a novel induction machine model

7/31/2019 A Novel Induction Machine Model

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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

International Journal of Electrical Engineering Education 37/3


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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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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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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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[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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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.



8/16


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.



10/16


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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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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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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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


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


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.


14/16


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.



15/16


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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


a novel induction machine model

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