Fast-individual-harmonic-extraction technique

Transcription

1 Fst-individul-hrmonic-extrction technique C.H. Ng, K. Buswon, G.A. Putrus nd L. Rn Astrct: Hrmonic distortion in power networks is continuously incresing owing to the incresed use of nonliner lods in power-distriution networks. Consequently, hrmonic monitoring nd control re ecoming prticulrly importnt for oth utilities nd consumers to reduce their hrmful effects. In some pplictions, such s rel-time hrmonic monitoring nd ctive filters, techniques for fst extrction of individul hrmonic components re required. The reltively long response time of conventionl techniques mke them less ppeling when the speed of response is importnt. The pper dels with the prolem of individul-hrmonic extrction for the purpose of hrmonic filtering nd compenstion. A common prolem ssocited with existing individul-hrmonic-extrction techniques is the speed of extrcting single hrmonic component, which determines the response of the compenstor. Stndrd hrmonic-extrction techniques re investigted nd new fst-individul-hrmonic-extrction (FIHE) technique is proposed. It is shown tht the proposed FIHE is cple of performing hrmonic extrction six times fster thn the Fourier trnsform nd provides etter filtering chrcteristics thn trditionl filters such s Butterworth nd FIR filters. Computer simultion nd lortory experimentl results re provided to illustrte the chrcteristics nd performnce of the proposed technique. Introduction Hrmonic mesurement is n importnt process when designing hrmonic filter or compenstor. A power ctive filter works s hrmonic source, which reproduces the hrmonic components of interest (synchronised to the hrmonic components in the system) to chieve the desired compenstion. In most cses, the desired hrmonic components re reproduced y power invertor, which constitutes the min prt of the ctive filter. To otin the reference signl for the invertor, it is necessry to extrct the desired hrmonic component(s) from the distorted current (or voltge) wveform in the system. In some pplictions, the fundmentl component is removed from the signl detected nd the remining (nonfundmentl) hrmonic components re used to perform totl hrmonic compenstion (THC) [, ]. In certin pplictions, the ctive filter is only required to del with selected dominnt hrmonic components of the distorted signl; this is known s individul hrmonic compenstion (IHC). Regrdless of the use (THC or IHC), hrmonic extrction is necessry process which is required for otining the desired reference signl for the invertor. Most existing hrmonic-extrction techniques re either sed on filtering techniques in the time domin or Fourier nlysis in the frequency domin. In this pper the possiility of speeding up n individul hrmonic extrction process is investigted. As result, fst-individul-hrmonic-extrction (FIHE) technique is r IEE, IEE Proceedings online no. 7 doi:.9/ip-gtd:7 Pper first received th June nd in finl revised form nd Mrch. Originlly pulished online: th June C.H. Ng, K. Buswon nd G.A. Putrus re with the School of Engineering, Northumri University, Newcstle upon Tyne NE 8ST, UK L. Rn is with the School of Engineering, Durhm University, Durhm DH LE, UK E-mil: ghnim.putrus@unn.c.uk proposed. It is shown tht the FIHE is cple of performing hrmonic-extrction six times fster thn the Fourier trnsform nd provides etter filtering chrcteristic thn trditionl filters such s Butterworth nd FIR filters. Computer simultion nd lortory experimentl results re provided to illustrte the performnce nd chrcteristics of the proposed technique. Hrmonic extrction in time nd frequency domins In this section, stndrd methods of hrmonic extrction in the time nd frequency domins re nlysed.. Hrmonic extrction in the time domin Hrmonic extrction sed on filtering cn e divided lrgely into two methods []. The first method uses filters to extrct selected hrmonic components directly (phse y phse in three-phse system). The second method mkes use of the generlised Prk trnsformtion to convert the selected hrmonic component from -phse sttionry reference frme to synchronous reference frme nd then perform hrmonic extrction y using simple lowpss filter [, ]. The most common filters re Butterworth, Cheyshev I/II, nd Bessel types. Throughout this pper, discussion is restricted to the Butterworth-type lowpss filter; s it is generlly cknowledged s good ll rounder filter which gives flt response in the pssnd nd n dequte rte of roll-off. The dvntges of time-domin methods re tht their design is strightforwrd nd fewer computtionl steps re required; hence smller processor memory is needed. The disdvntges re time dely, phse dely nd gin/ ttenution distortion. To ensure synchronistion etween the reproduced (y the ctive filter) nd the system hrmonic components, the ctive filter is required to e highly dynmic. As result, it is desirle to hve minimum dely in the process of regenerting hrmonic components used for compenstion. IEE Proc.-Gener. Trnsm. Distri., Vol., No., July

2 The inherent conflict etween the frequency chrcteristics nd time response of filter forces trde off etween good roll-off (etter ttenution) nd fster response. The higher the order of the filter the steeper is the roll-off in the frequency response nd the etter re the ttenution chrcteristics; ut t the expense of further dely in the response time. This phenomenon is demonstrted in Fig. where the frequency response nd the step response (time response) for Butterworth filters of different orders re shown. mgnitude Fig. output of secondorder filter Effect of filter order output of sixth-order filter step function 8 of the step function IH(jω)I.8.. n =. n = 8 n = n = n = n = n = 8 n = frequency, Hz Fig. Frequency nd step response for nth-order Butterworth lowpss filter Frequency response Step response Generlly, to permit extrction of the desired hrmonic component, the ttenution of the filter must e sufficiently high in order to eliminte the unwnted hrmonic components. This is not prolem in certin pplictions when only the fundmentl component is required. As the rtio of hrmonic quntities with respect to the fundmentl component is very low, hence low-ttenution (low-order) filter is enough to filter the hrmonic components. However, in the cse where one of the hrmonic components is of interest (e.g.: fifth, seventh or, th ), the hrmonic mgnitude is reltively low compred with the fundmentl component. Therefore, high-ttenution filter is required in order to eliminte ll other components (i.e. first nd fifth if seventh is of interest). This implies tht higher-order filter is required nd hence longer time dely is to e expected. For instnce, to filter seventh order hrmonic voltge in low-voltge system, where the fundmentl voltge is V, the seventh hrmonic voltge is out % ( V) of the fundmentl voltge. As mentioned ove, generlised Prk trnsformtion together with lowpss filters hve een used to extrct hrmonics. Figure demonstrtes the effects of filter s order when step function ws pplied to mplify the seventh-hrmonic voltge t t ¼ ms. The results show tht some unwnted hrmonic components remin in the filtered signl (s ripple) when second-order lowpss filter is used. When using higher-order filter (n ¼ ), smoother (ripple-less) output signl is produced, ut the time response of the filter is slower. In prctice, in ddition to the time dely introduced y high-order filter, n increse in the order of the filter rings with it other prolems, such s offset nd noise error to the pssnd region. Moreover, high-order filter often suffers from stility issues. It is fmilir to most engineers tht onord tweking of cpcitors, vrile resistors etc. to void oscilltion is lwys required for high-order filter.. Frequency-domin hrmonic extrction Hrmonic extrction in the frequency domin is sed on Fourier nlysis where informtion out the selected hrmonic component(s) is used to reconstruct the signl in the time domin. Generlly, the Fourier trnsform (FT) is used to convert signl f (t) from the time domin to the frequency domin [].Iff (t) is periodic with period T,its Fourier trnsform (Fourier series) is given y F ðno Þ¼ Z t þt f ðtþ expð jno tþdt ðþ T t where n ¼,,,, y is the hrmonic order, nd o ¼ p/t is the frequency of the fundmentl component (rd/s). For individul hrmonic extrction, n is the order of the required component (i.e.: st, th, 7th etc.). F(no ) contins the mgnitude nd phse informtion of the nth order component. The dvntges of using the Fourier trnsform for individul hrmonic extrction re tht it gives very good ttenution gin with no overshoot or oscilltion prolems. With these dvntges, the FT provides n excellent extrction profile, s cn e seen in the exmple shown in Fig.. However, the iggest disdvntge of using the FT is tht, minimum of one fundmentl cycle is required for the trnsformtion process. This is clerly shown in Fig. nd (), where integrtion over fundmentl period (T ) is required. Therefore, in Hz system, ms will e the minimum processing period for hrmonic component to e filtered. mgnitude Fig.. output of FT (mgnitude of the seventh-order hrmonic voltge) step function Time response of the Fourier trnsformtion 8 of the step function IEE Proc.-Gener. Trnsm. Distri., Vol., No., July 7

3 -phse non-liner lod AC source FIHE unit i h L i hm extrction process c D D D D D D R Fig. System used in the simultion Fst-individul-hrmonic-extrction (FIHE) technique In this Section, the fst-individul-hrmonic-extrction method is presented following some preliminry oservtions. As will ecome cler in this pper, the proposed method is intended for hrmonics tht re lnced etween the three phses. This is more usul in industril power systems such s those found in the steel industry nd on offshore oil rigs where the hrmonics re predominntly cused y lrge motor drives. Unlnce nd nonchrcteristic hrmonics my exist in some utility distriution systems comprising considerle single-phse lods. The performnce of the proposed method will decrese depending on the degree of unlnce. In the nlyses elow, note tht the set of nturl numers is denoted y N, the set of positive odd numers y N+ nd the set of positive even numers y E.. Preliminry oservtions Consider the following lnced three-phse signl contining hrmonics of the order nan+: S A ðt; nþ S ABC ðt; nþ ¼ S B ðt; nþ S C ðt; nþ M n cosðno tþ M n cos no t pn ¼ M n cos no t þ pn 7 ðþ The signl S ABC (t, n) cn e converted to single-phse signl y using the conversion function G d ðt; mþ ¼ sinðmo tþ sin mo t pn sin mo t þ pn where man+. The resultnt converted single-phse signl is given y ðt; m; nþ ¼ G dðt; mþs ABC ðt; nþ ¼ M n sinðmo tþ sinðno tþ þ M n sin mo t pn sin no t pn þ M n sin mo t þ pn sin no t þ pn ðþ ðþ Since sinðþ sinðþ ¼ fcosð Þ cosð þ Þg ðt; m; nþ¼ M n½cosfðm nþo tg cosfðm þ nþo tgš þ M n cos ðm nþo t pðm nþ cos ðm þ nþo t pðm þ nþ þ M n cos ðm nþo t þ pðm nþ cos ðm þ nþo t þ pðm þ nþ ðþ Depending on the vlues of n nd m, thevlueof(t, m, n) cn either e zero or nonzero. To hve etter nd cler insight of the vlues of (t, m, n), consider the three sequences {u k }, {v k }, {w k }; (k ¼,, y) defined s: u k ¼ k v k ¼ k þ ðþ w k ¼ k þ Let U ¼[ fu k gj k¼; ; ;::: V ¼[ fv k gj k¼; ; ;::: W ¼[ fw k gj k¼; ; ; ::: It cn redily e seen tht the sets U, V nd W form prtition of the set of nturl numers N, i.e. N ¼ U [ V [ W As mentioned ove, oth n nd m re positive odd numers. Consequently, the terms 7m n7 or 7m+n7 stted in () re even. Therefore, it is esier to explin nd understnd the lgorithm of the proposed technique in the even-numer domin. For the sequences descried in (), the set of positive even numers E cn e prtitioned into three disjoint sets, X, Y nd Z, defined s: X ¼[fx k gj k¼; ; ; ::: Y ¼[fy k gj k¼; ; ;::: Z ¼[fz k gj k¼; ; ; ::: ð7þ where the sequences {x k }{y k }{z k }; (k ¼,,, y) re defined y: x k ¼ u k ¼ k y k ¼ v k ¼ k þ z k ¼ w k ¼ k þ 8 IEE Proc.-Gener. Trnsm. Distri., Vol., No., July

4 Note tht for every numer u k AU, v k AV nd w k AW: p ðu kþ¼ p ðkþ ¼p ¼ ðzero-sequenceþ ð8þ p ðv kþ¼ p ðk þ Þ¼pþ p ¼ ðpositive-sequenceþ ð9þ p ðw kþ¼ p ðk þ Þ¼pþ p ¼ ðnegtive-sequenceþ ðþ Equtions (8), (9) nd () show tht, irrespective of the hrmonic order in U (rd, 9th, th etc.), there will lwys e zero-sequence component. Also, whichever hrmonic order is found in V (st,7th,thetc.),thiswillresultsin positive-sequence component. Similrly, the hrmonic order in W (nd, th, th etc.) will result in negtive-sequence component. This lso hppen in the even-numer domin, for every numer x k AX, y k AY nd z k AZ. Therefore, p ðx kþ¼ p ðkþ¼p ¼ ðzero-sequenceþ ðþ p ðy kþ¼ p ðk þ Þ¼pþ p ¼ ðnegtive-sequenceþ ðþ p ðz kþ¼ p ðk þ Þ ¼p þ 8p ¼ ðpositive-sequenceþ ðþ In the following, oth m nd n re positive nd odd, so tht 7n m7 nd 7m+n7 re oth even. To determine the vlue of (t, m, n), there re nine cses to e considered: Cse : 7m n7ax nd 7m+n7AX ðt; m; nþ¼ M n½cosfðm nþo tg cosfðm þ nþo tgš þ M n½cosfðm nþo tg cosfðm þ nþo tgš þ M n½cosfðm nþo tg cosfðm þ nþo tgš ¼M n cosfðm nþo tg M n cosfðm þ nþo tg ðþ Cse : 7m n7ax nd 7m+n7AY ðt; m; nþ¼ M n½cosfðm nþo tg cosfðm þ nþo tgš þ M n cosfðm nþo tg cos ðm þ nþo t þ p þ M n cosfðm nþo tg cos ðm þ nþo t p ¼ M n cosfðm nþo tg ðþ Similrly, the vlues of (t, m, n) in the following cses cn e verified: Cse : 7m n7ax nd 7m+n7AZ ðt; m; nþ¼m n cosfðm nþo tg Cse : 7m n7ay nd 7m+n7AX ðt; m; nþ¼ M n cosfðm þ nþo tg Cse : 7m n7ay nd 7m+n7AY ðt; m; nþ¼ Cse : 7m n7ay nd 7m+n7AZ ðt; m; nþ¼ Cse 7: 7m n7az nd 7m+n7AX ðt; m; nþ¼ M n cosfðm þ nþo tg Cse 8: 7m n7az nd 7m+n7AY ðt; m; nþ¼ Cse 9: 7m n7az nd 7m+n7AZ ðt; m; nþ¼. Appliction of the FIHE to Hz system For three-phse lnced Hz system, only five of these cses re relevnt; these re: cses,,, nd 7. The relevnt nonzero cses re summrised in Tle. Tle : Nonzero Cses Cse 7m n7 7m+n7 X X M n cos{(m n)o t} M n cos{(m+n)o t} X Y M n cos{(m n)o t} X Z M n cos{(m n)o t} Y X M n cos{(m+n)o t} 7 Z X M n cos{(m+n)o t} Note tht, when 7m n7ax (for cses nd ), (t, m, n) ¼ M n cos(ko t). Similrly, when 7m+n7AX (for cses nd 7), (t, m, n) ¼ M n cos(ko t). Finlly, when oth 7m n7ax nd 7m+n7AX (for cse ), ðt; m; nþ¼m n cosðko tþ M n cosðko tþ This mens tht the smllest frequency component tht ppers in (t, m, n) is lwys o ¼ p rd/s or Hz. Let t ¼ T ¼ ¼ seconds Also, since n is odd nd positive nd ssuming n ¼ p+ with pan, itisclertht: I d ðmþ¼ T Z ¼ t Z T Xþ p¼ t Xþ p¼ ðm; p þ ; tþdt ðm; p þ ; tþdt if n ¼ p þ ¼ m ¼ if n ¼ p þ ¼ m M m ðþ ð7þ where the closed intervl of length T is denoted y T ¼ [t, t +T ]witht eing rel numer. More specificlly, RT f ðtþdt ¼ R t þt t f ðtþdt. This mens tht there is no need to integrte (m, n, t) over the full period T to otin I(m); n integrtion over period of one-sixth of T is sufficient. This lst oservtion forms the sis of the FIHE method. In this method, it is cler tht, the time tken to extrct the hrmonic IEE Proc.-Gener. Trnsm. Distri., Vol., No., July 9

5 component is six times fster thn the originl method. Further, s will e shown, the proposed FIHE scheme hs the dvntges of eing overshoot-free, oscilltion-free nd it hs high ttenution gin chrcteristics, which cnnot e found in existing filtering technique.. Reconstruction of the extrcted hrmonic signl As explined in Section., hrmonic component cn e extrcted nd expressed in DC form s I(m). However, in some pplictions, the time-domin wveform of the hrmonic is importnt nd needs to e ville fter extrction; hence idirectionl trnsformtion is required. Therefore, to reconstruct the extrcted hrmonic signl, n equl weight mtrix is required which mens tht two extr sets of vriles re needed: g(m, n, t) ndd(m, n, t). For simplicity nd to provide ll phse informtion, g(m, n, t) hs to e set perpendiculr to the first set of vriles (m, n, t); i.e. where gðm; n; tþ ¼ G qðt; mþs ABC ðt; nþ G q ðt; mþ ¼ cosðmo tþ cos mo t pm cos mo t þ pm ð8þ ð9þ Using the previous nlysis, it cn e noted tht the smllest frequency component tht ppers in g(m, n, t) islsoo. On the other hnd, since R T sinðmo tþ cosðno tþdt ¼, for ll n nd m, this results in I q ðmþ ¼ Z T ¼ t Z t Xþ p¼ T Xþ p¼ Finlly, d(m, n, t) is defined y where gðm; p þ ; tþdt gðm; p þ ; tþdt ¼ dðm; n; tþ¼ G ðt; mþs ABC ðt; nþ C ¼ ðþ ðþ ðþ ðþ It is cler tht g(m, n, t) ¼ ndd(m, n, t) ¼, since S ABC (t, n) is lnced three-phse signl. Consequently, I ðmþ ¼ Z dðm; p þ ; tþdt ¼ ðþ t t Xþ p¼ Equtions (), (8) nd () cn e written in compct form s where Wðm; n; tþ ¼ Gðt; mþs ABCðt; nþ ðþ ðm; n; tþ Wðm; n; tþ¼gðm; n; tþ ðþ dðm; n; tþ nd Gðt; mþ sinðmo tþ ¼ cosðmo tþ sin mo t pm cos mo t pm Finlly, I(m) cn e otined s: IðmÞ¼ Z Cðm; p þ ; tþdt t t Xþ p¼ sin mo t þ pm cos mo t þ pm 7 ð7þ ð8þ To reconstruct the signl S c (t, m) the following eqution is pplied: S c ðt; mþ¼g ðt; mþiðmþ ð9þ where G (t, m) is the reverse of G(t, m) nd I d ðmþ IðmÞ¼I q ðmþ ðþ I ðmþ Evlution of the proposed FIHE technique Performnce of the proposed hrmonic-extrction technique is evluted y using computer simultion sed on typicl V three-phse supply system with three-phse diode-ridge rectifier (with kw R-L lod) cting s hrmonic current source. The circuit configurtion is shown in Fig.. Simultion work ws crried out in the Ansoft Simplorer environment [7, 8]. The hrmonic current i h produced y the nonliner lod ws detected t the mins input feeder. The current i h is fed to the proposed FIHE unit to extrct the selected hrmonic current i hm. The extrcted hrmonic current i hm is then converted ck to sinusoidl form. Figure shows the wveform of the extrcted current i h together with its hrmonic spectrum. Figure shows n exmple of the extrcted seventh-order hrmonic-current wveform where the detected current (given in Fig. ) is shown in the ckground. The hrmonic spectrum shown in Fig. demonstrtes tht only the seventh-hrmonic component exists fter extrction. The ove results demonstrte the stedy-stte hrmonic extrction-cpility of the proposed FIHE technique. However, the most importnt feture of FIHE is its good dynmic chrcteristic. Therefore, dynmic-response tests were crried out in oth computer simultion nd lortory experiments. A wveform with hrmonic content up to th-order hs een rtificilly creted for the test. A step function for the seventh-hrmonic component hs een dded into the wveform, which step up the seventh-hrmonic component y five times the originl vlue t t ¼. s. The resultnt wveform nd the step function re shown in Fig. 7. The signl shown in Fig. 7 ws fed to the FIHE unit nd the FIHE performnce is shown in Fig. 8. The signl I(m) of the FIHE is shown in Fig. 8 nd the extrcted signl is shown in Fig. 8. These results demonstrte tht the proposed FIHE hs very fst (dynmic) response nd provides zero overshoot s well s n oscilltion-free chrcteristic. For comprison, Fig. 9 shows the response otined when using stndrd hrmonic-extrction techniques, i.e. lowpss filter nd Fourier trnsform) nd the IEE Proc.-Gener. Trnsm. Distri., Vol., No., July

6 I h, A I h, A m m m 8m time, s 8 9 f, Hz Fig. Wveform nd frequency response of the extrcted system current i h Current Frequency spectrum I h7, A I, A extrcted hrmonic current, i h7 m m m 8m i h only seventh-order hrmonic component left 8 9 f, Hz Fig. Wveform nd frequency response of the extrcted seventhhrmonic current Extrcted seventh-hrmonic current nd i h wveform Frequency response of seventh-hrmonic current proposed FIHE technique. These results show the good performnce of the FIHE technique compred with tht of the stndrd hrmonic-extrction techniques. Lortory experimentl work ws conducted with the sme scenrio descried ove. Figure shows the stepfunction signl nd the signl I(m) of the FIHE with the step function. Figure shows the signl I(m), the distorted input signl, nd the extrcted seventh-hrmonic signl (with the step function dded). The results shown demonstrte clerly the functionlity nd ccurcy of the proposed FIHE technique. The proposed method is pplicle to oth odd nd even hrmonics. It hs een noted tht the ccurcy of the FIHE technique drops when extrcting hrmonic components Fig. 7 Wveform generted with the step function dded Step function for seventh-hrmonic Resultnt wveform Fig. 8 FIHE performnce for step chnge in the seventhhrmonic component I d (m) ofthefihe Extrcted seventh-hrmonic component (perphse) y FIHE y filter second-order filter y FT Fig. 9 Time-response plot of three different hrmonic-extrction techniques from three-phse signl tht is unlnced. For the system shown in Fig., unlnce in the supply voltge will cuse nonchrcteristic hrmonics in the system. Initil investigtion indictes tht, for moderte supply unlnce, the IEE Proc.-Gener. Trnsm. Distri., Vol., No., July

7 T error introduced into the extrcted hrmonics of ech phse is in the rnge %. If the hrmonics in the three phses re unlnced due to, for exmple, single-phse nonliner lods, the proposed method will derive the sme hrmonic level for ll three phses nd the error introduced will e proportionl to the degree of unlnce. A more thorough investigtion is underwy to quntify these errors. Conclusions T This pper presents the principles, chrcteristics nd performnce of new fst-individul-hrmonic-extrction (FIHE) technique. The principles of the proposed technique nd its performnce re compred with existing stndrd hrmonic-extrction techniques. The FIHE technique hs n excellent dynmic-response cpility (six times fster thn stndrd techniques, e.g. the Fourier trnsform) nd provides overshoot-free nd ripple-free chrcteristics. The results otined from the computer simultion nd lortory experimentl work show very high degree of confidence in the proposed technique. The good dynmic performnce of the proposed technique mkes it n excellent tool when the speed of response is importnt, s in modern hrmonic-control techniques such s, ctive power-hrmonic compenstors. References Fig. Experimentl results The step signl nd I(m) signl of the FIHE Chnnel : Step-function signl (V): V/division Chnnel : I(m) offihe(v):.v/division I(m) signl, distorted signl, nd the extrcted seventh-hrmonic component Time: ms/division Chnnel : I(m) offihe(v):.v/division Chnnel : Distorted signl (V): V/division Chnnel : Extrcted seventh-hrmonic component (V): V/division Peng, F.Z., Akgi, H., nd Ne, A.: A new pproch to hrmonic compenstion in power systems. Industry Applictions Society Annul Meeting, Pittsurg, USA, 988, Vol., pp Akgi, H., Fujit, H., nd Wd, K.: A shunt ctive filter sed on voltge detection for hrmonic termintion of rdil power distriution line, IEEE Trns., 999, ia-, pp. 8 Tn,P.C.,Holmes,D.G.,ndMorrison,R.E.: Controlofctivefilter in the kv AC trction systems. Power Electronic Puliction in AUPEC, Tnni, S., Mzudier, M., Berthon, A., nd Diop, S.: Comprison etween different rel-time hrmonic nlysis methods for control of electricl mchines. Power Electronics nd Vrile-Speed Drives, IEE Conf. Pul. 99, London, 99 Ng, C.H., Rn, L., Putrus, G.A., nd Buswon, K.: A new pproch to rel time individul hrmonic extrction. IEEE Power Electron. nd Drive Systems Conference, Singpore, Vldimir, A.K.: Computer sed hrmonic mesurement systems: discussion nd reliztion. ICHPS V Int. Conf., Atlnt, USA, Sep. 99, pp. 7 Ansoft. Co.: Simplorer v. multi domin simultion pckge user mnul (Ansoft Corprtion, ) English ed 8 in Fe IEE Proc.-Gener. Trnsm. Distri., Vol., No., July