Analysis of ECT Synchronization Performance Based on Different Interpolation Methods

Transcription

1 Sesors & Trasducers, Vol. 62, Issue, Jauary 24, pp Sesors & Trasducers 24 by IFSA Publsg, S. L. ttp:// Aalyss of ECT Sycrozato Performace Based o Dfferet Iterpolato Metods Yag Zx, J Jafe, Yua Yubo, Bu Qagseg Afflato Jagsu Electrc Compay Researc Isttute, 23, Ca E-mal: jjafesgcc@63.com Receved: 22 October 23 /Accepted: 9 Jauary 24 /Publsed: 3 Jauary 24 Abstract: Tere are two sycrozato metods of electroc trasformer IEC644-8 stadard: mpulsve sycrozato ad terpolato. We te mpulsve sycrozato metod s applcablty, te data sycrozato of electroc trasformer ca be realzed by usg te terpolato metod. Te typcal terpolato metods are pecewse lear terpolato, quadratc terpolato, cubc sple terpolato ad so o. I ts paper, te flueces of pecewse lear terpolato, quadratc terpolato ad cubc sple terpolato for te data sycrozato of electroc trasformer are computed, te te computatoal complexty, te sycrozato precso, te relablty, te applcato rage of dfferet terpolato metods are aalyzed ad compared, wc ca serve as gude studes for practcal applcatos. Copyrgt 24 IFSA Publsg, S. L. Keywords: Sycrozato, Iterpolato, Electrcal curret trasformer (ECT).. Itroducto Sce te sycrozato problem of electroc trasformer volves mult-levels: te sycrozato of every voltage ad curret te same terval, te sycrozato amog correlato tervals, te sycrozato amog correlato substatos, ad te wde-area sycrozed ad so o. IEC644-8 stadard specfes two metods of data sycrozato of te electroc trasformer: mpulsve sycrozato ad terpolato [9, ]. Te mpulsve sycrozato metod specfes tat every mergg ut must possess oe pps clock terface to receve te samplg sycrozato sgal of total substato ufed. Te data sycrozato of substato ca be realzed by usg t. Te data sycrozato amog eac measuremet te same terval, eac correlatve mergg uts, eac correlatve substatos, ad eve wde area sycrozato ca be solved by te mpulsve sycrozato. We te mpulsve sycrozato metod s used te mergg ut, te samplg sycroous start sgal wll be trasmtted to g potetal sde, wc creases te dffcultes for realzg te practcal egeerg. Besdes, for te two eds of trasmsso le usg te fber dfferetal protecto, te tradtoal metod by adjustg samplg tme to keep data sycrozato s ot sutable (we te substato as bee sycrozed but te tme amog substatos as ot bee sycrozed) or ecessary (we te tme amog substatos as bee sycrozed). We usg te terpolato metod, te mergg ut oly receves samplg values sgle drecto, ad te measurg values at te same tme ca be calculated by umercal terpolato metod. Te AD coverter of eac measuremet just carres o asycroous samplg, te samplg values of eac cael s voltage ad curret at te same tme mergg ut ca be calculated by Artcle umber P_779 25

2 Sesors & Trasducers, Vol. 62, Issue, Jauary 24, pp terpolato metod, ad te data ca also be sycrozed by terpolato metod f te samplg data amog substatos s ot sycroous []. After terpolato, te pase of samplg sequece ad orgal sequece ca be cosdered absolutely sycrozed, but tere are stll errors betwee te ampltude of lear terpolato pot ad real value. Commo terpolato metods are pecewse lear terpolato, quadratc terpolato, cubc sple terpolato, etc. dfferet terpolato metods ave dfferet precsos, computatoal complextes, relabltes, applcato rages. It s ecessary to aalyze varous terpolato metods quattatvely ad matematcally. 2. Prcple 2.. Lear Iterpolato Te matematcal model of lear terpolato ca be expressed as [2]: tt tt Lt t t t t t t I equato (),, values of curret t at t, () t t are te samplg t, respectvely. Te maxmum error of lear terpolato ca be expressed as [3]: R max N, (2) 2 I were s te t armoc, I s te ampltude of t armoc, N s te samplg pots of every perod Quadratc Iterpolato Te quadratc terpolato ca be descrbed as: tree dscrete pots of curret t equal tervals of tme t,( t ), t,( t ), t,( t ) 2 2 are kow, te quadratc terpolato polyomals ca be deduced by terpolato basc fuctos [4, 5]: ( tt )( tt ) ( tt )( tt ) L t t t ( t t )( t t ) ( t t )( t t ) ( tt )( tt ) t2 ( t t )( t t ) t t T t 2t t t tt ( )( ) 2 tt tt 2 2T, (3) were T.2 / N s te samplg terval, N s te samplg pots of every perod. Te maxmum error of quadratc terpolato ca be descrbed as: R max (4) N 3 [ I ] I equato (4), s te t armoc, I s te ampltude of t armoc Cubc Sple Iterpolato Te pecewse lear terpolato ad pecewse quadratc terpolato above ca oly make te coecto pots of every pecewse cotuous, but caot make te wole curve smoot. Te cubc sple terpolato ca preserve te advatages of pecewse low-order terpolatos, ad mprove te smootess of terpolatos. Te matematcal model of cubc sple terpolato ca be descrbed as: te dscrete samplg pots t, t, t2,, t of t terval a, b are kow, wt correspodg samplg tme a t t t t b ad samplg 2 tme terval,, 2,,, respectvely. Te curret (t) te t terval ca be approxmated as cubc sple terpolato polyomals s t. s t ca be defed as [6, 7]: t t tt s t M M M 2 t t t 6 M 2 t t t 6 t t, t ;,2,, (5) Takg all tese factors, oce + values M, M, M,, M are determed, te cubc sple 2 terpolato s() t ca be fxed. M, M, M,, M 2 ca be solved troug equatos below. 2 M g 2 M g 2 M g 2 M g g 6 t t ' ( t ) (6) (7) 252

3 Sesors & Trasducers, Vol. 62, Issue, Jauary 24, pp g t t t 6 ' 6 g f t, t f[ t, t ], 2,, (8) (9) M (,,2,, ) ca te be solved by solvg equatos (6). By substtutg M (,,2,, ) to equatos (5), te expressos s() t eac subterval ca be determed. Te maxmum error of cubc sple terpolato s [8]: R max () N 4 [ I ] 3. Smulatos 3.. Steady State Curret Te form of steady state curret s t I s t. I s te ampltude of steady state curret, 2 f s te agular frequecy, s tal pase, te umber of terpolato pots betwee two adjacet samplg pots s 4. Te samplg terpolato errors of lear terpolato, quadratc terpolato, cubc sple terpolato for steady state curret at samplg frequecy N 24, N 48, N 8 are as follows. We te samplg frequecy s N=24 (f s =.2 khz), N=48 (f s =2.4 khz), N=8 (f s =4 khz), te terpolato errors of lear terpolato, quadratc terpolato ad cubc sple terpolato are sow Fg., Fg. 2, Fg. 3, respectvely. Fg.. We frequecy s N=24, te terpolato error of lear terpolato, quadratc terpolato, cubc sple terpolato for steady state curret. Fg. 2. We frequecy s N=48, te terpolato error of lear terpolato, quadratc terpolato, cubc sple terpolato for steady state curret. 253

4 Sesors & Trasducers, Vol. 62, Issue, Jauary 24, pp Fg. 3. We frequecy s N=8, te terpolato error of lear terpolato, quadratc terpolato, cubc sple terpolato for steady state curret. As sow Fg., Fg. 2, Fg. 3, we te samplg frequecy s N=24 (f s =2 Hz), N=48 (f s =24 Hz), N=8 (f s =4 Hz), te maxmum errors of lear terpolato for steady state curret are less ta.86 %,.2 %,.8 %, te maxmum of quadratc terpolato for steady state curret are less ta.5 %,.4 %,.3 %, ad te maxmum of cubc sple terpolato for steady state curret are less ta.6 %,.4 %,.4 %, respectvely. All tese results before are agreed wt calculato results of equato () Traset Curret Te traset curret ca be expressed as: s t I t I e tt /, () were I s te ampltude of curret, 2 f s te agular frequecy, s te tal pase, T 2ms s te tme costat, te umber of terpolato pots betwee two adjacet samplg pots s 4. Te samplg terpolato errors of lear terpolato, quadratc terpolato, cubc sple terpolato for traset curret at samplg frequecy N=24, N=48, N=8 are as follows. We te samplg frequecy s N=24, N=48, N=8, te terpolato errors of lear terpolato, quadratc terpolato ad cubc sple terpolato are sow Fg. 4, Fg. 5, Fg. 6, respectvely. As sow Fg. 4, Fg.5, Fg.6, we te samplg frequecy s N=24, N=48, N=8, te maxmum error of lear terpolato for steady state curret s less ta.86 %,.2 %,.8%, te maxmum error of quadratc terpolato for steady state curret s less ta.5 %,.4 %,.3 %, ad te maxmum error of cubc sple terpolato for steady state curret s less ta.6 %,.4 %,.4 %, respectvely. Because te supermposed traset tt / decayg wave I e cages slowly, te / terpolato error of traset decayg wave I e tt s muc less ta se wave I s t, so te terpolato error of traset decayg wave s agreemet wt te terpolato error of se wave. Fg. 4. We frequecy s N=24 (fs=2 Hz), te terpolato error of lear terpolato, quadratc terpolato, cubc sple terpolato for traset curret. 254

5 Sesors & Trasducers, Vol. 62, Issue, Jauary 24, pp Fg. 5. We frequecy s N=48 (fs=24 Hz), te terpolato error of lear terpolato, quadratc terpolato, cubc sple terpolato for traset curret. Fg. 6. We frequecy s N=8 (fs=4 Hz), te terpolato error of lear terpolato, quadratc terpolato, cubc sple terpolato for traset curret Hg-order Harmoc Te g-order armoc ca be descrbed as: (2) s t I t I equato (2), s te t armoc wave, I s te ampltude of t armoc wave, s te tal pase of t armoc wave, 2 f s agular frequecy. Accordg to equatos (2), (4), (), f te samplg frequecy s N=24, N=48, N=8, te samplg terpolato errors of lear terpolato, quadratc terpolato ad cubc sple terpolato from te fudametal wave to te tet armoc wave are calculated respectvely. Te maxmum lear terpolato error s sow Table ad Fg. 7. As sow Table ad Fg. 7, we te samplg frequecy s N=24, te maxmum teoretcal error of lear terpolato creases from.856 % to % wt te cremet of armoc umber; we te samplg frequecy s N=48, te maxmum teoretcal error of lear terpolato creases from.24 % to 2.42 % wt te cremet of armoc umber; we te samplg frequecy s N=8, te maxmum teoretcal error of lear terpolato creases from.77 % to 7.7 % wt te cremet of armoc umber. Table. Te maxmum teoretcal error of lear terpolato metod for varous armocs. Harmoc order Samplg frequecy N=24 N=48 N=8.845 %.24 %.77 % %.857 %.38 % %.928 %.694 % % %.234 % % %.928 % % 7.7 % % %.5 % % % 3.7 % % % 7.35 % % 7.8 % 2.42 % 7.7 % 255

6 Sesors & Trasducers, Vol. 62, Issue, Jauary 24, pp Table 3. Te maxmum teoretcal error of cubc sple terpolato for varous armocs. Fg. 7. We te samplg frequecy s N=24, N=48, N=8, te lear terpolato error of armocs from te fudametal wave to te tet armoc wave. Te maxmum quadratc terpolato error s sow Table 2 ad Fg. 8. As sow Table 2 ad Fg. 8, we te samplg frequecy s N=24, te maxmum teoretcal error of quadratc terpolato creases from. 5 % to 5. % wt te cremet of armoc umber; we te samplg frequecy s N=48, te maxmum teoretcal error of quadratc terpolato creases from.4 % to.92 % wt te cremet of armoc umber; we te samplg frequecy s N=8, te maxmum teoretcal error of quadratc terpolato creases from.3 % to 3.7 % wt te cremet of armoc umber. Harmoc order Samplg frequecy N=24 N=48 N=8. %.7 %.4 % 2.2 %.2 %.79 % 3.4 %.63 %.4 % %.22 %.268 % 5.63 %.526 %.396 % %.42 %.649 % %.237 %.893 % 8.27 %.4389 %.229 % %.745 %.325 % 4.63 %.629 % % Te maxmum cubc sple terpolato error s sow Table 3 ad Fg. 9. As sow Table 3 ad Fg. 9, we te samplg frequecy s N=24, te maxmum teoretcal error of cubc sple terpolato creases from.6 % to 6.5 % wt te cremet of armoc umber; we te samplg frequecy s N=48, te maxmum teoretcal error of cubc sple terpolato creases from.4 % to % wt te cremet of armoc umber; we te samplg frequecy s N=8, te maxmum teoretcal error of cubc sple terpolato creases from.4 % to % wt te cremet of armoc umber. Table 2. Te maxmum teoretcal error of quadratc terpolato metod for varous armocs. Harmoc order Samplg frequecy N=24 N=48 N=8.2 %.4 %.3 % 2.87 %.2 %.24 % %.363 %.83 % %.87 %.9 % %.75 %.363 % % %.659 % % %.45 % % 6.87 %.545 % % % 2.96 % 77.3 % 2.99 % % Fg. 8. We te samplg frequecy s N=24, N=48, N=8, te quadratc terpolato error of armocs from te fudametal wave to te tet armoc wave. Fg. 9. We te samplg frequecy s N=24, N=48, N=8, te cubc sple terpolato error of armocs from te fudametal wave to te tet armoc wave. 4. Cocluso Te sycrozato terpolatos for steady state curret, traset curret ad armoc wave curret are separately calculated by usg lear terpolato, quadratc terpolato ad cubc sple terpolato. Te error betwee teoretcal values ad calculated values are aalyzed, ad te coclusos are as follows:. Te terpolato algortm for DC compoet does t produce error. 2. Te samplg terpolato error s te lear combato of varous armoc waves. Te larger te umber of te armoc, te larger te error of samplg terpolato. 3. Te error of cubc sple terpolatos s less ta tat of quadratc terpolato, ad te 256

7 Sesors & Trasducers, Vol. 62, Issue, Jauary 24, pp error of quadratc terpolato s less ta tat of lear terpolato. 4. We terpolatg for g-order armoc waves, f te umber of armoc s less ta 7 ( 7), te terpolato error of cubc sple terpolato s less ta tat of quadratc terpolato, ad te terpolato error of quadratc terpolato s less ta tat of lear terpolato. But f te umber of armoc s larger ta 8 (>8), te waveform cages rapd, te terpolato error of lear terpolato s less ta tat of quadratc terpolato te cotrary. 5. Altoug te terpolato error of cubc sple terpolato s less ta tat of quadratc terpolato, ad te terpolato error of quadratc terpolato s less ta tat of lear terpolato, tere as large dfferece amog te computatoal complexty of cubc sple terpolato quadratc terpolato, lear terpolato. For N pots terpolato, te computatoal complexty of cubc sple terpolato s: 2*N addtos ad subtractos, 3*N multplcatos ad dvsos ad N N matrx calculatos; te computatoal complexty of quadratc terpolato s: 4*N addtos ad subtractos, 2*N multplcatos ad dvsos s; te computatoal complexty of lear terpolato s: 5*N addtos ad subtractos, 4*N multplcatos ad dvsos. Terefore te computatoal complexty of cubc sple terpolato s larger ta tat of quadratc ad lear terpolato. Besdes, te calculato results of cubc sple terpolato are closely related to te tal values. Ackowledgemets Ts project s supported by Natural Scece Foudato of Jagsu Provce (BK2399), Natoal Natural Scece Foudato of Ca (62392), Natural Scece Foudato of Jagsu Provce (BK22326), te Fudametal Researc Fuds for te Cetral Uverstes. Refereces []. Cao Tuaje, Y Xagge, Zag Ze, L We, Data sycrozato of electroc strumet trasformers, Proceedgs of te CSU-EPSA, Vol. 9, No. 2, 27, pp [2]. Qao Hogx, Huag Saofeg, Lu Yog, Dscusso o data sycrozato of electroc curret trasducer based o quadratc terpolato, Power System Protecto ad Cotrol, Vol. 37, No. 5, 29, pp [3]. Dog Yua, Su Togjg, Xu Bgy, Data sycrozato based o cubc sple terpolato for electroc strumet trasformers, Electrc Power Automato Equpmet, Vol. 32, No. 5, 22, pp [4]. Xu Guagu, L Youju, Wag Welog, Xog Muwe, Desg of a sycrozato ad terpolato algortm of sampled values for dgtal substato IED, Electrc Power Automato Equpmet, Vol. 33, No. 4, 29, pp [5]. Xag Mjag, Gao Houle, A Yagqu, Du Qag, Lu Ka, Su Jagtao, A adaptve terpolato algortm mprove data sycrozato precso, Automato of Electrc Power Systems, Vol. 36, No. 8, 22, pp [6]. Ru Bal, Realzato of sycroous samplg by meas of terpolato, Joural of Au Uversty of Tecology, Vol. 22, No. 4, 25, pp [7]. Zeg Feg, Sua Peja, Guo Jwe, Study o electroc curret trasformer of acevg sycrosm wt terpolato metod, Electrc Swtcgear, No. 5, 28, pp [8]. Guo Jwe, Lag Ku, Dog Lka, Study o electroc curret trasformer of acevg sycrosm wt terpolato metod, Scua Electrc Power Tecology, Vol. 3, No. 5, 28, pp [9]. IEC644-8 Istrumet Trasformers part 8: Electroc Curret Trasformers, Iteratoal Electroteccal Commsso, 22. []. IEC Commucato Networks ad Systems Substatos. Part 9-: Specfc Commucato Servce Mappg (SCSM)-Sampled Values Over Seral Udrectoal Multdrop Pot to Pot Lk, Iteratoal Electroteccal Commsso, Copyrgt, Iteratoal Frequecy Sesor Assocato (IFSA) Publsg, S. L. All rgts reserved. (ttp:// 257