A Real-time Estimator of Electrical Parameters for Vector Controlled Induction Motor using a Reduced Order Extended Kalman Filter

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1 A Ral-m Emao o Elccal Paam o Vco Conolld Indcon oo ng a Rdcd Od Endd Kalman Fl Kywod Vcn, R Aaújo, Damanno Fa ESCOA SUPERIOR DE ECNOOGIA E DE GESÃO DO INSIUO POIÉCNICO DE BRAGANÇA Camp d Sana Apolóna, Apaado 4, Bagança Bagança, Pogal E-mal: avl@pb.p Phon n.º: , a n.º: 5-7-5, FACUDADE DE ENGENHARIA DA UNIVERSIDADE DO PORO Ra D. Robo Fa /n, Poo Poo, Pogal E-mal: aajo@.p.p, da@.p.p Phon n.º: , ; a n.º:, Indcon moo, maon chnq, vaabl pd dv, adapv conol. Abac h pap pn an applcaon o h ndd Kalman l EKF o h mlano on-ln maon o h dq oo l componn and all h lccal paam o a vco conolld ndcon moo. A m-dc dcd od modl c ddcd and pn a mpl and dcd a qaon and a cala op qaon. h appoach, combnd wh h o h oo nc am, o advanag o al-m dncaon, compad wh ll od modl, bca dc h compaonal co. h popod nw appoach q h mamn o moo pd, ao volag and cn gnal. Smlaon and pmnal d pnd n h pap hghlgh h mpovmn podcd by h nw appoach bad on h ndd Kalman l and a nw dczaon chnq, nd al opaon condon. Inodcon a dcad w vy podcv o conol o h ndcon moo. W hav wnd h dvlopmn o ncangly ophcad conol mhod, om mpl cala mhod o vco conol, and, mo cnly, o ldng mod conol ag. Alhogh h mhod a d cnly n ndal applcaon, many o hm q h nowldg o h moo modl paam, bca a mmach n h on pon o ca naccpabl conol o. In ac, h lcc paam chaaczng h ndcon moo modl can vay gncanly dng h nomal opaon o h moo. h cad by phnomna ch a ao and oo hang, magnc aaon and n c. h, ncany and paam vaaon can doa h achvd conol. h c o paam nvy on h pomanc n vco conol chm and h pobly o dncaon o h chang n moo paam whl h dv n nomal opaon hav bn gvn n []. ho, ncay o ma and ac h paam val n al-m opaon. any paam dncaon algohm bad on h la qa mhod amly, obv hoy, and o oh, hav bn popod n h lvan cnc la. Among hm, h Kalman l bad algohm hav bn dmonad o b h b o pocng noy dc mamn whl obanng acca ma [-9]. h ndd Kalman l a cv, opmal, al-m daa pocng algohm o nonlna ym o boh a and paam maon [] o a dynamc ym n a noy nvonmn. Whn a PW nv mployd o d h ndcon moo h ndamnal componn a condd a dmnc np o op dpndng on h modl c, and h wdband hamonc componn a

2 ncldd n h no vco o h a-pac modl c []. h no vco a amd o b wh, zo-man Gaan, and ndpndn. A lag amon o wo hav bn dvlopd on h bjc. any a ang ngl paam dncaon, namly oo m conan o oo anc, wh oo l maon o ndc vco conol ppo. wo appoach a d: ll od modl [-5] o dcd od modl movad by l compaonal o [5-9]. In [] h oo nc am d and h oh wo n h ao nc am. wo wo [, 8] a lad o h dncaon o all h paam. In h pap h aho pn a nw appoach o wo oo l componn and all h phycal paam o h h-pha ql cag ndcon moo wh an algohm whch bad on a dcd od modl o h ndcon moo n oo nc am, ng h ndd Kalman Fl and a pacla dczaon chnq o h conno-m modl. Indcon moo modl h wll-nown ablhd modl o h ndcon moo n h oo nc am pnd, l n [], by ao and oo pac phao volag qaon and ao and oo l qaon pd n m o ao and oo cn pac phao. Condng a ql-cag ndcon moo and lmnang ao l and oo cn pac phao wh om algbac manplaon, h ollowng qaon can b oband: j j a ω ω & & m m g m g and R R a Wh,,,, φ Fo compaonal convnnc h qaon abov a ally convd no a-pac-bad om o 4 h od wh only al qan a ollow. q d q d q d q d q d a a ω ω ω ω & & & & 4 h m-doman a-pac modl omd by h a qaon abov and an op qaon: C y and B A d d 5 [ ] q d q d q d y and Wh 6 A m-dc a-pac modl can n b oband by amng ha h panon o h ma ponnal ncon A pomd and only h m a condd []. Snc o a mall amplng nval,, mac A and B can b condd o b conan dng h ampl pod, only h lna m wll b condd: B A A B and A A A I A d d!!!! 7 W wll adop a dn agy o h modl c and modl dczaon by lcng h dcd od modl oband by condng qaon a a a qaon, nc ao cn a mad, and by omng h op qaon om qaon. Condng paa al and magnay pa o compl qaon h a qaon bcom:

3 & d d d 8 & q q d I mpoan o noc h ha boh qaon ha l om al and magnay pa o qaon hav h am c wh h am a paam and h am gnal volag and cn componn, b hd n m. ho, hy conan h am nomaon o dncaon ppo. So w wll h only on o hm, o nanc, h on. h op qaon bcom, a wng ho m: d ω ω a & 9 d q d Eqaon 8 and 9 om a nd od a-pac modl c nad o a 4 h od on, a ally wh 4 and 6. I woh nong ha h an mpoan and nw mplcaon anom h nal modl no a ngl and ho cala op modl c. Snc h a qaon 8 ha a dcd od, hal o ma lmn a zo and h op qaon ad paaly, bcom vy mpl and no ado o oban a nd od appomaon. Fo h conno a qaon 8 w can aly oban, by ng 7, h od m-dc a qaon a ollow, copondng o h El omla. d d d q q q h choc o maon o oo l componn nad o oo cn componn advabl bca n lgh load condon h oo cn a vy mall and bad convgnc and maon l can a a d n [5]. Fhmo, h magnd o h oo l componn a mch mo nom n dn load condon. Fo h dvav o h d componn ao cn n op qaon 9, nad o El omla, a b cv appomaon o h dvav wll b d and h op qaon bcom: d d ω q Endd Kalman Fl EKF a d d 4d d ω q h dc-m EKF a cv a and paam mao abl o wh ml-np, ml-op nonlna m-vayng ochac plan modl. By choong h val n h covaanc mac, h EKF can b nd o h no chaacc o h mad op and h nal ncan n h mad a and paam. h poc no covaanc ma allow h o modl, whch hav dbanc n h a and paam, o modl, whch a no q coc n h c. h mchanm ha ca h a and paam chang do no nd o b nown. In h pap, a pacla nalzaon o h poc covaanc ma pnd o mpov h l dynamc dng a-p. h EKF d, h, o ma h oo l dq componn and all phycal paam o h ndcon moo, modld a abov, h bng ally clad a a dcd od modl, wh a dn op qaon n h ca. h EKF can b d o boh a and paam maon by ang h phycal ndcon moo paam a addonal a and omng an agmnd a vco. A a l, vn h ognal a-pac modl lna, h agmnd on nonlna bca o n-mlplcaon o a. h EKF dal dcly wh h nonlna agmnd modl. o h EKF wh nonlna plan modl, h modl c m b lnazd aond an opang pon o podc a lna pbaon modl. H h applcaon o h EKF o h mlano maon o oo l ogh wh ndcon moo paam podc a h-od ndd a-pac modl wh h ollowng a vco: d q

4 [ ] [ ] [ ] q d R 4 6 wh h bcp dno h ndd o agmnd a vco. Fo h maon o h paam, o a qaon a oband by amng a andom wal o h adapaon a d n h la []: j j j Cond h ochac m-dc nonlna a-pac modl []:,, 4, h y m 5, 6 I ally amd ha h poc no and mamn no m a wh, zo-man Gaan and ndpndn qnc. h aocad EKF algohm gvn a ollow:. Dmnaon o h nal condon: { } { } P E E., ˆ ˆ,, ˆ ˆ.,,,,, I F 4. R F P F P ˆ 5. ˆ,, h h h H 6. [ ] ˆ ˆ m R H P H H P K 7. h y, ˆ 8. [ ] ˆ y y K 9. [ ] ˆ P H K I P. Go o p Condng h lna m n 7 and om qaon h ncon o h dcd od modl o h ndcon moo pnd h a a ollow:,, 5 d 7,, 5 q 8 4,,,,,, j j j j j 9 h op qaon :

5 y h, ω 6 4 ω d d d 5 q d 4 m Smlaon Rl h abov-popod algohm ha bn dvlopd n h alab wh Smln nvonmn and d nd a vco conol chm. h oo-d ao volag, ao cn and angla pd a ampld a,5hz. Ellpc low-pa p-l o h od wh a 5Hz co qncy w d. h mlaon condon w chon o b h am a h pmnal on n pvo wo [] and wll b d n pmnal l pnd n h n con. A al o mlaon how ha val mpoan and dcl apc a whn yng o ma mlanoly oo l and all ndcon moo paam wh a dcd od modl. On h on hand h no covaanc mac m b cocly, h a-vco and o covaanc ma nalzd, h a vco poply cald and, on h oh hand, h gnal m b pn and h modl c dnabl. All ndcon moo a-pac modl c wh a vco ncldng l o cn and dc o modd phycal paam o h machn, wll hav gncanly dn magnd o a vaabl. Whn h modl c conan paam wh dn od o magnd, mandaoy o cal h vaabl o ha hy all g oghly h am magnd. Oppoly, w do no cal h a vco a n h ca, gncan maon o occ o mall zd vaabl and om nmcal poblm a. o ovcom h poblm, h ollowng cald a vco ha bn lcd: [ R ] d q A a l, h dagonal nal val o h a covaanc ma and h dagonal lmn o poc and mamn no covaanc mac wll appomaly hav h am val and h am conclon applcabl o h nal val o h a vaabl. h EKF ha bn ad wh h ollowng nal condon: P R. m [......] dag[ ] I hold b nocd ha h mamn ma R m no mo a ma b a ngl val bca h ym op bcom a cala qaon nad o a ma on. A o h nalzaon o h poc no ym ma, om mpovmn w alo nodcd. o g a good ad o bwn ang pod o paam convgnc and acng o m vayng paam, h l dynamc and n h ql h no covaanc mac R m and R hold b poply nd dng h wo pha. I nown ha h lag h h dagonal ny o R, h mo qcly h l wll mody h ma o h h componn l o paam o h a vco n h lgh o h mamn. In oh wod, h lag h cocon gan, o any gvn R m, h j h bhavo o h l wll b, and mo dacally h a ma wll b modd o a nw mamn no accon. h bhavo p a lac o condnc n h pdcd a ma. Convly, o any gvn R, h lag h covaanc o mamn no R m, h mall h cocon gan wll b and h l h nw mamn wll b an no accon o pda h a ma. h p a lac o condnc n h nw mamn. So, pobl o g a aonably compom bwn h a pda dynamc whn ang h algohm n whch h a vaabl a a away o h al val, and h dynamc o acng h m vaan a a ha ann pod. In h wo, a mpl appoach wa d o do h, by ng an ponnal ncon o ach h dagonal lmn copondng o paam, a ollow: dag 8 8 g g g g [ ] R 4 p.8. and g 7 p.8. wh g,, 8 4 4

6 h dncaon con o a a-p ann wh a load oq o Nm and a a-p ann ollowd by a al o chang n pd nc bwn 5 and 5pm. h la wa alo nvgad wh no load oq. Smlaon and pmnal dncaon wh h a-p ann how ha lghly dn mad val o ao anc R S, can occ whn dn val o h pcv dagonal lmn o h a covaanc ma a lcd wha do no happn wh h oh paam. h q a dcl and cal nal nng o h EKF. h dcly l om h np gnal o h modl bng no pn nogh o mla h ao paam, manly R S, n h ady a opaon a h a-p ann. Inodcng om dynamc o h ndcon moo l a qa pd nc a pnd n h n g, ch a poblm do no appa. a b c d Fg. : Robn o mad a convgnc wh pc o nal a covaanc ma Pdag. lmn o P, wh a load oq o Nm. a Smlad lccal pd, b mad., c mad 5, d mad 5, mad.5 R S, mlad and mad d l componn.

7 Fg how h obn o mad l and paam convgnc wh pc o nal a covaanc ma, wh a load oq o Nm. A w can om plo b o h mplmnd algohm ha a good convgnc wh pc o dn nal a covaanc ma dagonal lmn. Fg a how h lccal pd gnad by h vco conol chm o h ndcon moo mplmnd n Smln. Fg how mlad and mad l wh only d componn pnd. Fg how h obn o mad paam convgnc wh pc o poc no covaanc ma, wh a load oq o Nm. a b c Fg. : Robn o mad paam convgnc wh pc o poc no covaanc ma Qdag. lmn o poc cov. ma, wh a load oq o Nm. a mad., b mad 5, c mad 5, d mad.5 R S. A good convgnc wh pc o dn poc covaanc ma dagonal lmn can b obvd. By man o h ponnal ncon 4 h daon o convgnc pha dcd d o h EKF dynamc mpovmn dng h aon n h way o h ady a dynamc o no jcon and paam acng. h obn o h algohm alo ad wh pc o dn val o mamn no covaanc R m. Addonally, convgnc pd o h a can b nd by h paam o h algohm. Smlaon howd ha h popod algohm wo vy wll wh dn load condon a w can n g wh mad paam wh a load oq o Nm and no load and ncldng a-p ann, a pnd. h dnc bwn a-p ann nclon o no ha no mpoan dnc n h mad paam nal val, a can b n n g 4, and only ac h m and h om o h paam convgnc, namly ao anc. h povdd ha a h m o maon a h am dynamc condon a waand. I n h aon ha h l hold b paclaly dynamc lavly o ao anc. h j h dnc bwn g,, and g 4 n 4. d

8 a b c Fg. : Emad paam wh a load oq o Nm and no load, ncldng a-p ann. a mad., b mad 5, c mad 5, d mad.5 R S. d abl I how h mad paam val, oband by h mlaon pnd abov. I can b n ha h ndcon moo paam a pcly mad whn compad wh h on oband by clacal mhod and d n h mlaon. abl I: Emad paam val, oband by mlaon. By clacal Incldng a-p ann Who a-p ann mhod Wh Nm Wh no load Wh Nm Wh no load m S mh mh..... R S Ω A h a ma val a poply ablzd h dynamc condon can b movd and h dynamc o EKF hold bcom l nv o nw mamn, b nv nogh o ac paam vaaon whch vy low o h ao anc, nc va wh mpa. Roo paam do no nd h qmn bng pcly mad wh a good pomanc convgnc. h am global l can alo b oband wh a qa wav pd nc componn wh magnd a low a pm, n h ang o o 5 pm o o 4 pm.

9 a b c Fg. 4: Emad paam wh a load oq o Nm and no load, who a-p ann. a mad., b mad 5, c mad 5, d mad.5 R S. d Epmnal Rl Fo mlaon and pmnal ppo a W ql-cag ndcon moo wa d ad volag 4V, cn 6.A, pd 4 pm, qncy 5Hz, pol pa. h ollowng o lccal paam wa oband by clacal mhod: ao anc.4ω DC wh h ndcon moo a ambn mpa, oo anc.7ω, ao and oo ndcanc 4.mH and magnng ndcanc mh. h load oq pogammabl by h modl ODEC om oy Som ha conol a powd ba. h volag, cn and pd gnal a avalabl n h ang o ±V n boh oo and ao nc am by ng h ADS analog vco poco. h hadwa [] ha an adjabl co qncy wh an acca analog llpc low-pa l, wh AX74, o avod alang o. Daa wa collcd a a amplng a o.5hz by ng a 6 b PCI-65E daa acqon cad and a SC-4 modl wh 8 SSH channl o mlano acqon, boh om Naonal Inmn. h EKF algohm w dvlopd n AAB langag and calclaon w mad o-ln. h ndcon machn conolld by an ndal qncy conv ACS-6-6- om ABB and h vco conol chm dvlopd n AAB and h on mplmnd by ABB a no h am. ho om dnc wh pc o h volon o h mad paam n h wo pha o h wo. h condaon o EKF algohm nalzaon pnd n h pcdng con hold b alo applcabl h. h dncaon con o a a-p ollowd by a al o commaon n pd nc bwn and 5pm wh a load oq o Nm, ha o ay, abo hal o ad val. Fg 5, om b o, how, n old ln, h mad a convgnc wh a

10 load oq o Nm. Fg 5 a how h mad lccal pd o h ndcon moo conolld by h ACS-6 qncy conv. Fg 5 how mad l dq componn. Alo pnd n g 5 b o wh a dod ln h volon o h mad paam, h ponnal ncon 4 w no d. Wh pmnal daa nconaly y mo mpoan o a h pd o convgnc o h a ma, whn h algohm a, by mpovng h l dynamc by man o hgh val o h poc no covaanc ma dagonal lmn. h val dca ponnally wh m o h ady a val o gaan l ably and paam acng. Anoh aon o mpov h dynamc a h bgnnng ha mad gnal a mch mo clo o nodal om han ho om mlaon. A a conqnc h m convgnc o h algohm a can b n n g 5. a b c d Fg. 5: Emad a convgnc wh a load oq o Nm. a ad lccal pd, b mad., c mad 5, d mad 5, mad.5 R S, mad dq l componn.

11 In boh mlaon and pmnal h and nd od o h qaon 7 wa d o h dczaon o a qaon 8 and h l howd ha h od pov o b cn, povdd ha h dvav o h ao cn gnal compd a n op qaon. abl II how h mad paam val, oband by pmnal. Addonal pmd o concld ha h maon can b pomd a low amplng a wh h am convgnc chaacc and h l o h amplng qncy o Hz a pnd n la colmn o abl II. abl II: Emad paam val, oband by pmnal. By clacal mhod Incldng a-p ann and Nm Who a-p ann and Nm Incldng a-p ann, Nm load and S Hz m S mh mh R S Ω o concld h dncaon pocd ncay o valda h dnd modl hogh om valdaon o vala h pomanc o h EKF algohm. h valdaon wa bad on h mlaon o a modd ndcon moo modl, wh h mad paam, by njcon o h mad volag and angla vlocy and bqn compaon o mlad and mad ao cn dq componn. h l a hown n g 6 a, o h mad paam and n g 6 b o h paam mad by clacal mhod. A can b n h mlad and pmnal cn a vy mla n h ca. So h EKF algohm o ll, paam and l, maon capabl o ng adqaly h np-op daa wh h dnd modl. a b Fg. 6: ad and mlad ao cn d componn. a Smlad cn wh mad paam by EKF, and b wh paam mad by clacal mhod. Conclon A nw appoach popod n h pap o dczaon o h ndcon moo modl o oban a dcd od modl wh o lccal paam and a ngl cala op qaon. h obn o h algohm nalzaon and convgnc pnd. A o nal qa pl n h moo pd nc ggd o mpov h pd o convgnc and aympoc bhavo o h ao paam, manly o ao anc maon. D o h dn magnd o h ognal a n h a vco h w cald o avod nmcal poblm and cay o coc maon.

12 A dagonal poc no covaanc ma, wghd by ponnal ncon, wa d o nca h pd o convgnc o h a ma whn h algohm a, mpovng h l dynamc by man o hgh val o h covaanc ma dagonal lmn. h val dca ponnally wh m o h ady a val o gaan l ably and paam acng. h oo nc am vy l nc h gnal qncy ang a low han n h ao nc am and conqnly, h amplng pod can b mad long a wll a compaon cadncy. h amplng qncy can b a low a Hz whl n h ao nc am ally n h ang o 5 o Hz. h, ogh wh h dcd od modl, allow cng down on h compaonal o, whch on man dawbac o h EKF. h algohm dcbd n h pap wa d o-ln b can b mplmnd and ppad o on-ln maon o all h phycal ndcon moo paam and no cd o ady a opaon bng capabl o opa n ann condon. oov, a h EKF can dal wh m-vayng lna plan modl c, h oo l and h paam can b mad whl h oo pd vayng ch a popod. ho, h appoach l o ao-nng and adapv dc ld-ond ndcon moo conol. Smlaon and pmnal d pnd n h pap hghlgh h mpovmn o h nw appoach bad on an ndd Kalman lng chnq nd al opaon condon. Rnc [] Khnan R., Bhaadwaj A. S.: A Rvw o Paam Snvy and Adapaon n Indc Vco Conolld Indcon oo Dv Sym, IEEE an. on Pow Elc., vol. 6, Ocob 99, n.º 4, pp [] Fnch J. W., Anon D. J., Acanly P. P.: Fll-Od Emao o Indcon oo Sa and Paam, IEE Elc. Pow Appl., ay 998, vol. 45, n.º, pp [] oon., albé G., Applcaon o h Endd Kalman Fl o Paam Emaon o Indcon oo, n Poc. EPE, Bghon, 99 pp [4] n F.-J.: Rob Spd-Conolld Indcon-oo Dv Ung EKF and RS Emao, IEE Poc.- Elc. Pow Appl., vol. 4, ay 996, n.º, pp [5] Anon D. J., Acanly P. P., Fnch J. W.: Obv o Indcon oo Sa and Paam Emaon, IEEE anacon on Indy Appl., vol. 7, Novmb/Dcmb 99, n.º 6, pp [6] Aqlla A. D., Cpno F., Salvao., Sa S.: Kalman Fl Emao Appld o Rob Conol o Indcon oo Dv, IECON 98, Aachn, Gmany, 998, pp [7] Wad S., Dnngan. W., Wllam B. W.: odlng and Smlaon o Indcon achn Vco Conol wh Roo Ranc Idncaon, IEEE an. on Pow Elc., vol., ay 997, n.º, pp [8] Aqlla A. D., Papa S., Salvao., Sa S.: A Dlayd Kalman Fl o On-n Emaon o Indcon oo Paam and Roo Fl Spac Vco Poon, EECON 96, pp [9] Aoy S., Ala I. H., E. K.: A Rdcd Od Roo m Conan and Fl Emao o Vco Conolld Indcon oo, EECON 96, pp [] Wal É., Ponzao.: Idncaon o Paamc odl om Epmnal Daa, aon, 997. [] Va P.: Paam Emaon, Condon onong, and Dagno o Elccal achn, Ood Scnc Pblcaon, 99. [] Aom, K. J., Wnma, B.: Comp Conolld Sym: hoy and Dgn, Pnc-Hall, 997. [] V., a H., Aaújo R., Fa D.: Sma Elcónco d Condconamno Pocamno, m mpo Ral, da nõ Con do oo d Indção áco Almnado po Convo d Fqênca, 7 a Jonada Hpano-a d Ingnía Elécca, add, Span, Jly,, vol. IV