frequency simple harmonic wave detection method
, frequency
and phase place
, and 10 ~ 50Hz frequency range between harmonic wave calculate all spectral lines of residue except first-harmonic spectral line according to the method for IEC, remove the root-mean-square value of harmonic amplitude after the intersegmental harmonic wave leakage contribution of ultralow frequency
;
4) in step 3)
, be 0 ~ 50Hz subharmonic frequency range between harmonic parameters, according to the regulation of IEC standard, be designated as harmonic wave between 0.5 time;
5) comprehensive step 2) and step 4) obtain full frequency band between harmonic parameters.
In an embodiment of the present invention, occur what harmonic wave between ultralow frequency produced
spectral line drops on frequency just
upper
,
spectral line drops on direct current spectral line, carries out every straight processing filtering DC component before carrying out FFT spectrum analysis
,
, make formula (6) can not be write as the secondly form of system of equations of ternary, now can select
be 1, corresponding
, wherein " * " represent get conjugation, at this moment, the spectral line sequence number of utilization be (1,1,2), correspondence spectrum value be (
,
,
), formula
can be reduced to:
(11).
The present invention is different from traditional interpolation correction algorithm, this algorithm has been considered the interference of negative frequency to Harmonic lines between ultralow frequency, become equation with three unknowns group to solve harmonic frequency between ultralow frequency by three spectral line value simultaneous, and solve amplitude and the phase place of a harmonic wave according to the spectrum value of maximum spectral line and conjugate thereof; Realize the relevant feature parameters of extracting fast and more exactly harmonic wave between ultralow frequency in short time, real-time and the validity requirement of algorithm while having ensured actual detection.Between the present invention has simultaneously derived and worked as, harmonic frequency is less than 1 frequency resolution
time parameter correction formula, actual can solve between harmonic frequency be low to moderate
below, greatly broken through traditional restriction that is subject to frequency resolution based on DFT harmonic analysis, shortened the length of analytic signal, this is that other algorithm institute is irrealizable.And algorithm of the present invention is not subject to the impact of DC component and to insensitive its accuracy and the robustness of having ensured of system noise.
Embodiment
Voltage and the frequency of considering electrical network in practical power systems are always changing, and the detection of a harmonic wave is always closely related with frequency with the voltage of electrical network, according to the voltage of electrical network and frequency change time speed, signal model in actual electric network is classified: stable state while 1) length, the voltage magnitude of power network signal and constant stationary random signal of frequency expectation value long period; 2) stable in short-term, the voltage magnitude of power network signal and constant stationary random signal of frequency expectation value short period; 3) unsteady state, the non-stationary signal becoming when the voltage magnitude of power network signal and frequency are when fault.And the foundation of model of the present invention is aimed at and analyzes under system normal operating condition, the constant expectation value of voltage magnitude and frequency voltage magnitude and frequency in actual electric network within a period of time, under a period of time steady state (SS)
,
; Consider that the noise signal in actual electric network is the white Gaussian noise signal separate with each frequency component of electrical network simultaneously
.
Set up the power network signal mathematical model containing harmonic wave between ultralow frequency:
(12)
In formula
,
with
for amplitude, frequency and the initial phase of each component in power network signal,
for the direct-flow offset weight that may comprise in actual electric network signal,
for the white Gaussian noise signal separate with each frequency component of electrical network, M is the frequency component number comprising in power network signal; Time
, the intercepting time period that T is signal analysis.
Consider that the signal collecting in actual electric network is discrete series
, therefore change the mathematical model in formula (12) into discrete form:
(13)
In formula
for the sample frequency of power network signal,
, N is the total sampling number of discrete signal intercepting.
It is pointed out that when the sampled signal of actual electric network is carried out to spectrum analysis, generally can first carry out carrying out filtering DC component every straight processing
, object be improve computing machine integer FFT do spectrum precision, prevent Frequency and interfering frequency that windowing (except rectangular window) produces.While carrying out the mathematical analysis of model, first ignore system noise part
, after analyze again the situation that affects on testing result when the system noise; Notice, the linear superposition that the signal in formula (12) is each component, carries out DFT(FFT to it simultaneously) still meet this overlaying relation when spectrum analysis.Therefore, for without loss of generality, analysis list frequency-distributed signal model:
(14)
Can find out from formula (14), the actual signal gathering from electrical network is discrete real sequence
,
, not single-frequency complex exponential signal sequence, the normalization discrete Fourier transform (DFT) (Discrete Fourier Transform, DFT) that it is added to rectangular window obtains
(15)
In formula
=0,1,, N 1.
If the frequency resolution of spectral line is
,
, by formula (14) substitution formula (15):
(16)
In actual conditions, general
be worth larger, therefore
.If normalized frequency is
,
, formula (15) is further reduced to:
(17)
In formula
.
From formula (17), can find out, actual electric network sampled signal is carried out after DFT spectrum analysis, there will be two frequency components on frequency spectrum
with
, wherein Section 2 is the negative frequency components composition of frequency spectrum.In the algorithm model of tradition based on DFT spectrum analysis, ignore the impact of negative frequency on spectral line, directly take Section 1 in formula (17), do so certain reason, reason be between reality harmonic component when enough words far away of distance 0 frequency spectrum, negative frequency is very little on the impact of harmonic component between reality, but for harmonic wave between ultralow frequency, negative frequency is very large on its impact, this is also that general algorithm is detecting the reason losing efficacy for ultralow harmonic wave, when detecting harmonic wave between power network signal ultralow frequency, reality must consider the impact of negative frequency on it.Negative frequency affects schematic diagram as shown in Figure 1 to frequency spectrum.
Consider the discrete spectrum of negative frequency impact, get adjacent three spectral lines
,
with
spectrum value
,
with
, here
for integer and to meet its corresponding spectral line be maximum spectral line.So
(18)
In like manner can verify
(19)
Therefore, note
,
, have according to formula (15):
(20)
In formula, three unknown numbers are
,
with
, wherein
=1,0,1, can become a ternary linear function group by simultaneous, and there is unique solution in this equation, therefrom can obtain
,
with
.For the convenience solving, formula (20) is considered as to system of homogeneous linear equations, untrivialo solution vector be
,
, 1}
^{t}, the unique conditional of its existence is the matrix of coefficients of system of equations
determinant be zero
.
Matrix of coefficients
for:
(21)
By
can solve frequency normalization to be asked
explicit Expression formula be
(22)
For the accuracy that ensures to calculate, need in guarantee formula (22)
spectral line is the highest spectral line,
with
be respectively the adjacent spectral line in its left and right, one of them must be time high spectral line so, so the frequency estimation of harmonic wave is between the ultralow frequency of consideration negative frequency impact
(23)
In formula
for frequency resolution.
The impact of taking into account system DC component: probably occur what harmonic wave between ultralow frequency produced in reality
spectral line drops on frequency just
upper
, like this
spectral line drops on direct current spectral line.Owing to originally may there being the interference of DC component in system, before carrying out FFT spectrum analysis, carry out every straight processing filtering DC component
,
, make formula (20) can not be write as the secondly form of system of equations of ternary, now can select
be 1, corresponding
, " * " represents to get conjugation.At this moment, the spectral line sequence number of utilization is (1,1,2), corresponding spectrum value be (
,
,
), formula (22) can abbreviation be:
(24)
In order to ensure signal to noise ratio (S/N ratio) and the computational solution precision of actual signal spectral line, carry out harmonic amplitude and phase estimation between ultralow frequency time and other interpolation algorithm similar, need to select the highest spectral line in part to calculate.By the highest spectral line of formula (17)
equation:
(25)
In formula
,
。
Wherein
.
Formula (25) is got to conjugation to be obtained:
(26)
Obtained by formula (22)
after, simultaneous formula (25) and formula (26) composition system of equations, this system of equations the inside only has
with
be unknown quantity, all the other are known quantity, therefore can solve:
(27)
Formula (22) and formula (27) are considers humorous wave frequency, amplitude and phase estimation formula between negative frequency and interpolation ultralow.It should be noted that above-mentioned model is based upon signal and intercepts by rectangular window (i.e. not windowing), this is also the testing requirement that meets harmonic wave between IEC61000-4-7 pair.
Harmonic wave between the ultralow frequency having detected for above-mentioned theory, and the regulation detecting in conjunction with harmonic wave between 24337-2009 couple of IEC61000-4-7 and GB GB/T, can form a whole set of be applicable to Whole frequency band between harmonic wave detection scheme, its FB(flow block) is as shown in Figure 2.Its detailed step is:
1) signal electrical network being collected, intercepts one section (as 10 cycles of IEC regulation, 200ms, respective frequencies resolution is 5Hz) with rectangular window and makes FFT(DFT) spectrum analysis.
2) for harmonic component between more than 50Hz, the regulation detecting according to harmonic wave between IEC61000-4-7 pair, with the form of a harmonic wave group or subgroup, measure frequency range between two harmonic waves and between first-harmonic and second harmonic between harmonic parameters, and frequency using the mean value of two harmonic wave sequence numbers as corresponding harmonic wave group or subgroup (as harmonic wave between between 2 subharmonic and 3 subharmonic be designated as 2.5 between harmonic wave group or subgroup).
3) for the necessary impact of considering negative frequency of harmonic wave between 0 ~ 50Hz frequency range (being subharmonic), the frequency of subharmonic is less, and the impact that is subject to negative frequency is more serious; Consider same frequency range between harmonic wave quantity can be not too many, there is the fluctuation of frequency and amplitude in electrical network simultaneously, the 5Hz frequency resolution specifying taking IEC is example, and between the ultralow frequency of 0 ~ 10Hz frequency range, harmonic wave calculates harmonic parameters (amplitude between ultralow frequency according to algorithm mentioned in this article
, frequency
and phase place
), and 10 ~ 50Hz frequency range between harmonic wave calculate all spectral lines of residue (containing first-harmonic spectral line) according to the method for IEC and remove the root-mean-square value after the intersegmental harmonic wave leakage contribution of ultralow frequency
.
4) the 3rd) in step
, be 0 ~ 50Hz subharmonic frequency range between harmonic parameters, according to the regulation of IEC standard, be designated as harmonic wave between 0.5 time.
5) comprehensive step 2) and step 4) obtain full frequency band between harmonic parameters.
The harmonic detection method proposing for the present invention carries out Digital Simulation, simulated environment is Matlab 2012b, the 0.2s sampled signal length (synchronized sampling) specifying with IEC61000-4-7 in emulation embodiment, corresponding 5Hz frequency resolution, if humorous wave frequency is 3.5Hz between ultralow frequency, amplitude coefficient with respect to fundamental frequency is 1%, and under noise circumstance, the testing result of various algorithms is as shown in table 1:
The accuracy of detection of each algorithm in table 1 noise circumstance
Note: while having noise, carry out 100 Monte Carlo simulations and test the mean value of asking for result; "---" represents that corresponding algorithm is without correct testing result.
As can be seen from Table 1, the window function and interpolation algorithm based on DFT lost efficacy in the time detecting between ultralow frequency harmonic wave, and reason is the restriction that is subject to frequency resolution; Prony algorithm has high accuracy of detection in the time of noiseless, but trickle noise can cause sharply dropping to of precision to detect, and its robustness is bad; The method of carrying herein can have higher accuracy of detection in larger noise circumstance, has ensured robustness and the real-time of algorithm in real system detects.
The comparison of the present invention and other algorithms:
(1) algorithm and the comparison of IEC canonical algorithm herein
Between IEC61000-4-7 regulation, harmonic wave group algorithm detects subharmonic, supposes that between low frequency, the relative amplitude of harmonic wave is
, frequency is
, phase place is 0.Sampling time is 0.2s, when
while variation from 0 ~ 40Hz with the step-length of 0.05Hz, corresponding testing result as shown in Figure 3
As seen from Figure 3, IEC algorithm is 5Hz in frequency resolution, range of allowable error is in ± 5% time, between the minimum that can detect, harmonic frequency is about 7.5Hz, consider the phase angle variations of harmonic wave and the stability of algorithm between ultralow frequency, between 0 ~ 10Hz frequency range, harmonic wave IEC algorithm cannot detect exactly simultaneously.
And herein algorithm shows as Fig. 4 the intersegmental harmonic wave testing result of ultralow frequency:
As seen from Figure 4, for the sampled signal of 0.2s length, doing frequency resolution corresponding to spectrum analysis is 5Hz, algorithm herein can IEC standard regulation ± detect in 5% error range low tone harmonic frequency be 1Hz(now between correspondence harmonic phase be 0), greatly shake off the restriction of frequency resolution, realized and in the short period, detected rapidly and accurately harmonic wave between ultralow frequency.
Under noiseless disturbed condition, in specify ± 5% range of tolerable variance of IEC, between the phase angle variations scope of harmonic wave be 0 ~ 2 π, obtain harmonic frequency between minimum that herein algorithm and IEC canonical algorithm can detect in the time of unlike signal length, the numerical value of enumerating and changing trend diagram thereof are respectively as shown in table 2 and Fig. 5.
Harmonic frequency between the minimum that in table 2 range of tolerable variance, algorithm can be identified
Can be found out by table 2 and Fig. 5, herein algorithm can detect harmonic wave between ultralow frequency in a short period of time, and without increasing signal length, thereby provides new way for harmonic wave fast detecting between ultra-low frequency.
(2) algorithm and Prony algorithm comparison herein
10 signal cycles that specify taking IEC standard are as example, between harmonic frequency change at 0 ~ 10Hz, its amplitude coefficient (with respect to fundamental frequency) is 1%, carries out 100 Monte Carlo simulation experiments as shown in table 3 with the contrast of Prony algorithm testing result under different noise circumstances:
Each algorithm accuracy of detection contrast situation when table 3 noise
Note: "---" represents that corresponding algorithm cannot detect result.
In different noise circumstances, the accuracy of detection variation tendency of Prony algorithm and herein algorithm is as shown in 6:
Can be found out by table 3 and Fig. 6, Prony algorithm, in noise-free environment, has quite high accuracy of detection and (reaches
the order of magnitude), and along with the increase (being that signal to noise ratio (S/N ratio) diminishes) of noise, its accuracy of detection sharply drops to and cannot detect (noise that is approximately higher than 50dB intensity).And in practical power systems because the voltage of electrical network and frequency are always in continuous variation, sampled signal is introduced random noise unavoidablely simultaneously, Prony algorithm is very sensitive to noise signal, is not suitable for the detection of harmonic wave between electric system ultralow frequency.It can also be seen that, algorithm noise immunity is better herein simultaneously, and in the time that noise strengthens, the variation of testing result precision not quite (is stabilized in
~
the order of magnitude), can meet the demand of the actual detection of electric system.
In sum, the short time fast detecting that can realize harmonic wave between ultralow frequency of the present invention, this algorithm is not subject to the impact of DC component and insensitive to system noise simultaneously, can ensure robustness and the real-time of this algorithm in reality detects.
The foregoing is only preferred embodiment of the present invention, all equalizations of doing according to the present patent application the scope of the claims change and modify, and all should belong to covering scope of the present invention.