Field Oriented Control of a Multi-Phase Asynchronous Motor with Harmonic Injection

Transcription

1 Field Oriened Conrol of a Muli-Phase Asynchronous Moor wih Haronic Injecion Robero Zanasi Giovanni Azzone Inforaion Engineering Deparen Universiy of Modena e Reggio Eilia Via Vignolese 95 4 Modena Ialy e-ail: roberozanasi@uniorei giovanniazzone@uniorei Absrac: The paper presens a new coplex dynaic odeling of uli-phase asynchronous oors and invesigaes he Field-Oriened conrol applied o his ype of achines The new dynaic odel is obained in he sae space using a coplex recangular ransforaion The obained odel is coplex reduced-order and akes ino accoun also he odd haronic injecion of he oor The Field-Oriened conrol is considered and discussed in he uli-phase general case and hen ipleened in Malab/Siulink considering a specific nueric case The siulaion resuls validae he obained coplex odel and he ipleened conrol design Keywords: Muli-phase asynchronous oors Haronic injecion Field Oriened conrol INTRODUCTION The ineres in uli-phase asynchronous achines has considerably increased during las years see Jones and Levi () and Levi e al (7) especially in racion and propulsion research fields where high-power perforances are needed Moreover he advanages of he odd haronic order injecion is well known in lieraure for is providing an higher orque densiy see Toliya e al (99) and Duran e al (8) Differen Field Oriened conrol sraegies have been ipleened and discussed in he specific case of 5-phases achines see for insance Duran e al (8) and Xu e al () This paper presens a new coplex and general dynaic odel of a uli-phase asynchronous oor wih an arbirary nuber of phases and he odd order haronic injecion and hen i exends he Indirec Roor Field- Oriened (IRFO) conrol o he uli-phase general case The dynaic equaions of he syse have been obained using a coplex and congruen sae space ransforaion and graphically represened using he Power-Oriened Graphs odeling echnique The paper is organized as follows: Secion briefly inroduces he basic properies of he POG echnique in he coplex case Secion 3 presens and describes he new coplex reduced dynaic equaions of he syse and he corresponding odel Secion 4 repors he IRFO conrol equaions in he general uliphase case Las Secion 5 shows soe siulaion resuls POWER-ORIENTED GRAPHS The Power-Oriened Graphs see Zanasi (99) and Zanasi () is a graphical odeling echnique suiable for odeling physical syses The POG are noral block diagras cobined wih a paricular odular srucure essenially based on he use of he wo blocks shown in a) elaboraion block x y G(s) x y b) connecion block x y K x K Fig POG: a) elaboraion block; b) connecion block Fig : he elaboraion block sores and/or dissipaes energy (ie springs asses dapers capaciies inducances resisances ec); he connecion block redisribues he power wihin he syse wihou soring or dissipaing energy (ie any ype of gear reducion ransforers ec) The POG schees can be used boh for scalar and vecorial syses and for real and coplex variables In he vecorial case G(s) and K are arices: G(s) is always a square arix of posiive real ransfer funcions; arix K can also be recangular ie varying and funcion of oher sae variables The circle presen in he eb is a suaion eleen and he black spo represens a inus sign ha uliplies he enering variable The ain feaure of he Power-Oriened Graphs is o keep a direc correspondence beween he dashed secions of he graphs and real power secions of he odeled syses: he real par of he scalar produc x y of he wo power vecors x and y involved in each dashed line of a poweroriened graph see Fig has he physical eaning of he power flowing hrough ha paricular secion Fro he POG schees one can direcly obain he sae space equaions of he syse: Lẋ = Ax +Bu y = B x The energy arix L is always syeric and posiive definie: y

2 R s I s I s I s3 I ss I r I r I r3 I rr L s V V M s M s3 V s V 3 V s V rr M s3s τ M sr J M r M r3 b τ e V r M r3r Fig Srucure of a uli-phase asynchronous oor L = L > When an eigenvalue of arix L ends o zero (or o infiniy) he syse degeneraes owards a saller dynaic syse The dynaic equaions Lż = Az+ Bu and y = B z of he reduced syse can always be obained fro he original one using a congruen ransforaion x = Tz (arix T can also be coplex and/or recangular) where L = T LT A = T AT = T LṪ and B = T B When arix T is recangular he syse is ransfored and reduced a he sae ie Noaions In his paper he following noaions are used o denoe respecively full diagonal colun and row arices: R R R R i j R R R R ij = :n : i Ri = :n R R n R n R n n i R i = R R R n T :n R r L r j R j = R R R : The sybol δ(n) k denoe he following funcion: { if n k k ± k ± δ(n) k = in he oher cases where nk Z The sybol I denoes an ideniy arix of order 3 COMPLEX DYNAMIC MODEL OF THE MOTOR The basic srucure of a uli-phase sar-conneced asynchronous oor is shown in Fig The elecrical and echanical paraeers of he syse are shown in Table All he elecrical paraeers of he achine have been obained connecing in series he p polar couples of he oor Le V s I s V r and I r denoe he saor and roor volage/curren vecors in he exernal erence frae Σ : V s = V s V s V ss I s = I s I s I ss V r = V r V r V rr I r = I r I r I rr s : nuber of saor phases; r : nuber of roor phases; p : nuber of roor and saor polar expansions; γ s : saor angular phase displaceen (γ s = π ); s γ r : roor angular phase displaceen (γ r = π ); r θ : roor angular posiion; : roor angular velociy; θ s : saor volage angular posiion; ω s : saor volage frequency; θ : elecric angle (θ = p θ ); R s : saor phases resisance; L s : saor phases self inducance; M s : axiu uual inducance of he saor phases; R r : roor phases resisance; L r : roor phases self inducance; M r : axiu uual inducance of he roor phases; M sr : axiu value of he uual inducance beween saor and roor phases; J : roor ineria oenu; b : roor linear fricion coefficien; τ : elecrooive orque acing on he roor; τ e : exernal load orque acing on he roor Table Elecrical and echanical paraeers of a uli-phase asynchronous oor where V si = V i V s for i { s } and V ri = V rr V r for i { r } Using he following generalized sae vecor q and exended inpu vecor V: I s I e V s V e q = I r = V = V r = and applying he Lagrangian approach discussed in Zanasi e al (9) one obains he following dynaic equaions of he uli-phase asynchronous oors: ( d L e ) I e R = e + F e K e I e V + e () d J L( q) q K T e b R + W q V The srucure of he arices L( q) R and W in () are given in Zanasi and Azzone () In order o ake ino accoun he odd order haronic injecion of he oor he self and uual inducance arices L s L r and M sr are supposed o have he following srucure: L s = L s I s + M s L r = L r I r + M r M sr (θ) = M sr i j s a s n cos(n(i j)γ s ) n=: : s : s i j r a r n cos(n(i j)γ r ) n=: : r : r i j a sr n cos(n(θ + iγ r jγ s )) n=: : r : s where sr = in{ s r } L s = L s M s and L r = L r M r The coefficiens a s n a r n and a sr n of he Fourier series are supposed o saisfy he following consrains: s n=: a s n r n=: a r n n=: a sr n

3 V e I e Re( ) T ω Ve ωn Re( ) Coplex/Real conversion T ωn Σ Σ ω 3 s ω L- e ω Ī e ω Re ω Ωe Elecrical par (coplex variables) ω Fe ω Ē e ω Ke Re( ) ω K e Re( ) 4 Energy Conversion Coplex/Real conversion 5 6 τ J s τ r b Mechanical par (real variables) Fig 3 POG graphical represenaion of a uli-phase asynchronous oor in he ransfored roaing frae Σ ω Le TωN C (+)/ ω denoe he following arix: Vs ω ω V= T TωN (θ) = Tω (θ)n = ω V= ω Vr = ω Ī s ω Ve Tω z N () τ ω q= T ω q= ω Ī r = Ī e ω e where Tω (θ) C ( )/ is a coplex arix: Tω (θ) = h k e j k(θ hγ) : :: wih γ = π and where vecor z R and arix N C (+)/ (+)/ are defined as follows: h z = I N = : Based on arix () he following ransforaion arix T ω C (s+r+) (s+r+) can be defined as: TωN ( s θ s ) T ω = TωN ( r θ p ) = T ωn Tω ( s θ s ) = Tω ( r θ p ) N s N r T = ω N = T ω N where θ p = θ s θ All he coluns of arix T ω are orhogonal coplex vecors The coplex arix T ω can be used o perfor a pseudo sae space ransforaion q = T ω ω q fro he original exernal frae Σ o a new coplex roaing frae Σ ω The dynaic equaions in he new coplex ransfored frae Σ ω are: ω Le J ω L ω ω Ī e Re+ = ω Fe+ ω ω Ω ωīe e Ke ω K e b ω q ω R+ ω W ω q ω Ve + ω V The sae space ransforaion is called pseudo because he ransfored arices ω L ω R and ω W are obained using arix T ω : ω L= T ω L T ω ω R= T ω R T ω (4) ω W= T ω W T ω while he coplex vecors ω V and ω q are obained using arix T ω : where ω Vr = because he roor phases are shorcircuied The ransfored vecor ω Ī e has he following srucure: ω Ī e = T ωn T ωn ( s θ s ) I e = ω Ī ω s Ī s I s T ωn ( r θ p ) = ω I ss I ω r Ī r = ω Ī r ω I rr where ω I ss = ω I rr = because he saor and roor phases are sar-conneced and vecors ω Ī s and ω Ī s are: ω Ī s = k ω Ī sk :: s ω Ī s = k ω Ī rk :: r = k = k I dsk + j I qsk :: s I drk + j I qrk :: r The ransfored vecor ω Ve has he following srucure: ω Ve = T ωn T ωn ( s θ s ) V e = ω ω Vs Vs V s T ωn ( r θ p ) = ω V ss = V r where ω V ss = because he inpu saor volages are balanced and where vecor ω Vs is defined as: ω L = ω Vs = k ω Vsk = :: s k V dsk + j V qsk :: s I can be easily proved ha he ransfored arix ω L has he following syeric consan srucure: L s + s M s a s M sre a T sr M sre a sr L r + r M r a r J where a s a r and a sr are real consan arices (funcion of he Fourier series coefficiens) defined as follows: k a s = k a r = a s k :: s k a sr = a r k :: r a sr :: r 7 k δ(k) l τ e l :: s The energy redisribuion arix ω W has he following skew-syeric srucure:

4 L s + s M sa s M sre a T sr ω Ī s R s +jω s k s (L s + s M sre a sr L r + r M M sa s ) jω s M sre k s a T sr ra r ω Ī r = jω p M sre k r a sr R r +jω p k r (L r + r M ra r ) J j p M sre ω Ī rk r a sr j p M sre ω Ī sk s a T sr b ω Ī s ω Ī r ω Vs + (3) Fig 4 The coplex dynaic equaions of a uli-phase asynchronous oor in he ransfored roaing frae Σ ω jω sk s (L s + s M sa s) j(ω s ω )MsreksaT ω sr Ks ω W= j(ω s ω )Msrekrasr jωpkr(l r+ r M ra r) ω Kr ω K s ω K r ω Lek J ω L k ω ω Ī Rek ek + ω Ω = ek ω Ī ω ek Vek + ω K ek b ω q ω k R k + ω W ω k q k ω V k (8) where k = k k In he new frae Σ ω he ransfored :: orque vecor ω K e has he following srucure: ω K e = ω ω K s K r = j p M sre ω Ī r k r a sr j p M sre ω Ī s k s a T sr The echanical orque τ can be expressed as follows: ( τ =Re( ω K ω e Ī e ) = Re ω K s ω ) ω Ī s K r ω Ī r = p ( M sre Re j ω Ī r k r a sr j ω Ī ) s k s a ω T Ī s sr ω Ī r = pm sre k=: ka sr k (I drk I qsk I dsk I qrk ) (5) A POG graphical represenaion of syse (4) is shown in Fig 3: secion - 3 represens he sae space ransforaion Σ Σ ω Funcion Re( ) denoes he coplex o real conversion of he inpu Secion 3-4 represens he Elecrical par of he syse ha in his case is described only by coplex arices and coplex variables (he lighly shaded secion of Fig 3) The Mechanical par of he oor is described by secion 6-7 which is characerized only by real values and real variables Secion 4-6 represens he energy and power conversion (wihou accuulaion nor dissipaion) beween he Elecrical and Mechanical pars I can be easily proved ha in (4) he er ω K e is siplified by he er ω Fe ω Ī e so he dynaics of he syse can be rewrien in he following siplified for: ω Re + ω Ω e ω Le J ω L ω Ī e } ω q {{} = ω K e b ω R+ ω W ω Ī ω e Ve + ω q ω V The expanded for of syse (6) is shown in Fig 4 where: M sre = M sr s r ω p = ω s ω Le us now consider he case s = r = sr and le P π denoe he following peruaion arix: P π = e h h e s+h (7) : sr where e h denoes a colun vecor of lengh sr wih in he h h posiion and in every oher posiion Applying he ransforaion ω Ī e = P ω π Ī ek o he elecrical par of syse (6) one obains he following reordered syse: (6) where ω Lek = P T π ω Le P π ω Rek = P T π ω Re P π ω Ω ek = P T π ω Ω e P π ω Ī ek = P T π ω Ī e ω Vek = P T π ω Ve : k ω Lek = Lsek M srek M srek L rek :: k ω Rs Rek = R r :: ω K ek = j p km sre ω k Īrk k ω ω Ī sk Ī ek = ω Ī rk :: k ω Vsk ω Vek = ω Vrk :: k ω jωs kl sek jω s km srek Ω ek = jω p km srek jω p kl rek j p km sre k ω Ī sk :: k :: and L sek = L s + s a s k M s L rek = L r + r a r k M r M srek = M sre a sr k Equaions (8) clearly show ha a uliphase asynchronous oor wih an odd order haronic injecion can be aheaically described by sr ses of decoupled equaions in he frae Σ ωk ie he syse can be seen as he se of sr hree-phase asynchronous oors respecively supplied by balanced volages a frequencies k ω s The oal echanical orque of he oor τ is he su of he orques generaed by he sr inernal decoupled oors: τ = τ k = k=: k=: = Re j p kmsre k k=: = pm sre k=: Re( ω K ek ω Ī ek ) ω Īrk j p k Msre ω k Īsk :: ka sr k (I drk I qsk I dsk I qrk ) k k ωīsk ω Ī rk :: Clearly his expression is equal o he one given in (5) 4 INDIRECT FIELD ORIENTED CONTROL Le ω Φ e = ω Le ω Ī e denoe he fluxes vecor in he frae Σ ω The seady-sae equaions of he elecrical par of syse (6) can also be wrien as follows: ω ω Vs Rs ω Ī s j ωs k s ω Φ s = + ω Rr j ω p k r ω Ī r ω Φ r The considered uli-phase asynchronous oor has been conrolled using he Indirec Roor Field-Oriened (IRFO) conrol echnique see Leonard () and Vas (99) obaining he following equaions:

5 ω PIω τ eq(9) ω I qs PIqk ω V ds Σω Vs τ Lre p Msre I qs PI q V qs Σ ω ω Φdr Field Weakening eq() ω I ds PIdk ω V qs ω Iqs Σ Σ Is Moor Φ dr Msre I ds I qs PI d k 3 V qs3 V ds V s ω Ids Σω I ds k 3 V ds3 Σ θs eq() ωp p ωs s ω Fig 6 Indirec Roor Field-Oriened conrol (see Fig 5) of a 5-phases achine: conrol of he fundaenal haronic and scaling of he hird one Fig 5 Indirec Roor Field-Oriened conrol including fundaenal plus odd haronic injecion τ = pm sre k=: k a sr k L rek Φ drk I qsk (9) ω Φ dr = M sre a sr ω I ds () M sr(:) H 5 5 Muual inducance: s + 3 rd haronic ω p = I qs () T r I ds where ω Φ dr = Re( ω Φ r ) ω I ds = Re( ω Ī s ) and T r = L re /R r is he roor consan The corresponding IRFO conrol schee is shown in Fig 5: he PI ω regulaor conrols he angular velociy generaing a orque erence τ and racking a defined speed erence ω The PI dk and PI qk conrollers regulae he roor flux coponens ω Φ dr and he echanical orque τ respecively according o he equaions (9) and () and generaing he volage erences ω V ds and ω Vqs In he previous secion i has been shown ha he uliphase asynchronous oor can be described as a se of sr decoupled achines These achines can be conrolled separaely or equivalenly one can conrol only he firs achine corresponding o he fundaenal haronic and hen scaling he obained volage erences V ds and Vqs by using a scaling coefficien see Duran e al (8) The firs soluion is ore flexible because one can define a cuso conrol for each achine bu is ipleenaion has an higher cos in ers of nuber of conrollers and uning The second soluion has a sipler srucure bu i is liied o he firs achine only The rade-off ha has o be considered is beween he conrol degrees of freedo and he conrol copuaional and ipleenaion coss 5 SIMULATION RESULTS The siulaion resuls presened in his secion have been obained in Malab/Siulink by ipleening he proposed coplex and reduced odel of he syse and using he IRFO conrol sraegy The following elecrical and echanical paraeers have been considered: s = 5 r = 5 p = L s = H M s = H R s = 3Ω L r = H M r = H R s = 3Ω M sr = 9 H J = 3 kg b = 5 N s/rad and V ax = V V ax is he axiu value of he saor volage vecor ω V s The following su of balanced volage inpus has been considered: rounds Fig 7 Muual inducance beween he firs saor phase and he roor phases V s = h V sh = :5 3 k=: :5 h V k cos(k (θ s (h )γ s )) where k indicaes he injeced haronic order and h he nuber of saor phases The following Fourier series coefficiens o define he shape of he self and uual inducances have been used: a s = 8 a r = 8 a sr = 8 The IRFO conrol schee used in siulaion is shown in Fig 6: only he conrol of he firs subspace corresponding o he fundaenal haronic has been considered while he hird haronic injecion conribuion has been obained by scaling he volage erence values V ds3 = k 3 V ds and V qs3 = k 3 V qs wih k 3 = 5 The uual inducance M sr beween he firs saor phase and he roor phases is shown in Fig 7: he s and 3 rd haronic coponens are sued using he coefficiens a sr as weighs The ie behaviors of he saor and roor currens I s and I r in he original erence Σ are shown in Fig 8: heir apliudes and heir frequencies change according o he velociy racking of Fig The generaed echanical orque τ and he applied load orque profile τ e are shown in Fig 9: for s and 9 s he orque τ seles o he corresponding value of he load orque τ e because a null velociy is racked For s and 89 s a posiive and negaive orque τ are generaed respecively following he corresponding increase and decrease of he velociy (see Fig ) The conrol variables I ds and I qs are repored in Fig : heir acual values (blue solid) opially overlap he corresponding erence values (black dashed) providing low and liied conrol errors shown in Fig

6 I s A I r A Currens I s Currens I r Tie s Fig 8 Saor and roor currens in he original erence frae Σ τ N τe N 4 5 Mechanical orque τ Load orque profile τ e Tie s Fig 9 Mechanical orque τ and load orque τ e ω rad/s Ids A Iqs A Angular velociy Curren I ds Curren I qs Tie s Fig IRFO conrol variables: acual values (blue solid) and erence values (black dashed) 6 CONCLUSION In he paper a new coplex dynaic odel of a uliphase asynchronous oor has been presened and a generalized for of he Field-Oriened conrol has been applied o he oor The coplex and reduced-order odel has been obained using a coplex recangular ransforaion and considering he odd haronic injecion The Field- Oriened conrol has been considered and discussed in he uli-phase general case and hen i has been ipleened in Malab/Siulink considering a oor wih 5 phases The siulaion resuls have shown he good eω rad/s eids A eiqs A Angular velociy error e ω Curren error e Ids Curren error e Iqs Tie s Fig IRFO conrol error variables e ω e Ids and e Iqs behaviors of he presened odel and he ipleened conrol design REFERENCES R Zanasi Power Oriened Modelling of Dynaical Syse for Siulaion IMACS Syp on Modelling and Conrol Lille France May 99 R Zanasi The Power-Oriened Graphs Technique: syse odeling and basic properies Vehicular Power and Propulsion Conference Lille France Sep M Jones E Levi A lieraure survey of sae-of-he-ar in uliphase AC drives in Proc UPEC Safford UK pp 55-5 E Levi R Bojoi F Profuo HA Toliya S Williason Muliphase inducion oor drives - A echnology saus review IET Elecr Power Appl vol no 4 pp July 7 HA Toliya TA Lipo JC Whie Analysis of a Concenraed Winding Inducion Machine for Adjusable Speed Drive Applicaions Par I Moor Analysis IEEE Trans Energy Conv vol 6 no 4 pp Deceber 99 MJ Duran F Salas MR Arahal Bifurcaion Analysis of Five-Phase Inducion Moor Drives Wih Third Haronic Injecion IEEE Trans Indusr Elecr vol 55 no 5 May 8 H Xu HA Toliya LJ Peersen Roor Field Oriened Conrol of Five-phase Inducion Moor wih he Cobined Fundaenal and Third Haronic Currens Proc IEEE APEC vol pp March R Zanasi G Azzone Coplex Dynaic Model of a Muli-phase Asynchronous Moor ICEM - Inernaional Conference on Elerical Machines 6-8 Sepeber Roe Ialy R Zanasi F Grossi G Azzone The POG echnique for Modeling Muli-phase Asynchronous Moors 5h IEEE Inernaional Conference on Mecharonics April Málaga Spain W Leonhard Conrol of Elecrical Drives Springer- Verlag Berlin Heidelberg NewYork 3rd Ediion ISBN P Vas Vecor Conrol of AC Machines Clarendon Press Oxford s Ediion 99 ISBN