Voltage-Behind-Reactance Model of Six-Phase Synchronous Machine Considering Stator Mutual Leakage Inductance

Transcription

1 Voltage-Behind-Reactance Model of Six-Phase Synchonous Machine Consideing Stato Mutual eakage Inductance Navid Amii, Seyyedmilad Ebahimi, Jui Jatskevich and Hemann W. Dommel Abstact Six-phase machines and the challenge of intefacing them with powe electonics devices and inductive netwoks have always been an inteesting topic of eseach. Coupled Cicuit Phase Domain (CCPD) model of the machine has been an altenative to the conventional qd equivalent cicuit to addess the machine intefacing issue. This pape pesents anothe suitable altenative in the fom of a Voltage-Behind-Reactance (VBR) model of the machine. Simila to the CCPD model, the VBR model can also be intefaced with any kind of netwok without using any snubbe cicuits. At the same time The VBR model achieves bette numeical efficiency compaed with the CCPD model due to educed size of the system matices. The machine model is simulated in a single-phase to gound fault scenaio while being connected to a 6 phase gid. The pesented model also consides machine mutual leakage inductances which ae shown to have consideable effects on the simulation esults. Keywods: Multi phase machines, voltage behind eactance, machine modeling, simulation. M I. INTRODUCTION ODEING of otating electical machines fo studying the system tansients has been an active eseach aea fo many decades [1], wheein many models have been poposed fo the electomagnetic tansient pogams (EMTP) [] and the state-vaiable (SV) -based pogams [3], [4]. Electical machines with high numbe of phases (moe than thee) on the stato in addition to having educed powe pe phase also open the possibility fo fault toleant opeation [5]. Six phase synchonous machines wee intoduced to ovecome the issue of high stato cuent in high powe geneatos. This makes the six-phase synchonous machine a suitable choice fo ships, aicaft, and vehicles whee high eliability is of citical impotance [6]-[7]. With the moden vaiable speed high powe dives, multi-phase machines ae also playing an inceasing ole in ovecoming the cuent limitation issues of semiconducto devices used in powe electonic convetes [8] [9]. The coupled cicuit phase domain (CCPD) [1] and conventional qd [11], [1] models have been used fo steady state and tansient analysis of multi-phase machines. In sixphase machines, the stato is typically fomed as two sets of This wok was suppoted by the Natual Science and Engineeing Reseach Council (NSERC) of Canada unde the Discovey Gant Pape submitted to the Intenational Confeence on Powe Systems Tansients (IPST15) in Cavtat, Coatia June 15-18, 15 thee phase windings. The CCPD model is established based on the magnetically-coupled cicuits of the stato and oto windings [13]. The advantage of such epesentation is that the stato and oto cicuits can be diectly intefaced with any extenal netwok, povided that the simulation pogam whee the model is developed allows the use to implement vaiable inductances. In addition to the stato, the oto will typically have seveal equivalent dampe windings and a field winding, all of which esults in inceasing the size of the machine s inductance matix. Moeove, since this model will have (vaiable) oto-position-dependent self and mutual inductances, the entie machine s inductance matix will also be vaiable. The changing inductances esult in a model that is computationally vey expensive fo eithe EMTP o SV tansient simulation pogams, as the model paametes have to be ecalculation at each time step. The synchonous machine qd models ae typically expessed in the otating oto efeence fame [14], [15], whee the equivalent cicuit is deived in two qd planes both otating with the same oto speed. Moeove, the dual qd plane cicuits have coss-coupling between each of the fou axes due to the angle shift between the two thee-phase sets. As expected, this fomulation achieves a constant paamete model in tansfomed qd vaiables and a coesponding numeically efficient implementation of the model. Howeve, simila to the conventional thee-phase qd models [16], the six-phase qd model is difficult to inteface with the extenal inductive powe netwoks which ae typically epesented in physical vaiables and coodinates. This pape extends the voltage behind eactance (VBR) fomulation that has been poposed fo the thee-phase machines fo both EMTP and SV tansient simulation [17], [18] pogams to the six-phase synchonous machine. The machine s equations in dual qd plane ae e-aanged to povide a standad state-space model with the oto flux linkages as the state vaiables, stato cuents as inputs and sub-tansient stato voltages as outputs. The equivalent subtansient back emfs ae fist expessed in dual qd plane and then tansfomed into the six-phase abc1abc coodinates. Moeove, the model takes into account the mutual leakage inductances between the two thee-phase stato winding sets [14] which ae shown to have a significant effect on the machine tansients. The model achieves a diect and convenient inteface of the stato equivalent cicuit with abitay extenal netwoks, which is paticulaly useful fo investigating the machine-convete tansients. The new

2 model is demonstated on an example six-phase synchonous geneato system that feeds a six phase gid though an inductive line. The poposed six-phase VBR model is shown to poduce vey accuate simulation esults and outpefom the conventional qd and CCPD models in tems of simulation speed. II. VBR FORMUATION AND EQUIVAENT CIRCUIT This pape models a six phase synchonous machine with two sets of symmetic thee phase windings on the stato. The stato windings ae aanged with a 3 degee displacement between the thee phase sets. The oto is salient and has two dampe windings along the oto qd axes and a field winding along d axis [14]. defined as: lm la 1acos la 1bcos( ) la 1ccos( ), (5) 3 3 ldq la 1asin la 1b sin( ) la 1csin( ). (6) 3 3 whee la1a, la1b and la1c ae the actual leakage inductances which ae pesent mostly in the stato slots between the conductos of one thee-phase set and each of the conductos on the othe thee phase set. In a symmetic winding configuation the following elation can be consideed: aa 1 bb 1 cc 1, ab 1 bc 1 ca 1,. (7) ac 1 ba 1 cb 1 Fig. 1. Two pole salient oto six-phase synchonous machine with 3 degee phase shift between the two thee-phase set. Having two sets of thee phase windings on the stato, two sepaate tansfomations to the oto fame ae equied fo each thee phase winding. This will ceate a dual plane stuctue, each epesenting one thee phase set and both otating with the same speed as the oto. Due to the angle shift between the thee phase windings, the paametes on each axis in one plane have magnetic coupling with both axes of the othe plane. This coupling is epesented by mutual leakage inductances in the machine s equivalent cicuit. Simila to a conventional thee phase synchonous machine, the six phase machine model is based on the equivalent cicuit in the oto efeence fame using the following tansfomation. q1 K( ) f abc1abc (1) whee f [ f, f, f, f, f, f, f, f, f ] T () abc1abc a1 b1 c1 a b c f [ f, f, f, f, f, f, f, f, f ] T (3) dq1 d1 q1 1 d q K( ) diag [ K( ), K( ), I 33 ] (4) whee K is the Clak s tansfomation matix. In Fig.. lm and ldq ae the mutual leakage inductances between the conductos of each thee phase set [14]. These mutual inductance is the esult of the magnetic flux that does not coss the ai gap and couples the conductos of each thee phase sets in the same stato slot. The mutual leakage inductances fo two thee-phase sets with ζ degee shift ae Fig.. Dual plane qd equivalent cicuit of a six-phase machine in oto efeence fame. It should be noted that the zeo sequence cicuit is not consideed in the VBR fomulation due to the fact that the VBR model is implemented in a six-phase phase domain system with no gound connection. Based on the equivalent cicuit in Fig.1. the following stato voltage equation can be defined: vsd1 i s sd1sq1 psd1, (8) vsq1 i s sq1sd1 psq1, (9) vsd i s sd sq psd, (1) v i p. (11) sq s sq sd sq In (8)-(11) the stato flux vecto on both qd planes can be defined as following: sd1 lm( isd1isd) i l sd1 ldqisq, (1) sq 1 lm( isq 1isq) i l sq1 ldqisd, (13) sd lm( isd1isd) i l sd ldqisq 1, (14) sq lm( isq 1isq) i l sq ldqisd1, (15) whee the magnetizing flux is defined as:

3 ( ( i i )), (16) sd1 sd l ( ( i i )). (17) sq1 sq In (16)-(17) the sub-tansient inductances ae defined as: ( ), ( ). l (18) Accoding to the machine equivalent cicuit and by using (16)-(17), the state space equations can be expessed as the oto flux vecto. p (( 1) ( isd1 isd )),(19) l l p (( 1) ( isd1 isd )) l v,() p (( 1) ( isq 1 isq)). (1) By substituting the stato flux in (8)-(11) with the ones fom (1)-(15) and (16)-(17), the stato voltage equations can be e-witten in the following fom: vsd1 isd1 isd1 e sd1 v sq1 i sq1 i sq1 e sq1 s v sd i sd i sd e sd vsq isq isq e sq () whee the sub-tansient inductance matix is defined as: l lm l lm lm ldq ldq lm lm ldq ldq lm l lm l lm (3) and the sub-tansient back-emf voltages as: e d1 q1 ( ) 3 l l l ( ) 3 l ( )( i sd1 isd ) v, (4) l e q1 d1 ( 1) e ( i i ) sq1 sq, (5) d q ( ) 3 l l l ( ) 3 l ( )( ) e i sd1 isd v l q d ( 1) ( i i ) sq1 sq, (6), (7) whee the sub-tansient fluxes ae defined as: d1 d l, (8) q1 q. (9) Using equations (19)-(1) and (4)-(7) the following state-vaiable model can be obtained: isd1 i sq1 p A Bi sd i sq v, (3) isd1 e sd1 i sq1 e sq1 C i sd e D sd i sq e sq v whee the system matices ae defined as the following: ( 1) l l l A ( 1), (31) l ( 1) l

4 l l B 1, (3) l ( ) 3 l l l l C ( ) 3 l l l l ( ) 3 l ( 1) ( ) 3 l ( 1), (33) l D l l l l l, (34) Applying invese tansfomation on () back to the sixphase system yields the following equation fo the machine which can be easily intefaced with an inductive o powe electonics netwok. 1 vabc 1abc siabc 1abc p( K( ) dq1k( ) iabc 1abc) e abc1abc III. CASE STUDY AND SIMUATION RESUTS (35) The pesented VBR model is applied to a six-phase machine with 3 degee phase shift between the two thee phase winding sets. The machine feeds a six-phase system though an inductive line. The detailed electical system in Fig. 3. is implemented in Matlab/Simulink [4] Using PECS [3] toolbox. The simulations use ode45 vaiable time step solve with elative and absolute eo toleances set at 1-4. Fo benchmaking pupose, thee machine models have been implemented: the conventional qd model, the coupled cicuit phase domain model and the pesented VBR model. The esults obtained with each model ae compaed with a efeence solution. The efeence solution uses a CCPD model with elative and absolute eo toleances set at 1-7 and the maximum step size of µs. It should be noted that snubbe esistos (R snubbe = 4 Ω) ae connected in paallel with the qd model to allow intefacing with an inductive netwok. The simulation uns fo a 1.5 second scenaio in which a single phase fault occus on phase a1 of the stato at t=.5 s. The modeling pocedue consides the effect of mutual leakage inductances pesent between the phase conductos in stato slots in the machine. In ode to bette show the effects of these inductances, tansient analysis is done both with and without the pesence of mutual leakage inductances in Fig. 6. Fig. 3. Cicuit schematics of the implemented system. Accoding to Fig. 4, all models ae able to epoduce the same machine wavefoms. Howeve the models ae diffeent in tems of accuacy and simulation pefomance. Fig. 5. depicts a magnified view on the machine phase cuents and toque immediately afte the fault. Accoding to Fig. 5. the qd model uses a significantly highe amount of time steps to achieve its solution. Due to the use of snubbe esistos, the qd model solution has a consideable amount of eo compaed to the efeence. At the same time, the othe two models achieve highly accuate esults with significantly less time steps with no intefacing issue. Moeove, in ode to show the impotance of modeling the mutual leakage inductance in the six-phase machine, the simulation is also done fo the same scenaio with and without consideing its effect. Fig. 6. show that the mutual inductance can have a damatic effect on the shape and peak of the machine cuent wave fom. While the cuent in the phase unde fault is not affected vey much, the cuent in the healthy phases ae affected consideably accoding to thei value of the mutual leakage inductance with the fault phase.

5 i a1s 3, See Fig. 5. a) -3, i as b) T e (N.m) - See Fig See Fig c) w (pm) d) Time (s) Fig. 4. The machine a) stato phase a1 cuent b) stato phase a cuent c) Electomagnetic toque and d) oto speed. i a1s 3 1 Refeence qd Model qd Model (high numbe of data points) CCPD a) VBR i as T e (N.m) b) Refeence qd Model CCPD VBR qd Model (high numbe of data points) c) qd Model (high numbe of data points) Refeence qd Model CCPD VBR Time(s).57 Fig. 5. Magnified view of a) stato phase a1 cuent b) stato phase a cuent and c) the machine toque using the implemented models. i a1s i b1s i c1s i as i bs i cs T e (N.m) 1-1 No Mutual leakage No Mutual leakage No Mutual leakage No Mutual leakage No Mutual leakage No Mutual leakage No Mutual leakage Fig. 6. Stato six phase cuents at the moment of a single phase fault on phase a1 at t =.5s, with and without consideing the mutual leakage inductances. The Simulation pefomance in tems of CPU time and numbe of steps ae pesented in table I. It can be obseved that the conventional qd model takes the highest numbe of steps (31,863) and CPU time (1.7s) due to high stiffness

6 ceated by the snubbe esistances. The CCPD model does not need snubbe esistances in ode to inteface with an inductive system and theefoe solves the system in the simulation scenaio in consideably less amount of CPU time (3.9s) and lowe numbe of time steps (6,39) due to slowe system eigenvalues. Howeve, the CCPD model still needs to solve a 9 by 9 matix at evey time step which has a negative effect on its pefomance. The VBR model on the othe hand inceases the simulation pefomance even moe by educing the size of the system to a 6 by 6 matix, esulting in a CPU time of 1.1s and 1,541 fo the total numbe of time steps. Table II also shows vey high accuacy of the CCPD and VBR model in compaison with the conventional qd model. Accoding to Fig. 6. it has been also shown that mutual leakage inductance between the conductos can affect the esults consideably, especially in high cuent tansients. TABE I. CPU TIME AND NUMBER OF STEPS FOR THE 1.5-SECOND STUDY Model / Pefomance Simulation Time Numbe of steps qd model with snubbes 1.7 s 31,863 CCPD model 3.9 s 6,39 VBR model 1.1 s 1,541 TABE II. Model / Pefomance COMPARISON OF -NORM ERRORS Stato Cuent IV. CONCUSIONS -nom of elative eos Electomagnetic Toque Stato Voltage qd model with snubbes 5.6 % 4. % 6.56 % CCPD model.4 %.5 % 1.8 % VBR model.4 %.5 % 1.8 % In this pape a voltage behind eactance model of a six phase synchonous machine along with oto dampe windings is pesented. The modeling pocedue accounts fo the mutual leakage inductances which ae shown to have consideable effects on the final esults duing tansients. The machine model is connected to a six-phase gid. The simulation is un fo a single-phase to gound fault scenaio in a 1.5 second time window. The VBR model is shown to be able to significantly outpefom the conventional qd model due to lack of snubbe esistances. The VBR model could also achieve bette pefomance compaed to the CCPD model due to educed size of the system matix. REFERENCES [1] E. evi, Multiphase electic machines fo vaiable-speed applications, IEEE Tans. Ind. Electon., vol. 55, no. 5, pp , May 8. [] EMTP-RV. Electomagnetic tansients pogam. CEA Technologies, Inc., Monteal, Canada. [3] Piecewise inea Electical Cicuit Simulation (PECS) Use Manual, Vesion 1.5, Plexim GmbH, 6. Available: [4] MATAB 7 (R9 a), The MathWoks Inc., Natick, MA, USA, 9. [5]. Pasa and H. A. Toliyat, Fault-toleant five-phase pemanent magnet moto dives, in Conf. Rec. IEEE IAS Annu. Meeting, Seattle,WA, 4, pp [6] G. J. Atkinson, B. C. Mecow, A. G. Jack, D. J. Atkinson, P. Sangha, and M. Benaous, The analysis of losses in high-powe fault-toleant machines fo aeospace applications, IEEE Tans. Ind. Appl., vol. 4, no. 5, pp , Sep./Oct. 6. [7] C.. Feeia and R. W. G. Bucknall, Modelling and eal-time simulation of an advanced maine full-electical populsion system, in Poc. IEE PEMD Conf., Edinbugh, U.K., 4, pp [8] E. evi, M. Jones, S. N. Vukosavic,"A seies-connected two-moto sixphase dive with induction and pemanent magnet machines," IEEE Tans. Enegy Conves., vol.1, no.1, pp , Mach 6. [9] C. Bandao Jacobina, V. F. Moua Bezea Melo, N. Rocha, E. R. Cabal da Silva, "Six-Phase Machine Convesion System With Thee- and Single-Phase Seies Convetes," IEEE Tans. Ind. Appl., vol.5, no.6, pp , Nov.-Dec. 14. [1] J. Figueoa, J. Cos, and P. Viaouge, Genealized tansfomations fo polyphase phase-modulation motos, IEEE Tans. Enegy Conves., vol. 1, no., pp , Jun. 6. [11] D. C. White and H. H.Woodson, Electomechanical Enegy Convesion. New Yok: Wiley, [1] R. H. Nelson and P. C. Kause, Induction machine analysis fo abitay displacement between multiple winding sets, IEEE Tans. Powe App. Syst., vol. PAS-93, no. 3, pp , May [13] J. R. Mati and K. W. ouie, A phase-domain synchonous geneato model including satuation effects, IEEE Tans. Powe Syst., vol. 1, no. 1, pp. 9, Feb [14] R. F. Schifel and C. M. Ong, Six phase synchonous machine with AC and DC stato connections, Pat I: Equivalent cicuit epesentation and steady-state analysis, IEEE Tans. Powe App. Syst., vol. PAS-1, no. 8, pp , Aug [15] R. F. Schifel and C. M. Ong, Six phase synchonous machine with AC and DC stato connections, Pat II: Hamonic studies and a poposed uninteuptible powe supply scheme, IEEE Tans. Powe App. Syst., vol. PAS-1, no. 8, pp , Aug [16] P. C. Kause, O. Wasynczuk, and S. D. Sudhoff, Analysis of Electic Machiney and Dive Systems, nd ed. Piscataway, NJ, USA: IEEE Pess,. [17] S. D. Pekaek, O. Wasynczuk, and H. J.Hegne, An efficient and accuate model fo the simulation and analysis of synchonous machine/convete systems, IEEE Tans. Enegy Conves., vol. 13, no. 1, pp. 4 48, Ma [18] A. M. Came, B. P. oop, and D. C. Alipantis, Synchonous machine model with voltage-behind-eactance fomulation of stato and field windings, IEEE Tans. Enegy Conves., vol. 7, no., pp , Jun. 1. APPENDIX TABE III. MACHINE PARAMETERS [14] Paametes Value Rated Phase Voltage / Powe 4 V / 1 kva Rated Roto Speed / Toque 18 pm / 53 N.m,, ls 3 mh, 1.4 mh, 15 µh, l, l 1 µh, 18 µh, 14 µh s,,, 16 mω,.5 mω,.3 mω, 1.6 mω a1a, a1b, a1c 43 µh, -43 µh, H line, line.1 Ω, 1 µh