FREQUECY DETECTIO METHOD BAED O RECURIE ALGORITHM Katsuyasu akano*, Yutaka Ota*, Hioyuki Ukai*, Koichi akamua*, and Hidki Fujita** *Dpt. of ystms Managmnt and Engining, agoya Institut of Tchnology, Gokiso-cho, howa-ku, agoya, 466-8555, JAPA Phon & Fax: +81-5-734-548 E-mail: katsu@fanklin.lcom.nitch.ac.jp **Elctic Pow Rsach & Dvlopmnt Cnt Chubu Elctic Pow Co.,Inc 0-1, Kitaskiyama, Ohdaka-cho, Midoi-ku, agoya, 459-85, JAPA Abstact :W hav dvlopd th highspd and high pcision PMU(Phaso Masumnt Unit by using DP(Digital ignal Pocsso and th GP(Global Positioning ystm. Th dvlopd PMU nabls to calculat th voltag and cunt phasos and th oth stat vaiabls in lctic pow systms fom th masud instantanous voltag and cunt in al-tim. In this pap, ou pupos is to popos a nw dtction mthod of fquncy in th pow systms basd on o analysis of th cusiv algoithm usd in th PMU. This mthod nabl s to dtct fquncy with fin accuat and highspd. Kywods : fquncy dtction, cusiv, phaso masumnt unit, o analysis 1 ITRODUCTIO Fquncy is on of th most impotant stat vaiabls in pow systms. It is typically masud by dtcting th zo-cossing piod of th voltag wav. Howv, sinc pow lctonics tchnology is incasingly applid in th pow systms, it is quid to dtct mo pcisly and fast in th psnc of hamonics and nois. On th oth hand, applications of digital signal pocssing tchnology to th pow systms hav civd a gat dal of attntion in cnt yas. Th hav bn many digital algoithm applid to calculat fquncy and phaso[1][][3][4][5]. In thos, i n paticula, th PMU is widly applid in od to dtct both.m.s. valu and phas angl of voltag/cunt at multi points in th pow systms. Th calculation algoithm of th voltag/cunt p haso in th PMU is basd on th cusiv (Disct Foui Tansfom. Howv, in th cas whn fquncy of pow systms is fluctuating and diffnt fom th fundamntal fquncy of algoithm, th calculatd.m.s. valu and phas angl hav th dtction o du to th dviation of fquncy[6][7]. Th pupos of this pap is to popos a nw dtction mthod of th fquncy and th phaso in th pow systms by analyzing th o containd in th voltag phaso du to th dviation of fquncy. Fistly, assumd th dviation of fquncy, th o quation of voltag phaso on th basis of th cusiv algoithm is divd. inc both th phas and th magnitud componnts of this quation contain oscillatoy o, th unning avag filt is applid in od to mov th o. Thn th dviation of fquncy is cusivly dducd fom th filtd voltag phaso. condly, sinc th voltag phaso, which th filt is applid to, has th off-st o causd by dviation of fquncy and th phas dlay du to th filt, this off-st o and phas dlay is compnsatd by applying th dtctd dviation of fquncy. On th oth hand, it is mad cla that th o quation which dos not contain oscillatoy o is dducd by using th positiv-phas-squnc componnt tansfomd by th mthod of symmtical coodinats in th cas of th phas balancd voltag. Finally, th numical simulation sults a shown to confim th ffctivnss of th poposd mthod compad with th zo-coss dtction mthod[8]. In paticula, th poposd mthod has obustnss in th cas whn th voltag wav contains hamonics.
FREQUECY DETECTIO METHOD BAED O RECURIE ALGORITHM.1 Rcusiv Algoithm oltag phaso is dtctd by th fundamntal fquncy componnt of (Disct Foui Tansfom algoithm fo th squntial data of voltag instantanous valu v as follows: j jπ v f0kt k+ k 1 (1 wh, j;unit of complx numb, f 0 ;fundamntal fquncy of, T ;sampling piod, (1/( f 0 Ts;th numb of sampling p on cycl of fundamntal fquncy. In Eq. (1 th sin componnt is takn as th fnc, bcaus th phas angl fom th zo-coss point of th sin wav should b dtctd. By intoducing otational coodinat systm, jπf 0 T ( th following cusiv lation is divd. j + 1 + (v + 1 v + 1 j πf0( + 1T (3 Compad with th usual algoithm as Eq. (1, th calculation numb of cusiv as Eq. (3 is indpndnt fom th sampling numb. In fact, th calculation numb of al pat and imaginay pat of th latt is only six fo any, whas th fom is 4+1. This implis that th cusiv is bnficial fo calculation tim.. Eo Analysis of Rcusiv Instantanous valu of singl phas voltag v is xpssd as follows: v sin(πft + φ ( j jφ jπft jφ jπft (4 wh, ; amplitud of th voltag i; initial phas angl of th voltag f; fquncy of pow souc. ow consid that th fquncy of pow souc fluctuats and is diffnt fom th fundamntal fquncy of algoithm f 0, that is f (f-f 0. By substituting Eq. (4 into Eq. (1 and (, th ptubd voltag phaso is obtaind, which consists of two vcto componnts. j vk k 1 j( { ( A φ 0 + π f ( T j π fkt k 1 jα jπ f0 kt + + Β wh, sin( π ft A sin( π ft jβ j πf0t j( φ 0 π ( f 0 + f ( T jπ ( f f 0 + kt k 1 sin( π ( f0 + f T B sin( π ( f + f T α φ π f( 1T + π ft 0 0 β π φ + π(f + f ( 1T π(f + f T 0 (5 (6 (7 f (f-f 0 ;dviation of fquncy of pow souc. Fig.1 shows th lation btwn A, B and fquncy f (f 0 + f by using Eq. (6. In th Fig. 1, if th dviation of fquncy is lss than.0 Hz, which is th ang of fquncy gnally considd in th pow systms, it is stimatd that A, B. Moov th phaso hav high quality fo signal-nois-atio in high fquncy band, in paticula fo hamonics. On th oth hand, both g and g in th Eq. (7 a lina function of f. Th phaso lationship of Eq. (5 is shown in Fig.. It is found that th scond tm of Eq. (5 bings about th oscillatoy o with th fquncy of about two tims of th fquncy of pow souc. Ths o a du to th fquncy dviation f. }
Magnitud 1. 0.9 0.6 0.3-0.3 0 60 10 180 40 300 360 Fig.1 Magnituds of vcto A and B Fig. Phaso lationship.3 Fquncy Dtction inc th phaso vcto is oscillating du to dviation of fquncy, th unning avag filt is applid in od to mov th ffct of th scond vcto in Eq. (5. Fig. 3 shows th fquncy chaactistic of th applid filt. Th filt is dsignd in od to mov th band aound 60 k Hz, k1,,3,, which includ th oscillatoy fquncy of th o vcto. Th filtd phaso is xpssd as follows: * A j( α + K f wh, K f ;phas dlay du to th filt. (8 Th dviation of fquncy is dducd fom th diffnc of th squntial phas angl of Eq. (8 as follows; ( α + K f ( α 1 + K f f f πt (9.4 High Pcision Phaso Dtction Fom Eq. (8, th.m.s. valu and th phas angl is computd as follows: * A ag( * α π f + K f φ π f ( Fquncy [Hz] A A 1 T + π B ft + K f B π ( f0 + f (10 (11 Gain 1. 0.8 0.6 0.4 0. Fig. 3 Fquncy spons of unning avag filt In th Eq. (10 and Eq. (11, both th.m.s. valu and th phas angl includ off-st o causd by th dviation of fquncy. By substituting th dviation of fquncy dtctd as pvious sction into A in Eq. (10, th.m.s. valu is compnsatd. On th oth hand, sinc th xact valu of th phas angl consists of th fist and th thid tm of Eq. (11, th phas angl is compnsatd by subtacting th dtctd dviation of fquncy and K f fom thm. As th sult, th following pcis phaso angl and.m.s. valu a obtaind. 0 60 10 180 40 300 360 * * * A (1 ag( ** ag( * + π f ( 1 T φ + π ft Fquncy [Hz] K f (13.5 Balancd Th-Phas oltag Cas Fo th cas of th-phas voltag, th phaso can b dtctd by th positiv-squnc componnt voltag. In th cas of balancd th-phas voltag, th phaso quation is obtaind as follows: 1 j π j π p 3 3 ( a + b + c 3 jα jβ ( A Β + 1 j π j π j π 3 jα 3 jβ 3 ( A Β 3 + + j π j π j π 3 jα 3 jβ + ( A + Β 3 jα A (14 It is notd that th phaso dos not contain th oscillatoy o. Thfo, th dviation of f-
quncy is dtctd without applying th unning avag filt. In this cas, th off-st o of th.m.s. valu and th phas angl is also compnsatd by Eq. (1 and Eq. (13 in th simila mann as th singl phas cas. Th flow chat of th poposd fquncy dtction mthod and compnsation mthod of th.m.s. valu and th phas angl is shown in Fig. 4(a fo singl-phas cas, and in Fig. 4(b fo th-phas cas, spctivly. 3 IMULATIO REULT In this sction, th ffctivnss of th poposd fquncy and phaso dtction mthod is confimd with numical simulations. Th poposd mthod is compad with th zo-coss dtction mthod. Th zo-coss mthod is th fquncy dtction mthod by masuing th tim intval btwn zo cossing points of th instantanous voltag. Th fquncy is dtctd vy 180 dg by using singl-phas voltag, and vy 60 dg by using th-phas voltag, spctivly. Howv, in th cas whn voltag wav contains hamonics, th dtctd fquncy contains th significant o. Thfo, in gnal, th low-pass filt is applid in od to mov th hamonics[8]. In this pap, two cass of simulations a caid out in od to compa fou kinds of fquncy dtction mthod as follows: imulation cas 1. Pow souc without hamonics. Pow souc with 5% fifth od hamonics Fquncy dtction mthod 1. ingl-phas zo-coss mthod. Th-phas zo-coss mthod 3. Poposd singl-phas mthod 4. Poposd th-phas mthod At any cass, th balancd th-phas voltag is usd, and th fquncy of voltag has stp chang fom 6 Hz to 60.10 Hz. Th numb of sampling is 4, in oth wods, sampling fquncy is 1440 Hz. v Rcusiv-.m.s. valu & phas angl ag( Running avag filt * ag( * Compnsation by Eq. (1 & (13 ** ag( ** Diffntiating By Eq. (9 f f (a ingl-phas cas v a, v b, v c, Rcusiv- Rcusiv- Rcusiv- a, b, c, Positivphassqunc componnt By Eq. (14 p,.m.s. valu & phas angl p, ag( p, Running avag filt p, * ag( Diffntiating By Eq. (9 p, Compnsation by Eq. (1 & (13 * f ag( p, * * p, f * * (b Th-phas cas Fig. 4 Flow diagam of fquncy and phaso dtction
3.1 Cas-1 Fig. 5 shows th simulation sults of th cas-1. Fig. 5(a shows th instantanous voltag wav. Fig. 5(b shows th fquncy dtctd by zocoss mthods. In Fig. 5(b, both singl- and thphas zo-coss mthods hav dad tim to dtct fquncy in stp wis. Howv, th poposd mthods in Fig.5(c and (d dtct in lump wis, and th changs of fquncy can b dtctd instantanously. In paticula, th poposd th-phas mthod shows good tansint pfomanc without unning avag filt. If th fluctuation of fquncy is within.0 Hz, th poposd mthods nabl to dtct fquncy in th sam way. Fig. 6 shows th phas angl compnsatd by d - tctd fquncy. In Fig. 6(a, it is found that whn fquncy of pow souc is diffnt fom th fundamntal fquncy, th phas angl dtctd fom singl-phas componnt contains oscillatoy os. Th off-st and oscillatoy o a compnsatd by poposd mthod within 1.5 cycls. On th oth hand, in Fig. 6(b, it is found that phas angl d tctd fom positiv-squnc componnt contains only off-st os. Th o is also compnsatd by poposd mthod within 1.5 cycls. Thfo, th pcis dtction of phas angl is alizd by th poposd compnsation mthod. Th.m.s. valu is compnsatd in th simila way. oltag [v] 1.5 0.5-0.5 - -1.5 6Hz 60.10Hz va vb vc (a Instantanous voltag Fquncy Dviation [Hz] 0.1 input zo1 zo3 (b Zo-coss mthod Fquncy Dviation [Hz] 0.4 0.16 0.1 (c Poposd singl-phas mthod (d Poposd th-phas mthod Fig. 5 tp spons of cas-1 Fquncy Dviation [Hz] 0.1 Eo of Phas Angl [dg] 0.10-0.10 - -0.30-0.40 bfo aft (a ingl-phas mthod (b Th-phas mthod Fig. 6 Compnsation sults of phas angl Eo of Phas Angl [dg] 0.10-0.10 - -0.30-0.40 bfo aft
3. Cas- Fig. 7 shows th simulation sults in th cas whn th voltag wav contains hamonics. Fig. 7(a shows th instantanous voltag wav. Fig. 7(b and (c shows fquncy dtctd by zocoss mthods. It should b notd that th both singl and th phas zo-coss mthods a vy snsitiv to hamonics. This is du to that th zocoss point of voltag wav is shiftd fom th zocoss point of th fundamntal fquncy wav by th hamonics. Thfo, in th cas whn th zo-coss mthods a usd, th low-pass filt is applid to mov hamonics. Th fquncy spons of th low-pass filt is shown in Fig. 8. Th phas dlay of this filt is about cycl. Fig. 9 shows th simulation sults by using zo-coss mthods with low-pass filt. Fig. 9(a shows th instantanous voltag wav aft filting. In Fig.9 (b and (c, th zocoss mthods nabl to dtct fquncy pcisly by applying th low-pass filt to th voltag. Howv, in this way, th dlay tim to dtct fquncy bcoms long than th cas without low-pass filt. Finally, dad tim bcom to b about.0 cycls. On th oth hand, in Fig. 7(d and (, th poposd mthods a lss affctd by hamonics and th dlay tim is within 1.5 cycls. Th a two asons why th poposd mthods hav ssntially obustnss against hamonics. Fistly, poposd mthods pick out fundamntal fquncy as Fig. 1. condly, th unning avag filt applid to - mov th oscillatoy o is also usful to mov th affct of hamonics. Compaing Fig. 9(b(c and Fig. 7(d(, it is found that th poposd mthods nabl to dtct fquncy mo accuatly and mo instantanously than zo-coss mthods whn th fluctuation of fquncy occus. oltag [v] 1.5 0.5-0.5 - -1.5 6Hz 60.10Hz va vb vc (a Instantanous voltag Fquncy Dviation [Hz] 0.1 (b ingl-phas zo-coss mthod Fquncy Dviation [Hz] 3.00.00 0-0 -.00-3.00-4.00 (cth-phas zo-coss mthod Fquncy Dviation [Hz] 0.4 0.16 0.1 (d ingl-phas poposd mthod ( Th-phas poposd mthod Fig. 7 tp spons of cas- Fquncy Dviation [Hz] 0.16 0.1
Gain 1. 0.8 0.6 0.4 0. 0 60 10 180 40 300 360 Fquncy Dviation [Hz] Fquncy Dviation [Hz] oltag [v] 0.1 0.1 Fquncy [Hz] Fig. 8 Fquncy spons of low-pass filt 6Hz 60.10Hz 1.5 0.5-0.5 - -1.5 va vb vc (a Instantanous voltag aft filting (b ingl-phas zo-coss mthod (c Th-phas zo-coss mthod Fig. 9 tp spons of cas- with LPF 4 COCLUIO This pap poposd a nw dtction mthod of fquncy in lctic pow systms. Th poposd mthod basd on cusiv algoithm alizs th accuat dtction of fquncy, in paticula, in th cas whn th voltag wav contains hamonics. Moov, this mthod nabls to dtct fquncy instantanously whn th fluctuation of fquncy occus. In addition, th.m.s. valu and th phas angl is compnsatd by using th dtctd fquncy. REFERECE [1] A.G.Phadk, J..Thop, Impovd Contol and Potction of Pow ystms Though ynchonizd Phaso Masumnts, Contol and Dynamic ystms, Advancs in Thoy and Applications, ol.43, Acadmic Pss, 1991, pp.335-376. [] P.J.Moo, R.D.Caanza, and A.T.Johns, Modl ystm Tsts on a w wmic Mthod of Pow ystm Fquncy Masumnt, IEEE Tans. on Pow Dlivy, ol.11, o., Apil.1996, pp.696-701. [3] J.Z.Yang and C.W.Liu, A Pcis Calculation of Pow ystm Fquncy and Phaso, IEEE Tans. on Pow Dlivy, ol.15, o., Apil.000, pp.494-499. [4] Yutaka Ota, Masatak Kawada, Hioyuki Ukai, Koich akamua, obuyuki Matsui, Mototaka on,.c.vma, Hidki Fujita, ynchonizd Phaso Masumnt of tat Paamts at Multi-points in Elctic Pow ystm by using DP and GP, Pocdings of Intnational Confnc on ignal Pocssing Applications and Tcnology(ICPAT, ovmb 1999. [5] Yutaka Ota, Hioyuki Ukai, Koich akamua, Hidki Fujita, Evaluation of tability and Elctic Pow Quality in Pow ystm by using Phaso Masumnts, in Pocdings 000 Intnational Confnc Pow ystm Tcnology(PowCon000, vol.3, Dcmb 000, pp.1335-1340. [6] Tsuyoshi Funaki, Knji Matsuua, hunsuk Tanaka, Eo Estimation and Coction of Dtctid Phas by Ral Tim, Tans. IEE of Japan, ol.10-b, o.1, Dcmb 000, pp.168-1690. (in Japans [7] Katsuyasu akano, Yutaka Ota, Hioyuki Ukai, Koich akamua, Hidki Fujita, Application of Phaso Masumnt Mthod to Fquncy Dtction of Pow ystm, Tans. IEE of Japan, ol.11-b, o.1, Dcmb 001, pp.1856-1857. (in Japans [8] Tomio Chiba, Mitsuyasu Kido, Junzou Kawakami, Katsuhiko Yonda, Tadao Kawai, Kotao Hiasawa, High-Pcision oltag Masuing Mthod using Fast ampld Data fo Elctic Pow ystms, Tans. IEE of Japan, ol.113-b, o.5, May 1993, pp.553-561. (in Japans