"" I! 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 synchro- nously 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 con- nected 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.