Estimation of Permanent Magnet Motor Parameters
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Transcript of Estimation of Permanent Magnet Motor Parameters
8/3/2019 Estimation of Permanent Magnet Motor Parameters
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IbEE Industry Application Society
Annual Meeting
New Orleans, Louisiana, Octcber 5-9, 1997
Estimation of Permanent Magnet Motor Parameters
S . Weisgerber A. Proca A . KeyhaniThe Ohio State University
Electrical Engineering Department
Columbus, OH 43210
Phone (614) 292-4430 E-Mail: [email protected]
Abstract - To properly design and optimize a control system
of a permanent-magncf (PM) machine, the machine model
and its parameters mu,Ft be known. This study presents a
method for developing a model and estimating the
parameters of a PM nraclzine from standstill time-domain
step response data. A dq-axis equivalent circuit model with
a general number of da'mper windings is defined to describetlze behavior of the E'M machine. Standstill simulation
studies were performed on a known PM machine to
generate syntlzetic test data to be used for estimation
procedure verification. The stand-still dq-axis model is
subjected to a step-inp,rct signal and the resulting output is
used fo r parameter estimation. Th e output-error estimation
algorithm is used to evtimate the unknown parameters ofthe dq-axis model, Stand-still tests were then performed on
a 4 pole, 1 kW PM machine to obtain data. The PM
machine parameters were estimated and the model verijied
against experimental ifiput and output test data.
I. INTRODUCTION
Several different papers have been published that study
techniques used to estimate the parameters of synchronous
machines [13. This study focuses on model identification and
parameter estimation of synchronous permanent-magnet
(PM) machines using time-domain estimation techniques.
PM machine modeling is seen in such works as [3],[4], and
[ 5 ] . The estimation method used in this study [1][5], consists
of fixing the rotor of the PM machine in a specific position,
applying a simple DC voltage source, and collecting input
and output data to be used for estimation.
The present study focuses on a dq-axis circuit model of thePM machine with a k;eneral number of damper windings,
experimental technique: used to collect data, and m ethod of
applying the measured data to identify the correct m odcl andto estimate the m achine parameters.
11. PROBLEM DESCRIPTION
The objective of this study is to develop a method that can be
used to model the stand-still dq-axis model of a PM machine,
0-7803-4067-1 97/$10.00 0 1997 IEEE.
obtain input and output data from a PM machine, and
estimate the standstill dq-axis machine parameters.
To study this problem a PM m achine with known parameters
is simulated to create synthetic input and output test data to
be used during the identification process. Next, to evaluate
the effect of noise on the estimation method the test data iscorrupted with noise of a known distribution. Once the
noise-corrupted test data is obtained it can be treated as if it
came from a PM machine with an unknown system model
and parameters. A system model is then assumed and th e
output-error estimation method is used to estimate the
parameters of the unknown system.
Once the simulation studies verify that th e technique used to
estimate the parameters is valid, tests are performed on a
working 4 pole, 1 kW PM machine. Experimental input and
output data is collected from the PM machine to be used for
system model identification and parameter estimation. A
system model is assumed and the output-error estimationmethod is used to estimate the parameters. The assumed
model and estimated parameters are then validated against
the actual experim ental data. Next, if required, a different
model with more damper windings (section 111.) is chosen
and the same procedure is executed until the identified model
and estimated parameters are validated.
111. PERMANENT MAGNET MACHINE MODEL
The first step in identifying the permanent m agnet m achine is
to develop a generic dq-axis model that can be applied to any
PM machine over any operational frequency range. This
proposed generic model is derived from the PM machineshown in fig. 1 and defined by the electrical circuits shown in
Fig. 2. The circuits in Fig. 2 were derived by first
considering a PM machine such as the one shown in Fig. 1,
writing the voltage equations that describe the machine in theabc reference frame, and then transforming the equations tothe dq-axis reference frame [2] . Fig. 1 shows a generic two-
pole three phase PM machine with an unknown number of
damper windings in the rotor. The damper windings in the
rotor represent induced currents circulating in the rotor (eddy
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sensor
cs axis1 \ axis
Figure 2b. PM machine q-axis equivalent circuit
The state-space equations that describe the circuits in Fig. 2
where j represents either d o r q and p = &/dt :
y l = C,ixjv ( k ) ( 2 )
where,
x = [id i,, . . . i M d ] xq = [iq iZq . . . i N q ] (3)
Figure 1. Brushless permanent magnet machine
M = number of d-axis damper windings
N = number of q-axis dam per windings
currents), which are induced by harmonics in the applied
voltages and/o r oscillations in the rotor speed. Damper
windings are no t required to be built physically into the rotor
for purposes of creating starting torque since the motors in
question are controlled to be in synchronism at all speed
values. The circuits in Fig. 2 show the dq-axis of a PM
machine with M unknown d-axis damper windings, and N
unknown q-axis dam per windings. The number of damperwindings in the d-axis and the q-axis are determined through
the identification schem e proposed in this paper.
A mathematical representation is derived from the circuits
shown in Fig. 2. The number of damper windings to be used
is determined by using testing and identification procedures
over the PM mac hine's operational frequency range. The
equations shown below are given in state-space form based
on a general number of d-axis and q-axis damper w indings.
_ _
Figure 2a. PM machine d-axis equivalent circuit
30
U , =[Vd 0 . . . olT Uq = [ v q . . . 0IT (4)
The system output is,
Yi = [ij] (7)
The inductance matrix Lj and the resistance matrix Rj are
generalized as,
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O l
identiped=f(gl
rrrimuted
Rd =
f
R, =
-
R. , 0 . . 0
0 RI,
. .
0 . . . R,,-
. .R n , I
The off-diagonal terms in the Lj matrices are all Lmj and the
off-diagonal terms in the Rj matrices are all zero.
In the above equations the variables w(k) an d v(k) represent
the process noise and i.he measurement noise respectively.
The process noise is created by a naturally occurring
disturbance to the input sequence. Whereas the measurement
noise is present because all measurements are inherently
subject to error, the intensity of the measurement noise is
based on the quality of the sensor being used
The unknown parameter sets Bj are the parameter values to
be estimated from the stand-still step response input and
output data using the output-error estimation algorithm.
During standstill cond tions th e term s o A, an d o , h , a r e
equal to zero hence the unknown parameter sets for the d-
axis and the q-axis are,
IV . OUTPUT ERROR PARAMETER ESTIMATION
ALGORITHM
As stated this paper will focus its study on the evaluation of
the output error method (OE) for estimating PM machine
parameters. Th e oulput error estimation algorithm is
described in the flow cl-,art show n in Fig. 3 . The output errorestimation algorithm is based on minimizing a cost function
defined as,
1 N
N =O
V ( 0 )=- ( e ' ( k ) e ( k ) }
where e(k) is defined as the difference between the measured
output Y(k) and th e estiinated output ? ( k ) , and N is the total
number of measured data points.
I I
Figure 3 . The output error estimation algorithm
V. STUDY PROCESS
The study process consisted of first verifying the estimation
procedure through simulation and synthetic data and then
applying the technique to an actual PM machine. A test was
set up to subject a PM machine to a DC step input voltage
and collect the input and output data. To collect the
experimental data the following procedure was used. Using a
four pole, 1 kW PM machine, a DC step voltage signal was
applied to the machine for a period of time long enough so
that the system reached steady-state. During this time, the
step response input and output is collected as experimental
data to be used during the estimation procedure. To be able
to execute the described test, the PM machine must be setup
up specifically either for the d-axis test or for the q -axis test.
Fig. 4 and Fig. 5 show how the rotor should be positioned for
each test. As shown in Fig. 4, for the d-axis step response
test the q-axis is aligned with th e as-axis which represents a
rotor angle 0, of zero degrees. For the q-axis step response
test, the rotor angle 0, is 90" as shown in Fig. 5. Here the d-
axis is aligned with the as-axis. Based on the test setup
shown, the measured variables from the expe rimental test are
I and V. Both I an d V are variables in the 3-phase abc
reference frame. To apply the experimental data to the dq-
axis circuit model shown in Fig 2 the measured data must
undergo a transformation [2]. Based on the rotor position for
each test, d-axis or q-axis, the measured variablesI
and V canbe transformed using the following equations.
For the d-axis step response test,
JsI ( k ) = - - i d ( k)
2
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Figurc 4. Rotor position for the d-axis step response test
and for the q-axis step respon se test,
cs a x i s &
Figure 5 . Rotor position for the q-axis step response test
J3I ( k ) = - i d ( k )
2
VI. ESTIMATION PROCEDURE
(17)As described previously and verified through simulation, the
estimation procedure can be generalized in the following
steps:
(i) Collect the standstill step response data using the testsetup described for both the d-axis and the q-axis.
(ii) Use the transformation equations (15) through (18) toconvert the test measurements to the dq-axis reference
frame.
(18)
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Assume model with minimum number of damper
windings.
Using the standst 111 dq-axis equivalent circuit mode l and
the OE estimation algorithm, identify the standstill
model and estimate model parameters.
Simulate identified model with estimated parameters
using measured input voltage.
Validate the identified model and estimated parameters
against the experimental test data.
Increase number of damper windings and estimate the
parameters to tr,y to obtain a better fit between the
simulated and measured response.
d-axis
v a l u earam e t e r
RS 1 3 . 9 m R
R l d 6 8 4 m RR2d 3.6 m R
LIS 38.0 nH
Lmd 89.86 pH
Lld 1 .20 m H
L2d 251.6 m H
(viii) Revalidate the simulated data against the measured
test data.
(ix) Continue adding damper windings until error betweenthe estimated and measured data is acceptable or does
not reduce significantly with the addition of new
dampers.
q-axis
parameter v a l u e
RS 73.9 m R
R l q 0 .4 1 3 R
R2q 67.0 m i 2
R3q 7.603 RLIS 38.0 n H
Lmq 92.34 p HLlq 0 .788 p H
L2q 0.108 m H
L3q 7.409 pH
VII. SIMULATION STUDIES AND RESULTS
rigma V a u e
parameter ,ialue
0.:!91612
0.80330
5.3900
0.0002H
Lmd 22450 H
L ld 0.11262H
L2d 0.0044H
The defined estimation procedure was first validated using
simulation studies. A PM machine was simulated to create
synthetic data with noise. Fig. 6 shows the step response for
the defined d-ax is test with synthetic noise added at a signal-
to-noise ratio o f 200:1. Table 1 shows the estimation results
using two damper windings. It can be seen from the table
that the simulation results validate the PM model
identification an d parameter estimation procedure.
Estimated Value
estimated value % error
0.2915 < l 0.03
0.77970 2.94
5.22180 3.12
0.0003 H 7 . 5 3
2.2 I70 H 1.25
0.0254 H 2.91
0.0042 H 5 16
"0 0.05 0.1 0.15
0oise,
Amps
10.05 0.1 0.15
Figure 6 . d-axis riynthetic test data w ith noise a dded
-0.5-0
time
VIII. EX PERIMENT AL STUDIES AND RESULTS
As mentioned, tests were performed on a 4 pole 1 kW PM
machine. The identification procedure was initialized,
including the inductances and resistances, based on apriori
information such as motor design data. The initial guess for
the motor stator winding resistance r , can be determined
based on the steady-state value of the current.
The PM machine q-axis model was identified using three
damper windings, whereas the d-axis model was identified
with two damper windings. The parameters of the model are
presented in Table 2.
TABLE 2. ESTIMATED DQ-AXIS PARAMETERS
The parameter estimation validation was performed for
several sets of data. Figs. 7 an d 8 represent the d-axis and q-
axis model validation for one of the sets. The input to the
model (top picture on each graph) differs from a typical dc
step, the cause being the use of a car battery for dc supply.
The input was intentionally kept in this form due to its
richness in frequencies. An alm ost perfect fit is observed in
the d-axis model. For the q-axis model, there is a m ismatchbetween the measured a nd estimated data that appeare d in all
data sets. Further study will be necessary to determine its
nature.
IX. CONCLUSIONS
The present paper focuses on the modeling of a permanent
magnet machine. First, simulation studies were conducted to
establish the modeling procedure and the effect of noise on
parameter estimation. An expe rimental setup was built to test
the machine. The testing consisted of aligning the rotor to
the d-axis and q-axis of the stator reference and applying
voltage steps to the stator windings. The win ding current and
the input voltage were recorded. Next, the model structurewas identified through several iterations on the number of
damper windings. Each iteration consisted of parameter
identification and comparison to the measured responses. In
this case, the number of damper windings for the d-axis andq-axis was different. The last part of the study was modelvalidation, in which different sets of input-output data were
used. The model was subjected to the measured input and the
model output was compared to the machine output.
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10 I I I
35
I
30-5 - - 1 1 .
- - measured1imulated
820 ; ;5 / I ’
z 1 5 - l l ‘ .
iime in seconds
lime in seconds
Figure 7. d-axis validation
14 .....,.. i.,.. ..... ...........j ....: . . . . ........ . : ............ . . . . . . . .
0 00 1 0.02 0.03 0.04 0.05 006 0.07 0.08
ilme ~n econds
40
35
30
$25
E$2 0
$ 1 5
10
5
0
0 001 OW 003 004 005 006 007 00 8
ilme in seconds
Figure 8. q-axis validation
Future work will be developed following two main paths.
First, a complete dynamic model will be built for working
conditions. The structure of the model will have to include
the permanent magnet as a source of magnetic flux and the
back emf on the d-axis and q-axis (for the stand still testbeing zero). The present model will serve as a starting point
for it. Second, Finite Element Analysis (F EA) will be used to
determine the machine parameters and compare them to the
parameters obtained through the present method.
X. ACKNOWLEDGMENTS
This work is supported in part by Delphi Saginaw Steering
Systems.
XI. REFERENCES
[4
15
34
[l ] Keyhani andS .
I. Moon, “Maximum likelihoodestimation of synchron ous mach ine parameters an d study
of noise effect from flux dec ay”, IEE Proc., vol. 139, no.
1, pp. 76-80, Jan. 1992.
[2] P. C. Krause and Oleg Wasynczuk, ElectromechanicalMotion Devices. N ew York: McG raw-Hill, 1989.
[3] T. Sebastian, M.A. Rahman, “Modeling of PermanentMagnet Synchronous Motors”, IEEE Transactions on
Magnetics, vol. MAG-22, no. 5 , pp. 1069-1071, Sept.
1986.
T. Sebastian, G.R. Slemon, “Transient Modeling and
Performance of Variable-Speed Permanent Magnet
Motors”, IEEE Transactions on Industry Applications,
vol. 25, no. 1, Jan/Feb 1989.I. Kamwa, P. Viarouge, M. Ferfra, “Modeling and
Identification of Permanent Magnet Synchronous
Machines from Standstill Time Response Tests Using a
Non-Linear Method”, IEEE 1993