Induction Machine Shaft Speed Estimation using Imaginary Power

Transcription

1 Induction Machine Shaft Speed Etimation uing Imaginay Powe Teence Summe, Robet Eic Betz, Bendan McGath, Gaham Cliffod Goodwin School of Electical Engineeing and Compute Science Univeity of Newcatle, Callaghan, NSW, Autalia, 238 Abtact Thi pape invetigate the ue of an imaginay powe baed peed etimation algoithm fo induction machine. The algoithm invetigated wa fit popoed by Peng and Fukao in 994, and ha ome vey deiable chaacteitic uch a inenitivity to knowledge of tato eitance, and doe not ue pue integation. Howeve, it ha been hown that the algoithm doe not opeate tably unde egeneation condition, which ha been the main eaon that it ha not een much application. Thi pape invetigate the eaon behind thi intability, and then develop a olution appoach. I. INTRODUCTION Etimation of the haft peed of induction machine ha been extenively eeached ove a numbe of yea 9. The motivation fo thi eeach ha been the deie to eliminate the fagile peed eno fom induction machine dive ytem, becaue it compomie the ugged chaacteitic of the machine. Alo in ome ituation it i not pactical to ue a peed eno becaue of a paticulaly hazadou envionment, o vey high haft peed. In addition, in ome application the cot of the peed eno i ignificant. The peed etimation technique dicued in the liteatue fall into thee boad categoie in tem of thei fundamental opeational pinciple: ) Uing the machine back emf o flux to detemine the otational peed of the oto 2, 9,. 2) Detection of oto lot hamonic in the tato cuent wavefom 5, 8. 3) Detection of the aliency in the oto of the machine 6, 7. Thi pape conide a peed etimation algoithm publihed by Peng and Fukao in 994. In tem of the categoie mentioned above it i a back-emf baed algoithm (i.e. Categoy ). The unique featue of the algoithm i that it ue intantaneou imaginay powe a the diving feedback ignal fo the etimato, which in pinciple mean that it would be inenitive the accuate knowledge of tato eitance. A futhe advantage of the algoithm compaed to, fo example ome aliency technique, i that it only equie a tandad machine. One of the ignificant poblem with Categoy peed etimation algoithm i that they implicitly ely on accuate knowledge of the machine tato eitance in ode to calculate the back-emf accuately. An accuate etimate i eential at low haft peed whee the tato eitance dop i vey ignificant compaed to the back-emf voltage itelf. One of the complication in etimating the tato eitance i that it i not contant but vaie ignificantly with the tempeatue of the machine. The ue of intantaneou imaginay powe in to etimate peed eliminate the tato eitance fom the etimation poce, ince imaginay powe doe not involve any eal powe component. Theefoe the algoithm, in pinciple, hould wok a well at low peed a it doe at highe peed. The Peng-Fukao algoithm ha a Model Refeence Adaptive Sytem (MRAS) tuctue, imila to 9. Howeve, it diffe in that it doe not ue pue integation, a doe 9 and many othe peed etimation technique. Conequently it avoid the initial condition and dift poblem aociated with pue integation baed appoache. Thi popety, coupled with the tato eitance independence, i the main motivation fo tudying the algoithm. In imulation and pactical eult wee peented that indicated that the algoithm woked vey well in all egion of opeation. The autho have caied out extenive imulation tudie of the algoithm and found that it poduce accuate peed etimate at all peed when the machine i motoing, but the peed etimate ae incoect unde egeneation condition. Thi pape invetigate and analye the poblem of incoect peed etimate unde egeneation fo the Peng- Fukao algoithm, and then popoe and imulate a olution appoach. The emainde of the pape i oganied a follow: Section II give a bief deivation of the Peng-Fukao etimato; Section III peent imulation eult of the etimato unde motoing and egeneation condition; Section IV peent ome analyi of the lineaied equation of the ytem; Section V peent a olution appoach and imulation eult; and finally Section VI peent the concluion. II. BRIEF DEVELOPMENT OF PENG-FUKAO ESTIMATOR Complete detail of the Peng-Fukao algoithm can be found in. In thi ection we hall develop the baic equation, uing tandad notation, that fom the bai of the etimato (which ae not deived in ). Conide the following definition fo the oto magnetiing cuent (expeed in a tationay efeence

2 fame) of an induction machine: e m Refeence Model i m = Ψ L m = i ( σ )i () whee σ L l L m (the oto leakage facto). Theefoe σ = L L m. Uing () the tandad tationay fame equation fo an induction machine can be manipulated into the following fom. Let u fitly conide the new fom of the oto equation fo a quiel cage machine (i.e. u = ): i i q i e m m di m i T L m F i m m H i K 2 ( ) ( I q i i ) L T i i m m m HG Adaptive Model F I KJ q m - q m PI di m = jω i m T i m T i (2) which can alo be witten a: di m = ω i m i m T i T (3) uing the fact the jω i m = ω i m. The tato equation can be witten a: di u = R i L L di m Reaanging () in tem of i one can wite (4) a: whee: di u = R i σl ( σ)l di m σ = L2 m L L The back-emf of an induction machine i the voltage behind the tanient eactance of the machine, and thi can be witten a: di e m = u R i σl By compaion with (5) one can ee that: di m e m = ( σ)l e m = L2 m di m L Clealy one can ubtitute (3) into () to give: ) e m = L2 m (ω L i m T i m T i The intantaneou imaginay powe i: (4) (5) (6) (7) (8) (9) () () q m = i e m (2) Subtituting () into (2) we can wite afte ome manipulation: ( q m = L2 m L ω (i m i ) ) (i m i T ) (3) Figue how the block diagam of the Peng-Fukao MRAS etimato with the elevant equation indicated. Remak : Note that in Figue and the following equation a ˆx ymbol indicate that x i an etimated vaiable. Fig.. MRAS tuctue of the Peng-Fukao haft peed etimato. III. SIMULATION RESULTS OF PENG-FUKAO ESTIMATOR The etimato hown in Figue wa pogammed into the Sabe R mixed mode imulation package. The imulation wa configued o that the induction machine wa diven fom a fixed fequency upply and the haft of the machine wa contolled by a contollable peed ouce. Thi allowed the induction machine to be manipulated into vaiou mode of opeation imply, with all the condition of opeation known. Pio to etablihing thi imulation configuation the etimato wa ued in conventional field oiented contol (FOC) imulation. Howeve the condition of opeation wee not well enough contolled, and it wa difficult to etablih what wa happening to the etimato. Figue 2 how the eult of a imulation. The top two plot ae the dq tationay fame oto magnetiing cuent fo the machine i.e component of i m. Vaiable, maked in the figue a imd and imq ae the tue value of the oto magnetiing cuent, and imd pime and imq pime ae the etimated value of the ame vaiable. The voltage and peed of the machine ae amped up fo the fit econd of opeation (not hown hee). Afte t = econd the peed i held at a contant value that i jut below the ynchonou peed of the machine (ynchonou peed i 377 ad/ec a the machine i being diven fom a contant 6Hz upply) i.e. the machine i motoing up to t =.4 econd. Remak 2: The etimated peed i vey accuate fom t = to t =.4 econd. At t =.4 econd the machine haft peed i inceaed though the ynchonou peed and dive the machine into egeneation. One can ee that the etimated and actual peed match fo a hot peiod afte thi tanition and then the etimated peed divege fom the tue peed and ettle back to a peed that indicate that the machine i again motoing. Meanwhile the tue peed emain contant until t =.6 econd, when it i again deceaed below ynchonou peed. One can ee a the tue peed dop below ynchonou peed the etimated peed elock to the tue peed. 2

3 it it in dq component: dî md dî mq = (ω ˆω )î mq T î md T i (6) = (ω ˆω )î md T î mq (7) Fig. 2. Cloe-up of cuent and peed etimate fo the Peng-Fukao etimato. Remak 3: When the etimated peed divege fom the tue peed thee i a phae hift in the etimated î m fom the tue i m cuent. If one view the i vecto at thi time it i contant. The phae change in the î m can be een in Figue 2. Remak 4: The Peng-Fukao algoithm wa extenively teted on a FOC ytem and the ame popetie wee obeved when the machine egeneate the etimated peed divege fom the tue peed. Remak 5: Alo in Figue 2 one can ee that the q m ˆq m eo duing the etimated peed divegence doe not change. Conequently thee i no feedback to coect the etimation eo that occu. IV. ANALYSIS OF PENG-FUKAO ESTIMATOR Fom equation (3) and (3) it can be een that etimato i non-linea it ha bilinea tem if one conide the input into (3) to be ˆω. Note that the algebaic output equation (3) ha even moe complex non-lineaitie. Theefoe in ode to examine the obevability, contollability and tability of the etimato equation one need to lineaie them aound the opeating point of inteet. In ode to implify the analyi of the equation we hall convet () and (3) into a ynchonouly otating efeence fame aligned with the tato cuent vecto. The eultant equation ae: dî m = j(ˆω ω )î m î m i T T (4) ( ˆω î md ) î mq i T (5) ˆq m = L2 m L whee ω i the otational angula velocity of the tato cuent vecto, and î md and î mq ae the dq component of the î m vecto. The ubcipt indicate that the quantitie ae with efeence to the tato cuent vecto. Equation (4) can be futhe manipulated to epeent Remak 6: The following lineaiation will aume that the i vecto i not changing i.e. d i = and dω =. Thi implifie the equation, and i epeentative of the condition whee the Peng etimato mibehave i.e. unde a condition whee the tato cuent i fixed and the oto peed i fixed, but unde egeneation. Applying tandad technique fo lineaiing non-linea diffeential equation, unde the contant i condition, allow u to wite: d(δî md ) d(δî mq) = (ω ˆω )(δî mq) = î md(δˆω ) T (δî md) î mq(δˆω ) (8) (ω ˆω )(δî md) T (δî mq) (9) δˆq m = L2 m L i î md (δˆω ) ˆω (δî md) T (δî mq) (2) whee δx denote a mall ignal tate vaiable o contol input o ytem output. The above equation can be witten in conventional tate pace fom a follow: whee: d(δî m) = m) B(δˆω ) (2) δˆq m = C(δî m) D(δˆω ) (22) δî (δî m) = md δî mq T A = (ω ˆω ) (ω ˆω ) T î B = mq î md C = L2 m L i ˆω T D = L2 m L i î md (23) (24) (25) (26) (27) To check fo contollability and obevability the fol- 3

4 lowing matice mut have full ank: CM: B AB A 2 B A n B (28) C CA OM: CA 2 (29). CA n whee n the numbe of tate. Let u conide the obevability matix. Subtituting in the value fo C and A and ealiing that n = 2, we get the following matix: C CA = L2 m L i ˆω T ˆω T (ω ˆω) T ˆω (ω ˆω ) T 2 (3) If thi matix doe not have full ank then deteminant of the matix will be zeo. The deteminant i: C det = L2 m i CA L (ω ˆω )ˆω 2 T 2 (3) Clealy thi can only be zeo if (ω ˆω ) = i.e. the lip i zeo. Thi i the only condition that the ytem i unobevable. Now let u check on contollability. î î mq B AB = mq T î md (ω ˆω ) î md î mq(ω ˆω ) îmd T (32) Thi matix hould be full ank fo the ytem to be contollable. Thi mean that the deteminant hould be non-zeo. Taking the deteminant we get: det B AB = (ω ˆω )(î 2 md î 2 mq) (33) Clealy the deteminant i equal to zeo when: (ω ˆω ) = (34) and theefoe the ytem i uncontollable unde thi condition. Let u now conide the mall ignal tability of the etimato. Uing the lineaied equation (2) and (22) with teady tate opeating point fo both motoing and egeneation we can plot the pole and zeo of the open loop and cloed loop tanfe function fo the etimato. Let u fitly conide the open tability of the etimato fom q to ˆq m. The pole-zeo plot fo PI gain of i hown in Figue 3. One can conclude that the ytem i table, with the zeo and pole nea to oigin due to the PI contolle. If the etimation loop i now cloed, with the machine motoing and gain of K p = K i = the pole-zeo plot of Figue 4 eult. Again the etimato i table and i in fact table fo all feedback gain in the motoing mode of opeation. Stictly peaking (ω ˆω ) i not the lip fequency, ince taniently ω i not the ame fequency a ω m. Howeve, unde the tet condition they ae the ame. Imaginay Axi Open Loop Pole Zeo Plot q q hat (ω = 34 ) Fig. 3. Etimato open loop pole-zeo plot fo K p = K i =, motoing mode. Imaginay Axi Cloed Loop Pole Zeo Plot q q hat (ω = 34 ) Fig. 4. Etimato cloed loop pole-zeo plot fo K p = K i =, motoing mode. If the poce i epeated fo egeneation vey diffeent eult ae obtained. Fo PI gain of (a ued in the motoing mode analyi) the open loop pole zeo plot of Figue 5 eult. The etimato i open loop table again with a pole and zeo nea the oigin due to the PI contolle. Remak 7: Note that the lineaied etimato i nonminimum phae with a zeo in the ight half plane. Cloing the loop give a pole-zeo plot a hown in Figue 6. The etimato in egeneation i untable with thee gain (ee Remak 8). In fact the etimato i untable in cloed loop fo all gain with negative feedback. Root locu plot have been geneated to confim thi but ae not hown hee fo conciene. Remak 8: One can ee that thee i an appaent pole zeo cancellation in the ight half plane of Figue 6. Howeve if one expand the plot hee one can ee that the pole and zeo ae not co-incident and theefoe thee 4

5 Open Loop Pole Zeo Plot q q hat (ω = 4 ) Cloed Loop Pole Zeo Plot q q hat (ω = 4 ) Imaginay Axi 5 5 Imaginay Axi Fig. 5. Etimato open loop pole zeo plot fo K p = K i =, geneating mode. Fig. 7. Etimato cloed loop pole zeo plot fo K p = K i = and poitive feedback Imaginay Axi Cloed Loop Pole Zeo Plot q q hat (ω = 4 ) Fig. 6. Etimato cloed loop pole zeo plot fo K p = K i =, geneating mode. would be a mall untable eidual hee. The cloed loop ytem in the geneating mode can be mall ignal tabilied if the ign of the feedback i changed and the gain i educed enomouly. Figue 7 how a cloed loop pole-zeo plot of the etimato with a feedback ign change and popotional and integal gain of.. With thi vey low gain howeve the etimato i almot completely uele due to vey poo dynamic pefomance. Remak 9: Oveall, one can conclude, at the opeating point of the lineaiation, that motoing i table fo all feedback gain if negative feedback i applied. In egeneation the etimato i untable fo all gain if negative feedback i utilied. The manifetation of intability in geneating mode i low if high gain ae ued becaue thee i almot a pole-zeo cancellation at the ight half plane pole. Howeve, fo modet gain (.) egeneation can be tabilied by eveing the ign of the feedback. The etimato cetainly ha tability poblem, and will equie additional feedback in ode to tabilie it. V. AUGMENTED ESTIMATOR AND SIMULATION RESULTS A numbe of technique wee invetigated to geneate the neceay feedback to tabilie the etimato. Obevation made fom the imulation tudie indicated that the intability in the etimato eulted in the î m pace vecto angle moving, unde egeneation condition, elative to the tato cuent pace vecto i. Theefoe all the technique developed attempted to tackle thi phae hift iue by poviding coective action. We hall only conide one technique in thi pape. The geneation of the neceay feedback ignal to implement the phae locking function i imple, and can be expeed a: ê m e m = ê m e m in( θ m ˆ θm ) ê m e m ( θ m ˆ θm ) = ê m e m δ θ m (35) fo mall θ m ˆ θm, whee θ m denote that thi i the angle of the back-emf, a oppoed to the i m cuent. Clealy one can geneate a nomalied veion of (35) a follow: δ θ m êm e m (36) ê m e m The popoed olution i to ue the phae diffeence between the back-emf deived fom the efeence model via (8) and that deived fom etimato equation (). The idea in thi cae i to phae lock the angle of ê m to e m, and theeby lock î m to i m. Remak : Thi appoach till ha the poblem that thee i an implicit dependence on knowledge of R, ince the phae angle of e m will vay conideable fom the tue veion of thi vaiable at low peed. Theefoe the ê m, and hence î m will be foced towad an incoect phae. On the plu ide, thi appoach ue expeion that ae 5

6 em Refeence Model qm q m = i Ä em q m i q m - qˆm S PI - q$ m em Ä Adaptive Model diˆm Ä em diˆm qm i é L ù ê 2 q m - (iˆm Ä i ) ú êl T ûú wˆ = ë m iˆm e i diˆm = wˆ Ä iˆm - (iˆm - i ) T i wˆ Fig. 8. iˆm S qˆm = wˆ wˆ L2m æ ö q$ ç wˆ (iˆ e i ) (iˆm Ä i ) m ø T L çè m wˆ Modified Peng peed etimato with phae locking. Fig.. R. Simulation eult fo phae lock feedback with 3% eo in the etimato till emain table in the ene that the etimated peed indicate that the machine i egeneating when indeed it i egeneating. Remak : Fom Figue 9 and one can ee that the i m cuent ae in phae with the tue im cuent. Remak 2: Simulation wee alo caied out uing the nomaliation indicated in (36) with imila eult. VI. C ONCLUSIONS Fig. 9. The following concluion can be dawn fom thi pape: A lineaied veion of the oiginal Peng-Fukao etimato ha been hown to be table fo all feedback gain in motoing mode and untable in egeneation unle the ign of the feedback i eveed and the feedback gain i vey low. Simulation of the full non-linea etimato confim the lineaied intability. The Peng-Fukao etimato i both obevable and contollable, which indicate that, in pinciple, the intability poblem can be emedied with appopiate feedback. The etimato can be tabilied by adding auxiliay feedback which phae lock the di m / vecto to em. Thi inevitably lead to dependence on knowledge of the tato eitance. An augmented algoithm uing thi appoach ha been devied and imulated, and the etimato i table. Simulation eult of phae lock feedback with actual R. aleady pat of the etimato, and the appoach doe not ely on pue integation. Thee ae a vaiety of appoache fo feeding back the phae infomation. In thi pape we will feed the output of (35) a an augmentation of the ω ignal back into the etimato. In addition thee i a feed-fowad tem of ω which i ummed into the feedback path. Thi allow the gain of the PI contolle to be lowe. The block diagam of thi tuctue i hown in Figue 8. A can be een the aw value of the appoximate phae eo between the di m / vecto and the em vecto i fed back into the peed feedback loop of the etimato. The imulation eult of thi tategy appea in Figue 9. A can be een fom thi figue the intability poblem evident in Figue 2 do not occu. One can ee that the phae of the etimated magnetiing cuent emain locked to the tue value. The gain of the feedback i low in the cae of the imulation hown it i.. In thi imulation the tato eitance ued to deive em i the tato eitance of the machine. A econd imulation wa caied out with the tato eitance ued fo em equal to.3r. The eult of thi imulation ae hown in Figue. A can be een fom thi figue the eo in the tato eitance lead to a ignificant eo in the etimated peed. Howeve, R EFERENCES F. Z. Peng and T. Fukao, Robut peed identification fo peed enole vecto contol of induction machine, IEEE Tanaction on Induty Application, vol. IA-3, no. 5, pp , J. Holtz and J. Juliet, Senole acquition of oto poition angle of induction moto with abitay tato winding, IEEE Tanation on Induty Application, vol. 4, no. 6, pp , Nov-Dec R. Betz and T. Summe, Speed etimation fo induction machine uing imaginay powe, in CDROM Recod of the IEEE-IAS Annual Meeting, Salt Lake City, Oct

7 4 G. Bottigliei, G. Scelba, G. Scacella, A. Teta, and A. Conoli, Senole peed etimation in induction machine, in Poceeding of Electic Machine and Dive Confeence, vol.. IEEE, June 23, pp A. Feah, P. Hogben-Laing, K. Badley, G. Ahe, and M. Woolfon, The effect of oto deign on enole peed etimation uing oto lot hamonic by adaptive digital filteing uing the maximum likelihood appoach, in Confeence Poceeding of the IEEE IAS Annual Meeting. IEEE, Oct. 997, pp M. Schoedl, Senole contol of ac machine at low peed and tandtill baed on the infom method, in Confeence Recod of the 3t Annual Meeting of the IAS, vol.. IEEE, Oct. 996, pp P. L. Janen and R. D. Loenz, Tanducele field oientation concept employing atuation-induced aliencie in induction machine, IEEE Tanaction on Induty Application, vol. 32, no. 6, pp , Novembe/Decembe K. Hut, T. Habetle, G. Giva, and F. Pofumo, Speed enole field-oiented contol of induction machine uing cuent hamonic pectal etimation, in Confeence Recod of Induty Application Society Annual Meeting, vol.. IEEE, Oct. 994, pp C. Schaude, Adaptive peed identification fo vecto contol of induction moto without otational tanduce, IEEE Tanaction on Induty Application, vol. 28, no. 5, pp. 54 6, Septembe/Octobe 992. J. Jiang and J. Holtz, High dynamic peed enole ac dive with on-line model paamete tuning fo teady-tate accuacy, IEEE Tanaction on Indutial Electonic, vol. 44, no. 2, pp , Apil