a novel induction machine model
TRANSCRIPT
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A novel induction machine model and its
application in the development of anadvanced vector control scheme
M. Sokola1 and E. Levi2
1Department of Mechanical Engineering, University of Bath, Bath, UK
2School of Engineering, Liverpool John Moores University, Liverpool, UK
E-mail: [email protected]
Abstract A novel induction motor model, that fully accounts for both the fundamental iron loss and main flux
saturation, is derived. The model is then applied to the design of a modified rotor flux oriented control scheme. A
rotor flux estimator and a rotor resistance identifier are both developed using the novel model, so that simultaneous
compensation of main flux saturation, iron loss and rotor resistance variation is achieved.
Keywords induction motor; iron loss; main flux saturation; modelling; simulation; vector control
Introduction
The standard, constant-parameter dq axis model of an induction machine
neglects both the variation of the main flux saturation level and the existence
of iron loss in the machine. Out of these two effects, main flux saturation is
undoubtedly the more important one and numerous efforts have been made
in the past to include this effect in the dq axis model.15 Among many
applications that require a model that accounts for main flux saturation,
induction motor vector control is one of the most important.5,6 This is also
the application that has initiated development of the model to be described
here, which simultaneously accounts for both the iron loss and the main flux
saturation.The impact on the accuracy of vector control of the omission of the represen-
tation of iron loss in control systems has been assessed recently and has been
found to be far from negligible.7,8 It therefore follows that design of a vector
control system for an induction machine should be based on an appropriate
dq axis model that accounts not only for main flux saturation, but for the
iron loss as well. The only such model available at present,9 is inappropriate
for this purpose since all the winding currents are selected as state-space
variables. As a consequence, the system matrix is full, with a large number of
saturation-dependent coefficients and with dq axis cross-coupling terms. A
novel induction machine model,10 that accounts for both main flux saturation
and the iron losses, is therefore proposed in this paper. It is shown that an
appropriate selection of the state-space variable set enables development of the
model with an extremely simple system matrix, in which cross-coupling terms
do not appear.
The developed novel model is further used, instead of the constant-parameter
one, in the design of a modified rotor flux oriented (RFO) control scheme for
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234 M. Sokola and E. Levi
an induction machine. The RFO scheme under consideration is the one in
which rotor flux estimation is based on measured stator currents and rotor
speed (isv estimator). A novel rotor flux estimator, that is aware of the
existence of the iron loss and the variation in the level of the main fluxsaturation, is derived. The control system therefore enables automatic adap-
tation to the actual magnetic conditions in the machine.
The vast majority of induction machine vector control schemes, including
the one considered here, require accurate knowledge of the rotor resistance for
achieving correct field orientation. Temperature-related variations of rotor
resistance can only be compensated by on-line identification. Among many
existing solutions,11 it appears that the model reference adaptive control
(MRAC) approach is the most popular one. MRAC-based identification
schemes differ with respect to which quantity is selected for adaptation pur-
poses. The method based on reactive power12,13 is probably the most frequently
applied one, due to its insensitivity to stator resistance variations and simplicity
of calculations as no integration is involved. The adaptation in MRAC schemesis operational in steady states only and is disabled during transients. The
identification scheme is then based on a steady-state machine model, which is
appropriate since thermal processes in the machine are much slower than
electromagnetic and mechanical transients.
The majority of rotor resistance (Rr) identification schemes have been devel-
oped from a simple induction machine model, in which all the other parameters
are assumed to be constant and existence of iron losses is neglected. These
simplifying assumptions have a negative effect on the accuracy of the Rr
identification and the response of the drive can become worse than with no
adaptation at all.14,15 The control system should therefore provide compen-
sation of other detuning sources as well, in order to achieve satisfactory oper-
ation of the rotor resistance identifier.14,15
The developed novel machine modelis for this reason used to design a modified reactive power based rotor resistance
identifier. The information regarding main flux saturation and iron losses are
passed from the rotor flux estimator to the rotor resistance identifier. The
identified value of the rotor resistance is returned to the rotor flux estimator.
Full compensation of main flux saturation, iron loss and rotor resistance
variation is enabled in this way. The only parameters whose variations remain
uncompensated are stator and rotor leakage inductance. Variation of these
parameters is in majority of cases the least important one,10 so that the
developed scheme can be regarded as providing complete compensation of the
relevant parameter variation effects.
The paper is organised as follows. The following section briefly reviews the
standard configuration of the RFO induction motor drive that will be modified
later on. Induction machine modelling, that accounts for both the main flux
saturation and the iron loss, is described next. The modified RFO scheme,
obtained utilising the novel model, is further developed. Simulation results that
illustrate performance of the proposed control system are finally presented.
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235A novel induction machine model
Basic configuration of the drive
The structure of the RFO induction motor drive, analysed in the paper, is
depicted in Fig. 1. Indices s and r denote stator and rotor quantities, index m
describes quantities associated with the main (magnetising) flux, whilesuperscript e denotes estimated values. Power invariant transformation between
phase domain and dq domain is assumed. Current control is performed in
the stationary reference frame. Ideal current feeding is assumed, so that reference
and actual stator phase currents are equal. In the field weakening region, the
field weakening block reduces the rotor flux reference inversely proportionally
to the speed of rotation. As shown in the upper part of the figure, rotor flux
estimation is performed in the rotor flux oriented dq reference frame, on the
basis of measured stator currents and rotor speed. This estimator neglects
existence of both the iron loss and the main flux saturation. Therefore the
inductance parameters in the estimator are constant and equal to their rated
value (denoted by index n). The identified rotor resistance Rer
is supplied from
the on-line identifier, which is explained next.Reactive power based rotor resistance estimator is shown in Fig. 2. The
reference value of the reactive power is calculated from measured stator currents
and stator voltages (that are usually reconstructed) using the following corre-
lation:
Q*= (nqs
idsn
dsiqs
)= (nbs
iasn
asibs
) (1)
where indices a, b identify stationary reference frame. Adaptive, rotor resistance
dependent, reactive power is found from the constant parameter induction
machine model, assuming constant rotor flux value ( i.e. steady-state operation)
and using measured stator currents in the rotor flux oriented reference frame
R re
Lmn idse
ia1+sLrn/Rr
e
r
e -jr
e
iqse
e ibRotor flux PLmn/Lrn andestimator 3/2 ic
re
sl
e
Lmn/Lr n
r
e
Rre
Tee
r
e
* Speed c. T e* - Torque c. iqs* 2 C P+I P+ I R ia
- - jr
e ibField
r* ids* e P I M
weakening, P+I W ic
rnFlux c. 3 M
Fig. 1 Rotor flux oriented induction motor drive with rotor flux estimation from
measured stator currents and rotor speed.
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236 M. Sokola and E. Levi
idse
iqse
vs
Calculation re
Calculation vs
of Qa of Q* is
(eqn. (2)) (eqn. (1)) is
Qa - Q*
Q
P I
Rre
(to flux estimator)
Fig. 2 Reactive power-based rotor resistance estimator of MRAC type.
from:
Qa= (v+vesl
)Lsn
(ie2ds+s
nie2qs
)=ver
Lsn
(ie2ds+s
nie2qs
) (2)
where sn=1L2mn/LsnLrn . The difference between the two reactive powers in(1) and (2) is assigned to discrepancies between the rotor resistance value used
in the controller and the actual one. This error signal is processed through a
PI controller. Its output is the identified rotor resistance value, which is passed
on to the rotor flux estimator of Fig. 1.
Novel induction machine model
An induction machine is represented with the dynamic space vector equivalent
circuit shown in Fig. 3, in an arbitrary frame of reference rotating at angular
speed va
. Iron loss and main flux saturation are accounted for. The circuit of
Fig. 3 can be described with the following set of equations (underlined variables
jaLsis jaLrir Rs Ls + Lr Rr +
is ir iFe im +
Lm vs jr
RFe +
jaLmim
Fig. 3 Space vector dynamic equivalent circuit of an induction machine in an arbitrary
reference frame.
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237A novel induction machine model
are space vectors):
n2s=R
si2s+L
ss
di2s
dt
+dy2
m
dt
+jva
(Lss
i2s+y
2m)
0=Rri2r+L
sr
di2r
dt+
dy2
mdt+j (v
av) (L
sri2r+y
2m)
RFe
i2Fe=jv
ay2
m+
dy2m
dt, i2
Fe+ i2
m= i2
s+ i2
r
RFe=f (v
e), L
m=f (y
m)
Te=P
1
Lsr
[(Lsr
idr+y
dm)y
qm(L
sriqr+y
qm)v
dm]
(3)
Index s identifies leakage inductances and ve
is the fundamental angular
frequency of the supply. The equivalent circuit of Fig. 3 and the system ofeqns (3) differ insignificantly from the machine representation used in the
existing model.9 An equivalent iron loss inductance,9 connected in series with
the equivalent iron loss resistance and used to describe rate of change of eddy-
current losses, is omitted. Detailed investigation of the importance of this
inductance in vector controlled induction machines10 has indicated that its
omission essentially does not affect the results.
Winding dq axis currents are selected as state-space variables in a further
development of the existing model.9 Consequently, the final model is of very
complicated structure, as the system matrix contains a large number of satu-
ration-dependent coefficients, including cross-coupling inductances. That model
is therefore inconvenient for design of a rotor flux estimator. Recent investi-
gation4
has shown that selection of the state-space variable set plays a decisiverole in determining the overall complexity of the resulting saturated induction
machine model. With this in mind a number of alternative machine models,
that all include both main flux saturation and iron loss, are derived.10 As the
ultimate goal is to design a rotor flux estimator of the isv type, stator dq
axis current components and rotor flux dq axis components (inputs and
outputs of the estimator, respectively) need to be selected as state-space vari-
ables. The most convenient selection of state-space variables under these con-
ditions is the set comprising i2s, y2 m
, y2 r
.10 Let the final model of the machine be
given in the form
[n]dq=[A] d [x]
dq/dt+[B][x]
dq(4)
Then
[n]dq=[n
dsnqs
0 0 0 0]t
[x]dq=[i
dsiqs
ydm
yqm
ydr
yqr
]t(5)
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238 M. Sokola and E. Levi
[A]=
tNN
NNNv
Lss
0 1 0 0 0
0 Lss
0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
0 0 1 0 0 0
0 0 0 1 0 0
uNN
NNNw
(6)
[B]=
tNNNNNNNNNNv
Rs
va
Lss
0 va
0 0
va
Lss
Rs
va
0 0 0
0 0 1
Tsr
01
Tsr
(vav)
0 0 0 1
Tsr
vav
1
Tsr
RFe 0 LrLm
1TsFe
va 1TsFe
0
0 RFe
va
Lr
Lm
1
TsFe
0 1
TsFe
uNNNNNNNNNNw
(7)
Time constants, introduced in (7), are defined as Tsr=L
sr/R
r, T
sFe=L
sr/R
Fe.
The torque is expressed as
Te=
3
2P
1
Lsr
(ydryqmy
qrydm
) (8)
Application of the model requires two non-linear functions, namely
Lm=f (ym)RFe=f (v
e)
(9)
The induction motor model, given with (4)(9), fully accounts for both iron
loss (it should be noted that the equivalent iron loss resistance in the model
(4)(9 ) represents only the fundamental iron loss component; as a vector
controller is essentially the fundamental harmonic controller,7,8 this is exactly
what is needed for development of a modified vector control system) and main
flux saturation. System matrix [A] however contains only constant coefficients,
in contrast to the system matrix of the model obtained when all the current
dq axis components are selected as state-space variables.9 Design of the vector
control system, described in the next section, is therefore significantly simpler.
Non-linear relationships, given in (9), have to be determined experimentally.
A 4 kW induction machine, whose data are given in Appendix, is used in the
study. Magnetising curve was obtained from no-load test with sinusoidal
supply, while equivalent iron loss resistance was calculated from variable fre-
quency no-load test with PWM supply.8 Figure 4 illustrates the magnetising
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239A novel induction machine model
Fig. 4 Magnetising curve of the 4 kW machine and the magnetising inductance
(experimentally identified points denoted by crosses; Lmsat=Lmn).
curve of the machine and the magnetising inductance as functions of the
magnetising current, while Fig. 5 shows measured values of the fundamental
component of the iron loss and approximation of the equivalent iron loss
resistance.
Apart from the model derived in this section, the dynamic equivalent circuit
of Fig. 3 enables derivation of a number of alternative induction machine
models, in which sets of state-space variables are selected in a different way.
All these models again account for both main flux saturation and iron loss
and utilise characteristics of the machine shown in Figs. 4 and 5. Induction
machine is for simulation purposes represented with one of these alternative
Fig. 5 Fundamental iron loss and equivalent iron loss resistance RFe
(experimental and
approximation) of the 4 kW induction machine.
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240 M. Sokola and E. Levi
models, in which the set of state-space variables consists of i2s, i2
r, y2m
, while the
model described with (4)(9) is used for construction of the novel rotor flux
oriented control scheme. The model with i2s, i2
r, y2 m
selected as state-space vari-
ables directly follows from ( 3) by elimination of the equivalent iron loss current:
[n]dq=[n
dsnqs
0 0 0 0]t
[x]dq=[i
dsiqs
ydm
yqm
idr
iqr
]t(10)
where
[A]=
tNNNNN
v
Lss
0 1 0 0 0
0 Lss
0 1 0 0
0 0 1 0 L sr
0
0 0 0 1 0 L sr
0 0 1 0 0 0
0 0 0 1 0 0
uNNNNN
w
(11)
[B]=
tNNNNNv
Rsv
aLss
0 va
0 0
va
Lss
Rs
va
0 0 0
0 0 0 (vav) R
r(v
av)L
sr0 0 (v
av) 0 (v
av)L
srR
rR
Fe0 1/T
Fev
aR
Fe0
0 RFe
va
1/TFe
0 RFe
uNNNNNw
(12)
and TFe=L
m/R
Fe. Torque remains to be given with (3) and the non-linear
functions are as in (9).
Stator voltage equations are omitted in the simulation from the machinemodel (10)(12) due to assumed ideal current feeding and are used only for
calculation of the reactive power in (1).
Modified rotor flux oriented control scheme with full compensation of parameter
variation effects
Rotor flux estimator
Omission of stator voltage equations from (4)(9) and application of the rotor
flux oriented control constraints (va=v
r, y
dr=y
r, y
qr=0, dy
qr/dt=0)
enables derivation of the following equations in the rotor flux oriented reference
frame (vsl=v
rv):
yr+T
sr
dyr
dt=y
dm(13)
yqm=v
slTsryr
(14)
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ydm+T
sFe
Lm
Lr
dydm
dt=
Lm
Lr
(yr+L
srids+v
rTsFe
yqm
) (15)
yqm+TsFeLmLr
dyqmdt =
LmLr
(LsriqsvrTsFeydm) (16)
Te=P
1
Lsr
yryqm
(17)
Lm=f (y
m), y
m=y2
dm+y2
qm(18)
This model is almost identical to the one that results when main flux saturation
is neglected and only iron loss is accounted for.8 The difference is that magnetis-
ing inductance and hence the rotor inductance as well are variable parameters,
dependent on the actual operating conditions that dictate the level of the main
flux saturation in the machine. Such a remarkable simplicity of the model is
the result of the convenient choice of the state-space variable set in the previous
section. It should be noted that relationship Lm
/Lr=f (y2m ) can be used instead
of Lm=f (y
m) in order to reduce the number of mathematical operations in
the estimator. This is so because only ratio Lm
/Lr
appears in the model given
with ( 13)(18).
Equations (15)(16) contain on the left-hand side time derivatives of the
magnetising flux components, scaled with the very small term TsFe
Lm
/Lr.
Detailed investigation10 has revealed that these terms can be ignored, without
practically any consequence on the accuracy of the rotor flux estimation. Using
this simplification, the modified rotor flux estimator is constructed by means
of (13 )(18). The structure of the estimator, using Lm
/Lr=f (y2
m) and
vrTsFe=f (v
r) as non-linear functions, is depicted in Fig. 6. The rotor leakage
idse
Lrn dme
1 re
1+sTre
Tre
re
TFee
m2
P Tee
Lm/Lr = f(m2)
Lrn
Tre
iqse qm
e sle r
e
Lrn
re
TFee
rTFe = f(r)
Fig. 6 Modified rotor flux estimator with compensation of main flux saturation, iron
loss and rotor resistance variation.
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242 M. Sokola and E. Levi
time constant Tesr
includes the estimated value of the rotor resistance that is
obtained from the modified reactive power based identifier. The modified rotor
resistance identifier is discussed next.
Rotor resistance identifier
Equations of the reactive power based rotor resistance identifier (1)(2) are
derived from the induction machine model in which iron loss is neglected and
all the inductances are assumed to remain constant. The modified identifier is
derived from the stator voltage equations of the novel machine model (4)(9)
that accounts for both the iron loss and the main flux saturation. Calculation
of the reference value of the reactive power remains to be given with (1). Stator
voltage dq axis components in the rotor flux oriented reference frame are
from (4)(7) in any steady state determined with
nds=R
sidsv
rLss
iqsv
ryqm
nqs=Rsiqs+vrLss ids+vrydm(19)
The adaptive, rotor resistance dependent, reactive power is then obtained in
the form
Qa= (v+vesl
)[Lss
(ie2ds+ ie2
qs)+ye
qmieqs+ye
dmieds
] (20)
Values of the estimated magnetising flux dq axis components are passed from
the rotor flux estimator to the modified rotor resistance identifier. They contain
information related to both the saturation level and the iron loss. The structure
of the modified rotor resistance identifier is depicted in Fig. 7. The value of the
identified rotor resistance is returned back to the rotor flux estimator and such
two-way communication between the two parts of the control system provides
complete compensation of parameter variation effects. Schematic layout of theproposed vector control system is given in Fig. 8.
idse
iqse
dm
e vs Calculation r
e Calculation
vs
qme of Qa
of Q*
is
(eqn. (20)) (eqn. (1)) is
Qa - Q*
Q
P I
R re
(to f lux estimator)
Fig. 7 Modified reactive power-based rotor resistance estimator.
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243A novel induction machine model
dme
qme Modified Motor
rotor signalsids
eresistance
iqse
estimatorTo vector r
e (Fig. 7) (voltagescontroller and currents)
re
Rotor flux
Tee estimator R
r
e
re
(Fig. 6)
Motor (stator currentssignals and rotor speed)
Fig. 8 Schematic illustration of the resulting control system with complete compensation
of parameter variation eVects.
Verification of the proposed vector control system
In order to verify the developed vector control system, the following approach
is adopted. The rotor resistance value in the rotor flux estimator is set to the
rated value, while the rotor resistance in the motor is set to a detuned value,
20% smaller or bigger than rated. The rotor resistance identifier is disabled. A
speed command is given, a load torque is applied and the drive is allowed to
settle in steady-state operation. Since the rotor resistance in the motor differs
from the value used in the controller, steady-state operation is characterised
with detuning, so that orientation angle error exists and rotor flux in the motor
differs from the reference value. At time instant t=0.1 s, the rotor resistanceidentifier is enabled and it starts to adapt Re
r. Compensation of parameter
variation effects takes place. Full compensation is achieved if the errors inorientation angle and in rotor flux amplitude are driven towards zero, while
the identified value of the rotor resistance is driven towards the value used in
the motor model. Simulation results are shown in Figs. 9 and 10, where
estimated rotor resistance, rotor flux error (percentage value of the difference
between reference and actual rotor flux) and orientation angle error are shown.
Figure 9 illustrates estimated rotor resistance, rotor flux error (percentage
value of the difference between reference and actual rotor flux) and orientation
angle error for operation with rated speed command. Traces for three load
torque values are shown (100%, 60% and 25% of the rated torque), for two
values of the rotor resistance in the motor, Rr=0.8 R
rnand R
r=1.2 R
rn. Initial
steady state is characterised with large orientation angle errors and substantial
errors in the rotor flux amplitude. When enabled, the rotor resistance identifier
updates the value of the rotor resistance and passes it to the rotor flux estimator.
This action neutralises the detuning, forcing both errors towards zero. Final
steady-state is characterised with rotor flux error of less than 1% and orien-
tation angle error less than one degree. Final values of the estimated rotor
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244 M. Sokola and E. Levi
0.7
0.8
0.9
1
1.1
1.2
Estim.rotorresistance(p.u.)
0 0.5 1 1.5 2
Time (s)
Rr = 1.2 Rrn
Rr = 0.8 Rrn
100% torque
100% torque
60% torque
25% torque
60% torque
-8
-4
0
4
8
12
Rotorfluxerror(%)
0 0.5 1 1.5 2
Time (s)
Rr = 1.2 Rrn
Rr = 0.8 Rrn
100% torque
100% torque
60% torque
25% torque
60% torque
-10
-5
0
5
10
Orientationangleerror(deg)
0 0.5 1 1.5 2
Time (s)
Rr = 1.2 Rrn
100% torque
100% torque
60% torque
25% torque
25% torque
60% torque
Rr = 0.8 Rrn
Fig. 9 Identification of Rr
at the rated speed, using control system of Fig. 8 with rotor
flux estimator of Fig. 6 and rotor resistance estimator of Fig. 7 (load torque in % as
parameter). Rotor resistance, rotor flux error and orientation angle error are shown.
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245A novel induction machine model
0.8
0.9
1
1.1
1.2
Estim.rotorresistance(
p.u.)
0 0.5 1 1.5 2
Time (s)100% load 60% load
Rr = Rrn
Rr = 1.2 Rrn
Rr = 0.8 Rrn
-12
-9
-6
-3
0
3
6
9
12
15
Rotorfluxerror(%)
0 0.5 1 1.5 2
Time (s)100% load 60% load
Rr = 0.8 Rrn
Rr = 1.2 Rrn
Rr = Rrn
-6
-4
-2
0
2
4
6
8
Orientationangleerror(deg)
0 0.5 1 1.5 2
Time (s)100% load 60% load
Rr = 1.2 Rrn
Rr = Rrn
Rr = 0.8 Rrn
Fig. 10 Identification of Rr
in the field weakening region (125% speed) using control
system of Fig. 8 with rotor flux estimator of Fig. 6 and rotor resistance estimator of
Fig. 7 (100% and 60% load torque). Rotor resistance, rotor flux error and orientation
angle error are shown.
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246 M. Sokola and E. Levi
resistance closely match the actual values in the motor. The convergence of
the rotor flux estimate is rather slow for light 25% load, this being a known
problem with this type of on-line rotor resistance identification.
Figure 10 illustrates results of the similar study, where operation in the field-weakening region is considered. Speed of operation is 125% of the rated speed
and the same traces are shown as in Fig. 9, for two load torque values (100%
and 60%). Apart from the cases Rr=0.8 R
rnand R
r=1.2 R
rn, the third one
when Rr=R
rnis included. The purpose of including the traces for R
r=R
rnis
to verify capability of the rotor flux estimator to fully compensate for main
flux saturation and iron loss in the absence of rotor resistance detuning. As
can be seen from Fig. 10, when Rr=R
rninitial steady state is characterised
with zero orientation angle error and zero rotor flux error. When enabled, the
rotor resistance identifier delivers at the output rotor resistance value equal to
the rated value. For other two values of the rotor resistance in the motor,
Rr=0.8 R
rnand R
r=1.2 R
rn, there is significant detuning in the initial steady
state, which is the consequence of combined impact of rotor resistance variation,
main flux saturation and iron loss. When the rotor resistance identifier is
enabled, it again drives the errors towards zero by updating the identified rotor
resistance value. Final steady state is once more characterised with practically
zero orientation angle error and zero rotor flux error. Rotor resistance estimate
in all the cases equals actual rotor resistance in the motor.
Figures 9 and 10 confirm that the control system of Fig. 8 provides full
compensation of iron loss, main flux saturation and rotor resistance variation.
Conclusion
The paper proposes a novel induction machine model, that accounts for both
the main flux saturation and the iron loss in the machine. As shown in the
paper, a convenient selection of the state-space variable set enables descriptionof the machine with a very simple set of equations. The model is further applied
in design of a modified rotor flux oriented control scheme, that provides
simultaneous compensation of rotor resistance variation, main flux saturation
and iron losses. The control scheme encompasses the rotor flux estimator of
the isv type and the reactive power based rotor resistance identifier. Both
are appropriately modified in order to enable complete compensation of param-
eter variation effects (except for the variations in leakage inductances). In this
way the estimator is informed about temperature-related changes in the rotor
resistance, while the identifier is made aware of the actual magnetic circum-
stances in the machine. Such an arrangement provides the complete compen-
sation of parameter variation effects, as verified by simulation.
The modelling procedure described in the paper and the model application
in the design of a RFO induction motor drive control system with full compen-
sation of parameter variation effects are believed to be well suited to the
teaching of the topics related to advanced electric machine modelling and
control.
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Appendix
Induction motor data
4 kW 380 V 50 Hz 8.7 A Y 2P=4, Ten=26.5 Nm
Rsn=1.37 V R
rn=1.1 V L
ssn=4.87mH
Lsrn=7.96 mH L
mn=0.143 H
Magnetising curve approximation (r.m.s. values):
Ym=C
0.1964285Im
Im2.2 A
Lm=Y
m/I
m
Iron loss resistance approximation:
RFe=C
128.92+8.242f+0.7788f2
(V) f50 Hz184155272/f (V) f>50 Hz
International Journal of Electrical Engineering Education 37/3