DIGIL MESUREMEN OF POWER SYSEM HRMONIC MGNIUDE ND PHSE NGLE R Micheletti (, R Pieri ( ( Departmet of Electrical Systems ad utomatio, Uiversity of Pisa, Via Diotisalvi, I-566 Pisa, Italy Phoe +39 5 565, Fax +39 5 565333, E-mail: RobertoMicheletti@dseauipiit ( Departmet of Electrical Systems ad utomatio, Uiversity of Pisa, Via Diotisalvi, I-566 Pisa, Italy Phoe +39 5 565, Fax +39 5 565333, E-mail: RezoPieri @dseauipiit bstract - he paper deals with the measuremet of harmoic magitude ad phase agle of a periodic voltage v(t i power systems he proposed procedure is based o Walsh spectrum measuremet followed by digital computatio to provide the Fourier spectrum of v(t i magitude ad phase Computer simulatio results are preseted to validate this method Keywords - Harmoic magitude ad phase agle, Power systems, Walsh spectrum, Fourier spectrum INRODUCION he usual represetatio of a periodic time-varyig sigal is the Fourier series he coefficiets of the equatio ca be obtaied accordig to two mai approaches I the first, the iput sigal is sampled, coverted ito digital form ad the Fourier itegral calculatio is used i order to obtai the coefficiets If the period is kow, this method requires oly oe cycle of the sigal: however the mai drawback is the awkward ad the time-cosumig multiplicatios by cosie ad sie coefficiets of Fourier processig Variatios have bee suggested processig the digitized data usig Fast Fourier rasform algorithms: they allow accurate measuremets but the disadvatage is the large amout of computatio ivolved [] I this paper a digital techique of measurig the phase agle of the harmoics as well their magitudes is preseted he method is firstly based o Walsh spectrum measuremet with successive digital computatio to provide the Fourier spectrum of the periodic voltage i magitude ad phase [] his measuremet method is fast ad accurate ad does t require expesive istrumetatio; moreover it is particularly well-suited to power-frequecy waveform measuremets [3]-[6] he Walsh method, i commo with Fourier method, requires that the periodic voltage v(t be frequecy limited, which implies that its Walsh spectrum is made up of a ifiite series of cal ad sal terms However, if the highest harmoic order of v(t is H, the measuremet of oly S cal compoets ad S sal compoets (with S H permits to obtai exactly the Fourier spectrum of v(t except for the roudoff ad trucatio errors i digital processig he extractio of the cosie ad sie coefficiets of v(t from the trucated measure of cal ad sal coefficiets is achieved by way of a Walsh to Fourier matrix coversio process; a subsequet matrix multiplicatio provides for compesatio for the Walsh spectral trucatio he compesatio matrix takes the form of a diagoal matrix with fixed elemets; each coverted Fourier spectral compoet requires to be multiplied by a correctio factor which is fixed for a give harmoic he Walsh method is attractive for use at power system frequecy, for the the sychroizatio of the Walsh waves with v(t is ot difficult It is evidet that Walsh spectra are superior to the Fourier spectra, i that the multiplicatio with sie ad cosie fuctios, respectively, is obviated by simple reversal of sigs (Walsh waves assume values of plus ad mius oe Moreover both odd ad eve harmoics ca be measured PRINCIPLE OF OPERION he Fourier series of a periodic waveform v(t is writte [ a cos t + b si t] v(t a + ω ω ( a b v(t v(t cos si ω t dt ω t ad π/ω is the period of the fudametal frequecy at 5 Hz Walsh fuctios, which are show i Fig up to sal 8 (t, ca be used as a alterative orthogoal family for the series represetatio of a periodic waveform accordig to dt ( (3
[ cal (t + sal (t] v(t + (4 v(t cal (t dt (5 v(t sal (t dt (6 v(t wal (t dt (7 S represet the cal spectrum ad sal spectrum of v(t, respectively he elemet of the Walsh to Fourier coversio matrices F a ad F b, are the Fourier coefficiets of the Walsh fuctios cal m (t ad sal m (t, respectively hus S (9 wal(t sal(t cal(t sal(t a, m cal m (t cos ω t dt ( - cal(t - cal4(t - cal6(t - - sal3(t - sal5(t - sal7(t - - cal3(t - cal5(t - cal7(t - - sal4(t - sal6(t - sal8(t - b, m sal m (t si ω t dt ( is the idex of row (correspodig to the harmoic order ad m is the idex of colum Moreover the elemets of the coversio matrix F a assume the same absolute value of the coversio matrix F b, but their sigs may differ he coversio matrices are essetially semiifiite; whe applied i (8 however they are trucated to SxS elemet square matrices lso a additioal matrix multiplicatio is required which compesated for the Walsh spectral trucatio K a a* a K b b* b ( Fig Walsh fuctios up to the eighth order he procedure for the Walsh series determiatio ad the correspodig Fourier derivatio is give below Cosider a periodic voltage v(t whose period is at the fudametal frequecy of 5 Hz Sigal v(t is sampled systematically with samplig frequecy f s he Walsh coefficiets ad are obtaied accordig to (5, (6, (7 o yield the Fourier spectrum of v(t, ad are first multiplied by Walsh-Fourier coversio matrices F a ad F b, respectively, to give approximatios a*, b* to the cosie ad sie coefficiets of v(t: a* F a b* F b (8 a ad b are the desired cosie ad sie coefficiets, respectively he simplest form of compesatio occurs whe S k -, ie, the highest harmoic order i v(t does't exceed the third, or the seveth, or the fifteeth, he compesatio takes the form of a diagoal matrix, with fixed elemets π / si( π / k + k + is the harmoic order 3 COMPUER SIMULION RESULS (3 he measuremet algorithm has bee verified by computer simulatio to ivestigate the validity of this
techique he program geerates a iput voltage that is sampled with samplig frequecy f s khz; its waveform is show i Fig 73 44 55 8 73 44 57 5 65 75 73 5 685 9 379 57 5 53 84 38 94 8 ge lta pu tvo I 6 4 - -4 3 546 6 343 546 5 6 38 578 3 65 44398 958 94 6 38 a * 4 863 585 997-6 4 6 8 4 6 8 time (s Fig Waveform of the iput voltage Let the highest harmoic i the iput voltage be the seveth (ie, k3 ad that the Walsh trucated spectrum, accordig to (4, (5, (6, (7 results 73 44 55 8 73 44 57 5 65 75 73 5 685 9 379 57 5 53 84 38 94-5wal (t+3546cal (t 6343cal (t+546cal 3 (t- 56cal 4 (t-38cal 5 (t-578cal 6 (t+365cal 7 (t +9593sal (t+964sal (t-463sal 3 (t+3959sal 4 (t -77sal 5 (t-5sal 6 (t-548sal 7 (t I applyig (8 the Fourier spectrum is obtaied i terms of Walsh spectrum 9593 964 463 3959 77 5 548 6735 6639 59965 49986 43 394 577 b * Next we apply the compesatio matrices (:
a 3 53 4 337 394 66 9638 v (t 8 6 4 - -4 4 4938 958 94 638 4863 585 997 4 3 96436 3 44 787 56969 947 958-6 4 6 8 4 6 8 time (s Fig 3 Waveform of v(t he results obtaied of magitude ad phase agle of the harmoics are illustrated i able I able I - Magitude ad phase agle of the harmoics b 3 53 4 337 394 66 9638 Harmoic order Magitude -5 5 9984-53 3 53-47 4-963 5 84 79 6 5 96 7 35-7 Phase agle (rad 6 735 6 639 59965 49986 43 394 577 7 83 7 55 674 6668 5699 4996 735 Fig 4 shows the istataeous error of v(t with respect to the iput voltage s we ca see, the istataeous error is withi 5 V hus 5 v(t 5 + 43 cos ωt 96436 cos ωt 344 cos 3ωt 787 cos 4ωt + 56969 cos 5ωt + 947 cos 6ωt 958 cos 7ωt + 783 si ωt + 755 si ωt + 674 si 3ωt + 6668 si 4ωt + 5699 si 5ωt + 4996 si 6ωt + 735 si 7ωt E ror 5-5 - -5 - -5 4 6 8 4 6 8 time (s Fig 4 Istataeous error of v(t with respect to the iput voltage Fig 3 shows the waveform of v(t
Results obtaied with the proposed method are the compared with those obtaied via the fast Fourier trasform (FF algorithm Fig 5 shows the frequecy cotet of the iput voltage y si t e d l pe ctra s w e r P o 6 5 4 3 Frequecy cotet of the iput voltage 5 5 5 3 35 4 Frequecy (Hz Fig 5 Frequecy cotet of the iput voltage Ispectio of Fig 5 cofirms that the proposed method based o Walsh spectrum gives good results compared with those obtaied via the FF algorithm 4 CONCLUSION his paper has described a digital techique of measurig the magitude ad phase agle of the harmoics of a periodic voltage v(t i power systems he method is based o Walsh spectrum measuremet followed by digital computatio for obtaiig the Fourier spectrum his measuremet method is fast ad accurate ad does t require expesive istrumetatio; moreover it is particularly well-suited to power-frequecy waveform measuremets, for the the sychroizatio of the Walsh waves with the iput voltage is ot difficult It is evidet that Walsh spectra are superior to the Fourier spectra, i that the multiplicatio with sie ad cosie fuctios, respectively, is obviated by simple reversal of sigs (Walsh waves assume values ± Moreover both odd ad eve harmoics ca be measured REFERENCES [] Hadbook of Measuremet Sciece Ed by P H Sydeham, New York, Joh Wiley, 98 [] R, Kitai: O-lie measuremet of power system harmoic magitude ad phase agle, IEEE ras Itrum Meas, Vol IM-7,, pp 79-8, 978 [3] DJ, Kish, GH, Heydt: itroductio to power systems aalysis usig Walsh fuctios, Proc North merica Power Symposium IEEE Comp Soc Press, Los lamitos, C, US, pp 48-9, 993 [4] J, Su, H, Grotstolle: Walsh fuctios method for the cotrol of active power filters, Proc Symposium Power Electroics Circuits Hog Kog Polytech, Hog Kog, pp -5, 994 [5] FHJ, ltuve, VI, Diaz, ME, Vazquez: Fourier ad Walsh digital filterig algorithms for distace protectio, Proc IEEE Power Idustry Computer pplicatio Coferece, New York, NY, US, pp 43-8, 995 [6] DV, Coury, HGF, rito: Digital filters applied to computer relayig (of power lies, Proc IEEE Iteratioal Coferece o Power System echology POWERCOM 98, New York, NY, US, pp 6-66, 998