research article sample data synchronization and harmonic...

Research ArticleSample Data Synchronization and Harmonic Analysis AlgorithmBased on Radial Basis Function Interpolation

Huaiqing Zhang Yu Chen Zhihong Fu and Ran Liu

The State Key Laboratory of Transmission Equipment and System Safety and Electrical New TechnologyChongqing University Chongqing 400044 China

Correspondence should be addressed to Huaiqing Zhang zhanghuaiqingcqueducn

Received 9 May 2014 Revised 10 August 2014 Accepted 12 August 2014 Published 25 September 2014

Academic Editor Jun-Juh Yan

Copyright copy 2014 Huaiqing Zhang et alThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The spectral leakage has a harmful effect on the accuracy of harmonic analysis for asynchronous sampling This paper proposeda time quasi-synchronous sampling algorithm which is based on radial basis function (RBF) interpolation Firstly a fundamentalperiod is evaluated by a zero-crossing technique with fourth-order Newtonrsquos interpolation and then the sampling sequence isreproduced by theRBF interpolation Finally the harmonic parameters can be calculated by FFTon the synchronization of samplingdata Simulation results showed that the proposed algorithm has high accuracy inmeasuring distorted and noisy signals Comparedto the local approximation schemes as linear quadric and fourth-order Newton interpolations the RBF is a global approximationmethod which can acquire more accurate results while the time-consuming is about the same as Newtonrsquos

1 Introduction

With the widespread use of conventional and modern non-linear loads harmonic and interharmonic currents are beinginjected into the power network and cause power quality(PQ) problems [1] Meanwhile the embedded generationand renewable sources of energy have also created new PQproblems such as voltage variations flickers and waveformdistortions [2] So the monitoring of harmonic and inter-harmonic distortions is an important issue that has beenaddressed in recent years

The discrete Fourier transform (DFT) with its highlyefficient algorithm that is the fast Fourier transform (FFT) isthe recommendedmethod in IEC 61000-4-7 [3] which entailsusing a rectangular window with time length of 200ms TheFFT algorithm has high precision for harmonic analysis inthe case of synchronous sampling However the fundamentalfrequency of power system signals may vary because of thegeneration and load mismatch even leads to fluctuationsSo the synchronous sampling is unattainable and conse-quently the spectral leakage caused by the FFT applied inasynchronously sampling reducesmeasurement accuracy Inpaper [4] the authors have shown that even a small error

in synchronization causes significant errors in amplitudeand phase estimations Although the groupingsubgroupingmethod introduced in IEC standard can mitigate the spectralleakages the low-frequency resolution remains a problem

Therefore numerous investigations for reducing spectralleakage errors in the case of asynchronous sampling havebeen carried out extensively recently Among them thewindowed interpolated FFT algorithm (WIFFTA) is themost popular because of its low computational burden andfeasibility for real-time monitoring In the WIFFTA thewindow functions and interpolation algorithms are adoptedto reduce leakage effect and the picket fence effect Thedrawback of this frequency-domain approach is that thecalculation accuracy is not particularly high and is deficient indetecting fluctuations in signalsThe reasonwhy the accuracycannot be improved is because a flattop window with anextremely small side lobe which can reduce both the short-and long-range leakage errors does not exist [5]

On the contrary the time quasi-synchronous samplingalgorithm (TQSA) [6] can modify the sampling rate andreproduce a synchronous sampling signal It includes thefollowing three steps as obtaining the fundamental periodof signal firstly then adjusting the sampling rate so as to

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 874945 7 pageshttpdxdoiorg1011552014874945

2 Mathematical Problems in Engineering

satisfy the integer-period sampling condition and finallyreproducing the approximate synchronous sampling pointsvia interpolation Hence the harmonic parameters can bedetermined accurately from the DFTFFT results due to thesignificant decrease in spectral leakage The performance ofsuch techniques is highly dependent on the accuracies of fun-damental period and interpolation samplings Generally thefundamental period can be determined by the zero-crossingtechnique and the approximate synchronous sampling canbe interpolated by the linear quadratic cubic spline [7] orNewtonrsquos interpolations However the above methods arelocal interpolationwhichmeans the interpolated point is onlydetermined by nearby sampling data and the interpolationaccuracy is limited

Thus a global interpolation scheme which is based onradial basis function (RBF) for data synchronization methodwas proposed in this paperThe RBF [8] specifically the mul-tiquadric (MQ) function adopted here is defined on distanceThe MQ function was initially proposed for scattered dataapproximation and it has been proved by Franke that theMQmethod has the superior comprehensive performance in 29kinds of scattered data interpolation methods [9]

The organization of this paper is as follows In Section 2the Newtonrsquos forward-difference interpolation for funda-mental period is addressed with comparison to linear andquadratic techniques Then in Section 3 the RBF interpo-lation method for sampling sequence synchronization wasproposed Test results are given to show the usefulness of theproposed method by comparing with others

2 Newtonrsquos Interpolation for theFundamental Period

The time interval between adjacent zero-crossing points canbe considered as the fundamental period of the signal for aperiodic signal So determining the zero-crossing points is ofgreat importance which is not only the basis for fundamentalfrequency but also has directly affection in absolute phasecalculation The linear and quadratic interpolations are thecommonly used methods to acquire the zero-crossing time

21 Linear and Quadratic Interpolations Schemes For lineartype one can firstly find two adjacent sampling points whichsatisfy 119909(119905

1) lt 0 and 119909(119905

2) gt 0 Then the 119905zero can be

determined with linear approximation as

119905zero = 1199051+

minus119909 (1199051)

119909 (1199052) minus 119909 (119905

1)

(1199052minus 1199051) (1)

Finally the next zero-crossing point can be found by the sim-ilar wayTherefore the fundamental period can be calculated

For quadratic situation one can find three adjacentsampling points which satisfy 119909(119905

0) lt 0 119909(119905

1) gt 0 and

119909(1199052) gt 0 just like Figure 1(a) Use the quadratic polynomial

approximation as

119909 (119905) = 119909 (1199050) + 119909 [119905

0 1199051] (119905 minus 119905

0)

+ 119909 [1199050 1199051 1199052] (119905 minus 119905

0) (119905 minus 119905

1)

(2)

where 119909[1199050 1199051] is the first-order divided difference defined

as 119909[1199050 1199051] = (119909 (119905

1) minus 119909(119905

0))(1199051minus 1199050) and 119909[119905

0 1199051 1199052] is

the second-order divided difference in terms of 1199050 1199051 and 119905

2

and defined as 119909[1199050 1199051 1199052] = (119909 [119905

1 1199052] minus 119909[119905

0 1199051])(1199052minus 1199050)

Find the zero-crossing point 119905zero by iteration method whichsatisfies 119909(119905zero) = 0 then denote the 119905zero as 119905zero 1 Andcorrespondingly another three adjacent sampling points arechosen which satisfy 119909(119905

0) lt 0 119909(119905

1) lt 0 and 119909(119905

2) gt 0

like Figure 1(b) and 119905zero 2 can then be worked out Becausethe two zero-crossing points were calculated with differentsampling points we took their average 119905zero = (119905zero 1 +

119905zero 2)2 as a more accurate approximation for zero-crossingpoint In the same way the next zero-crossing point can bedetermined and the fundamental period is obtained

22 Newtonrsquos Interpolation Scheme In above linear orquadratic interpolations only lower-order polynomials wereconsidered Therefore Newtonrsquos interpolation algorithm [6]is chosen for its low computational complexity and conve-nience for programming in this paper Since the error offourth-order polynomial is small enough in estimating thefundamental period only five adjacent sampling points weretaken as the interpolation nodes as 119909(119905

minus2) 119909(119905minus1) 119909(1199050) 119909(1199051)

and 119909(1199052) just like Figure 2

The fourth-order Newtonrsquos divided-difference formulacan be represented as

119891 (119911) = 119891 (1199110) + 119891 [119911

0 1199111] (119911 minus 119911

0)

+ sdot sdot sdot + 119891 [1199110 1199111 1199112 1199113 1199114]

3

prod

119895=0

(119911 minus 119911119895)

(3)

For the five pairs of numbers (119909(119905119894) 119905119894) where 119894 =

minus2 2 the actual zero-crossing time 119905zero is calculated byNewtonrsquos interpolation polynomial as follows

119905zero = 119905minus2

+ 119891 [119909 (119905minus2) 119909 (119905

minus1)] (0 minus 119905

minus2)

+ sdot sdot sdot + 119891 [119909 (119905minus2) 119909 (119905

minus1) 119909 (119905

0) 119909 (119905

1) 119909 (119905

2)]

times

1

prod

119895=minus2

(0 minus 119905119895)

(4)

where 119891[119909(119905minus2) 119909(119905minus1)] = (119905

minus1minus 119905minus2)(119909(119905

minus1) minus 119909(119905

minus2)) The

same process is implemented to calculate next zero-crossingtime If setting the first zero-crossing time as 119905start and thelatter as 119905end we can estimate the fundamental period andfrequency by

119879lowast= 119905end minus 119905start

1198910=

1

119879lowast

=

1

(119905end minus 119905start)

(5)

In practice the power signal contains higher harmonicsor noise the zero-crossing point may be not unique or itis sensitive to the noise influence So a filtering process isnecessary which can remove the harmonics and diminish theinterference by averaging the noise Specifically the finite-impulse response (FIR) filter is chosen for the determinationof the fundamental period due to its linear phase properties

Mathematical Problems in Engineering 3

The quadratic interpolation curve

t0t1 t2

tzero 1

Signal x(t)

(a)

t0 t1t2

tzero 2

The quadratic interpolation curveSignal x(t)

(b)

Figure 1 Quadric interpolation for zero-crossing points

t0

tminus1tminus2

t1 t2

tzero

Signal x(t)

Figure 2 Newtonrsquos interpolation for zero-crossing points

3 RBF Interpolation Method forSampling Synchronization

The synchronization is essentially a mapping from an asyn-chronous sampling set to a synchronous one The syn-chronous sampling points are defined on calculated syn-chronous time which is based on the fundamental periodand zero-crossing point calculation Usually the mappingis implemented by lower-order polynomial interpolationswhich are local approximations even for fourth-order New-tonrsquos scheme Therefore we proposed an RBF-based globalinterpolation method in this paper The basic idea of RBFmethod is briefly narrated as follows Firstly we constructan approximation signal 119909

119886(119905) to approach the original

signal 119909(119905) through RBF interpolation Then we reproduce

the approximate synchronous sampling for synchronizationtime

31 Principle of Radial Basis Function Interpolation Radialbasis function B 119877+ rarr 119877 (domain 119877119889) is defined as thefunction of distance 119903 =

10038171003817100381710038171003817119909 minus 119909119895

10038171003817100381710038171003817 The commonly used RBF

contains the thin plate splines function (B(119903) = 1199032 ln 119903) the

Gaussian function (B(119903) = exp(minus1205731199032)) Hardyrsquos multiquadric(MQ) functions and so forth The widely used MQ functionhas the following expression [8]

120601 (x) = radicx minus c2 + 1205722 (6)

In above c means the center of basis function 120572 is theshape parameter which generally associated with the distancebetween adjacent centers Taking 120572 = 120573

10038171003817100381710038171003817c119894minus c119895

10038171003817100381710038171003817 120573 is also

called shape parameter The principle of RBF interpolationis regarding the unknown functions as linear combinationof radial functions So the approximation function can beobtained after calculating the undetermined coefficients Forexample a series of known data points (119905

119899 119909(119905119899)) 119899 =

1 2 119873 and the approximation function can be con-structed as

119909 (119905) asymp 119909119886(119905) = 120582

11206011(119905) + 120582

21206012(119905) + sdot sdot sdot + 120582

119873120601119873(119905)

= sum120582119895120601119895(119905)

(7)

Then substituting the interpolation data points into aboveequation we can obtain the following

119909 (119899) = sum120582119895120601 (

1003817100381710038171003817119905119899minus 119905119888119898

1003817100381710038171003817) 997904rArr [Φ

119889] [120582] = [x

119889] (8)

where [x119889] = [119909 (119905

1) 119909 (119905

2) 119909 (119905

119873)]

T are the samplingpoints [120582] = [120582

1 1205822 120582

119873]T are the coefficients and [Φ

119889]


denotes the interpolation matrix whose element is B119899119898

=

[(119905119899minus 119905119888119898

)2

+ 1205722]

12

Solve ((8)) as

[120582] = [Φ119889]minus1

[x119889] (9)

Therefore the approximation signal 119909119886(119905) is

119909119886(119905) = [Φ (119905)] [120582] = [Φ (119905)] [Φ

119889]minus1

[x119889] (10)

It is obvious that the approximated value at any time isrelated to the entire interpolation sampling So the interpola-tion curve ismuch smoother and the approximation accuracyis higher

32 Sampling Data Synchronization with RBF In the afore-mentioned section two adjacent zero-crossing points denot-ed by 119905start and 119905end were determined by the fourth-orderNew-tonrsquos interpolation Therefore according to the fundamentalperiod 119879

lowast calculated by ((5)) the original sampling period119879119904 can be adjusted to 119879119904

lowast which satisfies119879lowast = 119873119879119904lowast In other

words we choose119873 equidistant sampling time as [119905start 119905start+119879119904lowast 119905start + 2119879119904

lowast 119905start + (119873 minus 1) 119879119904

lowast 119905end] and then the

integer-period sampling is achieved In practice number 119873

can be arbitrarily selected according to the harmonic analysisrequirements So the approximate synchronous sampling canbe obtained with RBF interpolation as

119909119886(119899) = 119909

119886(119905)

1003816100381610038161003816119905=119899119879slowast = [Φ (119899119879slowast)] [120582]

= [Φ (119899119879slowast)] [Φ119889]minus1

[x119889]

(11)

With synchronization processing of asynchronous sam-pling 119909(119899) spectral leakage is efficiently decreased and thenthe harmonic parameters (ie frequency amplitude andphase) can be accurately determined from the DFTFFTresults In addition the phase compensation should beimplemented as 1205931015840(120596

119896) = 120593(120596

119896) minus 120596119896119905start because of shifting

time 119905start

4 Harmonic Analysis Based onNewtonrsquos and RBF Interpolation forAsynchronous Sampling

So far we combined the fourth-order Newtonrsquos interpolationand RBF interpolation methods The former is utilized todetermine the signalrsquos zero-crossing points and its funda-mental frequency And then an approximate function 119909

119886(119905)

which is defined on the time range [119905start 119905end] can be obtainedby the RBF interpolation The synchronization data 119909

119886(119899)

for given synchronization time can be acquired Finallythe amplitude parameters can be calculated by FFT whilethe phase parameters can be obtained after compensationprocessing due to the beginning time 119905start The detailed flowchart is shown in Figure 3

5 Simulation Result

In this section two simulation tests to verify the estimationaccuracies of frequency amplitude and phase with the above

Start

End

Phase

Amplitude

Frequency

Input asynchronous sampling data x(n)

and corresponding sample time

Fourth-order Newtonrsquos interpolation fordetermining zero-crossing point tzero after filtering

Determining the fundamental period Tlowast andfundamental frequency f0

RBF interpolation for x(n) obtain approximation xa(t)

Obtain synchronization data xa(n) = xa(nTslowast)

Harmonic calculation of xa(n) by FFT

Phase compensation according to tstart

Figure 3 Harmonic analysis procedure based on Newton and RBFinterpolation

methods are performed For RBF method the centers arechosen at the asynchronous sampling time and the shapeparameter is 120573 = 50

51 Simulation for Power Harmonic Measurement In thisexample the input signal which is the same as that used in[10] is employed

119909 (119899) =

3

sum

119898=1

119860119898sin(

2120587119891119898119899

119891119904

+ 120593119898) (12)

The fundamental frequency 1198911equals 4985Hz The

amplitudes of all the harmonics are normalized and arerespectively 119860

1= 1 119860

2= 007 and 119860

3= 02 The initial

phases are respectively 1205931

= 09 rad 1205932

= 12 rad and1205933

= 075 rad According to the recommendation of IECStd 61000-4-7 the window width is set as 200ms Thereforethe sampling frequency 119891

119904and the length of the sampling

sequence119873 are set to be 2560Hz and 512 points respectively

511 Comparison of Fundamental Frequency CalculationFirstly the precision comparison of zero-crossing detec-tion among linear quadratic and fourth-order Newtonrsquosinterpolations was carried out All the 512 sampling pointsare applied for 50th-order FIR filtering However only thefiltered sampling points which are located behind the first51 points will be analyzed for zero-crossing detection Thecutoff frequencies of bandpass FIR filter are 35 and 65Hz


Table 1 Relative error comparison for fundamental frequency

Frequency (Hz) Linear Quadric Newtonrsquos method4950 82209119890 minus 004 16279e minus 004 14848e minus 0074960 90416e minus 004 10133e minus 004 41249e minus 0074970 32312e minus 004 37585e minus 004 23696e minus 0074980 87208e minus 004 37908e minus 004 36028e minus 0084985 92362e minus 004 31479e minus 004 47762e minus 0084990 86749e minus 004 22591e minus 004 11480e minus 0074995 73448e minus 004 12716e minus 004 16125e minus 0075000 55541e minus 004 33378e minus 005 18374e minus 0075005 36123e minus 004 40445e minus 005 17949e minus 0075010 18301e minus 004 79253e minus 005 14653e minus 0075015 51957e minus 005 67919e minus 005 83851e minus 0085020 54970e minus 007 87226e minus 006 82554e minus 0095030 63150e minus 004 11163e minus 004 13240e minus 0075040 57513e minus 004 57684e minus 006 22788e minus 0075050 10963e minus 004 19676e minus 004 25694e minus 007

respectively The fundamental frequency varies from 495Hzto 505Hz and the specific results were shown in Table 1

InTable 1 the relative error of fundamental frequencywasdefineds as Err

119891=

1003816100381610038161003816Δ119891

10038161003816100381610038161198910 It can be seen that the quadratic

interpolation is overall better than linear type and the fourth-order Newtonrsquos interpolation acquires much better accuracywhich is about 2 or 3 orders ofmagnitude less than the formertwo schemesHowever the average time-consuming of linearquadric and Newtonrsquos interpolations is about 11ms 44msand 62ms respectively

512 Interpolation Error Analysis for Asynchronous DataThen the interpolation error comparison at frequency4985Hz for asynchronous data in a fundamental periodamong the above methods was implemented The absoluteerror curves were shown in Figure 4 It is obvious that theRBF interpolation takes great advantage over other threemethods because of its global approximation characteristicThe RBF method can improve significantly the approxima-tion accuracy which is about 2 orders of magnitude than theNewtonmethodThe rootmean square error (RMSE) Err RMSis defined as

Err RMS = radic1

119873

119873minus1

sum

119899=0

[119909 (119899) minus 119909119886(119899)]2

(13)

The results showed that the RMSEof RBFNewton linear andquadric interpolation methods is 12555e-007 78289e-00620026e-003 and 32539e-004 respectivelyThe average time-consuming is correspondingly about 187ms 207ms 35msand 43ms And subsequently the fundamental frequencyvaries from 495Hz to 505Hz and the RMSE curves wereshown in Figure 5 So the RBF method can acquire muchbetter interpolation accuracy than the Newtonrsquos

Table 2 Relative error comparison of harmonic parameters

Parameters RBF Newton Hanning1198911

47762e minus 008 47762e minus 008 10523e minus 0061198601









Table 3 Relative error comparison of harmonic parameters withnoise (SNR = 35 dB)


15990119890 minus 006 15990e minus 006 28590e minus 0061198601









513 Harmonic Analysis Comparison Finally a harmonicanalysis procedure was executed which was based on the syn-chronization sequence The TQSA methods as the RBF andNewtonrsquos interpolation the Hanning windowed interpolatedFFT method were compared

It can be clearly seen that the time-domain methods asRBF and Newtonrsquos interpolation can significantly improvethe parametersrsquo accuracy when compared to the Hanningmethod Particularly for the phase parameter the accuracyimproves about 4 to 5 orders ofmagnitude inTQSACompar-atively speaking the RBF method possesses higher accuracyfor higher harmonic parameters than the Newton methodfor instance the third harmonic case The average time-consuming of RBF Newtonrsquos interpolation and Hanningmethod is about 224ms 236ms and 43ms respectivelyHowever the time-consuming of filter process in TQSA isabout 1225ms (Table 2)

514 Application to Signal with Noise In addition zero-mean Gaussian white noise is added to the clean signal Thesignal to noise ratio (SNR) is set as 35 dB and in orderto acquire a general conclusion the mean value of 500independent simulations is regarded as one measurementresultThe results of the Hanningmethod RBF andNewtonrsquosinterpolation methods are listed in Table 3

From Table 3 we can conclude that the TQSA methodshave obvious advantage in phase calculation more than theHanning method The accuracy is improved about 2 or 3orders of magnitude However the accuracy of frequency


0 0005 001 0015 002 0025Time (s)

Abso

lute

erro

rs

10minus9

10minus10

10minus810minus710minus610minus510minus410minus310minus2

RBFNewton

LinearQuadric

Figure 4 Absolute interpolation errors comparison

Root

mea

n sq

uare

erro

rs

10minus7

10minus6

10minus5

10minus4

10minus3

10minus2

495 496 497 498 499 50 501 502 503 504 505

Frequency (Hz)

RBFNewton

LinearQuadric

Figure 5 Root mean square errors comparison

and amplitude parameters is almost in the same magnitudeThis is mainly because the noise is relatively great and itsinfluence is taken into account in sampling interpolationprocess Although the RBF and Newtonrsquos methods maintainhigh accuracy in interpolation the interpolated curves basedon noisy data have relatively larger error compared to theclean signal

52 Analysis of Current Signal Measured from a Forge Inthis example we evaluate the performance of the TQSA inthe measurement of distorted signal which was recordedfrom a forge The current signal is greatly affected by theworking condition and the amplitude has obvious variationin Figure 6 Therefore the TQSA method and Hanningmethod are compared for distorted signal located in the seg-ment sampling points of [425501 426500] And the samplingfrequency 119891119904 and the length of the sampling sequence 119873 are5000Hz and 1000 points respectively So the time length ofsegment is 200ms which is recommended in IEC 61000-4-7

Firstly the Hanning method is implemented as perthe IECrsquos recommendation Then we can construct theapproximate signal from the harmonic parameters whichare calculated as aforementioned And consequently the

422 423 424 425 426 427 428 429 43 431minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

times105

Figure 6 Recorded current wave in a forge

0 100 200 300 400 500 600 700 800

Measurement signalHanning method

minus2

minus15

minus1

minus05

0

05

1

15

2

Figure 7 Approximate signal based on Hanning method

RBF and Newtonrsquos interpolation methods are performedThe fundamental period can be considered a constant in200ms time intervalThe asynchronous sampling pointswereinterpolated with TQSA However the calculations of FFTfor synchronization data were carried out for every signalfundamental period because of its amplitude variation Theapproximate signals can then be constructed based on theabove harmonic parameters Finally the comparison can becarried out betweenmeasurement signal and the approximatesignal in Figures 7 and 8 for Hanning method RBF andNewtonrsquos method respectively

From Figures 7 and 8 we can conclude that the TQSAcan calculate the harmonic parameters accurately even in afundamental period so the reconstructed signal by TQSAhas much better approximation results as shown in Figure 8While for Hanning method the window length is 10 timesthe fundamental period it cannot capture the amplitudevariation in a time window and the reconstructed signalhas the same amplitude If the time window is changed toa fundamental period the ability of following the amplitudevariation is sure to be improved However the harmonicparametersrsquo accuracymust be decreasedThe contradiction isessentially the problem that a time window which can reduceboth the short- and long-range leakage errors does not existThe results showed that the RMSE for Hanningmethod RBF


0 100 200 300 400 500 600 700 800

Measurement signalRBF methodNewton method

minus2

minus15

minus1

minus05

0

05

1

15

2

Figure 8 Approximate signal based on RBF Newtonrsquos method

and Newtonrsquos interpolation methods is 03598 01027 and01027 respectively

Therefore the above two simulation tests demonstratedthat the TQSA can avoid the contradiction between time andfrequency limitation raised from frequency-domain methodas Hanning The RBF method is a global interpolation andcan acquire more accurate results than the Newtonrsquos for cleansignal Even for noisy data case the RBF can also acquirea similar accuracy to the Newtonrsquos Moreover the time-consuming of RBF is about the same as Newtonrsquos

6 Conclusions

The spectral leakage problems due to the asynchronoussampling affect significantly the harmonic measurementaccuracy So a time quasi-synchronous sampling algorithmbased on RBF interpolation has been proposed in this paperThe principle of RBF interpolation and its application inharmonic analysis was discussed in detail Compared to thelocal approximation schemes as linear quadric and fourth-order Newton interpolations the RBF is a global schemewhich can acquire more accurate results while the time-consuming is about the same as Newtonrsquos The simulationsresults also showed that the RBF method is suitable for theapplicationwhere frequency or amplitude variation detectionis strictly required

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the Fundamental ResearchFunds for the Central Universities (no CDJZR11150008)

References

[1] GWChang andCChen ldquoAn accurate time-domain procedurefor harmonics and interharmonics detectionrdquo IEEE Transac-tions on Power Delivery vol 25 no 3 pp 1787ndash1795 2010

[2] J R de Carvalho C A Duque M A A Lima D V Couryand P F Ribeiro ldquoA novel DFT-based method for spectralanalysis under time-varying frequency conditionsrdquo ElectricPower Systems Research vol 108 pp 74ndash81 2014

[3] ldquoElectromagnetic compatibility (EMC)mdashPart 4ndash7 testing andmeasurement techniquesmdashgeneral guide on harmonics andinterharmonics measurements and instrumentation for powersupply systems and equipment connected theretordquo IEC 61000-4-72002(E) 2002

[4] D Gallo R Langella and A Testa ldquoOn the processing ofharmonics and interharmonics in electrical power systemsrdquoin Proceedings of the IEEE Power Engineering Society WinterMeeting vol 3 pp 1581ndash1586 Singapore January 2000

[5] T Yamada ldquoHigh-accuracy estimations of frequency ampli-tude and phase with a modified DFT for asynchronous sam-plingrdquo IEEETransactions on Instrumentation andMeasurementvol 62 no 6 pp 1428ndash1435 2013

[6] F Zhou Z Huang C Zhao X Wei and D Chen ldquoTime-domain quasi-synchronous sampling algorithm for harmonicanalysis based onNewtonrsquos interpolationrdquo IEEE Transactions onInstrumentation andMeasurement vol 60 no 8 pp 2804ndash28122011

[7] JWang F Yang and X Du ldquoMicrogrid harmonic and interhar-monic analysis algorithm based on cubic spline interpolationsignal reconstructionrdquo in Proceedings of the IEEE InnovativeSmart Grid TechnologiesmdashAsia (ISGT Asia rsquo12) pp 1ndash5 TianjinChina May 2012

[8] L Hardy R ldquoMultiquadric equations of topography and otherirregular surfacesrdquo Journal of Geophysical Research vol 76 pp1905ndash1915 1971

[9] R Franke ldquoScattered data interpolation tests of somemethodsrdquoMathematics of Computation vol 38 no 157 pp 181ndash200 1982

[10] J Wu and W Zhao ldquoA simple interpolation algorithm for mea-suring multi-frequency signal based on DFTrdquo MeasurementJournal of the InternationalMeasurement Confederation vol 42no 2 pp 322ndash327 2009

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Stochastic AnalysisInternational Journal of


satisfy the integer-period sampling condition and finallyreproducing the approximate synchronous sampling pointsvia interpolation Hence the harmonic parameters can bedetermined accurately from the DFTFFT results due to thesignificant decrease in spectral leakage The performance ofsuch techniques is highly dependent on the accuracies of fun-damental period and interpolation samplings Generally thefundamental period can be determined by the zero-crossingtechnique and the approximate synchronous sampling canbe interpolated by the linear quadratic cubic spline [7] orNewtonrsquos interpolations However the above methods arelocal interpolationwhichmeans the interpolated point is onlydetermined by nearby sampling data and the interpolationaccuracy is limited

Thus a global interpolation scheme which is based onradial basis function (RBF) for data synchronization methodwas proposed in this paperThe RBF [8] specifically the mul-tiquadric (MQ) function adopted here is defined on distanceThe MQ function was initially proposed for scattered dataapproximation and it has been proved by Franke that theMQmethod has the superior comprehensive performance in 29kinds of scattered data interpolation methods [9]

The organization of this paper is as follows In Section 2the Newtonrsquos forward-difference interpolation for funda-mental period is addressed with comparison to linear andquadratic techniques Then in Section 3 the RBF interpo-lation method for sampling sequence synchronization wasproposed Test results are given to show the usefulness of theproposed method by comparing with others

2 Newtonrsquos Interpolation for theFundamental Period

The time interval between adjacent zero-crossing points canbe considered as the fundamental period of the signal for aperiodic signal So determining the zero-crossing points is ofgreat importance which is not only the basis for fundamentalfrequency but also has directly affection in absolute phasecalculation The linear and quadratic interpolations are thecommonly used methods to acquire the zero-crossing time

21 Linear and Quadratic Interpolations Schemes For lineartype one can firstly find two adjacent sampling points whichsatisfy 119909(119905

1) lt 0 and 119909(119905

2) gt 0 Then the 119905zero can be

determined with linear approximation as

119905zero = 1199051+

minus119909 (1199051)

119909 (1199052) minus 119909 (119905

1)

(1199052minus 1199051) (1)

Finally the next zero-crossing point can be found by the sim-ilar wayTherefore the fundamental period can be calculated

For quadratic situation one can find three adjacentsampling points which satisfy 119909(119905

0) lt 0 119909(119905

1) gt 0 and

119909(1199052) gt 0 just like Figure 1(a) Use the quadratic polynomial

approximation as

119909 (119905) = 119909 (1199050) + 119909 [119905

0 1199051] (119905 minus 119905

0)

+ 119909 [1199050 1199051 1199052] (119905 minus 119905

0) (119905 minus 119905

1)

(2)

where 119909[1199050 1199051] is the first-order divided difference defined

as 119909[1199050 1199051] = (119909 (119905

1) minus 119909(119905

0))(1199051minus 1199050) and 119909[119905

0 1199051 1199052] is

the second-order divided difference in terms of 1199050 1199051 and 119905

2

and defined as 119909[1199050 1199051 1199052] = (119909 [119905

1 1199052] minus 119909[119905

0 1199051])(1199052minus 1199050)

Find the zero-crossing point 119905zero by iteration method whichsatisfies 119909(119905zero) = 0 then denote the 119905zero as 119905zero 1 Andcorrespondingly another three adjacent sampling points arechosen which satisfy 119909(119905

0) lt 0 119909(119905

1) lt 0 and 119909(119905

2) gt 0

like Figure 1(b) and 119905zero 2 can then be worked out Becausethe two zero-crossing points were calculated with differentsampling points we took their average 119905zero = (119905zero 1 +

119905zero 2)2 as a more accurate approximation for zero-crossingpoint In the same way the next zero-crossing point can bedetermined and the fundamental period is obtained

22 Newtonrsquos Interpolation Scheme In above linear orquadratic interpolations only lower-order polynomials wereconsidered Therefore Newtonrsquos interpolation algorithm [6]is chosen for its low computational complexity and conve-nience for programming in this paper Since the error offourth-order polynomial is small enough in estimating thefundamental period only five adjacent sampling points weretaken as the interpolation nodes as 119909(119905

minus2) 119909(119905minus1) 119909(1199050) 119909(1199051)

and 119909(1199052) just like Figure 2

The fourth-order Newtonrsquos divided-difference formulacan be represented as

119891 (119911) = 119891 (1199110) + 119891 [119911

0 1199111] (119911 minus 119911

0)

+ sdot sdot sdot + 119891 [1199110 1199111 1199112 1199113 1199114]

3

prod

119895=0

(119911 minus 119911119895)

(3)

For the five pairs of numbers (119909(119905119894) 119905119894) where 119894 =

minus2 2 the actual zero-crossing time 119905zero is calculated byNewtonrsquos interpolation polynomial as follows

119905zero = 119905minus2

+ 119891 [119909 (119905minus2) 119909 (119905

minus1)] (0 minus 119905

minus2)

+ sdot sdot sdot + 119891 [119909 (119905minus2) 119909 (119905

minus1) 119909 (119905

0) 119909 (119905

1) 119909 (119905

2)]

times

1

prod

119895=minus2

(0 minus 119905119895)

(4)

where 119891[119909(119905minus2) 119909(119905minus1)] = (119905

minus1minus 119905minus2)(119909(119905

minus1) minus 119909(119905

minus2)) The

same process is implemented to calculate next zero-crossingtime If setting the first zero-crossing time as 119905start and thelatter as 119905end we can estimate the fundamental period andfrequency by

119879lowast= 119905end minus 119905start

1198910=

1

119879lowast

=

1

(119905end minus 119905start)

(5)

In practice the power signal contains higher harmonicsor noise the zero-crossing point may be not unique or itis sensitive to the noise influence So a filtering process isnecessary which can remove the harmonics and diminish theinterference by averaging the noise Specifically the finite-impulse response (FIR) filter is chosen for the determinationof the fundamental period due to its linear phase properties



t0t1 t2

tzero 1

Signal x(t)

(a)

t0 t1t2

tzero 2


(b)


t0

tminus1tminus2

t1 t2

tzero

Signal x(t)








10038171003817100381710038171003817119909 minus 119909119895




120601 (x) = radicx minus c2 + 1205722 (6)


10038171003817100381710038171003817c119894minus c119895

10038171003817100381710038171003817 120573 is also


119899 119909(119905119899)) 119899 =


119909 (119905) asymp 119909119886(119905) = 120582

11206011(119905) + 120582

21206012(119905) + sdot sdot sdot + 120582

119873120601119873(119905)

= sum120582119895120601119895(119905)

(7)


119909 (119899) = sum120582119895120601 (

1003817100381710038171003817119905119899minus 119905119888119898

1003817100381710038171003817) 997904rArr [Φ

119889] [120582] = [x

119889] (8)

where [x119889] = [119909 (119905

1) 119909 (119905

2) 119909 (119905

119873)]


1 1205822 120582


119889]



=

[(119905119899minus 119905119888119898

)2

+ 1205722]

12

Solve ((8)) as

[120582] = [Φ119889]minus1

[x119889] (9)


119909119886(119905) = [Φ (119905)] [120582] = [Φ (119905)] [Φ

119889]minus1

[x119889] (10)










119909119886(119899) = 119909

119886(119905)


= [Φ (119899119879slowast)] [Φ119889]minus1

[x119889]

(11)


119896) = 120593(120596


time 119905start



119886(119905)


119886(119899)


5 Simulation Result


Start

End

Phase

Amplitude

Frequency












119909 (119899) =

3

sum

119898=1

119860119898sin(

2120587119891119898119899

119891119904

+ 120593119898) (12)



1= 1 119860

2= 007 and 119860

3= 02 The initial


= 09 rad 1205932

= 12 rad and1205933










119891=

1003816100381610038161003816Δ119891




Err RMS = radic1

119873

119873minus1

sum

119899=0

[119909 (119899) minus 119909119886(119899)]2

(13)





























0 0005 001 0015 002 0025Time (s)

Abso

lute

erro

rs

10minus9

10minus10


RBFNewton

LinearQuadric


Root

mea

n sq

uare

erro

rs

10minus7

10minus6

10minus5

10minus4

10minus3

10minus2

495 496 497 498 499 50 501 502 503 504 505

Frequency (Hz)

RBFNewton

LinearQuadric





422 423 424 425 426 427 428 429 43 431minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

times105


0 100 200 300 400 500 600 700 800


minus2

minus15

minus1

minus05

0

05

1

15

2





0 100 200 300 400 500 600 700 800


minus2

minus15

minus1

minus05

0

05

1

15

2




6 Conclusions




Acknowledgment


References


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











t0t1 t2

tzero 1

Signal x(t)

(a)

t0 t1t2

tzero 2


(b)


t0

tminus1tminus2

t1 t2

tzero

Signal x(t)








10038171003817100381710038171003817119909 minus 119909119895




120601 (x) = radicx minus c2 + 1205722 (6)


10038171003817100381710038171003817c119894minus c119895

10038171003817100381710038171003817 120573 is also


119899 119909(119905119899)) 119899 =


119909 (119905) asymp 119909119886(119905) = 120582

11206011(119905) + 120582

21206012(119905) + sdot sdot sdot + 120582

119873120601119873(119905)

= sum120582119895120601119895(119905)

(7)


119909 (119899) = sum120582119895120601 (

1003817100381710038171003817119905119899minus 119905119888119898

1003817100381710038171003817) 997904rArr [Φ

119889] [120582] = [x

119889] (8)

where [x119889] = [119909 (119905

1) 119909 (119905

2) 119909 (119905

119873)]


1 1205822 120582


119889]



=

[(119905119899minus 119905119888119898

)2

+ 1205722]

12

Solve ((8)) as

[120582] = [Φ119889]minus1

[x119889] (9)


119909119886(119905) = [Φ (119905)] [120582] = [Φ (119905)] [Φ

119889]minus1

[x119889] (10)










119909119886(119899) = 119909

119886(119905)


= [Φ (119899119879slowast)] [Φ119889]minus1

[x119889]

(11)


119896) = 120593(120596


time 119905start



119886(119905)


119886(119899)


5 Simulation Result


Start

End

Phase

Amplitude

Frequency












119909 (119899) =

3

sum

119898=1

119860119898sin(

2120587119891119898119899

119891119904

+ 120593119898) (12)



1= 1 119860

2= 007 and 119860

3= 02 The initial


= 09 rad 1205932

= 12 rad and1205933










119891=

1003816100381610038161003816Δ119891




Err RMS = radic1

119873

119873minus1

sum

119899=0

[119909 (119899) minus 119909119886(119899)]2

(13)





























0 0005 001 0015 002 0025Time (s)

Abso

lute

erro

rs

10minus9

10minus10


RBFNewton

LinearQuadric


Root

mea

n sq

uare

erro

rs

10minus7

10minus6

10minus5

10minus4

10minus3

10minus2

495 496 497 498 499 50 501 502 503 504 505

Frequency (Hz)

RBFNewton

LinearQuadric





422 423 424 425 426 427 428 429 43 431minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

times105


0 100 200 300 400 500 600 700 800


minus2

minus15

minus1

minus05

0

05

1

15

2





0 100 200 300 400 500 600 700 800


minus2

minus15

minus1

minus05

0

05

1

15

2




6 Conclusions




Acknowledgment


References


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











=

[(119905119899minus 119905119888119898

)2

+ 1205722]

12

Solve ((8)) as

[120582] = [Φ119889]minus1

[x119889] (9)


119909119886(119905) = [Φ (119905)] [120582] = [Φ (119905)] [Φ

119889]minus1

[x119889] (10)










119909119886(119899) = 119909

119886(119905)


= [Φ (119899119879slowast)] [Φ119889]minus1

[x119889]

(11)


119896) = 120593(120596


time 119905start



119886(119905)


119886(119899)


5 Simulation Result


Start

End

Phase

Amplitude

Frequency












119909 (119899) =

3

sum

119898=1

119860119898sin(

2120587119891119898119899

119891119904

+ 120593119898) (12)



1= 1 119860

2= 007 and 119860

3= 02 The initial


= 09 rad 1205932

= 12 rad and1205933










119891=

1003816100381610038161003816Δ119891




Err RMS = radic1

119873

119873minus1

sum

119899=0

[119909 (119899) minus 119909119886(119899)]2

(13)





























0 0005 001 0015 002 0025Time (s)

Abso

lute

erro

rs

10minus9

10minus10


RBFNewton

LinearQuadric


Root

mea

n sq

uare

erro

rs

10minus7

10minus6

10minus5

10minus4

10minus3

10minus2

495 496 497 498 499 50 501 502 503 504 505

Frequency (Hz)

RBFNewton

LinearQuadric





422 423 424 425 426 427 428 429 43 431minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

times105


0 100 200 300 400 500 600 700 800


minus2

minus15

minus1

minus05

0

05

1

15

2





0 100 200 300 400 500 600 700 800


minus2

minus15

minus1

minus05

0

05

1

15

2




6 Conclusions




Acknowledgment


References


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119891=

1003816100381610038161003816Δ119891




Err RMS = radic1

119873

119873minus1

sum

119899=0

[119909 (119899) minus 119909119886(119899)]2

(13)





























0 0005 001 0015 002 0025Time (s)

Abso

lute

erro

rs

10minus9

10minus10


RBFNewton

LinearQuadric


Root

mea

n sq

uare

erro

rs

10minus7

10minus6

10minus5

10minus4

10minus3

10minus2

495 496 497 498 499 50 501 502 503 504 505

Frequency (Hz)

RBFNewton

LinearQuadric





422 423 424 425 426 427 428 429 43 431minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

times105


0 100 200 300 400 500 600 700 800


minus2

minus15

minus1

minus05

0

05

1

15

2





0 100 200 300 400 500 600 700 800


minus2

minus15

minus1

minus05

0

05

1

15

2




6 Conclusions




Acknowledgment


References


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 0005 001 0015 002 0025Time (s)

Abso

lute

erro

rs

10minus9

10minus10


RBFNewton

LinearQuadric


Root

mea

n sq

uare

erro

rs

10minus7

10minus6

10minus5

10minus4

10minus3

10minus2

495 496 497 498 499 50 501 502 503 504 505

Frequency (Hz)

RBFNewton

LinearQuadric





422 423 424 425 426 427 428 429 43 431minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

times105


0 100 200 300 400 500 600 700 800


minus2

minus15

minus1

minus05

0

05

1

15

2





0 100 200 300 400 500 600 700 800


minus2

minus15

minus1

minus05

0

05

1

15

2




6 Conclusions




Acknowledgment


References


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 100 200 300 400 500 600 700 800


minus2

minus15

minus1

minus05

0

05

1

15

2




6 Conclusions




Acknowledgment


References


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article sample data synchronization and harmonic...

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