Steady-State Analysis and Control of Double Feed Induction Motor

Transcription

1 ntnational Jounal of Elctonic and Elctical Engining Stady-Stat Analyi and Contol of Doubl Fd nduction Moto H. Sdiki, Dj. Ould Abdlam, T. Otman-chif, A. Bchouch, K. Mbah Abtact Thi pap xplo tady-tat chaactitic of gid-connctd doubly fd induction moto DFM in ca of unity pow facto opation. Bad on th ynchonizd mathmatical modl, analytic mination of th contol law i pntd and illutatd by vaiou figu to unand th ffct of th applid oto voltag on th pd and th activ pow. On oth hand, unlik pviou wok wh th tato itanc wa nglctd, in thi wok, tato itanc i includd uch that th quation can b applid to mall wind tubin gnato which a bcoming mo popula. Finally th wok i cownd by intgation of th tudid induction gnato in a wind ytm wh an opn loop contol i popod conf a makabl implicity of implmntation compad to th known mthod. Kywod DFM, quivalnt cicuit, induction machin, tady tat D. NTRODUCTON URNG th lat ya th induction machin i th mot ud to poduc lctical ngy in wind pow pojct. A common configuation fo lag wind tubin i bad on doubly fd induction gnato DFG with back to back convt btwn th AC gid and th oto winding. Th man advantag of th DFG i th ability of vaiabl pd opation with only 20-30% of th gnatd pow having to pa though th pow convt. Thi yild conidabl duction in th convt iz which tanlat to ubtantial ytm cot bnfit Thfo, th pow xtaction fom th wind can b optimizd inc th otating pd can b changd popotionally to th wind pd. Many contol tatgi of th DFG hav bn popod in th litatu. Mot a bad on th tanint tat modl of th machin and lad to an adquat contol of th activ and activ pow [1]-[2]. Th mthod intndd to th contol main complx if no implification a mad in th tanint modl and do not contibut to unand how th DFM opat. Uful analytical mthod fo th mination of tady tat contol law hav alo bn pntd. Th mot a uually pfomd uing nonlina containd optimization outin [3] - [4]. Many autho how that i poibl to div analytical olution fom th tady tat modl in tm of oto voltag and contol angl on th whol pd ang opation [5]-[6]. But in th popod appoach it i aumd that th mchanical lo and th tato itanc a nglctd. H. Sdiki, T. Otman-chif, A. Bchouch and K. Mbah a with th Dpatmnt of Elctical Engining, Mouloud Mammi Univity, BP 17 RPTizi-Ouzou, Algia-mail: dikihamid2004@yahoo.f; otman_chif@yahoo.com,ali_bchouch@yahoo.f;mbahkouila@yaho o.f; DJ. Ould Abdlam i with th MPS Laboatoy, Univity of Haut Alac, 4 u d Fè Lumiè, F Mulhou, Fanc -mail : djaffa.ould-abdlam@uha.f. Futhmo, many autho wa xamind th Stability of th DFM by coniding only th tady tat modl [7]. n thi pap, w pnt a aild analyi of th ovall pfomanc of th DFG opating at th tady tat and in th ca of unity pow facto. Stato itanc i includd uch that th quation can b applid to mall wind tubin gnato. Thi ach of th tady tat chaactitic i uful to unand th DFM bhavio in a boad pctum. Epcially it how how th pd and lctomagntic toqu a affctd by th injctd oto voltag. Moov ult obtaind fom th tationay analyi can allow initializing lctical vaiabl in th dynamic modl. Thi tudy can b ud a a bai to analyz thi gnato and hd mo light on th opation point. Thi wok includ appopiat way to find th limit quantiti and afty aa, accoding to th applid oto voltag and th lip, nuing afty opation of th machin. At th nd, analytical xpion, divd fom th tady tat modl, a givn and which lad to an ay opn loop contol of th DFM. Th popod contol don t nd any no but nu a zo activ pow. Th quivalnt cicuit i obtaind fom th convntional dq tato and oto quation. Th tady tat quation dcibing th modl of th DFM a function of th indpndnt vaiabl. Th a: th pd of th gnato which i intoducd though th lip, th two oto xcitation voltag componnt. All th quantiti a fd to th tato voltag. Thi pap i oganizd a follow: - Sction giv a mthodology which lad to th ud DFM quivalnt cicuit. - Sction dal with th diffnt chaactitic of th DFM und unity pow facto opation. - Sction dfin th afty opation aa accoding to lip, oto voltag and th accptabl maximal cunt. - Sction pnt an opn loop contol of th DFM o ay to implmnt, without any no and nuing unity pow facto opation. - Th concluion of thi tudy i givn in ction.. STEADY STATE DFM MODELS A. Dynamic Equation of DFM Th DFM conit of a wound oto induction machin connctd to a convt. Th tato i upplid by th gid o that th oto id fquncy upimpod with th oto pd ult a ynchonouly otating fild. Only th fundamntal componnt of th voltag and cunt a conidd fo tato and oto. Co lo a nglctd in th gnal analyi. 25

2 ntnational Jounal of Elctonic and Elctical Engining Uing th dq fnc fam, th gnal full od dynamic modl of DFM i givn by: dφd d R d ωφq 1 dφq q R q ωφd 2 dφd d R d ωφq 3 dφq q R q ωφd 4 Th oto and tato flux a latd to th cunt by: φ L L 5 d d m d φ L L 6 q q m q φ L L 7 d d m d φ L L 8 q q m q Th lctomagntic toqu Г can b xpd accoding to th tato flux and oto cunt by: Lm Γ p qφd dφq 9 L Th vaiation of th mchanical pd i givn by th following diffntial quation: dω Γ J f Ω Γ l 10 B. Stady Stat Equation of DFM Th tady tat quation of th DFG a obtaind by canclling th tim divativ in th dynamic quation 1 to 4. Moov, by placing th diffnt flux by thi pctiv quation givn accoding to th tato and oto cunt, w obtain: R L ω L ω 11 d d q m q R L ω L ω 12 q q d m d R L ω L ω 13 d d q m q R L ω L ω 14 q q d m d Th fou quation 11 to 14 can b ducd to two: on at th oto and anoth at th tato by intoducing th voltag and cunt pac vcto,, and dfind by: d jq 15 d jq 16 d jq 17 By noting: d jq 18 L ω 19 L ω 20 L ω 21 m m ω ω 22 Eay intmdiat calculation mak poibl to lad to two quation, on at th tato and th oth at th oto, which minat th tady tat opation of th DFM. Th two quation a givn by: R j j 23 m R j m j 24 n matix fom w got: R j j m R j m j 25 Th quation 23 and 24 can b tanlatd in quivalnt cicuit calld: tanfom quivalnt cicuit of DFM givn by Fig. 1. Fig. 1 Tanfom quivalnt cicuit of DFM Th quivalnt cicuit of Fig. 1 i aly ud in th litatu. t i pfabl to adopt th quivalnt cicuit wh all th iz a fd to th tato. Thi pntation would b obtaind by intoducing on th oto quantiti th tanfomation atio dfind by: a / Thn w po: : Roto voltag fd to th tato. a / a : Roto cunt fd to th tato. By intoducing thi vaiabl chang into 23 and 24 it giv: R j 26 R j j 27 m 26

3 ntnational Jounal of Elctonic and Elctical Engining With: 2 a : Lakag Ractanc fd to th tato 2 R a R : Roto itanc fd to th tato Th quation 26 and 27 can b wittn in th following matix fom: Fom Fig. 3 it i ay to wit: 1 R j R j // / R j j R j R j j 2 29 R j j R j j j 28 Thn it poibl to lad to an quivalnt cicuit which diff fom th induction machin convntional cicuit by th pnc of th oto voltag / : 2. 2 j // R / j 1 1 R / j j R j R j j and 2 2 Computation 30 By hot-cicuiting th tato id ouc, on lad to th quivalnt diagam of Fig. 4. Fig. 2 Equivalnt cicuit of DFG fd to th tato id C. Analytical Expion of and To find th xpion of th tato and oto cunt, and, accoding to th voltag and th lip, it would b ai to apply th thom of uppoition. Th cunt will b th uppoition of two cunt 1 and 2 which a pctivly catd by th ouc with / null and by th ouc / whil kping null. Th am pocdu will b applid to th cunt and 1 Computation By hoting-cicuit th oto id ouc, on lad to th quivalnt diagam of Fig. 3. Fig. 4 Suppod thom applid to th quivalnt cicuit fo 2 and 2 computation By uing fig. 4, it can b aily dducd that: 2 / R / j R // j R j R j R j j R 31 R // j 32 j R j R j j 2 Finally th cunt xpion: and a givn by th following Fig. 3 Suppod thom applid to th quivalnt cicuit fo 2 and 2 computation R j j j R j R j j 1 2 j R j 2 R j R j j

4 ntnational Jounal of Elctonic and Elctical Engining Modifid Equivalnt Cicuit of th DFM Th quivalnt diagam of Fig. 3 can b modifid, to bcom mo alitic, by giving a phyical intptation fo th fictitiou ouc /, fictitiou itanc R / and by aoning on th pow having to pa though th oto. Accoding to th quivalnt cicuit of Fig. 2, w can conclud that th lctomagntic pow P, tanmittd to th tato, i qual to th activ pow povidd by th / ouc minu th joul lo diipatd in th fictitiou itanc R / : R 2 P R 35 On oth hand, th lctomagntic pow P i th um of th activ pow P, povidd by th al ouc, plu th mchanical pow P m minu th Joul lo diipatd in th al itanc R : 2 R m P P R 36 Making mmb to mmb th quality btwn 35 and 36, w lad to th following mchanical pow xpion: m P R R P R 37 t can b n accoding 37 that th mchanical pow P m i mad up of two tm: th fit tm, 1 P / i pat of th activ pow povidd by th ouc which convtd 2 into mchanical pow. Th cond tm 1 R / pnt th mchanical pow povidd fom th outid. By taking account of th lo in th itanc R, w lad finally to th modifid quivalnt cicuit hown at Fig. 5: via a th-pha tanfom. Moov, th contol of th gid id ctifi nu a unity pow facto o that th oto id activ pow can b conidd a null. Th DFG activ pow Q gn i dfind a only that paing though th tato. Q Q gn Fig. 6 DFG connctd to th gid A. Stato Ractiv Pow Chaactitic Th activ pow Q wappd btwn th tato and th gid i givn by th following lation: Q By taking th tato voltag vcto m. 38 a oigin of th pha w a abl to wit d jq d, moov, by ubtituting th cunt activ pow Q i givn by: Q.m. givn by 33 th R q R R R R R R R R d R R 2 2 R R R R R To hav Q contantly null, th following quality hould b atifid: R q R R R 40 d R R 0 Fom 40 w dduc that if w want to opat at zo activ pow, th oto voltag componnt q mut b calculatd accoding d by th following lation: Fig. 5 Modifid quivalnt cicuit of th DFM. CHARACTERSTCS OF THE DFG N THE CASE OF UNTY POWER FACTOR OPERATON Th oto of a DFM i gnally upplid by a PWM invt. Th DC bu voltag i maintaind contant by an adquat contol of a PWM ctifi which i fd by th gid q R R R 41 d R R R Fig. 7 how th DFM oto voltag q againt th lip a vaiation in d. Not that th opating chaactitic nu a zo activ pow at th tato id. 28

5 ntnational Jounal of Elctonic and Elctical Engining Fig. 7 q componnt vu lip fo diffnt valu of B. Stato Cunt Chaactitic d By uppoing th activ pow qual to zo, it ult: d q q Q m. m j t i dducd that th tato cunt do not hav an imaginay componnt; th pac vcto al componnt d : mg with th d 43 By uing th lation givn by 33 combind with th fundamntal lation of 40 w find that th cunt givn by: d R R R i 44 Fig. 8 how th tato cunt with pct to th lip a inca fom -0.4p.u to 0.4p.u. Not that chaactitic hav a d lin fom whn not xcd th atd cunt. t i poibl to giv an xplanation if th itanc i mad qual to zo in 44: d 46 R R Th chaactitic hav a contant lop / R with a mino influnc of th d / R tm. C. Roto Cunt Chaactitic Th oto cunt, with th adoptd Convntion, i th diffnc btwn th magntizing cunt m paing though and th tato cunt: R R j d jq 47 j With: d d 48 R R R q R R d R R R 49 Th activ pow P paing though th tato i givn by th following lation: R. P 45 Fig. 8 vu lip fo diffnt valu of d d Fig. 9 vu lip fo diffnt valu of d Fig. 9 how th modulu of th oto cunt vu th lip a inca fom -0.4p.u to 0.4p.u. t i notd that th cunt i till diffnt of zo bcau of th ndd magntizing activ pow paing only though th oto. D. Elctomagntic Toqu Chaactitic Th lctomagntic pow P paing though th oto towad th tato, aco th ai gap, i qual to th pow povidd by th tato to th gid incad by th joul lo in th R itanc. With th adoptd convntion, it i wittn: 2 2 P P R R P 50 2 d d R R R R R R R 29

6 ntnational Jounal of Elctonic and Elctical Engining Th lctomagntic toqu i conquntly givn by: P R P Γ Ω Ω 2 51 Fig. 10 Γ vu lip fo diffnt valu of d On can obv, in Fig.10, th lctomagntic toqu chaactitic vu th lip a vaiation in. t i notd d that th lctomagntic toqu ha a lin quation if th toqu not xcd th atd valu. t i ay to giv an xplanation if R i mad qual to zo in 50: Fig. 11 P vu lip fo diffnt valu of d F. Global DFM Activ Pow Chaactitic Th activ pow of th gnato P DFM wa dfind a bing th um of th pow xchangd with th gid id tato and th invt id oto: P P P R 55 DFM q d Γ 2 R Ω RΩ d 52 Fom 52, not that lctomagntic toqu vay with a 2 ngativ and contant lop: / R Ω. Thi jutifi th chaactitic givn on Fig. 10. E. Ractiv and Activ Roto Pow Chaactitic Th activ pow Q povidd by th oto id invt i ud to magntiz th machin inc th tato activ pow Q i impod null. Thi lat i givn by: R Q m m d jq j R d q 53 Th oto id invt povid to th oto an activ pow givn by: R P R R d jq j R q d 54 Fig. 12 P DFM vu lip fo diffnt valu of. SAFETY OPERATNG RANGES OF THE DFM d To avoid th xciv hating of th machin winding, it hould b takn ca that tato and oto cunt a blow o qual to thi ating valu. Owing to th fact, th xpion of th tato cunt i impl; w giv th limitation tatgy by acting of th tato cunt. A. Limitation by th Roto oltag W uppo that th machin opat at a givn lip, w thu calculat th voltag to b applid to th oto o that d th machin tay within th accptabl limit of opation. To anw th put qution, it hould b olvd th following inquality: 30

7 ntnational Jounal of Elctonic and Elctical Engining R R R d 2 2 max 56 Th Rolution of thi inquality nabl to find th ang of to apply to th oto to avoid ovloading th machin. d t i found that: With: 57 d min d d max With: min max R R d max max R R d max max R R 58 d min max max 1 R R 59 d max max max Fig. 14 Slip limit with gad to th oto voltag Fig. 14 how that th accptabl opating aa i dlimitd by two lin: on coponding to th maximum lip not to xcd fo a givn voltag d min max. And th oth lin copond to th point of minimal lip min of which on hould not go down. Fig. 13 Roto voltag limit with gad to th lip Fig.13 how th accptabl opating ang of th DFM accoding to voltag and th lip. Thi ang aa i d dlimitd by two lin: on copond to th maximum voltag d max not to xcd and which coincid with gnato opation. Th oth lin dlimit th minimal voltag d min fom which on hould not go down and copond to moto opation. Th afty opating ang i ubdividd into two pat by a lin coponding to a null valu of th tato cunt which admit a quation: d 0 60 B. Limitation by th Slip Th afty opating aa of th DFM can alo b dfind by acting on th lip intad of th voltag. By olving th inquality givn by 60 it i found: min max d 61. OPEN LOOP CONTROL OF THE DFM n thi ction w popo an opn loop contol of th DFM. Th contol ha a a main aim to anw th following qution: which i th valu of th voltag to b applid to th oto if w want to opat und a known applid toqu and a did mchanical pd? Moov, th additional containt to opat und a null activ pow mut b vifid. Thi opn loop contol don t qui any no a in th ca of oth known contol. To chck th validity of th popod contol, numical imulation ult a givn wh th contol i applid to th dynamic dq modl of th DFM. Containt uch a th cunt limitation and mchanical fiction will b takn into account to upplmnt thi tudy. A. Contol oltag Computation Did opating point can b impod to th machin accoding to th oto applid voltag. Whil placing in th ca of th wind pow ytm, to hav an MPTT opation, th did pd of th DFG can b givn by th maud wind pd. Th applid toqu can b known fom th mchanical pow availabl on th tubin haft. To find th voltag to b applid to th oto, w pocd in two tp: to calculat initially th tato cunt accoding to th applid toqu thn w calculat th voltag accoding to th lip and. 31

8 ntnational Jounal of Elctonic and Elctical Engining Th lctomagntic toqu quation i fit wittn a: P P R R Γ m Ω Ω Ω 2 2 aig 64 To find th valu of coponding to ach valu of Γ m, w olv th cond od quation 64 and w lad to th following ult: R R Γ Ω R 65 Γ Ω R 66 B. Simulation and Rult A numic imulation wa caid out uing MATLAB- Simulink oftwa und th following condition: th machin bing at a tanill 1, thn w want that th DFM tat up and opat a gnato and in ubynchonou ang. Th applid toqu i of 15 N-m and th did lip i 0.3. Th machin i connctd to th gid; it i upplid with 220/380 balancd th-pha ytm and 50Hz of fquncy. To avoid ovflow of th cunt, at th tim of tating up, th fnc lip i applid gadually uing a wak ngativ lop. Rmmbd th popod contol nu a null tato activ pow. Th imulation ult hown at th Fig. 16 to 18 dmontat that th did opating point i wll obtaind and th activ pow i qual to zo. Th popod contol i thn validatd. Th cond olution, 2 givn by 66 i to b jctd inc it giv only ngativ valu. Knowing th valu of th cunt, givn by th quation 65, it i poibl to dduc fom 44 th voltag to b applid to th oto accoding th did mchanical pd: R R 67 d By intoducing th mchanical pd Ω : R R d Ω Ω Ω 68 Fig. 16 Simulatd fnc pd and DFM pd Obtaining th upply oto voltag and th tady tat contol can b ummaizd by th following diagam: ω Gid a, b, c d d, q q Rlation 58 Г m Rlation 60 d Dynamic dq modl of th DFM Fig. 17 Simulatd tato activ and activ pow Wind tubin f Rlation 40 q Г m f Fig. 15 Stady tat contol of th DFM auing a null activ pow and without any no Fig. 18 Simulatd tato cunt 32

9 ntnational Jounal of Elctonic and Elctical Engining CONCLUSON n thi wok, it hav bn tablihd th tady tat chaactitic of a DFM und unity pow facto opation. Divn fom th foth ynchonizd modl, it i givn th lation btwn th oto voltag componnt nuing a zo activ pow at th tato. Thi aumption lad to vy intting analytical xpion of th lctomagntic toqu, th pd and th oth iz, pmitting to unand th DFM bhavio and to dfin th afty opating aa. Moov, th analytical xpion lad to a vy intting and ay opn loop contol of th DFM without any no. Th implicity hold pomi of gat liability. TABLE STUDED DFM CARACTERSTCS Nominal pow P n 3Kw Nominal upply voltag /Y 220/380 Stato atd cunt /Y 11A/6.3A Nominal pd 1415 pm Stato itanc R 1.5Ω Stato cyclic inductanc L 260mH Lakag cofficint Roto contant T Roto itanc R 0.7Ω Tanfomation atio al /L m 2.02 Hamid Sdiki wa bon in Tizi-Ouzou, Algia, on Mach 21, H obtaind th Engin, th Magit, and th Ph.D. dg in Elctical Engining fom th Univity 2000, and 2010, pctivly. H joind th Univity of Tizi-Ouzou in 1993, wh h i cuntly woking a Aociat Pofo. Hi main intt a pow lctonic and lctical div. Djaffa Ould Abdlam wa bon in Tizi-Ouzou, Algia, on Apil 20, n 2002, h civd th M.Sc. dg in Elctical Engining fom th Univity of Fanch-Comté, Bançon, Fanc. n 2005, h obtaind th Ph.D. dg fom th Univity of Haut-Alac, Mulhou, Fanc, wh h i cuntly woking a Aociat Pofo. Hi wok concn atificial nual ntwok applid to pow activ filt and pow lctonic. Taha Otman-chif wa bon in Tizi-Ouzou, Algia; on Fbuay, 20, H civd th Magit and th PHD dg in Elctical Engining fom th Univity of Tizi-Ouzou, Algia in 1994, and 2008, pctivly. Cuntly, h i Aociat Pofo with th Dpatmnt of Elctical Engining, at th am univity. Hi ach intt includ lctical machin and wind ngy convion ytm. Ali Bchouch wa bon in Tizi-Ouzou, Algia, on Dcmb 09, H civd th Engin and th Magit dg in Elctical Engining fom th Univity of Tizi-Ouzou, Algia in 2007, and 2009, pctivly. Cuntly, h i a Lctu and a Rach with th Dpatmnt of Elctical,Engining, at th am univity. Hi ach intt includ lctical div and advancd tchniqu applid to wind ngy convion ytm. K. Mbah, wa bon in Tizi-Ouzou, Algia, on Januay 13, n 2011, h civd th M.Sc. dg in Elctical Engining fom th Univity Of Tizi-Ouzou, Algia in Cuntly h i a PHD tudnt at th am univity. Hi aa of intt a lctic machin wind pow gnation. REFERENCES [1] F. Poiti, T. Bouaouich, M. Machmoum. Advancd contol of a doubly-fd induction gnato fo wind ngy convion Elctic Pow Sytm Rach 79, pp , 2009 [2] Balduino Cza Rablo, Wilfid Hofmann, João Luca da Silva, Rodigo Gaiba d Olivia and Slênio Rocha Silva: Ractiv Pow Contol Dign in Doubly Fd nduction Gnato fo Wind Tubin EEE Tan. nd. Elcton., ol. 56, No. 10, pp , Oct [3] Joé Fnando Mdina Padón and Andé Elía Fijóo Lonzo: Calculating Stady-Stat Opating Condition fo Doubly-Fd nduction Gnato Wind Tubin EEE Tan. Pow Syt, ol. 25, No. 2, pp May [4] A.E. Fioo, J. Cida: Modlling of wind fam in th load flow analyi EEE Tan. On Pow Syt, ol. 15, No. 1, pp Fbuay [5] D. Aguglia1, P. iaoug, R. Wamku, J. Co: Analytical mination of tady tat convt contol law fo wind tubin quippd with doubly fd induction gnato ET Rnw. Pow Gn., 2008, ol. 2, No. 1, pp [6] Z.Wang, Y. Sun, G. Li, B.T. Ooi: Magnitud and fquncy contol of gid connctd doubly fd induction gnato bad on ynchonid modl fo wind pow gnation, ET Rnw. Pow Gn, ol. 4, PP ,. 3, Jan [7] Chin-Hung Liu and Yuan-Yih Hu: Effct of Roto Excitation oltag on Stady-Stat Stability and Maximum Output Pow of a Doubly Fd nduction Gnato EEE Tan. nd. Elcton, ol. 58, No. 4, pp , Ap