D,Q Reference Frames for the Simulation of Induction Motors

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t...... Electric Power Systems Research, 8 (1984/85) 15 -26 15

D, Q Reference Frames for the Simulation of Induction Motors

R. J. LEE, P. PILLAY and R. G. HARLEY Department of Electrical Engineering, University of Natal, King George V Avenue, Durban 4001 (South Africa) (Received April 9, 1984)

SUMMARY reference frames are most frequently used, the particular reference frame should be This paper presents the equations of three chosen in relation to the problem being inves- preferable reference frames for use in the tigated and the type of computer (analog or simulation of induction machines when using digital) that is used. The purpose of this paper the d, q 2-axis theory. It uses case studies to is therefore to provide guidelines (by means demonstrate that the choice of the reference of case studies) in order to choose the most frame depends on the problem to be solved suitable reference frame. It also shows that and the type of computer available (analog or there are instances when the rotor reference digital). frame, which appears to have been avoided by most authors, is the best choice.

1. INTRODUCTION 2. THEORY

Induction motors are being used more than Figure 1 shows a schematic diagram of a 3- ever before in industry and individual ma- phase induction motor with the d,q axes chines of up to 10 MW in size are no longer a superimposed. The q-axis lags the d-axis by rarity. During start-up and other severe motor 90°. A voltage Vasis applied to stator phase A operations the induction motor draws large while the current flowing through it is ias. currents, produces voltage dips, oscillatory Phases Band C are not shown on the diagram torque and can even generate harmonics in in an attempt to maintain clarity. In the d,q the power system [1 - 4]. It is therefore im- model, coils DS and QS replace the stator portant to be able to model the induction phase coils AS, BS and CS, while coils DR and motor in order to predict these phenomena. QR replace the rotor phase coils AR, BR and Various models have been developed, and the CR. d,q or two-axis model for the study of tran- Although the d,q axes can rotate at an sient behaviour has been well tested and arbitrary speed, there is no relative speed be- . proven to be reliable and accurate. tween the four cdils DS, QS, DR and QR. The It has been shown [1] that the speed of physical significance of showing the D,Q coils rotation of the d,q axes can be arbitrary in Fig. 1 is to illustrate that in effect the 3- although there are three preferred speeds or phase induction motor with its six coils is re- reference frames as follows: placed by a new machine with four coils. In (a) the stationary reference frame when the order to predict the mechanical and electrical d,q axes do not rotate; behaviour of the original machine correctly, (b) the synchronously rotating reference the original ABC variables must be trans- frame when the d,q axes rotate at synchro- formed into d,q variables, but this transfor- nousspeed; mation d~pends on the speed of rotation of (c) the rotor reference frame when the d,q the D,Q coils, hence each reference frame has axes rotate at rotor speed. its own transformation. Most authors [2 -4] use one or other of In general, for any arbitrary value of 0', the these reference frames without giving specific transformation of stator ABC phase variables reasons for their choice. Whilst either the sta- [F ABcJ to d,q stator variables [FOdq]is carried tionary [3] or the synchronously rotating [2] out through Park's transform as follows:

0378-7796/84/$3.00 @ Elsevier Sequoia/Printed in The Netherlands +d axis

rotor phase A axis

stator phase A axis

+q axis

Fig. 1. D, Q axes superimposed onto a three-phase induction motor.

[FOdq] = [Po][F ABcJ (1) 1/;ds Lss Lm 0 0 ids where 1/;ejr= Lm Lrr 0 0 ejrI (4) 1/2 1/2 1/2 1/;qS 0 0 Lss Lm lqs 1/;qr 0 0 Lm Lrr iqr [Po] = 2/3 cose cos(e - A) cos(e + A) sin e sin(e - A) sin(e + A) [ ] At this stage e has not been defined and is quite arbitrary. However, certain simplifica- (2) tions in the equations occur if e is restricted A new variable called the zero sequence com- to be one of the following three angles: ponent is included with the d,q variables in (a) Stationary referenqe frame order to handle unbalanced voltages and to Now e = 0; this is called the stationary invert Park's transform. reference frame because the d, q axes do not The transformation of rotor ABC variables rotate. In addition, the +d-axis is chosen to to rotor d,q variables is again carried out us- coincide with the stator phase A axis. Hence, ing Park's transform, but this time the angle {3 from Fig. 1, in Fig. 1 is used instead of e. The voltage balance equations for the d,q W =pe = 0 (5) coils are as follows [5]: {3=e - er =-e r (6)

Vds = Rsids + P1/;ds + 1/;qSpe (3a) p{3 = -per = -Wr (7)

Vqs = Rsiqs + p1/;qS - 1/;dsPe (3b) Substituting eqns. (5) - (7) into eqn. (3),

Vejr Rriejr + p1/;ejr + 1/;qrp{3 (3c) = Vds = Rsids + P1/;ds (8a)

Vqr Rriqr + p1/;qr -1/;ejrP{3 (3d) = Vqs = Rsiqs + p1/;qS (8b) where Vejr = Rriejr + p1/;ejr- wr1/;qr (8c) Vqr = Rriqr + ptJ;qr + wrtJ;dr (8d) following two differential equations: pv~ = ids/Cs (17) Replacing the flux linkages in eqn. (8) by currents (using eqn. (4», pv~ = iqs/Cs (18) [v] = [R][i] + [L]p[i] + wr[G][i] (9) When there are shunt capacitors connected across the terminals of the motor but the where series capacitors are absent, [v] = [Vds, vdr, Vqs, Vqr]T (10) pidL = {V rn cos(wst + 7) - RLidL - vds}/LdI9) [i] = [ids' idr, iqs, iqr]T (11) piqL = {- Vrn sin(wst + 7) - RLiqL - Vqs}/LL and the other matrices appear in Appendix A. (20) Equation (9) can be rearranged in state where space form for solution on a digital computer as follows: PVds = (idL - ids)/C (21) PVqs =(iqL - iqs)/C (22) p[i] = [B]{[v] - [R][i] - wr[G][i]) (12) It is possible to formulate the equations where [B] is the inverse of the inductance where shunt and series capacitors are con- matrix and is given in Appendix A. [B] is a nected simultaneously to the motor. How- constant matrix and needs to be inverted only ever, such a system is rather unlikely in prac- once during the entire simulation. tice and is therefore not considered here. If the busbar voltages are (b) Synchronously rotating reference frame vas = Vrn cos( wst + 7) (13a) Now 0 = wst; this is called the synchronously rotating reference frame because the Vbs = Vrn cos(wst + 7 -;\) (13b) d,q axes rotate at synchronous speed. From Fig. 1, Vcs = Vrn cos( wst + 7 + ;\) (13c) and the motor terminals are connected direct- w = pO = Ws (23) ly to the busbars, then using Park's transform {3=O-Or=wst-Or (24) of eqn. (2), p{3= Ws- wr (25) Vds = Vrn cos( wst + 7) (14) Substituting eqns. (23) - (25) into eqn. (3), VqS =- Vrn sin(wst + 7) Note that Vds,the voltage applied to the stator Vds = Rsids + PtJ;ds + wstJ;qS (26a) d-axis coil, is the same as the stator phase A VqS=Rsiqs + ptJ;qS- WstJ;ds (26b) voltage. This means that the stator d-axis cur- Vdr =Rridr + ptJ;dr+ (ws - wr)tJ;qr (26c) rent idsis exactly equal to the phase A current ias, and it is not necessary to compute ias Vqr = Rriqr + ptJ;qr - (ws":"'- wr)tJ;dr (26d) separately at each step of the integration pro- Following the same procedure as for the cess through the inverse of Park's transform. stationary reference frame, the state space This saves computer time and hence is an equation for simulation on a digital computer advantage of the stationary reference frame. is If there are compensating capacitors con- p[i] = [B]{[v] - [R][i] - ws[F][i] nected between the busbar and the motor, then for the case when series capacitors are - (ws - wr)[Gs][i]) (27) present in the line (but shunt capacitors are where the [v] and [i] vectors are the same as absent (Appendix B»: in eqns. (10) and (11), and the other matrices Vds = Vrn cos(wst + 7) - v~ - RLids - LLPids appear in Appendix A. Without any capacitors, the terminal (15) voltages in eqn. (14) become

Vds = Vrn cos 7 VqS=- Vrn sin(wst + 7) - v~- RLiqs- LLpiqs (28) (16) VqS=-Vrn sin 7 The voltage components v~ and v~ across the This means that the stator d,q voltages are series capacitor are found by integrating the DC quantities and this has advantages in the field of feedback controller design when the Vqr = Rriqr + pl/Jqr (40d) motor equations are linearized around a steady operating point. It is also possible to Following the same procedure as for the use a larger step length in the digital integ- stationary reference frame, the state space ration routine when using this frame, since equation for simulation on a digital computer the variables are slowly changing DC quanti- IS ties during transient conditions. p[i] = [B]{[v] - [R][i] - wr[Gr][i]} (41) If there are compensating capacitors connected to the motor, then for the case when where the [v] and [i] vectors are the same as only series capacitors are present (Appendix in eqns. (10) and (11), and the other matrices B), the terminal voltage eqns. (15) and (16) appear in Appendix A. become: Without any capacitors, the terminal voltage eqn. (14) becomes Vds = Vm cos "1 - Vd- RLids - LLPids - 0:lsLLiqs (29) Vds = Vm cos(S0:lst + "1) (42) Vqs =- Vm sin(swst + "1) (43) Vqs =- Vm sin"-y- v~- RLiqs- LUJiqs The d,q voltages are therefore of slip fre- + 0:lsLLids (30) quency and the d-axis rotor current behaves exactly as the phase A rotor current does. The voltage components Vdand v~ across the series capacitor are found by integrating the This means that it is not necessary to com- following two differential equations: pute the phase A rotor current at each step of the digital integration process through the in- PVd = ids/Cs - wsv~ (31) verse of Park's transform. This saves computer pv~ =iqs/Cs + 0:lsVd (32) time and hence is an advantage of the rotor reference frame when studying rotor quanti- When there are shunt capacitors connected ties. across the terminals of the motor (without If there are compensating capacitors con- series capacitors), eqns. (19) - (22) become nected to the motor, then for the case when pidL = (V m cos "1 - RLidL - Vds - w.LLiqd/LL only series capacitors are present, the terminal voltage eqns. (15) and (16) become (33)

Vds m cos(S0:lst + "1) Vd RLids piqL = (- Vm sin "1 - RLiqL - VqS + WsLLidd/LL = V - - (34) - LUJids - w.LLiqs (44)

PVds = (idL - ids)/C - WsVqs (35) Vqs =- Vm sin(s0:lst + "1) - v~ - RLiqs PVqs = (iqL - iqs)/C + WsVds (36) - LUJiqs + 0:I.LLids (45) where (c) Rotor reference frame Now (J = f Wr dt; this is called the rotor ref- PVd= ids/Cs - wrv~ (46) erence frame because the d,q axes rotate at the rotor speed, but in addition the d-axis pv~ = iqs/Cs + 0:lrVd (47) position coincides with the rotor phase A When there are shunt capacitors connected axis. Hence, from Fig. 1, across the terminals of the motor (without W = p(J = 0:lr (37) series capacitors), eqns. (19) - (22) become

{3= (J - (Jr = 0 (38) pidL = {V m cos(S0:lst + "1) - RLidL p{3= 0 (39) - 0:I.LLiqL- vds}/LL (48) Substituting eqns. (37) - (39) into eqn. (3), piqL = {- Vm sin( S0:lst + "1) - R LiqL Vds = Rsids + P I/Jds+ 0:lrI/Jqs (40a) + 0:I.LLidL - VqJ/LL (49) VqS =Rsiqs + pl/JqS - wrl/Jds (40b) PVd = (idL - ids)/C - WrVqS (50) Vdr =Rridr + pl/Jdr (40c) pv~ = (iqL - iqs)/C + WrVds (51) Thus far, the voltage equations for three The calculated rotor d-axis variables of this reference frames have been presented with reference frame are, however, also trans- their advantages. The electrical torque is given formed at 50 Hz frequencies and are therefore by the following expression which is indepen- not the same as the actual rotor phase A dent of the reference frame: variables which are at slip frequency. Figure 3(c) and (d) show this difference between (52) Te = wbLm(idIiqs - iqricts)/2 the rotor phase A current and the rotor DR The mechanical motion is described by current. It would therefore be an advantage when PWr = (Te - Td/J (53) studying transient occurrences involving the stator variables to use this reference frame.

3. RESULTS 3.3. Rotor reference frame Since in this reference frame the d-axis of In order to evaluate the usefulness of each the reference frame is moving at the same frame of reference, the equations of § 2 are relative speed as the rotor phase A winding used to predict the transient behaviour of a and coincident with its axis, it should be 22 kW motor; its parameters appear in expected from the considerations of § 3.2 Appendix C. that the behaviour of the d-axis current and the phase A current would be identical. 3.1. Frames of reference Figures 4(c) and (d) show how the rotor Figure 2 shows the predicted start-up phase A current and the rotor d-axis current characteristics against no load, using each one are initially at 50 Hz, when the rotor is at of the above three reference frames. standstill, but gradually change to slip fre- The electrical torque (Fig. 2(b» exhibits quency at normal running speed. If this the familiar initial 50 Hz oscillations, and full reference frame is used for the study of rotor speed is reached within 0.2 s. Without any variables, it is therefore not necessary to go soft-start control being used, and considering through the inverse Park's transform to com- the worst case, the motor starting current pute the actual rotor phase A current. Since reaches 9.5 p.u. Despite the fact that the the rotor d-axis variables are at slip frequency, behaviour of the physical variables as this reference frame is useful in studying predicted by each reference frame is identical, transient phenomena in the rotor. the d,q variables in the respective frames of The transformation of the stator variables reference are different; this difference can be at slip frequency in this frame clearly shows exploited by careful matching of the refer- its unsuitability for studying stator variables, ence frame to the problem being solved, as which are at 50 Hz frequencies. illustrated in the following sections. 3.4. Synchronously rotating reference frame 3.2. Stationary reference frame When the reference frame is rotating at In this reference frame the d-axis is fixed to synchronous speed, both the stator and rotor and thus coincident with the axis of the stator are rotating at different speeds relative to it. phase A winding. This means that the mmf However, with the reference frame rotating at wave of the stator moves over this frame at the same speed as the stator and rotor space the same speed as it does over the stator phase field mmf waves, the stator and rotor d,q A windings. This reference frame's stator variables are constant quantities, whereas the d-axis variables therefore behave in exactly actual variables are at 50 Hz and slip frequen- the same way as do the physical stator phase cies respectively. Figure 5(b) shows how the A variables of the motor itself. Figures 3(a) stator d-axis current gradually reduces from and (b) show the identical nature of the stator 50 Hz frequency at the instant of switching to phase A current and the stator d-axis current. the equivalent of a steady DC when at rated It is therefore not necessary to go through the speed. inverse of Park's transform to compute the The steady nature of this stator d-axis cur- phase A current, thus saving in valuable com- rent makes this reference frame useful when puter time. an analog computer is used in simulation TERMINAL VOLTAGE 1.~ 6. TORQUE

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Fig. 2. Predicted starting up performance of a 22 kW induction motor. D-AXIS STATOR CURRENT -+

i ~ as ~ ids 0.: 0.:

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[ D-AXIS ROTOR CURRENT ll!l. 11!1.

i idr ar ~ 0.: 5. 0.: 5.

I!I.

I- I- Z -5. Z -5.1!I1 W W It: It: It: It: ::J ::J U U -ll!l. I!I.I!II!I 5111.0111 11!11!1.1!11!1 l51!1.1!I1!I 21!11!1.1!11!1 251!1.1!I1!1 5IZI.1!I1!1 ll!ll!l.1!II!I l51!1.1!I1!I 21!11!1.1!I1!I 251!1.1!11!1 TIME (c) milli-SEC TIME (d) milli-SEC

Fig. 3. Predicted starting up current using the stationary reference frame. D-AXIS STATOR CURRENT 11/1. +--

i I r as . I ::5 ::5 ~ds I 0: 0: T i I

...... z -5. 1111 Z -5. //I UI w 0:: 0:: 0:: 0:: t :J :J U U I I -1111.//I I I I Sill. 111111 1111111.//1//1 15111.//1//1 2//1//1.//1//1 25111.//1//1 13.//1//1 Sill."" 1//10.//I" . 15".111111 2111111.111//1 25//1.111111 TIME (a) mi 11 i-SEC TIME (b) mi IIi -SEC PHASE A ROTOR CURRENT D-AXIS ROTOR CURRENT 1//I. //I I I

i ::5 ar ::5 idr 0: 0:

-1//1. //I 13.//1111 5//1.//1//1 1//1111.//1//1 15//1.//1111 2111//1.//1//1 25//1. //1111 1"//1.111" 15111."" 2//1//1.//1111 25//1.//1" TIME (c) mi 11 i -SEC (d) mi 11 i-SEC

Fig. 4. Predicted starting up current using the rotor reference frame. PHASE A STATOR CURRENT D-AXIS STATOR CURRENT

i ::5 as ids 0:

I- I- Z ffi -5. IIJI 0:: ~ -8.11J 0:: 0:: ;:) ;:) U U -1"-" 511J.1IJ11J l11J0. IIJIIJ 1511J.00 20"- IIJ0 250.0B e. BIIJ 50. "0 10m.IIJ0 15m.IIJ0 200. 00 250.00 TIME (a) mi 11 i-SEC TIME (b) mi 1 Ii-SEC D-AXIS ROTOR CURRENT 1". PHASE A ROTOR CURRENT

i ::5 ar ::5 idr 0: 0:

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50.00 l11J0.1IJ11J 15". 00 20"- DB 250. BB 50.011J 1011J.00 150."0 200.00 250.00 TIME (c) mil Ii-SEC TIME (d) milli-SEC

I>:> Fig. 5. Predicted starting up current using the synchronously rotating reference frame. "" studies, since the input to the machine equa- ration step length may be lengthened without tions are constant prior to any disturbance. affecting the accuracy of results; Also, because the stator and rotor d-axis (e) should a multimachine system be variables vary slowly in this frame, a larger studied then the advantages of the synchro- integration step length may be used on a nously rotating reference frame appear to out- digital computer, when compared to that used weigh those of the other two reference with the other two reference frames, for the frames. same accuracy of results. It is also preferable to use this reference frame for stability of controller design where ACKNOWLEDGEMENTS the motor equations must be linearized about an operating point, since in this frame the The authors acknowledge the assistance of steady state variables are constant and do not D.C. Levy, R.C.S. Peplow and H.L. Nattrass vary sinusoidally with time. in the Digital Processes Laboratory of the Department of Electronic Engineering and M.A. Lahoud of the Department of Electrical Engineering, University of Natal. They are 4. CONCLUSIONS also grateful for financial support received from the CSIR and the University of Natal. This paper has taken. the three preferred frames of reference within the d,q two-axis theory and compared the results obtained with each frame. From this comparison it is concluded that: (a) when a single induction motor is being NOMENCLATURE studied, anyone of the three preferred frames of reference can be used to predict transient C shunt capacitor behaviour; Cs series capacitor (b) if the stationary reference frame is H inertia constant = Jws/2 used, then the stator d-axis variables are ids, Vds stator d-axis current and voltage identical to those of the stator phase A icb:, Vcb: rotor d-axis current and voltage variables. This eliminates the need to go iqs, v qs stator q-axis current and voltage through the inverse of Park's transform to iqr> Vqr rotor q-axis current and voltage obtain the actual stator variables, so saving in idL d-axis line current computer time. This would be useful when iqL q-axis line current interest is confined to stator variables only, as J inertia of motor for example in variable speed stator-fed induc- LL line inductance tion motor drives; Lss stator self inductance (c) if the rotor reference frame is used then Lrr rotor self inductance the rotor d-axis variables are exactly the same Lm mutual inductance as the actual rotor phase A variable. This p derivative operator d/dt again saves computer time by eliminating the RL line resistance need to go through the inverse of Park's Rs stator phase resistance transform to obtain the rotor phase A Rr rotor phase resistance variables. This would be useful when interest Te electrical torque is confined to rotor variables only, as for TL load torque example in variable speed rotor-fed induction 'Y phase angle at which phase A voltage motor drives; is applied to motor (d) when the synchronously rotating refer- A 21T/3 rad ence frame is used, the steady DC quantities 1/J flux linkage both of the stator and rotor d,q variables Wb nominal speed at which p.u.torque = make this the preferred frame of reference p.u. power, rade S-1 when employing an analog computer. In the Ws synchronous speed,rade S-1 case of a digital computer solution, the integ- Wr rotor speed, rade s- 1 REFERENCES Bs Bm 0 0 B Br 0 0 1 P. C. Krause and C. H. Thomas, Simulation of [B) = I m symmetrical induction machinery, IEEE Trans., 0 0 Bs Bm PAS-84 (1965) 1038 -1053. 0 0 2 D. J. N. Limebeer and R. G. Harley, Subsynchro- Bm Br nous resonance of single cage induction motors, Proc. Inst. Electr. Eng., Part B, 128 (1981) 33 - where 42. 3 C. F. Landy, The prediction of transient torques Bs = Lrr/D, Br = Lss/D produced in induction motors due to rapid recon- Bm = -Lm/D, D =Ls"Lrr-Lm 2 nection of the supply, Trans. S. Afr. Inst. Electr. Eng., 63 (1972) 178 - 185. 4 P. Pillay and R. G. Harley, Comparison of models 0 0 0 0 for predicting disturbances caused by induction ° 0 Lm Lrr motor starting, SAIEE Symposium on Power Sys- [Gs] = I tems Disturbances, Pretoria, September 1983, 0 0 0 0 SAIEE, Kelvin House, Johannesburg, pp. 11 - 18. 5 B. Adkins and R. G. Harley, The General Theory -Lm -Lrr 0 0 of Alternating Current Machines: Application to Practical Problems, Chapman and Hall, London, 1975, ISBN 0 412 12080 1. 0 0 Lss Lm 0 0 0 0 [Gr]=1 -Lss -Lm 0 0 0 0 0 0

APPENDIXA

D, Q axis model matrices APPENDIX B

Rs 0 0 0 Derivation of expressions for voltage drops i, 0 Rr 0 0 the d, q model [R] =I ° 0 Rs 0 The voltage drop across an inductor in AB( phase variables is given by 0 0 0 Rr [VABCh = Lp[iABcJ (B-1 = Lp[Per1[iOdq] (B-2 Lss Lm 0 0 The voltage drop in the Odq reference frame j L Lrr 0 0 given by [L] = I m 0 0 Lss Lm [Pe] [VABcJL = [Pe]Lp {[Perl [iOdq]} (B-3 0 0 Lm Lrr [VOdq]L =L[Pe]p{[por1[iOdq]) Lp[iOdq] + L[Po]p[Por1[iodq] (B-4 0 0 0 0 = Now, substituting for 0 0 -Lm -c-Lrr [G] = I ° 0 0 0 o ° ° Lm Lrr 0 0 [Pe]p[por1 = 0 0 w (B-5 r 0 -w 0 1 hence 0 0 Lss Lm ° 0 0 0 VdL= Lpid + wLiq (B-6 [F] = I -Lss-Lm 0 0 VqL= Lpiq - wLid (B-7 0 0 0 0 The voltage drop across a capacitor in AB( phase variables is given by 26

APPENDIX C [UABcJc = (l/C) f[iABcJ dt (B-8)

In the Odq reference frame the voltage drop becomes 22 kW Induction motor parameters [Per1[UABcJc = (1/C)j[Per1[iOdq] dt (B-9) Base time 1 s Differentiation gives Base speed 1 rade S-1 Base power 27.918kVA [Per1p[UOdq] + p[Per1[UOdq] Base stator voltage 220 V (phase) = (1/C)[Per1[iodq] (B-1 0) Base stator current 42.3 A (phase) Base stator impedance 5.21 S1 Multiplication by Pe gives Base torque 177.8 N m P[UOdq]+ [Pe]p[Per1[UOdq] Number of poles 4 = (l/C)[ iOdq] (B-ll) Rated output power 22 kW Therefore Stator resistance Rs 0.021 p.ll. Rotor resistance Rr 0.057 p.u. P[UOdq] Stator leakage 0.049 p.u. expressed = (l/C)[iodq] - [Pe]p[Per1[UOdq] (B-12 ) reactance Xs Hence Rotor leakage 0.132 p.u. at 50 Hz reactance Xr PUd = id/C - WUq (B-13 ) Magnetizing 3.038 p.u. PUq = iq/C + WUd (B-14 ) reactance Xm