Reference frames for simulation of electric motors and drives
C.W. Brice
brice@engr.sc.edu
E. Santi
X. Kang
santi@engr.sc.edu kangxi@engr.sc.edu
Department of Electrical Engineering
Swearingen Engineering Center
University of South Carolina
Columbia, SC 29208, USA
ABSTRACT
Simulations of AC electric machines are usually performed
in a coordinate system, or reference frame, with two axes
(often denoted by d and q, or represented as a complex
space vector). A third axis, the zero-sequence axis, is a
homopolar axis that is necessary for handling certain
unbalanced conditions. Historically, the transformation
was essential to getting reasonable results, since
computations were expensive and the transformation
results in great simplification of the electrical and
mechanical differential equations.
Today, however,
computation has become so incredibly cheap that the
computing power of imbedded controllers in many
appliances exceeds that of the early mainframe computers.
Consequently, this paper revisits the issue of
transformation of coordinate frames for performing
simulations.
Keywords: Motors, drives, simulations, reference frames.
1. INTRODUCTION
Analyses of electric machines have been made in rotating
reference frames since the early days with outstanding example
in the work of Park [1] on synchronous machines, which states
in the first paragraph that: "This paper presents a generalization
and extension of the work of Blondel, Dreyfus, and Doherty
and Nickel." Apparently, the early work of Blondel [2], which
established the two-reaction theory of the salient-pole machine,
provided the initial motivation for consideration of the
transformation to a rotating reference frame. Park transformed
to a frame fixed in the rotor, which is reasonable for a
synchronous machine, since the field winding (or the magnets
in the case of a permanent-magnet machine) are fixed in that
frame, and since some degree of saliency often exists.
Induction (or asynchronous) machines usually have a
symmetrical rotor structure, consisting of cages or polyphase
windings. Thus, the choice of reference frame is not so
obvious. Stanley [3], for example used a stationary reference
frame, while many authors (see Krause [4], for example) have
used frames that rotate either at synchronous speed or that are
fixed in the rotor. Recently Pekarek [5] studied the use of a
hybrid approach that uses stationary frame for the stator
electrical quantities and a rotating frame for the rotor electrical
quantities.
Electronic power converters may be modeled either by detailed
representation of the switching of the power electronic devices
(switching model) or by a state-space averaged modeled
(average model). In the case of the latter, it is possible to
L. U. Gökdere
gokdere@engr.sc.edu
convert the model into a dq frame [6], but the most natural
representation is in the polyphase circuit variables.
We consider two systems, one composed of an induction motor
fed from a stiff ac voltage source and the second of an inverter
(fed from a stiff dc voltage source) plus an induction motor
driving a synchronous generator, which supplies an impedance
load. In the first case, a series of several different reference
frames are studied in detail, while in the second case, we
consider the motor-generator set starting and operating under
closed-loop speed control. Appropriate reference frames are
used for each apparatus in the second case, with
transformations applied as needed.
2. DEVICE MODELS
2.1 Induction motor model
The Np-pole three-phase symmetrical induction motor is easily
represented in a general dq reference frame as depicted in
Figure 1. Notice that the frame may rotate at an arbitrary speed
ω [rad/sec] with respect to the stator phase a axis, or as a
special case ω may be equal to zero (stationary reference
frame). A common choice of a rotating frame is the
synchronously rotating frame ω = ωs = 4πf/Np. However, in
general, the frame need not rotate at a constant speed, and is
arbitrary. The angle of the transformation is θ [rad] which is
the time integral of ω [rad/sec], or
ω=
dθ
dt
(1)
Figure 1 shows the general reference frame and the conventions
used for the transformation, while equation (2) gives the
corresponding mathematics.
é
sin(θ)
é v as ù êê cos(θ)
ê v ú = êcos(θ − 2π ) sin(θ − 2π )
ê bs ú ê
3
3
êë v cs úû ê
2π
2π
cos(θ + ) sin(θ + )
êë
3
3
π
2
2
2
é
ê cos(θ) 3 cos(θ − 3 )
é v qs ù ê 3
ê v ú = ê 2 sin(θ) 2 sin(θ − 2π )
ê ds ú ê 3
3
3
êë v 0 úû ê 1
1
êë 3
3
ù
1ú é v ù
ú qs
1ú êê v ds úú
úê ú
ë v0 û
(2)
1ú
úû
2
2π ù
cos(θ + )ú
3
3 é v as ù
2
2π ú ê ú
sin(θ + ) ú ê v bs ú
3
3 ú
1
ú êë v cs úû
úû
3
Note that the frame speed ω may be constant (e.g., 0 or ωs) or
variable. If the frame speed is zero, the angle θ may also be
taken to be 0 for simplicity. The same transformation is used
for all circuit quantities: voltage, current, and flux linkage.
The variable labeled with the zero subscript is the zerosequence variable, which is clearly not needed (except to
prevent the inverse transformation from being undefined)
unless there is an unbalanced condition that results in ground
current (i.e., unless i a + i b + i c ≠ 0 ). We consider motors in a
three-conductor connection in this paper, so the zero-sequence
variables are identically zero.
b axis
q axis
θ
Figure 2. The induction motor model in a purely arbitrary
reference frame. The reference frame speed is ω and the rotor
speed is ωr.
2.2 Synchronous machine model
For the sake of comparison, we consider the synchronous
machine with a conventional model representing a single cage
damper (amortisseur) winding on each axis, and a single field
winding on the d axis. The rotor parameters are referred to the
stator.
Later, a synchronous generator will be another
component of the system under study to provide a shaft load on
the motor.
In contrast to the induction motor, which was symmetrical,
even a cylindrical-rotor synchronous machine has a certain
asymmetry, due to only having one field. Consequently, we
model this machine in a d-q frame that rotates with the rotor
and is aligned with the field along the d axis. Since ω = ωs =
ωr, the rotor circuits have no speed voltages. Details are shown
in Figure 3.
ω = dθ/dt
a axis
isq
Rs
ωsλsd
vsq
Lls
λsq
Llrq
Lmq
Rrq
irq
λrq
d axis
c axis
Llf
Figure 1. General reference frame with the q-axis at an
arbitrary angle with respect to stator phase a.
Note that converters may produce zero-sequence voltages that
tend to flow in stray capacitances and perhaps through
bearings. Since we use only low-frequency models here, that
has been omitted from consideration.
Figure 2 gives the induction motor model in the arbitrary
reference frame. The speed of the reference frame is ω and the
rotor speed is ωr, both in electrical rad/sec (i.e., actual speed
multiplied by the number of pole pairs). The rotor consists of a
single cage winding, and all rotor circuit parameters are
referred to the stator.
isq Rs
ωλsd
vsq
Lm
λsq
isd Rs
Llr (ω−ωr)λrd Rr
Lls
ωλsq
λrq
Llr (ω−ωr)λrq Rr
Lls
irq
isd
vsd
λsd
Lm
λrd
ωsλsq
Lls
λsd
Llrd
Lmd
λfd
Rrd
ifd
ird
Vf
λrd
Figure 3. Synchronous machine model. Note the field circuit
is referred to the armature (stator).
2.3 Converter model
The induction motor input is fed from an electronic drive that
implements a PWM inverter fed from a stiff DC voltage. The
inverter produces a constant volts-per-hertz output, so that as
the frequency varies from nominal, the voltage is changed
proportionally. To model this device, shown in Figure 4, an
average model is used. That is, an instantaneous duty factor is
used to determine the average voltage applied to each phase.
a
ird
Vd
vsd
Rs
Rf
b
c
IM
Figure 4. PWM inverter used to supply the induction motor.
2.4 System description
The system under consideration is an induction motor driving a
synchronous generator. The motor is connected to the PWM
inverter using a speed regulation loop (proportional control).
Figure 5 shows the block diagram of the controller. The shaft
speed is measured and compared to the desired speed (the
reference input) and used to control the duty factor of the
inverter in proportion to the error between the measured and
desired speeds.
ωref
Duty
K
Communication Interval = 1 msec
V,f
PWM
Figure 7 shows plots of the motor developed torque plotted
versus time for the duration of the simulation. In every case,
plots similar to this one showed the same results for torque,
speed and terminal electrical quantities in the simulations for
each reference frame.
MG
ω
Figure 5. Block diagram of proportional speed control loop.
PWM = inverter, MG = induction motor-synchronous generator
set, K = controller gain.
3.00
Execution time
Err
Pentium processor using the Windows NT 4.0 operating
system. Results will vary with other operating systems and
should be taken as relative information only.
2.50
2.00
1.50
1.00
0.50
0.00
Run
Simulation of the converter is most naturally accomplished in
the abc coordinate frame.
Stationary
Execution time
3.1 Induction motor simulation
A number of simulation runs were performed using the
simulation language ACSL to represent the induction motor
starting across a stiff voltage source with no load on the shaft.
After the motor was started, a step change in shaft load torque,
from 0 to 2000 N m, was applied. The whole simulation time
was 20 seconds. The execution times were tabulated for each
of three test cases, each consisting of four runs of the
simulation in reference frames that were stationary, rotating
with the rotor, and synchronously rotating. The execution
times are plotted in Figure 6.
Rotor
Synchronous
Communication Interval = 10 msec
3. SYSTEM SIMULATION RESULTS
This simulation was a 3-phase, 60-Hz, 460-V, 4-pole, 1764
rev/min, 300 kW (mechanical) single-cage induction motor.
The shaft load was a flywheel with inertia of 90 kg m2 and
friction coefficient of 0.088 N m sec. Rs = 3.527 mΩ, Rr =
4.232 mΩ, Lss = Lls+Lm = 3.994 mH, Lrr = Lss, Lm = 3.929
mH.
1a 1b 1c 1d 2a 2b 2c 2d 3a 3b 3c 3d Avg
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
1a 1b 1c 1d 2a 2b 2c 2d 3a 3b 3c 3d Avg
Run
Stationary
Rotor
Synchronous
Figure 6. Execution times for induction motor simulation in
three different reference frames for two different
communication intervals.
The communication interval is the time step between output
points, but the numerical integration algorithm was a fifth-order
variable step size Runge-Kutta. The whole experiment was
repeated for two different communication intervals. At the top
of Figure 6 is an interval of 1 msec while at the bottom is 10
msec. Note that 10 msec is too long for display of waveforms
at 60 Hz, but is suitable for torque and speed plots. The results
show that the effect of the reference frame is almost totally
insignificant at the shorter interval, but that the rotating
reference frames show approximately 2:1 speed advantage at
the longer interval. Thus, if the simulation is intended to show
details of 60 Hz waveforms of currents and voltages, one might
as well use the stationary frame. Otherwise, if the time
intervals can be chosen long enough, the rotating frames show
the greatest promise.
It is interesting to note the statistical variation in the execution
times, which we hypothesize is simply due to running the
simulations on an operating system that has other tasks running
in the background. The simulations were run on a 400 MHz
Figure 7. Electrical torque Te [N m] and mechanical torque
Tm [N m] plotted versus time t [sec].
3.2 System simulation
The system shown in Figure 8, which is the same as the one
shown by the block diagram in Figure 5, was simulated with
the same simulation program. The generator was loaded with a
resistive load after the motor starting sequence.
Generator ratings: 3-phase, 460 V, 60 Hz, 4 poles, 1800
rev/min, 300 kW (electrical). Parameters: Rs = 5.643 mΩ, Rf
= 0.444 mΩ, Rrd = 0.019 Ω, Rrq = 0.012 Ω, Lsd = 2.713 mH,
Lrd = 3.373 mH, Lfd = 3.222 mH, Lrd = 3.373 mH, Lmd =
2.563 mH, Lsq = 2.657 mH, Lrq = 2.947 mH, Lmq = 2.507 mH
During the starting sequence, the motor frequency is ramped up
from 10 Hz to 60 Hz as shown in Figure 9. The actual shaft
speed is shown in Figure 10, which follows the desired speed
trajectory. There is a change in the slope at about t = 8 sec,
where the input frequency begins to track the reference input
very closely. After this, the system is in a pseudo-steady-state
condition as the frequency ramps up. A slightly different
starting sequence is often used, wherein the voltage is held
constant at a small value while the frequency is ramped up, and
then both voltage and frequency are ramped together. We
intend to run more simulations to better tune the start-up
sequence with a view to reducing the transients.
The speed control uses a proportional controller with a gain of
20. This figure was derived by trial and error by running
several simulations.
Figure 10. Shaft rotational speed [rad/sec]. Notice that this
tracks the reference frequency input closely.
Figure 11. Electrical torque of induction motor [N-m]. Notice
the startup transients and transient due to the generator being
suddenly loaded at t = 10 sec.
The following scenario was studied on the generator: (1) The
motor was started as described above, while the generator is
excited to allow its voltage to build up. Initially, the generator
is feeding an open circuit. (2) After motor reaches operating
speed, the generator is suddenly loaded to its rating. Figures 11
and 12 show the electrical torque of the motor and the
generator.
PWM
Inverter
Voltage
Source
Synchronous Generator
Figure 12. Electrical torque [N m] of the generator. Generator
is loaded suddenly at t = 20 sec.
Induction
Motor
Resistance
Load
Figure 8. Schematic diagram of system studied.
These plots were obtained using the stationary reference frame
for the induction motor, but the same results were obtained for
the other two frames. Simulations of runs where all three
frames were used successively have variations that lie within
the width of the line used in the plot.
4. CONCLUSIONS
Reference frames for induction machines can be chosen
arbitrarily. The effects of the choice on the speed of the
simulation are not very large if it is desired to plot electrical
quantities such as current and voltage at the machine terminals.
In our system simulations, we used a stationary reference frame
for the cases plotted, but the same results were obtained for the
other reference frames.
Figure 9. Reference frequency [Hz] input to the PWM inverter.
The constant volts/Hz control causes the voltage rms value to
track this input.
One of the most interesting conclusions to be drawn from the
present work, is that the recording of data and its display seems
to dominate the processor time devoted to the calculations. On
a prototype of the VTB software, running several simulations
of different systems, similar conclusions were drawn. For
systems that use more advanced visualization tools, such as 3-d
renderings, the results are more startling: almost all the
computational time goes into the graphics.
5. PLANS FOR FUTURE WORK
This project is a part of the Virtual Test Bed (VTB), sponsored
by the US Office of Naval Research (Grant N00014-96-10926), which is also developing a new simulation environment.
Our immediate plans are to implement simulations of this
system in the VTB software, primarily to provide a test case
that shows realistic performance, in order to provide a basis for
making comparisons between different simulation programs
and the one under development. Unfortunately, these efforts
are sufficiently mature to present results yet, but we anticipate
some in the near future.
The system studied is just a part of an overall system with other
loads, transformers, converters, etc., representing a small or
medium scale power system. Eventually, we plan to report on
the results of simulating the larger system. As part of this
study, a more systematic means of designing the controllers
will be used.
The VTB software will allow advanced visualizations,
including 3-d renderings, which will allow some novel ways to
display results besides line plots and oscilloscope-type displays.
REFERENCES
1.
R. H. Park, "Two-Reaction Theory of Synchronous
Machines," AIEE Transactions, Vol. 48, pp. 716-730, July
1929.
2.
A. Blondel, Synchronous Motors and
McGraw-Hill, 1913.
3.
H.C. Stanley, "An Analysis of the Induction Machine,"
AIEE Transactions, Vol. 57, pp 751-759, 1938.
4.
P.C. Krause, Analysis of Electric Machinery, McGrawHill, 1986.
5.
S.D. Pekarek, O. Wasynczuk, H.J. Hegner, "An efficient
and accurate model for simulation/analysis of
machine/converter systems," IEEE Transactions on
Energy Conversion, Vol. 13, pp. 42-48, March 1998.
6.
D.W. Novotny and T.A. Lipo, Vector Control and
Dynamics of AC Drives, Oxford University Press, New
York, 1998.
Converters,