Dynamical models for fault detection in squirrel cage induction motors

Transcription

1 Int. J. Citical Infatuctu, Vol., No. 1/, Dynamical modl fo fault dtction in quil cag induction moto H. Rodíguz-Coté* Dpatmnt of Elctical and Comput Engining Nothatn Univity Boton, A 115, USA cot@c.nu.du *Coponding autho C.N. Hadjicoti Dpatmnt of Elctical and Comput Engining Univity of Illinoi Ubana, I 6181, USA chadjic@uiuc.du A.. Stankovi! Dpatmnt of Elctical and Comput Engining Nothatn Univity Boton, A 115, USA tankov@c.nu.du Abtact: Induction moto a th dominant componnt in indutial poc involving lctomchanical ngy convion. Safty, liability and fficincy a majo concn in modn induction moto application. Sinc dtcting fault on tim could avoid cotly unchduld hutdown, in cnt ya th ha bn an incad intt in induction moto fault dtction and diagnoi. In thi pap, w popo monitoing chm to olv fault dtction poblm of induction moto. W bgin with a monitoing chm to dtct dtuning opation in Indict Fild Ointd Contol (IFOC) divn induction moto. Scondly, w pnt a monitoing chm to dtct bokn oto ba on IFOC-divn induction moto. Th popod monitoing chm do not ly on pctal mthod; intad, it monito a cafully lctd induction moto tat, uing an onlin obv. Th ky to fault dtction i th dvlopmnt of a implifid dynamic modl of a quil cag induction moto with bokn oto ba. Numical imulation validat both monitoing chm. Kywod: fault dtction; dtuning; bokn oto ba; induction moto; diffntial gomty. Rfnc to thi pap hould b mad a follow: Rodíguz-Coté, H., Hadjicoti, C.N. and Stankovi!, A.. (7) Dynamical modl fo fault dtction in quil cag induction moto, Int. J. Citical Infatuctu, Vol., No. 1/, pp Copyight 7 Indcinc Entpi td.

2 16 H. Rodíguz-Coté, C.N. Hadjicoti and A.. Stankovi! Biogaphical not: H. Rodíguz-Coté obtaind hi BSc dg in Aonautic fom th National Polytchnic Intitut of xico, S dg in Automatic Contol fom CINVESTAV-IPN, xico and PhD in Automatic Contol fom Pai-Sud Univity, Fanc in 1995, 1997 and, pctivly. Sinc Januay, h ha hld a viito chola poition at Nothatn Univity. Hi ach intt a in th fild of adaptiv non-lina contol. C.N. Hadjicoti civd hi SB dg in Elctical Engining, in Comput Scinc and Engining, and in athmatic; Eng dg in Elctical Engining and Comput Scinc in 1995, and PhD dg in Elctical Engining and Comput Scinc in 1999, all fom th aachutt Intitut of Tchnology, Cambidg, A. In Augut 1999, h joind th Faculty at th Univity of Illinoi at Ubana-Champaign wh h i cuntly an Aociat Pofo with th Dpatmnt of Elctical and Comput Engining and a Rach Aociat Pofo with th Coodinatd Scinc aboatoy. Hi ach intt a in fault diagnoi and tolanc, o and contol coding and dict vnt ytm. A.. Stankovi! obtaind hi Dipl Ing dg fom th Univity of Blgad, Yugolavia in 198, S dg fom th am intitution in 1986 and PhD dg fom th aachutt Intitut of Tchnology in 199, all in Elctical Engining. D. Stankovi! intt a in modlling and contol poblm in pow lctonic and pow ytm. 1 Intoduction Taditionally, fault dtction poblm in ngy pocing ytm hav bn addd in a modl-f famwok. A cafully lctd ignal i monitod in tim o fquncy domain in od to tack dviation fom it xpctd valu. Only mo laboat monitoing chm includ th idntification of fal alam. Sinc fal alam could cau cotly hutdown, thi tat of affai i not compltly atifactoy. Impovmnt that hav bn conidd, motly in th ach litatu, includ th u of dynamical modl. Two main tumbling block main on thi appoach: on i th nd to tailo th lvl of dtail fo componnt modl (o a to kp th ovall modl tactabl) and th oth i th nd to diciminat th ouc of poibl miclaification in a ytmatic fahion. Thi pap pnt a viw of modl-bad fault dtction via analytic dundancy to dvlop a comphniv famwok fo fault dtction in ngy pocing ytm. Thi famwok i thn mployd a a tool to dign monitoing chm fo th induction moto, which i th dominant componnt in indutial poc involving lcto-mchanical ngy convion. Givn th main impdimnt to ou appoach, w intoduc dynamic modl with two impotant chaactitic: impl nough to b tactabl and dtaild nough to captu th fault ffct of intt. W plac a pcial mphai on th limination of th ffct of th ouc of fal alam on th monitoing chm.

3 Dynamical modl fo fault dtction in quil cag induction moto 16 Th pap i oganid a follow. Sction viw modl-bad fault dtction tchniqu. Th viw cov failu dtction via analytic dundancy, tating fom th pioning wok of Bad and Jon, and nding at th dtction chm bad on diffntial gomtic tchniqu. Sction popo a monitoing chm to dtct dtund opation of Indict Fild Ointd Contol (IFOC) divn induction moto. Sction 4 pnt a monitoing chm to dtct bokn oto ba in IFOC-divn quil cag induction moto, and i followd by th concluding mak. odl-bad fault dtction Hadwa dundancy appa to b th taditional ngining mthod to achiv fault dtction in dynamic ytm. Rpatd no, actuato and ytm componnt a placd aound th ytm to achiv fault dtction. Such chm oftn opat in a tiplx o quaduplx configuation, and th dundant ignal a compad fo conitncy. Th implicity of thi appoach account fo it wid u. Howv, th implicity i paid fo in tm of th conomic cot of implmntation, and th quid xta pac to accommodat dundant componnt. oov, it i cognid in pactic that imila componnt tnd to fail at th am tim of opation. Sinc th aly 197, nw mthod fo fault dtction hav mgd to ovcom th dawback of hadwa dundancy,.g., functional dundancy and analytic dundancy. In th functional dundancy appoach, th dundant hadwa lmnt a contuctd fom diimila componnt and th nw output a compad with mo ophiticatd chm. Analytical dundancy chm a baically ignal pocing tchniqu, uing tat timation, paamt timation, tatitical dciion thoy and logical and combinational opation. All ignal pocing tchniqu a pactically implmntd in lctonic cicuit and comput. Th baic function of a fault dtction chm i to poduc an alam whn a fault occu in th monitod ytm, and in a cond tag to idntify th faild componnt. Idally, th alam hould b a binay ignal announcing that a fault i affcting th ytm o that no fault i hamping th ytm. Anoth impotant chaactitic fo th alam i th pd of dtction. It i diabl to hav a fault dtction chm that poduc an alam immdiatly aft th occunc of th fault. Finally, th mot impotant chaactitic of th alam i th at of fal alam, a fal alam mak th pfomanc of fault dtction chm dtioat. Thu, a gat ffot in th dign of fault dtction chm focu on th poblm of gnation of alam poviding ffctiv dicimination btwn diffnt fault, ytm ditubanc and modlling unctainti..1 Fault dtction via analytic dundancy Ridual, alo dnotd a alam, a quantiti xping th diffnc btwn th actual plant output and tho xpctd on th bai of th applid input and th mathmatical modl. Thy a obtaind by xploiting dynamic o tatic lationhip among no output and actuato input. An impotant chaactitic fo idual i that thy nd to b obut with pct to th ffct of nuianc fault, othwi, nuianc fault will obcu th idual pfomanc by acting a a ouc of fal alam.

4 164 H. Rodíguz-Coté, C.N. Hadjicoti and A.. Stankovi! Th gnal pocdu of Fault Dtction, Iolation and Accommodation (FDIA) in dynamic ytm with th aid of analytical dundancy conit of th following th tp (Patton t al., 1989): 1 gnation of function that cay infomation about th fault, o-calld idual dciion on th occunc of a fault and localiation of th fault, o-calld iolation accommodation of th faulty poc, and tanition to nomal opation. Thi pap focu attntion on th gnation of idual uing th analytic dundancy famwok with pioity on th obutn with pct to nuianc fault. Robutn of th idual gnato with pct to nuianc fault undtood a a complt dcoupling fom th idual to nuianc fault i on of th mot difficult poblm in fault dtction. To dat, much of th wok on th gnation of idual pfomd in th analytic dundancy famwok obv two impotant tndnci: failu dtction filt and paity lation. Th imilaiti btwn th two appoach lad to th am idual fo lina ytm and fo om non-lina ytm (Gtl, ).. Failu dtction filt Bad (1971) wa th fit to popo a fault dtction filt, find lat by Jon (197). In thi filt, known a th Bad-Jon Dtction (BJD) filt, th achabl ubpac of ach fault a placd into invaiant indpndnt ubpac. Thn, whn a non-zo idual i dtctd, th fault i idntifid by pojcting th idual onto th achabl ubpac of fault and compaing th pojction with a thhold. Evn though multipl fault can b dtctd with thi filt, th appoach i vy tictiv a fault hav to atify a mutual dtctability condition (aoumnia, 1986). Futh impovmnt to th BJD filt w uggtd in (aoumnia t al., 1989) with th Rtictd Diagonal Dtction (RDD) filt. In thi appoach, fault a dividd into fault that nd to b dtctd (o-calld tagt fault) and nuianc fault (uch a paamt unctainti, chang in ytm paamt and noi). Nuianc fault a pojctd onto th unobvabl pac of idual whil maintaining obvability of tagt fault; thn, tagt fault a idntifid a in th BJD filt. Whn vy fault i dtctd, BJD and RDD filt a quivalnt. Th mot cnt vion of th BJD filt, o-calld Unknown Input (UI) obv appoach, wa popod in (Patton t al., 1989) uing th igntuctu aignmnt and th Konck canonical fom a dign mthod, and in (aoumnia t al., 1989) uing gomtic tchniqu. In th UI obv appoach, nuianc fault a pojctd onto th unobvabl ubpac o that idual a influncd only by th tagt fault, in thi way, implifying th dciion tak. It hould b pointd out that vn though th u of UI obv in fault dtction and iolation wa fit popod in (aoumnia t al., 1989; Patton t al., 1989), UI obv hav civd ignificant attntion only aft th pioning wok of Baill and ao, pntd fo intanc in (Baill and ao, ). W conid th UI obv appoach with gomtic tchniqu. In th non-lina tting, th idual gnation poblm uing analytic dundancy ha bn addd in (Hammoui t al., 1998) fo tat-affin ytm and lat in (D Pi and Iidoi, 1) fo input-affin ytm. In th wok, idual gnato contuction i bad, und mild additional aumption, on th xitnc of an

5 Dynamical modl fo fault dtction in quil cag induction moto 165 unobvability ubpac (ditibution) lading to a ubytm unaffctd by all fault ignal but th fault of intt; thn an aymptotic obv fo uch a ubytm, which in th non-lina ca may not xit, yild th idual gnato. Conid th following non-lina ytm: wh x!! f ( x) " g( x) u " l ( x) m " l ( x) m y! h( x) n x # " i th tat, l 1 m t # ", l n n n t t k y # " i th mauabl output, m u # " (1) i th input and m # " a abitay unknown function of tim, pnting th tagt failu mod and th nuianc failu mod, pctivly. Duing fault-f opation, m t and m n a qual to zo. Th column of f(x), g(x), l m (x), l t (x) and h(x) a mooth vcto fild with l m (x) and l t (x) dnoting th actuato failu ignatu. Th idual gnation poblm may b tatd a follow. Poblm 1 Conid th non-lina ytm dcibd by Equation (1). Dign, if poibl, a dynamic idual gnato with tat x # R of th fom: wh xˆ!! F( xˆ, y) " E( xˆ, y) u! ( xˆ, y), F( xˆ, y ) and th column of E( xy ˆ, ) a mooth vcto fild and () ( xˆ, y ) i a mooth mapping, that tak y and u a input and gnat th idual with th following local popti: 1 Whn th tagt failu m t i not pnt, th idual gnato Equation () i aymptotically tabl and dcay aymptotically to zo, that i, th tanmiion fom th input u and th nuianc fault m n to th idual i zo. Fo a non-zo tagt fault, th idual i non-zo. Condition 1 conid th tability of th idual gnato and au that th input ignal u and th nuianc fault m n do not affct th idual. Condition guaant that th tagt fault affct th idual. In D Pi and Iidoi (1), a ncay condition fo th xitnc of a olution to Poblm 1 i givn. Thi condition, und mild aumption, lad to a ubytm divn only by th fault of intt. Thu, a olution to Poblm 1 can b found povidd uch a ubytm admit an obv. Spcifically, aum that th minimal unobvability ditibution of Equation (1), dnotd by!*, containing th imag of th nuianc fault ignatu n = pan{l n (x)}, i locally non-ingula. In D Pi and Iidoi (1), it i hown that if:!* # pan{ l ( x )}! {} () t thn it i poibl, und ctain condition, to find a tat diffomophim and an output diffomophim: $ z1 % & $ w % z! & z! ( x! ) y & z * + 1 ( ), & ( ) * w + (4)

6 166 H. Rodíguz-Coté, C.N. Hadjicoti and A.. Stankovi! uch that in th nw coodinat, th ytm Equation (1) i dcibd by quation of th fom: z! 1! f$ 1( z1, z ) " g$ 1( z1, z ) u " l$ t1( z) mt z!! f$ ( z) " g$ ( z) u " l$ ( z) m " l$ ( z) m z!! f$ ( z) " g$ ( z) u " l$ ( z) m " l$ ( z) m w! h$ ( z ) t t n n t t n n w 1 1 1! z, fom wh it i poibl to xtact a ubytm divn only by th tagt fault a: z! 1! f$ 1( z1, w ) " g$ 1( z1, w ) u " l$ t1( z) mt w! h$ ( z ) (5) Claly, whn it i poibl to contuct an obv fo Equation (5), Poblm 1 i olvabl. Th computation of th minimal unobvability ditibution!* containing n can b computd a th lat lmnt of th following qunc (D Pi and Iidoi, 1): S! W * " K{ dh}, S! W * "[ f, S # K{ dh}] i i, 1 i, 1 "[ g, S # K{ dh}], i! 1,..., k, (6) wh k " n 1 i dtmind by th condition S k = S k 1. Concning W*, it i computd a th lat lmnt of th following qunc: W! P, W1! Wi, 1 " $ * f, Wi, 1 # K{ dh} % + " $ g, W # K{ dh} %, i! 1,..., k, * i, 1 + (7) with k " n 1 dtmind by th condition W i + 1 = W i. In Equation (6) and (7), [.,.] i th i poduct, X dnot th involutiv clou of X, and P = pan {l n (x)}. Dtuning dtction and accommodation on IFOC divn induction moto In mot high-pfomanc application of lctic div (lik pd and poition vo), th contol tuctu utilid i th o-calld fild-ointd (vcto) contol. It allow fo (almot) dcoupld contol of toqu and flux, and yild a vy fat tanint pon. In th ca of a wll-tund contoll, th main pfomanc limitation com fom th cunt bandwidth of th div. With modn pow lctonic witching dvic (lik IGBT) that bandwidth i wll abov a khz, ulting in outtanding lcto-mchanical pon. In th ca of a dtund opation, howv, th pfomanc can dgad ubtantially, both in tanint (toqu command following) and in tady tat

7 Dynamical modl fo fault dtction in quil cag induction moto 167 (fficincy). If th dtuning i caud by a typically non-monitod poc (lik oto tim contant vaiation, o a low dgadation of th haft poition no infomation), it may go undtctd fo a whil, and lad to datic fficincy duction, o vn a had fault. In thi pap w popo modl-bad chm to dtct th dtuning poc in a gnal pupo fild-ointd induction div moto. It i bad on diffntial gomtic conidation, and it i immun to a numb of tanint that nomally occu in a highpfomanc div, lik load toqu vaiation..1 Dtund opation of cunt-fd indict fild-ointd contolld induction moto It i wll known that mchanical commutation implifi ignificantly th contol tak in DC moto. Th action of th commutato i to v th diction of th amatu winding cunt a th coil pa th buh poition o that th amatu cunt ditibution i fixd in pac gadl of th oto pd. Thu, th fild flux poducd by th tato and th agnto-otiv Foc (F) catd by th cunt in th amatu winding a maintaind in a mutually ppndicula ointation indpndnt of th oto pd. Th ult of thi othogonality i that th fild flux i pactically unaffctd by th amatu cunt, o that, whn th fild flux i kpt contant, th poducd lcto-mchanical toqu i popotional to th amatu cunt. Highly dynamic pfomanc can b obtaind uing two lina contol loop, on (low) contolling th fild flux and th oth (fat) on contolling th amatu cunt. In induction moto fild flux and amatu F ditibution a not othogonal, nding th analyi and contol of th dvic mo complicatd. Howv, th action of th commutato of a DC machin in holding a fixd, othogonal patial angl btwn th fild flux and th amatu F can b mulatd in induction machin by ointing th tato cunt with pct to th oto flux o a to attain pactically indpndnt contolld flux and toqu. Such contoll a calld fild-ointd contoll and qui indpndnt contol of both magnitud and pha of th AC quantiti. An undtanding of th dcoupld flux and toqu contol ulting fom fild ointation can b attaind fom th modl of an induction machin in th fixd tato fam 1 (Kau t al., 1995; ohan, 1; Novotny and ipo, 1996): d 4 5 / i i 1 1 dt 8 9 m m m -!, 6 " 7 - ". " - " - d 4 5 / i i 1 dt 8 9 1! 1 m 1-!, 1-, 1. " i - m m m.!, 6 " 7. " - ". ".! 1 m 1!, 1 " 1 " i.. -. J P m!! ( 1- i., 1. i- ), P (8)

8 168 H. Rodíguz-Coté, C.N. Hadjicoti and A.. Stankovi! wh i -. a th tato cunt, 1 -. a th oto flux linkag, -. a th tato P voltag, m! i th mchanical vlocity, P i th numb of pol in th machin, J i th otational intia, i th toqu load,, a th tato and oto itanc,, m, m a th tato, oto and mutual inductanc,! and /!,. Th fild ointation concpt impli that th cunt componnt upplid to th machin hould b ointd in pha (flux componnt) and in quadatu (toqu componnt) to th oto flux vcto 1-.. Thi can b accomplihd by chooing to b th intantanou pd of 1-. and locking th pha of th fnc ytm to th diction of th magntiing flux 1, that i: $ 1 % $ 1 J: - %, &! & * + * 1. + (9) wh: Jx $ co( x), in( x) %! &. in( x) co( x) * + Auming that th machin i upplid fom a cunt gulatd ouc, th tato quation can b omittd; th oto dynamic, in tm of tato cunt and oto flux, in a oto fild-ointd fam i dcibd by th following quation:! 1 1 (1) 1!, " i d i m q! " (11) 1 J P m!! 1iq, (1) P, J: with i! i. dq -. Equation (1) (1) dcib th dynamic pon of a fild-ointd induction machin and a ntially paalll th DC machin dynamic. Equation (1) copond to th fild cicuit on a DC machin. Equation (11) dfin what i commonly calld th lip fquncy!,, which i inhntly aociatd with th diviion of th input tato cunt into th did flux and toqu componnt. Th lcto-mchanical toqu in Equation (1) how th did toqu contol popty of poviding a toqu popotional to th toqu command cunt i q. Th implmntation of fild ointation can b aily caid out, povidd th poition angl of th oto flux : i known. Th a two baic appoach to dtmin : : dict chm, which dtmin th angl fom flux maumnt, and indict chm, which mau th oto vlocity and utili th lip fquncy to comput th

9 Dynamical modl fo fault dtction in quil cag induction moto 169 angl of th oto flux lativ to th oto. Th indict mthod u th fact that a ncay condition to poduc fild ointation i to atify th lip lation. An indict fild-ointd contoll i dcibd by th following quation: ˆ! 1 ˆ 1!, 1 " i ˆ ˆ ˆ iˆ m q ˆ! " ˆ ˆ 1 d (1) wh ˆ i th timatd ynchonou vlocity, ˆ 1 i th timatd oto flux linkag, ˆ i th timatd oto tim contant and i ˆdq a th maud tato cunt. Th indict fild-ointd contoll togth with th cunt contoll and th pd contoll a hown in Figu 1. Fom Equation (1), (11) and (1) w obv that ˆ = and ˆ 1 = 1, povidd all paamt a accuatly known. A a ult, it follow that : ˆ! : and th maud tato cunt i ˆdq a qual to th actual tato cunt idq. It i aonabl to aum that w hav a good timat of m ; howv, th oto tim contant i uually not xactly known a it chang bcau of moto hating, mimatch in manufactuing o oth vaiation. A mimatch in th oto tim contant ult in a lo of th coct fild ointation, lablld a dtuning of th contoll. Th main conqunc of dtuning a: ;< Th flux lvl i not poply maintaind. ;< Th ulting tady tat i not th commandd valu. ;< Th toqu pon i dgadd. ;< Th fficincy i dgadd and th moto hating inca. Figu 1 Indict fild-ointd contol diagam ω * SPEED REGUATOR i * q i * d CURRENT REGUATOR vdq IFOC θ dq αβ i dq αβdq i αβ ω INDUCTION OTOR vαβ

10 17 H. Rodíguz-Coté, C.N. Hadjicoti and A.. Stankovi! Nxt, w obtain a dynamic modl that account fo th dtuning ffct. Although th tu fild-ointd induction moto dynamic Equation (1) (1) xit in th moto, it tat cannot b maud dictly. In fact, th only mauabl output of th induction moto a th tato cunt in th abc fam and th mchanical oto pd. Sinc it i aumd that th induction moto i wy connctd, th cunt i -. can b obtaind fom i abc a follow: i-! ia 1 i.! ( ib, ic ). A obvd in Figu, th lation btwn cunt, J: idq i-. (14) i -. and cunt i dq i dfind by:! (15) Not now that in a dtund condition (: = ˆ : ), th maud tato cunt a givn a: ˆ ˆ, J: i! i (16) dq -. thu, fom Equation (15) and (16), w conclud that:, J $ : i! iˆ dq dq (17) with $ :!:, ˆ :. Figu Fixd fam -. and otating fam dq and dq ˆ. q β d q θ d θ α Rplacing Equation (17) into Equation (1) (1) and -oding tm, w gt an indict fild-ointd contolld induction moto modl that includ dtuning ffct dcibd by th following quation:

11 Dynamical modl fo fault dtction in quil cag induction moto 171! 1 m 1 co( ) ˆ in( ) ˆ!, 1 " $ $ : id " $ : i % q * + co( $ : ) ˆ in( ) ˆ ˆ m iq, $ : id i!$ m q :!, 1 ˆ ˆ 1 ˆ! 1 ˆ m 1!, 1 " i ˆ ˆ ˆ d J P m! co( ) ˆ in( ) ˆ! 1 $ $ : iq, $ : i % d, P * + (18) wh!$ :!, ˆ. Not now that in tady tat, that i whn: ˆ! 1!! 1!!! $! :!, fom Equation (18), w hav that: 1 co( $ : ) iˆ, in( $ : ) iˆ 1 iˆ! co( $ : ) iˆ " in( $ : ) iˆ ˆ iˆ q d q d q d (19) fom which w obtain: 4 5iˆ q 61, ˆ 7ˆ tan( $ i : )! 8 9 4iˆ 5 q 1" ˆ 6ˆ i 7 8 d 9 d thu, $ :! povidd! ˆ.. Dtction of th dtund opation Nxt, w dign a idual gnato to dtct dtund indict fild-ointd contoll following th modl-bad-bad fault dtction mthod outlind in th pviou ction. W conid $ : in Equation (18) a an xtnally gnatd ignal and, fo fault dtction, w conid th following ytm:! 1 m 1 co( ) ˆ in( ) ˆ!, 1 " $ $ : id " $ : i % q * + J P m! co( ) ˆ in( ) ˆ! 1 $ $ : iq, $ : i % d, P * + Not now that to xp th dynamic in Equation () in tm of th ytm Equation (1), w nd to idntify th tagt and nuianc fault. Sinc th load toqu i alo an unknown quantity that may vay ov a wid ang dpnding on th moto application, w conid it a a nuianc fault, that i: ()

12 17 H. Rodíguz-Coté, C.N. Hadjicoti and A.. Stankovi! ln $ %! & P. &, &* J + Not now that th infomation about dtuning ( $ : = ) i containd on th tigonomtic function in Equation (), o by dfining: w hav: > $ :?, > $ :? $ co 1% mt! & & in * + $ m ˆ m ˆ % & id iq lt! &, & P m ˆ P m 1 ˆ iq, 1iq &* 4J 4J + $ 1 %, 1 f! &, & &* + $ m % & g! & & P m 1 &* 4J + " and u! $ iˆ iˆ % * +. d q A tatd pviouly, th only mauabl quantity in Equation () i th oto pd that i:, y! (1) Staightfowad computation how that fo Equation (1) th minimal unobvability ditibution i givn a:! 1 A B B! pan C D B P, B BE J BF and condition Equation () i not atifid. Conid now th magntiing flux a th output of th ytm, that i: y1! 1 () Analogou computation how that:

13 Dynamical modl fo fault dtction in quil cag induction moto 17! * A B B! pan C P D, E B J F B and condition Equation () i atifid. By inpction, w not that in thi ca, th ubytm Equation (5), with z1! 1 and w, 1! y 1 ad a:! 1 ˆ ˆ 1!, 1 " i " i m " iˆ m () m m m d d t1 q t Hnc, w hav a olution to Poblm 1 by digning an obv fo ubytm Equation (). Not that in Equation () th oto tim contant i unknown. Howv, th idual gnato:! 1 m 1 ˆ!, 1 " i d, G 1, 1 ˆ ˆ! 1, 1 >? wh GH olv Poblm 1. To vify that a olution to Poblm 1 i givn by Equation (4), not that th dynamic of th idual i dcibd by th following quation: > 1? ˆ m ˆ m!!,g, 6, 7 ˆ, mid, idmt, iqmt (5) 8 ˆ 9 Th cond ight-hand tm in Equation (5) i omwhat unxpctd; howv, w notic that thi tm will b zo povidd m! and m!. In fact, thi tm alo pnt th dtuning poblm xpd a th diffnc btwn th timatd oto tim contant and th actual oto tim contant. Hnc, w hav tablihd that Equation (4) i a olution to Poblm 1.. Etimation of th magntiing flux Not that to comput th idual, w nd to hav acc to th magntiing flux 1, which i not typically availabl. Howv, a tatd in ijalkovic (), thi flux can b computd a follow. ultiplying th fit and th cond quation in Equation (8) by i. and, i - pctivly, and adding th ulting quation w hav: $ d d % / & i i, i i! $ * 1 i " 1 i % + " $ * 1 i " 1 i % + " i, i * + t1 t (4) m m dt dt Simpl computation how that: 1 i " 1 i! 1 i d 1 i, 1 i! 1 i q

14 174 H. Rodíguz-Coté, C.N. Hadjicoti and A.. Stankovi! thu, w hav: $ d d % $ m 1 % / & i i, i i 1 i i i i dt dt! & " ", * + * d q (6) Auming that th oto dynamic i in tady tat, w hav: 1! i m d 1 i ˆ!, i q d Rplacing Equation (7) into Equation (6), on ha: $ d d % / & i i, i i! 1 " q * + ˆ dt dt (7) (8) wh q! i, i. Dfin now: i 5 - :! actan 6 i thu, Equation (8) can b wittn a: (9) ˆ 1 " q! :! / () > i- " i.? Und th abov aumption 1 i a contant in Equation (). In od to timat 1, w follow th gnal ult pntd in Kaagianni t al. (). Dfin th timation o: z! I, 1 ".: ( ) (1) thu, w hav: ˆ J. ( I, z ".: ( )) " q z!! I! " J : / " > i- i.? Dfining:.(: )! K/: I!!, K ˆ ( I " K/: )" q i " i -. () with K >, th timation o dynamic i dcibd by: K z!!, i ˆ - " i. z

15 Dynamical modl fo fault dtction in quil cag induction moto 175 thu, z convg xponntially to zo and: > 1 K/:? lim I, "! () K t Finally, fom Equation (), w hav that: 1! I " K/: W can alo timat th magntiing flux fom tato tady tat valu. Fo, not that: d d d d i. i-, i- i.! iq id, id iq, $ i- " i %. dt dt dt dt * + (4) thu, placing Equation (4) into Equation (8) and auming that th tato dynamic i in tady tat, w hav: 1!,, / " ˆ >. i- - i.? > i- i.?.4 Accommodation of th dtuning opation Fom Equation (7) w hav that in tady tat: i d 1! m moov, not that Equation (17) impli: i! iˆ dq dq thu, w can comput i q a: i! iˆ " iˆ, i (5) q d q d Finally, fom Equation (19), w hav: i ˆ q id ˆ! (6) i iˆ d q Fo accommodation of th dtuning opation, w dfin a thhold m > fo th idual in uch a way that th IFOC i dtund povidd > m. Thi thhold duc th ffct of noi and oth non-modlld dynamic on th dciion. Onc it i dcidd that th IFOC i dtund, th accommodation pocdu wait fo th ytm to achiv a tady tat opation. Thn it comput th nw oto tim contant fom Equation (6), and nd it to th IFOC chm.

16 176 H. Rodíguz-Coté, C.N. Hadjicoti and A.. Stankovi!.5 Simulation W now validat th idual gnato and th computation of th actual tim oto contant via numical imulation. W conid an induction moto with th following paamt: Paamt oto (HP), (H).86,.88 m (H).687, () 1.4, 1.77 P 4 J (kg m ).5 (Nm) 1 In th imulation w lct G = 1. In th following imulation, th accommodation pocdu i not compltd, w only comput th actual tim oto contant without cocting it on th IFOC chm. Th idual bhaviou i hown in Figu. In od to dmontat that th idual i not aymptotically affctd by chang of th load toqu, at t = c th load toqu i ducd by 5%. Th idual convg to zo a hown in Figu. Figu Ridual bhaviou.5.5 Ridual Tim [c] To how that th idual dtct chang of th oto tim contant at t = c, th tim oto contant i changd to twic th nominal valu. Not that th idual dtct th dtuning condition by going to a non-zo quilibium. At t = 1 c th computd oto tim contant i =.5686, Figu 4.

17 Dynamical modl fo fault dtction in quil cag induction moto 177 Figu 4 Computd oto tim contant Roto tim contant τ Tim [c] At t = 6 c, th tim oto contant i changd to half of th nominal valu, a w obv th dtuning i dtctd. W comput th oto tim contant a =.148. Finally, at t = 8 c, th tim oto contant i tod to it nominal valu. W obv that th idual go back to zo; in Figu 5 w pnt th mchanical oto pd. Figu 5 chanical oto pd m.5.5 Ridual Tim [c]

18 178 H. Rodíguz-Coté, C.N. Hadjicoti and A.. Stankovi! 4 Bokn oto ba dtction in IFOC divn quil cag induction moto Indutial xpinc ha hown that bokn oto ba can b a iou poblm with ctain induction moto with dmanding wok cycl. Although bokn oto ba do not initially cau an induction moto to fail, thy can hav iou conday ffct. Th fault may ult in bokn pat of th ba hitting tato winding at high pd. Thi in tun can cau a iou damag to th induction moto; thfo, faulty oto ba nd to b dtctd a aly a poibl. Bokn oto ba cau ditubanc of th flux pattn in induction machin. Th non-unifom magntic fild componnt affct machin toqu and tato tminal quantiti, and a thu dtctabl, in pincipl, by monitoing chm. To dat, diffnt mthod hav bn popod fo bokn oto ba dtction. Th mot wll-known appoach i th non-modl-bad oto Cunt Signatu Analyi (CSA) mthod (Thomon and Fng, 1). Thi mthod monito th fquncy pctum of a ingl pha of th tato cunt fo fquncy componnt aociatd with bokn oto ba. Th main diadvantag of th CSA mthod i that it li on th intptation of th fquncy componnt of th tato cunt pctum, which a influncd by many facto, including vaiation in lctic upply, and in tatic and dynamic load condition. Th condition may lad to o in th fault dtction tak (Bnbouzid and Kliman, ). On th oth hand, a pactical advantag of CSA i that only tato cunt nd to b maud. Effot to liminat th influnc of load condition hav bn pntd fo intanc in Schon and Thoma (1997), wh it i hown that th dict componnt of th tato cunt in a ynchonou fam i not affctd by load condition; thu, it i popod to monito th pctum of that tato cunt componnt. It tun out that in ou popod monitoing chm, w monito a tat cloly latd to th dict componnt of th tato cunt. Howv, w dicovd ou ignal lction uing gomtic tchniqu. Fuzzy logic (Ritchi t al., 1994) and nual ntwok (Filipptti t al., 1995) tchniqu hav alo bn popod to handl load-latd ambiguou fquncy componnt. Th CSA mthod ha bn th main appoach ud fo dtcting bokn oto ba on induction moto opating in opn-loop; howv, pctal analyi tchniqu applicabl und vaiabl pd condition hav alo bn pntd in th litatu (.g., Buntt t al., 1994; Waton and Eld, 199). In pit of th xtniv wok on bokn oto ba dtction, modl-bad tchniqu hav not civd much attntion. Th main aon a that fault-latd induction moto paamt a not wll known, and availabl modl a quit complicatd to b tactabl with modl-bad fault dtction tchniqu. Howv, by making a compomi btwn a btt tacking of th fault-latd ignal (by uing dynamic modl) and a ducd domain of applicability of th ult (du to aumption about th induction moto paamt), modl-bad bokn oto ba dtction tchniqu hav bn cntly popod. On uch xampl i th Vinna onitoing thod (V) pntd in Kal t al. (). Th V i bad on th compaion of th computd lcto-mchanical toqu fom two al-tim machin modl. A halthy induction moto lad to qual valu computd by th two modl, wha a faultd induction moto xcit th modl in a diffnt way, lading to a diffnc btwn computd toqu valu. Thi diffnc i ud to dtmin th xitnc of bokn oto ba. Th

19 Dynamical modl fo fault dtction in quil cag induction moto 179 V ha on diadvantag, which i alo pnt in ou popod monitoing chm: vaiation on th tim oto contant cau th pfomanc of th fault dtction chm to dtioat. 4.1 Squil cag induction moto modl with bokn oto ba W now pnt an induction moto modl with bokn oto ba. Th popod modl i l dtaild than th modl pntd, fo intanc, in anola and Tgopoulo (1999) and Williamon and Smith (198). A novl fatu i that th ffct of bokn oto ba i takn into account by adding only on additional tat to th claical induction moto modl (pntd fo intanc, in Kau t al., 1995). In thi way, th tactability quimnt i achivd. Th popod modl i bad on th ida that th up-impoition of an xta t of oto cunt on tho nomally found in a halthy moto may account fo th ffct of bokn oto ba (Williamon and Smith, 198). Ou main aumption a ummaid a follow, Figu 6. ;< Du to high pmability of tl, magntic fild xit only in th ai gap g and hav adial diction a, inc th ai gap i mall lativ to th inid diamt of th tato. ;< Th tato winding a, an, b, bn and cn, cn a idntical in that ach winding ha th am itanc and th am numb of tun. Th oto winding a, an, b, bn and c, cn a idntical in th am n. All winding hav inuoidal ditibution. ;< Th xta t of oto cunt (pnting th bokn ba) i includd by adding an xta winding, dnotd by b, bn, to th oiginal oto winding. b b ;< agntic atuation, ddy-cunt and fiction lo a not includd in ou analyi. Figu 6 Dvlopd diagam of th co-ctional viw c b a φ c a b c a b c b a c a b c a b g π b b π b b α θ φ

20 18 H. Rodíguz-Coté, C.N. Hadjicoti and A.. Stankovi! In Figu 6, a, b, c and a, b, c dnot th poitiv diction of th magntic flux poducd by ach winding. O indicat th poitiv diction of cunt. Th angula diplacmnt of th oto lativ to a i dnotd by :, th tato angula diplacmnt lativ to a i dnotd P, whil th oto angula diplacmnt lativ to a axi i dnotd P. Th angula diplacmnt :, P and P a latd a: P! P ":. Following th modlling pocdu of Kau t al. (1995), w hav that th dynamic modl of a quil cag induction moto with bokn ba i dcibd by:! 1!, R i " v! 1abc!, Ri abc! 1!, i, abc abc abc b b b wh 1, 1 a th tato and oto flux linkag, i abc, i abc a th tato and oto abc abc cunt, v abc a th tato voltag, 1, i a th bokn ba-latd flux linkag and b b (7) cunt, R = diag{ } i th oto itanc matix, with th oto winding itanc, and R = diag{ } i th tato itanc matix. Flux linkag and cunt a latd a: wh: $ iabc % $ b % $ 1abc % & i & " & abc b 1 &! & & abc, " " &* i & & b + * b b b + * 1 b + (8) b b! $ b * co( :. ) co( :., c) co( :. " c) % +,! co( -) co( -, c) co( - " c), b Q $ ",, %, * l m m m & 1 1! &, m l " m, m & 1 1, m, m l " m 1 1 $ l " m, m, m % & 1 1! &, m l " m, m, & 1 1 *, m, m l " m + $ co( : ) co( : " c) co( :, c) %! & co( : c) co( : ) co( : c) &, " &* co( : " c) co( :, c) co( : ) + R (9) with :!:,- and c S!.. In Equation (8) and (9), l, m a th tato lakag and lf inductanc, l, m a th oto lakag and lf inductanc, = m th tato-oto mutual inductanc, b, b = b i th bokn ba-latd winding to tato and to oto mutual inductanc

21 Dynamical modl fo fault dtction in quil cag induction moto 181 pctivly, b i th bokn ba-latd winding lf-inductanc and - i th angula poition of th bokn ba-latd winding. Finally, th mchanical dynamic i dcibd by: P d " " J! m! iabc iabc " iabc bib, d: >?, (4) Not that th bokn ba-latd winding inductanc b and b, th itanc of th bokn oto ba-latd winding b and th angula poition - a unknown paamt, inc th numb and th poition of th bokn oto ba a unknown in advanc. 4. Numical validation of th popod modl To validat th popod induction moto with bokn ba modl, w nd to compa ou ult with xpimntal maumnt. Howv, in th abnc of xpimntal tudi, w will compa th ult of th popod modl with th ult of th quil cag induction moto modl with bokn oto ba pntd in Wlh (1988). It i wll known that bokn ba ult on idband componnt aound th fundamntal of th tato cunt at fqunci givn by Bnbouzid and Kliman (). f b > 1?! T f (41) wh i th p unit lip and f i th upply fquncy. oov, xpimntal vidnc ha hown that whn th amplitud of th bokn oto ba hamonic Equation (41) i ov 5 db mall than th fundamntal fquncy componnt amplitud, th oto may b conidd halthy (Hivonn, 1994). Thu, in ou numical tudi w conid that th xitnc of hamonic at fqunci givn by Equation (41) with amplitud blow 5 db mall than th fundamntal a an indication that th modl i abl to poduc th ffct of bokn oto ba. W hav conidd two induction moto with paamt a follow: Paamt oto 1 ( HP) oto (1HP) l, l (H).4,.1.4,.6 m (H) , () 1.4, ,.5 P 4 4 J (kg m ).5.86 (Nm) 1 7 Input voltag (V) Ratd cunt (A) 4 18 Th induction moto modl popod in Wlh (1988) conid th oto cag a a ntwok of qually pacd loop, Figu 7. Auming that th oto cag i ymmtic, i.., ach oto ba ha th am itanc R b and lakag inductanc lb. Th oto ba paamt a obtaind in tm of th tandad ingl-pha quivalntcicuit modl via a lationhip that pv th pow tanfd aco th ai-gap. Thu, th oto ba paamt a givn a follow:

22 18 H. Rodíguz-Coté, C.N. Hadjicoti and A.. Stankovi! $ 4 5 % 4 Nb P S lb! & 6 l 1 7 m S &,, PS & 6Nb in > N? 7 * 8 b 9 + P! 8 N, 1 R >? mb b m Nb bb 4 4 PS 5! in 6 7 S 8Nb 9 44N 5 8 S 9 b b! 6 7 m with mb, th tato oto loop mutual inductanc, bb, th oto loop mutual inductanc and N b, th numb of oto ba. Thfo, to implmnt thi induction moto modl, th numb of oto ba i ndd. Not that in Equation (4), fo givn valu of l, P and m, th xit a minimum numb of ba that giv poitiv valu fo lb, mb and bb. In ou ca, w hav: oto inimum numb of oto ba (4) Now, in od to obtain a mo alitic numb of oto ba, w pfom a limitd uvy of data fom moto with imila chaactitic. Th collction of moto with chaactitic imila to oto 1 i hown in th nxt tabl: Ratd pow (HP) Ratd cunt (A) Input voltag (V) Numb of oto ba /6.1 8/ /4.9 8/ Although th amount of data i not ufficint to mak gnal concluion, w can dvi an timat. Fom th tabl abov, w can obv that th i a colation btwn th atd cunt and th numb of ba, thu, w lct 8 ba fo oto 1. Fo oto, w hav: Ratd pow (HP) Ratd cunt (A) Input voltag (V) Numb of oto ba o that w lct 58 ba fo oto.

23 Dynamical modl fo fault dtction in quil cag induction moto 18 Figu 7 Roto cag tuctu R b I b lb I lp 1 I b 1 I lp lb R 1 b 1 lb R b I lp I b In od to bak th ba in th modl ud in Wlh (1988), th loop cunt of two adjacnt loop a containd to b qual, o that in th common ba, th total cunt i zo. With th conidd paamt, w hav fo oto 1: =.11 and f b = {58.64, 61.5}, whil fo oto, w hav: =.74 and fb = {59.11, 6.88}. A obvd in Figu 8, th tato cunt ha componnt at tho fqunci with amplitud coponding to a non-halthy induction moto. Anoth way to bak th ba, pntd in nac t al. (4), i to inca th itanc of th ba that bak. Idally, th bokn ba itanc hould tnd to infinity; howv, thi cau tat continuity poblm in th modl. A wok aound i to gadually inca th bokn ba itanc. It i, howv, difficult to comput th maximal bokn ba itanc. In od to ovcom thi poblm, w inca th itanc until a maximal valu that do not cau numical poblm in ou imulation. Fo intanc, w inca th oto ba itanc of ba 8 of oto 1 fom to 44.9, which mak an inca of mo than 7 tim. Thi giv =.17 and f b = {57.96, 6.}, a hown in Figu 9. Finally, w pfom om imulation with ou popod modl with th following valu fo th bokn ba-latd paamt: Paamt oto 1 (HP) oto (1HP) b = b (H).7.17 b (H) b ().9.9

24 184 H. Rodíguz-Coté, C.N. Hadjicoti and A.. Stankovi! Figu 8 Spctal contnt of tato cunt i a, th ba i bokn by th containt of qual adjacnt loop cunt db Fquncy [Hz] Fquncy [Hz] oto 1 oto Figu 9 Spctal contnt of tato cunt i a in oto 1, th ba i bokn by incaing it itanc db db Fquncy [Hz] Fo ou modl, w hav =.164, f b = {58., 61.97} fo oto 1 and =.96, f b = {58.85, 61.14} fo oto. Not that fo th chon paamt, th magnitud of th idband fo oto, baly ach 5 db. On of th diffnc w obv in ou imulation concning th mthod to includ th bokn ba i that th avag mchanical pd chang. Thu, baking th ba a in Wlh (1988) inca th avag mchanical pd (l than 1 d/c); incaing th ba itanc dca th avag mchanical pd in th am popotion, whil adding th xta winding alo dca th avag mchanical pd. Thi cau th idband componnt to b locatd at diffnt fqunci a hown in Figu 11. Th magnitud of th idband alo hav diffnt valu, but thy a in th ang of th non-halthy ca.

25 Dynamical modl fo fault dtction in quil cag induction moto 185 Figu 1 Spctal contnt of tato cunt i a, th ba i bokn by adding th xta winding db db Fquncy [Hz] Fquncy [Hz] oto 1 oto Figu 11 Spctal contnt of tato cunt i a in oto 1, (dahd lin) th ba i bokn by th containt of qual adjacnt loop cunt, (continuou lin) th ba i bokn by incaing it itanc and (dottd lin) th ba i bokn by adding th xta winding db Fquncy [Hz] 4. Bokn oto ba dtction Now, w dign a idual gnato to dtct bokn oto ba on an IFOC-divn quil cag induction moto. To thi nd, by coniding i b a an xtnally gnatd ignal, w xp th induction moto dynamic Equation (1) in tm of a fam with pha lockd with th diction of th oto magntiing flux otating at ynchonou pd. Thu, w hav:

26 186 H. Rodíguz-Coté, C.N. Hadjicoti and A.. Stankovi!! 1!, i, 1 " v q q d q! 1!, i " 1 " v d d q d! 1 m b co( :,-) 1!, 1 " i, i q b i m q b ib! ", co( :,-) 1 1 A m J!! C iq1 " b $ co> : "-? id, in > : "-? id % ib D, T, 8 * + E F (4) l " m wh! i th oto tim contant, m! i th oto angula pd and P :!:,:. Auming now that th induction moto i fd by cunt invt with fat cunt contoll and coniding an IFOC chm, th induction moto dynamic that w conid fo fault dtction ad a:! 1 m b 1!, 1 " i, in > :,-? i! :! d d b A m J!! C i 1 " $ co>? i in >? i i T 8 * : "-, : "- % + D, E F i! m q b ib :! ", co> :,-? 1 1 q b d q b wh i d and i q a th tato cunt componnt contolld by th cunt contoll. To wit th oto flux dynamic Equation (44) in tm of Equation (1), fit w idntify th tagt and nuianc fault. Sinc w want to dign a bokn oto ba dtcto that i not influncd by load condition, and i b in Equation (44) a idntifid a th nuianc and tagt fault mod, pctivly, that i: $ b % &, in > :,-? $ % & & & & & l l P > : "-? i, > : "-? i & * + J & & & 8J &* + & b 1 &, co> :,-? 1 * + n! &, 1, t! & $ co in %. b d q (44)

27 Dynamical modl fo fault dtction in quil cag induction moto 187 oov, w hav: $ 1 % f! &, 1, * + g $ m % & &.! & P m 1 & 8J 1 * + " " Fom a pactical point of viw, it i diabl to dign a idual gnato uing th oto pd, a it i an aily mauabl tat. Howv, it can b hown that with th oto pd a th output of Equation (44), th coponding minimal unobvability ditibution intct th imag of th nuianc fault ignatu, that i, th load condition ffct cannot b movd fom th idual. Not now that coniding th oto flux 1 a th output of Equation (44), th minimal unobvability ditibution!* i computd A 1 B J B!*! pan C. 1 D (45) B J B B 1 1B EJ F Thu, on ha that Equation () i atifid fo :,- = and w can go futh to find th diffomophim Equation (4). By inpction, w not that Equation (5), with w 1 = y, ad a:! 1 m b 1!, 1 " i, in > :,-? i d b y! 1 and a a ult, w hav that, Poblm 1 i olvabl with th idual gnato dynamic dcibd by: (46) ˆ! ˆ m 1!,G 1, 6, G 71 " i 8 9! ˆ 1, 1, d (47) wh GH. Futhmo, fom th dynamic of dcibd by: b!!,g " in > :,-? ib, (48) it i poibl to vify that Condition 1 and a atifid, inc fo i b =, th idual go xponntially to zo, and i not affctd by th nuianc fault (load toqu). oov, fo i b # th idual will mov away fom zo.

28 188 H. Rodíguz-Coté, C.N. Hadjicoti and A.. Stankovi! In Schon and Thoma (1997), it i hown that an induction moto tat that i not influncd by load condition i th cunt i d, o that it i uggtd to monito th pctum of i d. Not that in tady tat 1! i. Thu, w aivd at th vy am concluion uing non-lina gomtic tchniqu. Now, w vify th pfomanc of th bokn oto ba dtcto via numical imulation. In all imulation, w conid th induction moto modl intoducd in Wlh (1988). Th idual bhaviou fo oto 1 i hown in Figu 1. In od to vify that th idual i not affctd aymptotically by chang on th load toqu, at t = c, w inca th load toqu by 5%. Not that th idual i not aymptotically affctd. Now to how that th idual actually dtct th ffct of bokn oto ba, at t = 4 c w bak on oto ba. Not that th idual, a pdictd, dtct thi ffct. Th idual bhaviou fo oto i hown in Figu 1. At t = c, w duc th load toqu by 5%. Not that th idual i not aymptotically affctd. At t = 4 c, on oto ba i bokn. A pdictd by ou computation, th idual act to th tagt fault. Howv, not that if th oto tim contant i not xactly known, dviation fom th valu ud in th idual gnato will poduc a action of th fault dtcto. Sinc chang on th oto tim contant a mainly du to th i of th tmpatu of th moto, th action of th fault dtcto to thi mimatch hould b low. Thi poblm alo occu in th V, a it i aumd that th oto tim contant i known xactly. Noiy maumnt can alo ditub th dtcto pfomanc. Howv, ou initial analyi indicat that it i poibl to ditinguih btwn noiy maumnt and bokn oto ba-divn idual. Figu 1 Ridual bhaviou fo oto 1 m d Ridual ( ) Tim [c]

29 Dynamical modl fo fault dtction in quil cag induction moto 189 Figu 1 Ridual bhaviou fo oto Ridual ( ) Tim [c] Not that th limitation of th dvlopd induction moto modl affct th fault dtcto chm. Fo intanc, inc w conid idally ditibutd tato and oto winding, it i not poibl to dtmin th influnc of oth cunt hamonic on th idual. 5 Concluion Th pap viwd modl-bad fault dtction tchniqu in th analytic dundancy famwok. Ou viw covd appoach fom th dtction filt of Bad and Jon to th non-lina unknown input obv. W outlind th non-lina unknown input obv tchniqu in od to olv fault dtction poblm in induction moto. Nxt, w pntd a monitoing chm to dtct dtuning opation on indict fild-ointd contolld cunt-fd induction moto. Th ky fo fault dtction wa th dvlopmnt of a modl, bad on th dtuning intptation intoducd in ijalkovic (), that xp th dtuning ffct in tm of th diffnc btwn th al and th timatd otating fam, and th lction, bad on tchniqu fom diffntial gomty thoy, of th induction moto tat to monito. It i alo hown how th fault can b accommodatd. Numical imulation validat th monitoing chm and how that th monitoing chm i not affctd by chang on th load toqu. W alo dvlopd a implifid modl fo a quil cag induction moto that includ bokn oto ba ffct. Rlying on diffntial gomty tchniqu, w hav popod a modl-bad olution to th bokn oto ba dtction poblm on IFOC-divn quil cag induction moto. W how that load toqu condition will not lad to o in th dtction, a th fault dtcto i not affctd by uch condition. Numical imulation of vy diffnt induction moto validat th modl and th pfomanc of th monitoing chm. Ou analyi uggt that modl-bad fault dtction tchniqu will hav an impotant ol to play in mging fault-tolant lctic div ytm. In uch application, thy will complmnt th modl-f tchniqu in ca wh accuat but tactabl modl a availabl.

30 19 H. Rodíguz-Coté, C.N. Hadjicoti and A.. Stankovi! Rfnc Bad, R.V. (1971) Failu accommodation in lina ytm though lf-oganization, PhD Ditation, Fbuay, Dpt. Ao. Ato., IT, Cambidg. Baill, G. and ao, G. () Contolld and Conditiond Invaiant in ina Sytm Thoy, Dpt. of Elcton. and Comp. Scinc, Univity of Bologna, Italy. Bnbouzid,.H. and Kliman, G.B. () What tato cunt pocing-bad tchniqu to u fo induction moto oto fault diagnoi?, IEEE Tan. on Engy Convion, Jun, Vol. 18, No., pp Buntt, R., Waton, J.F. and Eld, S. (1994) Th dtction and location of oto fault within th pha induction moto, Poc. Int. Conf. on Elct. ach., pp D Pi, C. and Iidoi, A. (1) A gomtic appoach to nonlina fault dtction and iolation, Tan. on Automatic Contol, Jun, Vol. 46, No.6, pp Filipptti, F., Fanchcini, G. and Taoni, C. (1995) Nual ntwok aidd on-lin diagnoi of induction moto oto fault, IEEE Tan. on Indutial Application, Vol. 1, No. 4, pp Gtl, J. () All lina mthod a qual-and xtndibl to (om) nonlinaiti, Int. J. Robut and Nonlina Contol, Vol. 1, pp Hammoui, H., Kinnat,. and El Yaagoubi, E.H. (1998) Fault dtction and iolation fo tat affin ytm, Euopan Jounal of Contol, pp. 16. Hivonn, R. (1994) On-lin condition monitoing of dfct in quil cag moto, Poc. Int. Conf. on Elct. ach.,, Pai, Fanc, pp Jon H.. (197) Failu dtction in lina ytm, PhD Ditation, IT. Kaagianni, D., Atolfi, A. and Otga, R. () Two ult fo adaptiv output fdback tabilization of nonlina ytm, Automatica, Vol. 9, No. 5, pp Kal, C., Pik, F. and Pacoli, G. () Dtction of oto fault in quil-cag induction machin at tandtill fo batch tt by man of th Vinna monitoing mthod, IEEE Tan. on Induty Application, ay Jun, Vol. 8, No., pp Kau, C.P., Waynczuk, O. and Sudhoff, D.S. (1995) Analyi of Elctic achiny, Nw Yok, USA: IEEE P. anola, ST.J. and Tgopoulo, J.A. (1999) Analyi of quil cag induction moto with bokn ba and ing, IEEE Tan. on Engy Convion, Dcmb, Vol. 14, No. 4, pp aoumnia,. (1986) A gomtic appoach to th ynthi of failu dtction filt, IEEE Tan. on Automatic Contol, Sptmb, Vol. 1, No. 9, pp aoumnia,., Vgh, G. and Willky, A. (1989) Failu dtction and idntification, IEEE Tan. on Automatic Contol, ach, Vol. 4, No., pp nac, A., Nait-Said,., Bnakcha, H. and Did, S. (4) Stato cunt analyi of incipint fault into aynchonou moto oto ba uing foui fat tanfom, Jounal of Elctical Engining, Vol. 58, No. 5 6, pp.1 1. ijalkovic,. () Snitivity Analyi fo Nonlina agntic, Nothatn Univity, ay. ohan, N. (1) Advancd lctic div, NPERE, innapoli, USA. Novotny, D.W. and ipo, T.A. (1996) Vcto Contol and Dynamic of AC Div, Oxfod, USA: Clandon P. Patton, R., Fank, P. and Clak, R. (1989) Fault Diagnoi in Dynamic Sytm Thoy and Application, 1t d., UK: Pntic Hall Intnational. Ritchi, E., Dng, X. and Jokinn, T. (1994) Diagnoi of oto fault in quil cag induction moto uing a fuzzy logic appoach, Poc. Int. Conf. on Elct. ach., Pai, Fanc, pp.48 5.

31 Dynamical modl fo fault dtction in quil cag induction moto 191 Schon, R.R. and Thoma, H.G. (1997) Evaluation and implmntation of a ytm to liminat abitay load ffct in cunt-bad monitoing of induction machin, IEEE Tan. on Induty Application, Novmb Dcmb, Vol., No. 6. Thomon, W.T. and Fng,. (1) Cunt ignatu analyi to dtct induction moto fault, IEEE Induty Application agazin, July Augut, pp.6 4. Waton, J.F. and Eld, S. (199) Tanint analyi of th lin cunt a a fault dtction tchniqu fo -pha induction moto, Poc. Int. Conf. on Elct. ach., pp Wlh,.S. (1988) Dtction of bokn oto ba in induction moto uing tato maumnt, PhD Ditation, ay, Cambidg: Dpt. EECS, IT. Williamon, S. and Smith, A.C. (198) Stady-tat analyi of -pha cag moto with oto ba and nd ing fault, Poc. Int. Elct. Eng., ay, Vol., No. B, pp.9 1. Not 1 All th way though vcto a dnotd a f! Q f f f R. abcx ax bx cx "