A Predictive Torque Control for Inverter-fed Induction Machines

Transcription

1 A Predicive Torque Conrol for Inverer-fed Inducion Machines Mario Pacas Universiy of Siegen Hölderlinsr. 3, D-5768 Siegen, Germany Pablo Correa Universiy of Siegen Hölderlinsr. 3, D-5768 Siegen, Germany José Rodriguez Universidad Técnica Federico Sana María, Valparaíso, Chile Absrac-A predicive conrol scheme suiable for he orque and flux conrol of mulilevel inverer-fed inducion machines is presened in his paper. The conrol sraegy combines he use of a PI conroller o obain a good seady-sae behavior and a predicive conroller o achieve a fas dynamic orque response. In his way, orque and saor flux references can be reached in he nex sample period. Wih he use of Mulilevel Space Vecor Modulaion (MSVM, low orque and flux ripple are possible wih fixed sample rae. Experimenal and simulaion resuls are presened in order o demonsrae he effeciveness of he proposed sraegy sofware. Udc Udc Cell U T T Cell U T T T3 T4 T3 T4 Cell V Cell V N Cell W Cell W u WN u,i, : NOMENCLATURE Saor volage, curren and flux space phasors i, : Roor curren and roor flux space phasors R,L: Saor winding resisance, saor self-inducance R,L : Roor winding resisance, roor self-inducance L h : Magneizing inducance M,M L : Elecromagneic orque, load orque Ωγ, : Mechanical roor speed, elecrical roor angle J,p : Toal ineria, number of pole pairs INTRODUCTION The Direc Torque Conrol (DTC mehod is considered as a high performance conrol sraegy for AC drives, wih many advanages like simple implemenaion and fas dynamic orque response especially in he field weakening range. However, i also has some drawbacks like variable swiching frequency, and needs a very high sampling frequency of he conrol (Ts < 5 µs o achieve a good performance []. Several auhors have proposed conrol sraegies in order o improve he performance of he classic DTC, including predicive conrol schemes and he use of a Space Vecor Modulaor [], [], [3]. Only few works address he problem in he Mulilevel Converer conex [4], [5]. Alhough many of hese schemes are presened as members of he DTC family, he real DTC sraegies are characerized by he absence of PI conrollers, which limis he bandwidh of he sysem [6]. U Fig. : Simplified scheme of a 5-level cascaded mulilevel inverer opology. This work presens a orque and flux conrol in combinaion wih a predicive conrol o ge rid of his problem. The conrol sraegy uses a PI conroller o obain a good seady-sae behavior in combinaion wih a predicive conroller o achieve a fas dynamic orque response. In his way, orque and saor flux references can be reached in he nex sample period. By using a mulilevel opology, exra advanages can be achieved, like lower orque disorion and currens harmonics, in comparison wih a sandard wo level opology. Problems like he saor winding insulaion breakdown, bearing failures, and high EMI are also drasically reduced, increasing in his way he reliabiliy of he sysem [7], [8]. Experimenal and simulaion resuls on a mulilevel inverer are presened in order o evaluae he effeciveness of he described mehod. I. THE INVERTER TOPOLOGY A cascaded muli-cell inverer opology is considered in his work as i is shown in Fig.. Each cell is composed of a diode recifier, a DC-link capacior and a single phase H- inverer. I can be easily verified ha an inverer wih cell synhesizes sepped waveforms wih 3 volage levels U dc,, -U dc and each exra cell adds wo exra levels per phase [8]. V iw W /5/$. 5 IEEE. 976

2 II. MATHEMATICAL MODEL For he modeling of he inducion machine, he following assumpions were made: The neural poin is no conneced. There are no eddy currens or core losses in saor and roor Only he fundamenal wave of he air gap field is considered for he calculaion of he inducances. Using hese assumpions, common in he modeling of elecrical machines, he inducion machine will be described by he following well-known se of complex equaions in he saor reference frame: d u = i R ( d d = i R j γ. ( d The fluxes and he currens are relaed by: = L i L i, (3 h = L h i L i, (4 and he elecromagneic orque is given by 3 L M = p h sin( ϑ. (5 σ L L Wih he mechanical equaion d J d M ML J Ω γ = =, (6 d p d he mahemaical model of he inducion machine is fully described. β ( k u k ϑ ( k ϑ ( k ϑ ( k ( ( k Fig. : Saor and roor flux space phasors for = k and = k. The saor flux space phasor in he nex sample ime corresponding o he desired orque value, is deermined by: j( ϑ ( k ϑ( k ( k ( k e =, (7 wih ϑ ( k defined as, ϑ ( k = ϑ ( k ϑ ( k. (8 Finally he appropriae saor volage phasor u can be found using he eq. ( afer discreizaion: ( ( u( = R i(. (9 ( k ( k k k k k Ts α III. A. Basic Operaion Principle THE CONTROL STRATEGY The presened conrol sraegy, like all Direc Torque Conrol mehods, is based on he conrol of he magniude and phase of he saor flux space phasor rough he proper choice of he saor volage space phasor. From eq. (5 and under he assumpion of consan magniude of saor and roor flux, i is clear ha a fas orque change can be achieved wih a change of he angle ϑ beween boh fluxes. Since he sample ime of he conroller is smaller han he roor ime consan, he variaion in he rajecory of he roor flux during he sample ime inerval can be negleced. Hence, i is he saor flux space phasor he main quaniy influencing he orque developmen. An example of he conrol principle can be seen in fig.. If an increase of he orque is needed, he angle variaion of he saor flux ϑ mus be greaer han he angle variaion of he roor flux ϑ in a sample ime and vice versa. M ϑ ( ϑ ( k k M ϑ ( Calculaion of ϑ ( k k γ ( Calculaion of he Saor Volage u( k Esimaion of ( k ( k Torque Esimaion M i( k Fig. 3: Simplified orque conrol diagram. k ( k ( k u( k k γ ( i( k 977

3 M i ϑ ( k ϑ ( k Calculaion of he Saor Volage u ( k Mulilevel Space Phasor Modulaor Predicion of ϑ ( M i k Predicion of ϑ ( k Predicion of ( k ( k u ( k ( k ( k Saor and Roor Flux Calculaion i ( k /3 Torque Predicion M ( i k k γ ( Fig. 4: Scheme of he predicive orque conrol. Finally he calculaed volage space phasor is synhesized using a Space Phasor Modulaor. Since a modulaor does no work wih he insananeous values, bu wih he mean values over a sample ime, his mehod should no be considered as direc conrol. Fig. 3 shows he basic srucure of he conroller. The orque reference is subraced from an esimaed value calculaed wih he phase currens and elecrical roor angle. This error is supplied o a PI conroller which delivers he variaion of he angle ϑ. The ask of he conroller is relieved by he addiion of he erm ϑ ( k as a feedforward [9], R M ϑ i ( k = ( γ T s 3p ( k, ( Alhough a PI conroller has advanages like zero seady sae error and robusness, his conroller has only a moderae dynamic performance in comparison wih a DTC sraegy. In order o improve he dynamic behavior, he difference ϑ ( k can be prediced o reach he orque reference in he quickes way possible. B. Dead-bea Operaion A dead-bea conrol aims o he quickes possible dynamic response of a sysem. Tha means, afer a ime equal o he sample period, he conrolled variables should reach he sepoin and remain here []. In order o obain a dead-bea like sysem response, eq. (5 can be used o calculae he slip angle ϑ ( k which fulfills he requiremen of orque in he nex sample ime as follows: M( k σ L L ϑ ( k = arcsin. ( 3 p L h ( k ' ( k Thus, he oupu for he dead bea conrol is deermined by: ϑ ( k =ϑ( k ϑ ( k ( By assuming a good model of he inverer-moor sysem, he orque reference should be reached in he nex sample ime. However, if orque variaion is big, especially for high roor speeds, he saor volage calculaed by equaion (9 could have a magniude beyond he maximum oupu volage of he inverer. In his case he conroller operae wih he maximum volage possible, o obain a orque in he nex sample ime as close as possible o he reference. As soon as he orque reach a window wih size ε around he orque reference: ε M M ε, (3 he operaion of he PI conroller can be resumed. A sof ransiion beween booh operaing modes can be achieved if he inegral par of he PI conroller is loaded wih a prediced angle variaion for he nex sample ime. I should be noed ha he equaions were expressed only one sep in he fuure, assuming ha he calculaion ime can be negleced. Considering ha he calculaion of he conrol algorihm needs some ime, he value prediced in he real algorihm correspond acually o k. Thus, he oupu volage is calculaed for he nex sample ime k, and he references values of orque and flux are reached in k. A diagram of he 978

4 conrol sraegy is presened in fig 4. The conrol principle can be applied in a converer of any number of levels. C. Mulilevel Modulaor The las block in he conrol algorihm is a modulaor which synhesizes he reference space volage phasor u using hree neares inverer space volage phasors. As illusraed in fig. 5 for a 3 level inverer, hese space phasors corresponds o he verices of he riangle which conains he reference u. Due o he high number of available space volage phasors in mulilevel opologies, his elecion can be a ime consuming process. Based on he algorihm presened in [], he ransformaion in eq. 4 was used o locae u in a new coordinae sysem, where he posiion of each inverer volage space phasor is described only by ineger numbers. 3/ 3/ u( k = u( k 3 (4 Thus, he coordinaes for he hree neares inverer space volage phasors u a, u b, u c can be easily found using he elemenary approximaion funcions ceil(x and floor(x wih he real and imaginary par of u. Once hese space phasors are deermined, he duy cycle is calculaed using he sandard mehod for wo level inverers. Finally, he hree-phase oupu volage combinaion (u UN, u VN, u WN corresponding o each volage space phasor were calculaed considering he crierion of common mode volage minimizaion. Tha means, he modulaor selecs only a combinaion which produces he lower common mode volage defined by uun uvn uwn ucm =. (5 3 The reducion of he common mode volage allows he miigaion of problems like bearing failures, and EMI [7]. Considering Fig. 5 as example, from he wo alernaives o generae he volage phasor u a, he second one (,, -U dc β u b u c u u a ( Udc, Udc, (,, U dc generaes a common mode volage wih lower ampliude. Despie he advanages ha he common mode volage reducion offers, he number of commuaions for some volage phasor ransiions wih his algorihm is no minimum. Furhermore i mus be used in combinaion wih an over-modulaion sraegy, if he performance of he DTC in he high speed range wans o be reached. IV. SIMULATION AND EXPERIMENTAL RESULTS A 5kW prooype of a mulicell inverer was developed in order o verify he proposed conrol mehod. Each cell is composed of a hree-phase recifier, a DC-link of 5V and 4 IGBTS IRG4PH4UD of A nominal curren wih heir respecive Drivers (fig. 6. A floaing poin DSP plaform ADSP-6 of Analog Devices was used for he implemenaion of he digial conrol. The modulaor, he imers and he incremenal encoder were programmed in a FPGA included in he board. The sample period was se up in µs, far more han a convenional DTC. The mulicell inverer was conneced o a 5.5kW inducion machine for he firs ess. (a Dead-Time Igb Drivers Opical links o he conroller Signals 5V Power Supply Dc-Dc Converers Opical Error Signals (b IGBTs α Fig. 5: Space phasor diagram of a hree level inverer Fig. 6: A cell of he mulilevel Inverer, a Top Side b Boom side. 979

5 Fig. 7a presens a simulaion of he dynamic performance of he conrol algorihm. A sep in he orque reference of Nm was applied wih a DC-link volage of V and wih he machine running a 4min -. I can be observed ha he proposed conrol sraegy has a shor orque rise ime and he magniude of he saor flux presens no change. Fig. 7b shows he same es in he experimenal se-up wih a hree level inverer. The resul is similar o he simulaion bu a lile overshoo due o an imperfec predicion of he fluxes can be seen in he orque. The dynamic behavior of he dead bea conroller depends srongly on he capaciy of he converer o deliver enough volage. For some operaing condiions, like in he high speed range, he conroller could deliver a reference volage u wih a modulaion index greaer han he maximum. In his case a saor volage phasor wih maximum ampliude is uilized. Fig. 8 illusraes his siuaion. For a roor speed of 8min -, he conrol sraegy uses he maximum oupu volage bu he orque reference value is no reached in one sep. Therefore an exra sample period is necessary o reach he reference value. In his sauraion condiion, he dead bea sraegy offers no special advanages respec o he convenional PI-Conroller soluion. V. CONCLUSION An applicaion of a predicive conrol sraegy for inducion machines in mulilevel inverers has been presened. The proposed conrol echnique combines he use of a PI conroller o obain a good seady-sae behavior and a predicive conroller o achieve a fas dynamic orque response, like he DTC sraegies. Experimenal resuls for a 3 level converer were obained, showing low orque ripple, fas Μ[Nm] Ψ [Vs] M Ψ ( k M( k a b Scale: Nm/div M M( k Scale:. Vs/div Ψ ( k [sec] Scale:.5 ms/div Μ[Nm] Ψ [Vs] Fig. 7: Torque sep of Nm and Saor flux ampliude for a roor speed of Ω =4min - in a 3 level inverer. a Simulaion; b Measuremen M Ψ ( k M( k a b Scale: Nm/div M Ψ ( k M( k Scale:.Vs/div [sec] Scale:.5 ms/div Fig. 8: Torque sep of Nm and Saor flux ampliude for a roor speed of Ω =8 min - in a 3 level inverer. a Simulaion; b Measuremen. 98

6 dynamics and almos fixed swiching frequency. The sraegy can also be applied o a n-level inverer. Since he orque conroller uses prediced values of he orque and fluxes o feed he conrol loop, discrepancies will be observed if his predicion is no performed wih enough precision. I should be noed ha he use of a modulaor reduces he performance of he proposed mehod in he high speed range, compared wih a DTC sraegy. For his reason, he sraegy mus be used in combinaion wih an over-modulaion, if he performance of he DTC in he high speed range has o be reached. In furher works his aspec will be considered. REFERENCES [] Jun-Koo Kang, Seung-Ki Sul New Direc Torque Conrol of Inducion Moor for Minimum Torque Ripple and Consan Swiching Frequency, IEEE Transacions on Indusry Applicaions, Vol, 35, No. 5, Sepember/Ocober 999. [] L. Tang, L. Zhong, M. F. Rahman, Y. Hu, An invesigaion of a modified direc orque conrol sraegy for flux and orque ripple reducion for a inducion machine drive sysem wih fixed swiching frequency in conf. Rec. IEEE-IAS Annual Meeing, CD-ROM. [3] Dariusz swierczynski, Marian P. Kazmierkowski, Frede Blaabjerg, Direc Torque Conrol wih Space Vecor Modulaion (DTC-SVM for Permanen Magne Synchronous Moor (PMSM, in Proc. of he European Power Elecronics Conference, EPE, CD-ROM. [4] C. Marins, X. Roboam, T. Meynard, A. Carvalho, Swiching frequency imposiion and ripple reducion in DTC drives by using mulilevel converer, IEEE Transacions on Power Elecronics, Vol. 7, N, March, pp [5] Trounce, J.C., Round S.D., Duke R.M., Evaluaion of Direc Torque Conrol Using Space Vecor Modulaion for Elecric Vehicle Applicaions, Ausralasian Universiies Power Engineering Conference, Perh, Ausralia,, pp [6] D.Casadei, F. Profumo, G, Serra, A. Ti., FOC and DTC: Two Viable Schemes for Inducion Moors Torque Conrol, in Proc. of he 9 h Inernaional Conference on Power Elecronics and Moion Conrol, EPE-PEMC, CD-ROM. [7] [R. Teodorescu, F. Blaabjerg, J. Pedersen, E. Cengelci, S. Sulisijo, B. Woo, P. Enjei, Mulilevel Converers-A survey, in Proc. of he European Power Elecronics Conference, EPE 99, Lausanne, Swizerland, 999, CD-ROM. [8] J. Rodriguez, J.-S. Lai, F.-Z. Peng, Mulilevel inverers: a survey of opologies, conrols and applicaion, IEEE Transacions on Indusrial Elecronics, Vol 49, N 4, Augus, pp [9] M. Depenbrock, A. Seimel, Speed-Sensorless Saor-Flux-Oriened Conrol of Inducion Moor Drives in Tracion, in Proc. of he Annual Conference of he IEEE Indusrial Elecronics Sociy (IECON 3, Roanoke, Virginia, Nov. -6, 3, pp [] Föllinger, O, Lineare Abassyseme, 5h ed., Oldenbourg-Verlag, München. [] Nikola Celanovic, Space Vecor Modulaion and Conrol of Mulilevel Converers, Disseraion, Virginia Poliechnic Insiue and Sae Universiy, Sepember,. Available: 98