Controllability Analysis of an Inverter Fed Induction Machine

Transcription

1 Controllability Analyi of an Inverter Fe Inuction Machine Henrik Mokull Bombarier Tranportation, SE Väterå, Sween S3, Automatic Control, KTH, SE-1 44 Stockholm, Sween Abtract: A controllability analyi of an inuction machine fe by a voltage ource inverter i performe to gain unertaning of funamental propertie of the proce from a control point of view. For example the effect of input contraint on reference tracking a well a iturbance rejection are coniere an ome of the reult are verifie through imulation. I INTRODUCTION Due to their rugge mechanical contruction, inuction machine are commonly ue in inutrial an traction application an everal metho to control the torque an flux of uch motor have appeare in the literature, ee e.g. [1] an []. When evaluating ifferent control metho, it i valuable to know the limit of performance. What actually et the boun on the achievable performance are funamental propertie of the proce itelf a well a contraint on the input. Uncertainty of the proce, incluing actuator, alo retrict performance a robutne an performance are uually conflicting requirement. Thee iue are rarely aree in connection to icuion of torque control algorithm. Thi paper however perform a controllability analyi a ecribe in [3] of the inuction motor fe by a voltage ource inverter. Some imilar reult are preente in [4]. A the inuction motor i a non-linear ytem it i linearize an examine at a number of operating point. Data for the motor examine in thi work are given in the appenix. II INDUCTION MACHINE DRIVE Moern rive often ue voltage ource inverter to fee the inuction motor, a hown in Figure 1. The input filter i ue to uppre iturbance on the line voltage, E(t), caue by the inverter. The coupling vector, k(t), etermine the tate of each converter phae a a function of time an ω m (t) i the mechanical rotor pee. Although the coupling vector contain a lot of high frequency component, only the funamental will be coniere in thi work. Hence, the inverter i imply moele by the following complex-value pace vector equation u () t u () t = k () t U () t U () t. (1) U Here pace vector are inicate by upercript an u (t) repreent the requete tator voltage, with magnitue an angle enote m u (t) an χ u (t) repectively. Furthermore, U (t) an U enote the actual an nominal DC-link voltage of the inverter. E - R L C i U - k Voltage Source Inverter Figure 1: Inuction machine rive. Motor The inuction machine can be ecribe by [4] 1 1 R ψ& µ ( t) = R ψµ ( t) ψr( t) u( t) () Lµ L L R r R r ψ& r( t) = ψ µ ( t) jpωm( t) ψr( t) (3) L L Here ψ µ (t) an ψ r (t) repreent the tator an rotor fluxe an p the number of pole pair of the inuction motor. R an R r tan for the reitance in the tator an rotor wining an the machine inuctance are enote L µ an L. Thi repreentation, where all leakage inuctance i put in the rotor meh, i calle the Γ- moel. Uing a polar notation with magnitue an angle for the fluxe, i.e. j () t j r () t () t m () t e χ µ χ ψµ = µ, ψr () t = mr () t e, (4) then the electrical torque can be expree a [4] 3 p T = mµ () t mr () t in ( χµ () t χr () t ). (5) L III INPUT CONSTRAINTS The maximum funamental tator voltage magnitue epen on the inverter DC-link voltage, U, ee e.g. [6]. A a matter of fact, the magnitue of the requete tator voltage i retricte to mu () t = u () t U. (6) π Aitionally, the frequency of the generate voltage houl be retricte to prevent the teay tate ω m

2 lip frequency, ω lip, from exceeing the pull-out lip frequency [6] which prouce a lot of current. That i ω = ω ω (7) 1 lip u p m T where the tator frequency, ω u (t), i the time erivative of χ u (t) an T = L /R r i the rotor tray time contant. IV LINEAR PROCESS MODEL The inverter fe inuction machine i ecribe by equation (1), (), (3) an (5). Input are the three tator voltage, repreente by the magnitue an frequency of the tator voltage pace vector u (t), an output are the control variable torque an, in thi cae, the tator flux magnitue. The DC-link voltage an the mechanical rotor pee alo affect the plant an are coniere a iturbance. To obtain a linear moel of the proce, the equation are rewritten in the following way 1 1 R muu m& µ = R mµ mr coδ coδ (8) uµ Lµ L L U 1 1 m& r = mµ coδ m (9) r T T r u uµ uµ u L mµ mµ U & R m m U δ = inδ inδ ω (1) R m 1 m & r µ mu U δ = inδ inδ uµ pω (11) m L mµ T m r mµ U where δuµ () t = χu() t χµ () t, δ () t = χµ () t χr () t. (1) The linearize moel will be enote by y() t = Gu() t G () t (13) where y(t) = (T(t) m µ (t)) T an (t) = (U (t) ω µ (t)) T. The controllability analyi will be performe at zero torque an at nominal flux at three ifferent tator frequencie, namely 1% (OP1), 5% (OP) an 9% (OP3) of bae pee ω. From (5) an (8)-(11) it now follow that at an operating point with torque, tator flux an mechanical rotor pee pecifie by T, m µ an ω m, the tationary tate are given by m µ mµ coδ m R µ in co ωu δ δ L mr x (14) = = arctan δ uµ 1 1 R in δ δ L µ L 1 4 L arcin T 3 pm µ an the tationary input by mr 1 1 µ in δ m u coδ uµ Lµ L u = = ω u 1 tan pωm δ T (15) V POLES AND ZEROS The pole an zero of the linear proce moel at the three operating point are hown in Figure. The ytem i table at all operating point with one LHPzero an four LHP-pole. A the tator frequency increae, two pole approach the zero wherea the remaining two pole move along the imaginary axi with the tator frequency. Conequently the ytem get le ampe with increaing rotor pee. The maller the tator reitance the cloer the pole get to the imaginary axi. Imag Axi Pole-zero map Real Axi Figure : Pole-zero map of the inuction machine. VI TIME DELAYS To repreent a realitic controlle rive, a time elay houl be ae at the ytem input. Auming that the time elay, T, i equal in all irection, the control banwith ω B i limite by ω B <1/T [3]. VII RELATIVE GAIN ARRAY (RGA) The relative gain array (RGA) i an inicator of enitivity to uncertainty. Large RGA element mean poor robutne for invere bae controller in preence of inepenent input uncertainty [3], inevitable in practice. On the other han, large RGA element alo mean that there i trong coupling an a iagonal controller i not likely to work atifactory. By large i meant value above 1 [3]. The maximum RGA element for the three operating point are plotte in Figure 3 a oli, ahe an otte line. A een, there are peak in the RGA element correponing to the

3 operating point tator frequency an the height increae with the tator frequency. Mot problematic are large RGA element aroun the eire banwith. One poibility to avoi potential problem i to et the banwith above the operating point tator frequency, where the RGA-element are mall. Due to banwith contraint for example ue to time elay, thi may not be poible for higher tator frequencie an the RGA inicator limit the achievable banwith for thoe operating point. With a banwith inepenent of the operating point, it follow that it probably ha to be le than the tator frequency at OP Maximum element of RGA(G) Angular Frequency [ra/] 1 4 Figure 3: Maximum RGA element. Many control algorithm aim at ecoupling the control of torque an flux, for example the claical fiel-oriente controller. Such controller are invere bae an conequently are retricte to the limitation given by the RGA analyi above. VIII SCALING To eaily examine performance requirement an limitation caue by input contraint, the ytem houl be cale. Introucing iagonal caling matrice with the maximum expecte value of the ignal along the iagonal, the tranfer function change to 1 ˆ 1 G = D, ˆ e GDu G = De GD (16) where the original tranfer function are enote with hat. Note that in the cale repreentation the input contraint become u i (t) 1 an the an performance requirement become e i (t) 1, where the control error, e(t), i the ifference between a reference r(t) an the output, y(t). The limit on the tator voltage magnitue follow from (6) an although (7) only wa et a a teay tate retriction on the tator voltage frequency, it will ue to cale thi quantity. Hence, chooing the mallet eviation of the aymmetric limit we get 1 u1max = min U mu, mu, u max = ω (17) lip π T We require the control error to be le than 5% of rate torque an flux (ee the appenix) an aume that the maximum iturbance on the input are % of nominal DC-link voltage an 5% of the pull-out lip frequency, repectively. Reference value are cale a r = Rr%, where R=D -1 e D r an the new input i upper limite by one. The matrix D r efine the maximum expecte reference change, which we et to 1% of rate torque an 1% of rate flux. IX SINGULAR VALUES The larget an mallet ingular value how the highet an lowet gain of the ytem over all irection. Small ingular value mean that large input are neee in the correponing input irection to effect the output an thi might of coure be a problem ince there are contraint on the input. If for example iturbance or reference act in the low gain output irection we might be in trouble. In Figure 4 the maximum an minimum ingular value at OP are hown a oli line. The maximum ingular value how reonance peak at the operating point tator frequencie, which coul be expecte from the polezero map in Figure. The minimum ingular value on the other han are mooth but ecreae with the operating point tator frequency. Singular Value [B] Singular Value Angular Frequency [ra/] Figure 4: Singular value of the inuction machine at OP. The ahe line in the figure how the gain in the output torque an flux irection, i.e. ( ω) G j ( ω) y = (18) ( ω) ( ω) 1 G j y 1 1 G j y G j y where y i et to (1 ) T an ( 1) T, repectively. Finally, the otte line how the gain in the iturbance irection, i.e. plotting (18) with the column of G intea of y. We ee that the gain in the output flux irection follow the minimum ingular value. Thi mean that flux control require large input. The gain in the output torque irection i higher although not the

4 maximum gain. It alo turn out that the gain in the irection of the rotor pee iturbance coincie with the gain in the output torque irection an, at leat for frequencie aroun an above the reonance, the gain in the irection of the DC-link voltage iturbance i very high. Hence the chance to uppre iturbance without violating input contraint o not eem too ba. Although the conition number, the ratio between the larget an mallet ingular value i large at many frequencie, the RGA, which i inepenent of caling, inicate robutne problem only aroun the reonance frequencie. A. Flux Control Robutne A an example to illutrate the robutne iue icue in connection to the RGA analyi, conier pure flux control at OP. With the preent caling, the input irection for pure flux control at the reonance frequency i given by (.8.997e -i.1 ) T, which i the input irection giving minimum gain. Now, the gain of the ytem in thi irection i 4B compare to the gain in the input irection ( 1) T, which i 33B. Hence a very mall error in the input irection of pure flux control at the reonance caue large error. The critical component i the error in amplitue, which correpon to only 19 V. Singular Value [B] Singular Value of G[1 ] T an G[ 1] T Angular Frequency [ra/] Figure 5: Gain in the tator voltage magnitue an frequency irection. The iue i alo viible in Figure 5, where the gain in the input irection (1 ) T (otte line) an ( 1) T (oli line) are hown together with the maximum an minimum ingular value (ahe line). At the reonance frequency, the gain in the input irection ( 1) T i een to be much larger than the minimum gain, correponing to pure flux control. We alo ee that the tator voltage magnitue act in the irection with minimum gain an the tator voltage frequency in the irection giving maximum gain for low frequencie. For high frequencie the ituation i the oppoite. X LIMITATIONS IMPOSED BY INPUT CONSTRAINTS Conier perfect control, i.e. u(t) i choen a 1 1 u () t G = Rr% () t G G () t (19) which give e(t) =. The quetion ake i if electing u(t) thi way till atifie the input contraint u 1. A the contraint on the tator voltage frequency may not be vali for tranient, we will alo conier the moifie input contraint u 1 1. In the following two ubection, limitation caue by the input contraint are eparately examine for iturbance an reference. A. Diturbance Rejection If only iturbance rejection i coniere, it follow that the available input are ufficient to perfectly cancel the effect of the iturbance if the following conition i met 1 ( HG ( jω) G ( jω) ) < 1 () where H i the ientity matrix in cae of the input contraint u 1 an the vector (1 ) in cae of u 1 1. Examining one iturbance at the time we replace G in () by it column. A the DC-link iturbance actually a to the tator voltage magnitue, which follow from equation (1), it turn out that () i atifie a long a (if we ue m u m µ ω µ ) U mµ ωµ (1) π 1. Relation (1) i atifie with equality for ω µ = 44 ra/, which mean that the DC-link voltage iturbance can not be completely rejecte at OP3. For rotor pee iturbance it turn out that even with the maybe too conervative contraint u 1, iturbance can be rejecte for frequencie up to 15 ra/, which i ufficient, a rotor pee iturbance uually are low frequency iturbance. B. Reference Tracking To perfectly track reference change without violating the input contraint, we nee the following conition to hol HG 1 ( jω) R ( jω) < 1, ω ω () r ( ) where ω r i the frequency up to which reference tracking i require an the contant matrix H i efine a in the previou ection. Now, if banwith i actually only require for torque control, one houl tuy the input require by the torque reference alone an imilarly for pure flux control. Thi mean replacing the matrix R in () by it column. From Figure 4 it i clear that flux control require larger input than torque control. It follow that with the contraint u 1 an with reference tep height of 1% of rate torque an 1% of rate flux, the frequencie where the input

5 reach their limit for pure torque control become 16 Hz, 18 Hz an 17 Hz at the three operating point. The correponing value for flux control are 17 Hz, 38 Hz an 36 Hz. If we intea conier the contraint u 1 1, the maximum torque banwith increae to 43 Hz, 16 Hz an 44 Hz, wherea the banwith for flux control become uncontraine. Thi reult can partly be unertoo from Figure 5. If the retriction on the tator voltage frequency i mae weaker the oli line in Figure 5 i imply hifte upwar (together with the maximum an minimum gain). The gain in the output torque irection turn out to be upper boune by the gain in the input irection (1 ) T an the torque control banwith i conequently upper boune by the tator voltage magnitue contraint. The flux control banwith however increae with the limit on the tator voltage frequency. XI PERFORMANCE CONSTRAINTS IMPOSED BY DISTURBANCES Aume that the iturbance are cancelle with feeforwar compenation a in (19). In a real control ytem there are alway time elay an the feeforwar compenation oe not completely remove the influence of the iturbance but only reuce it from G to G (1- e -T ), where T i the time elay. Hence, the bet we can o with linear feeback i (here enote erivative) e t = Sr t SG 1 e T t (3) () () ( ) () where S i the enitivity function. Coniering one iturbance at the time, we get requirement on the enitivity function in the irection where the iturbance act. The performance requirement i to keep the control error le than one an in orer to reject iturbance the following relation mut be fulfille, where G i enote the i th column of G max 1 j ω S jω G T i jω e < 1, i = 1, (4) ω Hence we at leat nee ( ( )) ( ) ( )( ) 1 S j ω <,, i j T G j 1 e ω ω i ( ω) (5) The limit (5) for all three operating point are hown in Figure 6 a oli curve for the DC-link voltage iturbance an a otte curve for the rotor pee iturbance at the three operating point. The time elay, T, i et to 1.5 time the ampling time, T, which i aume to be.5 m. To atify the performance requirement, a very large banwith i require to uppre the DC-link voltage iturbance. A the banwith are limite by robutne an input contraint, we conclue that the performance requirement a tate above can not be atifie, at leat not for high rotor pee. Norm(1/G /(1-exp(-jωT )) [B] Norm(1/G /(1-exp(-jωT )) Angular Frequency [ra/] Figure 6: Performance contraint ue to iturbance. XII SIMULATIONS Thi ection how imulation reult to verify ome of the theoretical reult obtaine through the controllability analyi. Throughout the imulation the Inirect Self Controller (ISC) ecribe in [7] an [8] i ue. Thi i a tator flux oriente controller aiming at ecoupling the control of flux an torque. All imulation reult are preente in the cale repreentation efine by (16). A. Conequence of Large RGA Element The analyi of the RGA element inicate poor robutne for invere bae controller. In [9] it wa hown that ieally the ISC perfectly ecouple the control of torque an tator flux at zero torque Torque an flux repone at flux reference tep Time [] Figure 7: Repone to a flux tep. In Figure 7, the repone in torque an flux ue to a tep in the flux reference are hown for the ieal cae with no time elay (oli line) an with a realitic controller with a ample time of.5 m (ahe line) at OP. The controller are tune to give banwith of 4 Hz with phae margin of 45 accoring to the tuning rule given in [9]. Note that the controller banwith coincie with the frequency where the RGA element are large. The reference flux jump from 1% to 9% of Ψ at time.5. The ieal controller prouce a flux

6 tep with no iturbance on torque wherea the realitic controller with time elay generate large torque error. B. Diturbance Rejection The invetigation on iturbance rejection howe that uppreion of DC-link voltage i limite by inputa well a banwith contraint wherea rotor pee iturbance houl not be a problem. Thee reult have been tete with the Inirect Self Controller, which baically oe iturbance rejection accoring to (19) Torque an flux repone at DC-link voltage tep Torque Flux Time [] Figure 8: Control error ue to DC-link voltage tep Torque an flux repone at rotor pee tep Torque Flux Time [] Figure 9: Control error ue to rotor pee tep. The ampling time wa et to.5 m an the controller were tune to give maximum banwith with 45 phae margin accoring to [9], i.e. 8 Hz for torque control an 16 Hz for flux control. Figure 8 how the repone in torque an flux at a jump in the DC-link voltage from 75V to 6V. The repone at OP1 are hown a the oli line, the repone at OP are hown a the ahe line an finally the repone at OP3 are hown a the otte line. A een the torque control error i large at OP3. It alo follow that the tator voltage magnitue aturate, a the controller i not able to keep the flux at it reference value. Hence, both the input contraint a well a the performance contraint are violate a preicte above. High frequency component of the tator voltage magnitue, caue by the DC-link voltage mainly effect the torque wherea low frequency component effect the flux, c.f. Figure 5. The repone in torque an flux at a rotor pee iturbance of half the pull-out lip frequency are hown in Figure 9 for the three operating point. Jut a preicte, thi iturbance can be uppree. XIII CONCLUSIONS The controllability analyi of the inverter fe inuction machine among other thing howe that the proce ha large RGA element for large operating point tator frequencie, which caue problem with cro-coupling at flux control. If poible the controller banwith houl be et higher than the operating point tator frequency to avoi problem. Thi may not be poible for higher tator frequencie ue to for example long time elay. In thi cae at leat the change rate of the flux reference houl be limite to avoi exciting the critical frequencie aroun the operating point tator frequency. It wa alo hown that the DC-link voltage iturbance enter the ytem in a irection giving large gain aroun the reonance frequency. Due to input contraint an limite control banwith it cannot be completely uppree for high tator frequencie. XIV APPENDIX: ΓMODEL MOTOR DATA Stator reitance R = 18.5mΩ Rotor reitance R r = 17.3mΩ Stator inuctance L µ = 6.mH Leakage inuctance L =.79mH Number of pole pair p = Rate DC-link voltage U = 75V Bae pee ω = 58ra/ Rate flux Ψ =.9V Rate torque T = 6Nm XV REFERENCES [1] B. K. Boe, High Performance Control an Etimation in AC Drive, in 3r International Conference on Inutrial Electronic, Control an Intrumentation, Volume:, 9-14, [] J. A. Santiteban an R. M. Stephan, Vector Control Metho for Inuction Machine: An Overview, IEEE Tranaction on Eucation, 1. [3] S. Skogeta an I. Potlethwaite, Multivariable Feeback Control, John Wiley & Son, [4] L. Harnefor, On Analyi, Control an Etimation of Variable- Spee Drive, Ph D thei, KTH, Stockholm, [5] A. Steimel, Stator-Flux-Oriente High Performance Control in Traction, in 35 th Annual Meeting IEEE IAS Inutry Application Society. Rome,. [6] K. P. Kovac, Tranient Phenomena in Electrical Machine, Elevier, [7] M. Jänecke, R. Kremer, G. Steuerwal, Direct Self Control, A Novel Metho Of Controlling Aynchronou Machine In Traction Application, in Recor of the EPE-Conference, Aachen, [8] D. Maichak, Schnelle Drehmomentregelung im geamten Drehzahlbereich eine hochaugenutzten Drehfelantrieb, VDI Verlag, [9] H. Mokull, Tuning of Fiel-Oriente Controller for Inuction Machine, in IEE-PEMD, Bath UK,.