基于加汉宁窗的FFT高精度谐波检测改进算法-王刘旺

4024Vol.40No.2420121216PowerSystemProtectionandControlDec.16,2012AnimprovedprecisealgorithmforharmonicanalysisbasedonHanning-windowedFFTWANGLiu-wang1,HUANGJian-cai1,SUNJian-xin2,WANGQiang2,ZHUYong-li1(1.SchoolofControlandComputerEngineering,NorthChinaElectricPowerUniversity,Baoding071003,China;2.JinzhongPowerSupplyCompany,Jinzhong030600,China)Abstract:Duetothewideuseofnon-linearcomponents,harmonicproblembecomesincreasinglyserious.ItisdifficulttoperformaccurateharmonicanalysiswithfastFouriertransform(FFT)undertheunsynchronizedsamplingconditions.WindowedinterpolationmethodscanimprovetheaccuracyofFFTforharmonicanalysis.AnimprovedalgorithmwithhighaccuracyisproposedbasedontheattenuationcharacteristicsoftheHanning-windoweddiscretespectrumofsignal.Viaaspecificpolynomialtransformofthespectralsequence,thealgorithmreducesthespectralleakagefurther,andthentheinterpolationisappliedtoderivethepracticalcorrectionformulasoftheharmonicfrequency,amplitudeandphase.ComparisonwithHanningandBlackman-HarrisinterpolationFFTmethodsiscarriedoutbyMatlabsimulations,whichverifiesthehigheranalysisaccuracyoftheproposedalgorithm.Anexperimentoncapacitorharmoniccurrentdemonstratesitsvalidityfurther.Keywords:harmonicanalysis;fastFouriertransform;Hanningwindow;frequencyfluctuation;spectrumleakageTM71A1674-3415(2012)24-0028-06[1]FFTFFTV.K.Jain[2]Hanning[3]Blackman-Harris[4-5]Nuttall[6-7]Rife-Vincent[8-9][10-12]FFT[13]FFT[14]Hanning31dBBlackman-Harris92dB6~7HanningFFT-29-HanningBlackman-HarrisHanning)]π2()π2([41)(21)(RRRHNwN  (1) )(RwDirichlet2/)1(jRe)2/(sin)2/(sin)(NNw)(tx)π2(cos)(mmmtfAtx(2)mAmfmsf(2))(nx)(cos)()(mmmsnATnxnx(3)ss/1fTsmm/π2ff)(nxmmjjmmmm()e()e()22AAX(4))(HanningH()wn)(nx)(Hnx)(HnxHH()()*()XXW(5))(HnxFFT)(Hnx)(HkX)(HkX)(HX]π2,0[N/π2Nff/smmmHmm2m22mjπ()jsin[π()]1()4sin[π()/]sin(π/)cos[π()/]sin(π/)sin[π()/]eekkkkXkAkkNNkkNNkkN(6)N(6)mmmmH2mmjπ()jsin[π()]()4π()[1()]eekkNAkkXkkkkk(7)mjmmHme,()40,NAkkXkkk(8)(8)HanningFFTmkkmkk0mkkfkkf)(mmm(9)Nff/smk)1,0(mk)1()(2HBkX(10)mmkkk(11))π(jmmmme)π(sinπ4kkNAB(12)(10)3()xn11ss320cos(2π)10cos(2π)ffnnff1(a)1(b)1s50.5Hz,400HzffFFTHanningFFTFFT1(a))HanningFFT1(b)(10))(HkXk)1(2mkk1mkk)1(/12-30-Fig.1ThespectrumofsignalsHanningFFT5H5HHHHH()()[(1)(1)][(2)(2)]XkaXkbXkXkcXkXk(13)(10))9)(4)(1()(2225HBkX(14)(13)(14)60/1a90/1b360/1c(14)mmkk)1(/12)9)(4)(1(/12221(c)1(b)HanningFFTHanningFFT)(HkX(13))(HkX)(5HkXmmkkmkmkmkmfmAmm1)mk0(mf15)Hz0(15)Hzmfakkbkkmabmin(,)kkk0f50Hz2)m)1()(m5Hm5HmkXkX(15)3)mk(14)(15)mmm134k(16)4)(9)mfmkk(14)mmH5m222mmmmj(π)mm1()(1)(4)(9)sin(π)e4πkXkkkkkNAk(17)mAm222mmmmmH5mm(1)(4)(9)4π()sin(π)AkkkkXkNk(18)π))((phasemm5HmkkX(19)FFTFFT-31-mms1kkfNfmtemp(20))(floormtempk(21)mmktempk(22))·(floorHanningBlackman-HarrisFFTMatlab(23)1211ms1m)π2(cos)(mnffmAnx(23)Table1Simulationparameters1234567/V220.00.410.03.06.01.53.0/rad0.10.20.30.40.50.60.7891011121314/V1.32.10.81.10.70.650.15/rad0.80.91.01.11.21.31.415161718192021/V1.00.060.40.020.030.0030.01/rad1.51.61.71.81.92.02.11f50.5Hzsf2200HzN5122Hanning-5HanningBlackman-HarrisFFTHanningB-H4.25e-05.2510-58.04e-033.34e-0105.84e-101.49e-07HanningFFT3~40.0031/73333202.92e-061.88e-0510-2(b)Fig.2Comparisonofrelativeerrorsofharmonicamplitudeandphase10.00180.04249.5~50.5Hz2323310kHz16008410450.00099Hz-32-Table2Comparisonofrelativeerrorsofamplitudeinfluctuationofthefrequencyf/Hz23456789101149.87.26e-051.54e-082.12e-079.85e-088.47e-081.01e-071.67e-075.28e-084.32e-072.33e-0849.97.59e-052.73e-081.57e-071.20e-071.94e-075.70e-085.15e-072.28e-085.43e-078.79e-0850.07.61e-054.06e-081.55e-079.81e-085.27e-073.27e-086.36e-075.69e-082.46e-072.73e-0750.17.35e-055.24e-082.24e-077.28e-081.10e-064.49e-083.67e-071.70e-071.42e-071.27e-0750.26.84e-054.63e-083.70e-075.18e-081.14e-069.63e-082.11e-071.62e-074.22e-076.71e-08Table3Comparisonofrelativeerrorsofphaseinfluctuationofthefrequencyf/Hz23456789101149.85.98e-034.09e-071.56e-054.28e-068.00e-072.67e-066.21e-061.55e-061.02e-053.36e-0749.96.64e-034.16e-079.95e-063.58e-069.67e-062.04e-061.32e-052.86e-079.17e-061.88e-0650.07.19e-031.15e-062.01e-063.53e-062.22e-058.10e-071.27e-051.56e-066.00e-063.49e-0650.17.62e-031.73e-067.60e-063.05e-063.53e-058.14e-079.63e-063.60e-065.46e-072.43e-0650.27.92e-033.48e-061.82e-052.21e-062.83e-052.58e-064.03e-062.97e-069.25e-062.98e-0702040608010012014016060402002040t/msFig.3WaveformofcapacitoroperatingcurrentTable4Analysisresultsofcapacitorcurrentf/Hz/mA/()150.0009931.04771124.783572100.001982.1649292.224603150.002977.58269110.37434200.003951.4980370.393855250.004943.7715529.549256300.005930.53261314.359247350.006920.64367113.520688400.007910.31392110.619499450.008901.8502688.692851~50402040608010012014016060402002040t/msFig.4FittingresultsbyproposedmethodFFTFFTHanningFFT,,.[J],2010,38(15):157-160.LIANGZhi-rui,YEHui-qiang,ZHAOF