NEW METHOD FOR DYNAMICAL CORRECTION OF FREQUENCY INSENSITIVE IMPEDANCE MEASURING ALGORITHMS

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1 NEW METHOD FOR DYNAMIAL ORRETION OF FREQUENY INENITIVE IMPEDANE MEAURING ALGORITHM Doiik Bak, Jausz zafra Waldear Rebizat Wroclaw Uiversity of Techology Wroclaw, Polad Abstract New ethod for dyaical correctio ad siultaeous frequecy adaptatio of easureets of protectio criterio values is preseted i the paper. Thaks to the ew algoriths worked out ad their special arrageets it is possible to get very fast ad frequecy isesitive estiators of ipedace ad its copoets. The siulatio results cofired theoretical cosideratios ad show advatages of the ew algoriths as well as their possible applicatio i fast digital protectio ad cotrol systes. Keywords: digital protectio protectio criterio values, dyaical correctio frequecy adaptatio INTRODUTION afe ad reliable operatio of power syste ad cotiuous delivery of electrical eergy require applicatio of adequate cotrol ad protectio systes. Especially iportat are protectios used to fast ad reliable eliiatio of faults occurrig i power syste. They protect power syste equipet fro daage or destructio ad prevet fro fault propagatio. Proper operatio of protectio schees depeds o their two ai fuctioal blocks: the easureet of criterio values ad decisio akig oes. Requireets cocerig easureets are ofte cotradictory for istace speed ad accuracy. This coflict is solved by usig dyaical correctio or, i geeral, corrective ad adaptive eas []. No doubt that ew protectio systes require very fast estiators of criterio values. It ca be realised with recogisig fault iceptio istat ad usig fast estiatio algoriths, aog the those applyig dyaical correctio. With such a approach flexible solutios ca be obtaied assurig both fast estiatio durig trasiets ad high accuracy i the steady state [,3,4]. Additioal proble appears whe the sigal paraeters chage that are usually assued to be costat for desiged estiators. A exaple could be sigal frequecy. Whe it chages certai easureet errors appear due to oorthogoality of sigal copoets (issued at the outputs of used fiite ipulse respose filters (FIR filters)) ad worse oise reductio. To avoid the proble it is ecessary to work out adaptive estiatio schees, where the estiator paraeters are atched to curretly easured frequecy variatio [5]. There are, however, soe cases whe it is ecessary to apply very fast estiators, e.g. whe sigal paraeters chage dyaically. I such situatios it is ecessary to develop special algoriths, which allow for siultaeous dyaical correctio ad frequecy adaptatio. The ethods ad algoriths of such type have bee worked out ad are preseted i the ext sectios of the paper. FUNDAMENTAL OF IMULTANEOU DYNAMIAL ORRETION AND FREQUENY ADAPTATION OF RITERION VALUE ETIMATOR. Kow ethods of dyaical correctio Methods for dyaical correctio etioed i itroductio are based o oe of the two fudaetal priciples: uique respose of reactive power estiator or cosiderig trasiet behaviour of applied digital filters [,3,4]. The first approach is siple ad attractive, however, it requires phase shift delay of π or equivalet uber of saples i order to realise differet tha reactive power estiators. The secod ethod is uerically coplex. I both cases it is ecessary to idetify the fault istat. Filterig ad easureet procedures are the started at this oet. It is iportat to recall here the priciples of the ore geeral ethod, sice its features will be applied later. Let saples of a sigal be give by the equatio: x cos( Ω + ϕ), () where: Ω πf / f, f frequecy of the sigal, f saplig frequecy. A FIR filter stiulated with such a iput sigal () produces the followig output sigal: y x( i) h ( i). () i where: h (i) si((i+,5) filter coefficiets. Oe ca otice that: x( i) cos[( i) Ω + ϕ] x cos( i + x si( i where: x cos( Ω + ϕ) x si( Ω + ϕ) ad the the output sigal of the filter ca be give i the for: y a( x + b( x, (3) 5th P, Liege, -6 August 5 essio 9, Paper 5, Page

2 where: a( b( i i h ( i)cos( i h ( i)si( i It is see that coefficiets a ad b deped o the tie istat ad frequecy Ω. Takig the other digital filter (usually orthogoal) with the ipulse respose h (i) oe ca get the output of this filter siilar to eq. (3) (say with coefficiets c, d). This allows calculatig orthogoal sigal copoets x, x, for each startig fro tie istat equal to oe. As a result oe ca estiate agitude ad phase of the sigal x (). Whe tie idex reaches value of filter widow legth N the usually oe of the coefficiets a, b i eq. (3) becoes costat ad the other is equal to zero; at the output of oe filter we get oe orthogoal copoet ad i the secod filter the other oe. It is the possible to calculate sigal agitude ad phase directly ad i quite siple way. It is see that the coefficiets a, b i eq. (3) are fuctios ot oly of the tie istat but also of the frequecy of the sigal. Havig two differet filters it is possible to estiate orthogoal sigal copoets for ay iput sigal frequecy at every tie istat. No doubt, however, that such calculatios would be uch ore coplex tha with use of stadard easurig procedures. That is why we are lookig for differet sipler ethods. oe of the are preseted i [3, 4].. Adaptatio of estiators to chages of sigal frequecy Lookig at eq. (3) we ca otice that eve at steady state, i.e. whe tie idex is equal to filter widow legth N, the coefficiets a, b deped o frequecy. The very fact of frequecy depedece ca also be see o frequecy respose of ay filter. It eas that whe the sigal frequecy differs fro oial value the outputs of a pair of orthogoal filters are ot orthogoal. It is the possible to calculate these orthogoal copoets fro liear cobiatio of the filter outputs accordig to eq. (3) ad siilar equatio for the other filter. With that i id we ca further estiate agitude ad phase of the sigal. However, the proble of adaptatio of agitude estiator to frequecy variatios has uch sipler solutio [3, 4, 6]. This result we obtai usig followig algorith of agitude easureet: y y ( k) y ( k) y, (4) F ( Ω ) F ( Ω ) si( kω ) where: y, y outputs of orthogoal filters i steady state, Ω πf / f, f frequecy of fudaetal copoet, F, F gais of the filters at frequecy f. However, whe the frequecy of the sigal chages, the the easured value is give by the equatio: F ( F ( si(, (5) F ( Ω ) F ( Ω ) si( Ω ) where: easured agitude, real, proper value of agitude, ad it is possible to calculate real value of sigal agitude (squared) as follows: where: W or F ( F F ( Ω ) F, (6) W ( or ( si( ( Ω ) si( ) Ω For large frequecy chages it is ecessary to use both adaptatio coefficiet W or ad adequate filter widow N (to get better oise filterig). The block schee of adaptive easureet is preseted i Fig.. v() x() Orthogoal Filters Frequecy easureet Algorith of agitude easureet Figure : chee of agitude easureet with adaptatio to frequecy variatio.3 Dyaical correctio of frequecy adaptive estiators Theoretical cosideratios ad siulatio results clearly show that algorith (4) is the best to realise both frequecy adaptatio ad siultaeous dyaical correctio. The siplest way to reach that is additio of dyaical correctio to frequecy adaptive schee described above see eq. (4). To get very siple for of fial result it is ecessary to solve the proble of adequate realisatio of delayed sigals of the filters: y ( k), y ( k). It ca be ade either at the iput or at the output of the filters as show i Fig.. x() x() z - h (i) z - h (i) h (i) y() y(-) y() y(-) Figure : Two possibilities of realisatio of delay of the filter output. a) b) 5th P, Liege, -6 August 5 essio 9, Paper 5, Page

3 Durig steady state it gives the sae result, however, durig trasiets we have followig equatios (describig delayed output sigal of FIR filter): for the schee fro Fig. a y( ) a(, x ( ) + b(, x ( ) (7a) for the schee fro Fig. b y( ) a( x ( ) + b( x ( ). (7b) olutio fro Fig. b is advatageous i such a eaig that coefficiets a, b are the sae for sigals at give istat ad for sigals delayed. It results i especially siple correctio factor. If such switch cotrol is used the switches are closed i tie equivalet to fixed delay value. It is couted fro fault istat. The followig dyaical correctio factor ca be foud: si ( ) ( ) Ω ( + ) for 9 N si ( ) Ω. (8) for > 9 Now the equatio of agitude of voltage or curret easureet with applicatio of dyaical correctio ad adaptatio to frequecy chages ca be give i the for (. (9) ( ) W ( v() x() Frequecy Measureet ( Filters ad switch cotrol outig or alculatio of W or ( Algorith of easureet alculatio of () Figure 3: chee of siultaeous frequecy adaptatio ad dyaic correctio for agitude easureet. Block schee of its realisatio is show i Fig. 3. Oe ca otice that dyaical correctio factor (8) is equal to oe i steady state. The sae is valid for the frequecy adaptatio factor at oial frequecy. Otherwise they are ot equal to uity ad either correctio or adaptatio is active (or both of the).4 Ipedace estiator If the sae algorith (4) is applied to easureet of voltage ad curret ad whe i additio the sae ethods for dyaical correctio ad frequecy adaptatio are used the very siple algorith of ipedace estiatio is obtaied: Z V I V I (. () ( It eas that i the case of ipedace easureet we ca obtai very fast frequecy isesitive algorith applyig either correctio or adaptatio (uder coditio that eq. (4) ad switch cotrol fro Fig. b are used). Of course, to suppress sigal oise adequately the filter widow legth should be cotrolled by atchig it to actual frequecy..5 Estiators of active power ad ipedace copoets It was proved that oe of kow algoriths of power ca be adapted ad siultaeously corrected i such a siply way as agitude. However, it was foud that siilar possibility exists whe we use a su of chose algoriths of active power. Let the equatios of active power calculatios be give i the for: u i ( ) u ( ) i P (, (a) F ( Ω ) F ( Ω ) si( Ω ) P u ( i ( ) u ( ) i. (b) F ( Ω ) F ( Ω ) si( Ω ) The the sae coefficiets of dyaical correctio ad frequecy adaptatio as i agitude easureet ay be used: P ( + P ( P. () ( ) W ( It allows calculatig resistace accordig to equatio: P P ( + P ( R. (3) I I ( It is see that coefficiets () ad W or (Ω ) were cacelled. The i this case we aage to obtai very fast frequecy isesitive resistace estiator siilarly to ipedace easureet. Havig such estiators of ipedace ad resistace it is possible to get reactace estiator: or Z R. (4) Ad fially we ca say that certai special faily of criterio value estiators was worked out. Estiators of ipedace ad its copoets have distictive features. They are very fast ad frequecy isesitive without applicatio of dyaical correctio ad frequecy adaptatio. 3 IMULATION TUDIE 3. cope of siulative aalyses Described above algoriths of ipedace copoets easureet with applicatio of dyaical correctio ad frequecy adaptatio were put to thorough siulative tests. The tests were perfored i two stages. Firstly, the sigals geerated with MATLAB package were used for testig of algoriths (), (3) ad (4). The cosidered sigals did ot cotai ay distur- 5th P, Liege, -6 August 5 essio 9, Paper 5, Page 3

4 baces, therefore they could serve as a base for checkig cofority of the results with theoretical expectatios. Next the other sigals cotaiated with higher haroic ad decayig D copoets were prepared (currets ad voltages), with the ai to deterie the ifluece of such disturbaces o accuracy of easureet. I the secod stage of aalyses the voltage ad curret sigals geerated with use of EMTP-ATP prograe were used. The odel of test power syste is show i Fig. 4. It cosists of two subsystes A ad B represeted by electrootive forces behid certai ipedaces (both positive ad zero sequece) coected with 4 kv 5 k overhead trasissio lie. Paraeters of the lie ad subsystes are give i Fig. 4. Nuerous fault cases o the lie coectig substatios A ad B were siulated, with such paraeters beig chaged as fault type, fault resistace, fault duratio etc. The purpose of this stage of aalyses was to check correctess of the ipedace copoets easureet ( Z, R, ) as well as perforig coparative studies with the stadard algoriths, i.e. whe correctio of easureet equatios is ot applied. A Z A,66+j,35 Ω Z A,7+j4,8 Ω l5k, R ',8 Ω/k L ', H/k '3 F/k 4 kv R ',75 Ω/k L '3,7 H/k '8,5 F/k B Z B Z A Z B Z A Figure 4: EMTP-ATP odel of test power trasissio syste 3. Ivestigatio results The ivestigatio results obtaied i the fist stage, i.e. for pure siusoidal sigals defied i MATLAB, fully cofired the expectatios as to accuracy ad dyaical features of the easureet algoriths with cobied dyaical-frequecy correctio. Thaks to applied aedets, as described i previous sectio all the algoriths proposed were able to deliver proper values of easured paraeters as fast as i the secod saplig step after sudde chage of sigal shape. uch results were a effect of copesatio of frequecy depedece ad correctio of dyaical trajectory of FIR filters used. I Fig. 5 the easureet outcoes of ipedace Z ad its copoets R, for the sigals with siultaeous sudde chage of aplitude ad frequecy are show. It is see that the steady state of algorith respose after sudde sigal jup was obtaied alost iediately, which is ot the case for algoriths without correctio applied. I Fig. 6 a copariso of easureet results (ipedace absolute value Z ) is preseted for the followig algoriths: with correctio eq. (), without correctio usig siple relatioship with orthogoal copoets of voltage ad curret sigals, i the for: U u + u Z, (5) I i + i without correctio usig the algorith (4) for both curret ad voltage agitude calculatio which yields the followig algorith: U u u ( ) u ( ) u Z (6) I i i ( ) i ( ) i Z, R, [Ω] f5hz Z R Tie [s] f55hz Figure 5: Results of ipedace absolute value Z (solid lie), resistace R (dashed lie) ad reactace (dotted lie); easureet with use of eq. (), (3), (4); at tie istat ts sudde chage of both sigal aplitude ad frequecy took place. Z [Ω] f5hz Tie [s] f55hz Figure 6: Results of ipedace absolute value Z easureet with use of algoriths: with correctio () (solid lie), stadard algorith (5) (dotted lie), stadard algorith (6) (dashed lie); at tie istat t4s sudde chage of both ipedace ad frequecy took place. 5th P, Liege, -6 August 5 essio 9, Paper 5, Page 4

5 Ipedace Z [Ω] Ipedace Z [Ω] Figure 7: Results of easureet of Z with use of algoriths: with correctio () (solid lie), without correctio (6) (dashed lie); at tie istat ts phase-to-groud fault begis; easureet i faulty phase (at subsyste A side) Figure : Results of easureet of Z with use of algoriths: with correctio () (solid lie), without correctio (6) (dashed lie); at tie istat ts phase-to-groud fault begis; easureet i healthy phase (at subsyste B side). Resistace R [Ω] Resistace R [Ω] Figure 8: Results of easureet of R with use of algoriths: with correctio (3) (solid lie), without correctio (dashed lie); at tie istat ts phase-to-groud fault begis; easureet i faulty phase (at subsyste A side) Figure : Results of easureet of R with use of algoriths: with correctio (3) (solid lie), without correctio (dashed lie); at tie istat ts phase-to-groud fault begis; easureet i healthy phase (at subsyste B side) Reactace [Ω] Reactace [Ω] Tie [s] Figure 9: Results of easureet of with use of algoriths: with correctio (4) (solid lie), without correctio (dashed lie); at tie istat ts phase-to-groud fault begis; easureet i faulty phase (at subsyste A side) Figure : Results of easureet of with use of algoriths: with correctio (4) (solid lie), without correctio (dashed lie); at tie istat ts phase-to-groud fault begis; easureet i healthy phase (at subsyste B side). 5th P, Liege, -6 August 5 essio 9, Paper 5, Page 5

6 I the Fig. 6 sigificat advatages of the algorith with correctio () over the other stadard easureet ethods are visible. Thaks to itroduced correctio both dyaical ad frequecy variatio iduced estiatio errors were eliiated, thus eablig fast ad sooth aswer of the algorith (). The algoriths (5) ad (6) delivered outcoes characterised by easureet trasiets lastig ca. s. Additioally (i case of eq. (5)), ot fadig oscillatios i the steady state are observed, which is a effect of FIR filter widows ad gais beig uatched to curret sigal frequecy. O the basis of siulative studies with use of the sigals fro EMTP-ATP (cotaiated with various disturbace copoets), the ifluece of sigal disturbaces o easureet accuracy was deteried. It was affired that higher haroics preset i voltage sigal cause appearig of fadig oscillatory copoets i trasiet state of easureet. I case of curret sigal, presece of decayig D copoet is resposible for uderestiatig of easured ipedace copoets (lower values are delivered tha they are i reality). oe results of Z, R ad easureet that were obtaied for sigals geerated with use of EMTP-ATP (phase-to-groud fault case i the iddle of the lie) are preseted i Figs. 7. It is see that proposed algoriths with correctio assure uch faster ad ore accurate results whe the easureet is perfored i the phase ivolved i the fault (Figs. 7 9). However, because of high relative cotet of disturbace copoets i the sigals fro healthy phases the results delivered with algoriths (), (3), (4) uder such coditios are worse tha with algoriths without correctio (Figs. ). uch features of the algoriths with correctio result fro the very fact that their equatios have bee derived for a siple siglecopoet sigal odel ad that dyaical correctio coefficiets () have very strog aplifyig effect o all disturbace copoets. The best operatio of the algoriths with cobied dyaical-frequecy correctio was observed for easureet of ipedace copoets for the cases of phase-to-phase faults. 4 ONLUION I the paper a ew ethod for cobied dyaical ad frequecy depedece correctio iteded for iproveet of the trasiet ad steady state features of easureet algoriths is preseted. Takig advatage of the approach proposed ew effective algoriths of ipedace easureet have bee desiged. The algoriths developed are characterised by very fast, yet accurate respose to dyaically chagig paraeters of the sigal to be easured, especially for currets ad voltages fro the phases ivolved i the fault. The ost iportat advatage of the correctio ethod described i the paper is its very siple realisatio with correctio coefficiets () that are quite easy to be deteried for give set of used filters ad easureet algoriths. The egative feature of preseted ethod is its high sesitivity to sigal copoets (disturbaces) that have ot bee icluded i the easureet odel, which is characteristic for all algoriths with itroduced corrective eas. REFERENE [] H. Ugrad, W. Wikler ad A. Wisziewski, Protectio Techiques i Electrical Eergy ystes, Marcel Dekker, Ic., New York, 995 [] E. Rosołowski ad J. zafra Dyaically corrected fast estiators of curret ad voltage agitude, IEE Proc, Geer. Tras. Distrib. 995 vol. 4 r 3, pp.3-36 [3] W. Rebizat, J. zafra ad Moftah Abubaker Moftah, A ew algorith of fast estiatio of fault ipedace ad its applicatio, 6th Iteratioal yposiu o hort-ircuit urrets i Power ystes, Liege (Belgiu), 6-8 epteber 994, pp [4] J. zafra ad A. Wisziewski, Measurig ad decisive algoriths of digital power cotrol ad protectios, WNT, Warsaw, (i Polish) [5] J. zafra W. Rebizat ad M. Michalik, Adaptive easureet of power syste currets, voltages ad ipedaces i off-oial frequecy coditios, Proceedigs of the 6th IEEE Istruetatio ad Measureet Techology oferece (IMT'99), Veice, Italy, May 999, Vol., pp [6] W. Rebizat, J. zafra ad M. Michalik, Frequecy isesitive syetrical copoets estiators for power syste relayig, Proc. of the 3th Power ystes oputatio oferece, P 99, Trodhei, Norway, Jue/July 999, pp th P, Liege, -6 August 5 essio 9, Paper 5, Page 6