A NOVEL ANALYSIS AND MODELLING OF AN ISOLATED SELF-EXCITED
INDUCTION GENERATOR TAKING IRON LOSS INTO ACCOUNT
D. Seyoum, C. Grantham and F. Rahman
School of Electrical Engineering and Telecommunications
The University of New South Wales
Sydney 2052, Australia
c.grantham@unsw.edu.au
Abstract
This paper presents a novel analysis for the dynamics of a self-excited induction generator (SEIG) driven
by a variable speed prime mover taking iron loss into account. It is important to mention that for stable
operation of the SEIG, the generator has to operate in the region of magnetic saturation where the
magnetizing reactance is at its minimum value. As a consequence in any accurate analysis, iron loss must
be included. This is particularly the case for small induction machines, where the magnitude of current in
the resistance representing iron loss is almost equal to the magnitude of the magnetizing current.
Neglecting the iron loss in this type of induction machine will cause large error in the analysis. Even if
the iron loss is to be neglected the approximate percentage error should be known. The dynamic analysis
and simulation of the SEIG are included in this paper.
1. INTRODUCTION
The principle of self-excitation is well known in that
when capacitors are connected across the stator
terminals of an induction machine, driven by an external
prime mover, voltage will be induced at its terminals [16].
Once self-excitation has initiated the induced emf and
current in the stator windings will continue to rise until
steady state is attained, influenced by magnetic
saturation of the machine. At this steady state operating
point the voltage and current through the capacitor will
continue to oscillate at a given peak value and
frequency. In order for self-excitation to occur, for a
particular speed there is a corresponding minimum
capacitance value [6-9]. The different models that have
been used to analyze SEIGs are the steady state model,
which include the loop-impedance method [4, 10-11],
the nodal admittance method [12], and the D-Q axes
model based on the generalized machine theory [5-6,13].
The steady state analysis of the SEIG including iron loss
has already been reported [14-15]. However, the steady
state analysis is not able to show the dynamics of the
SEIG. In all analyses using the D-Q axes model of the
SEIG based on the generalized machine theory reported
in the literature, iron loss has been neglected. It is
important to note that for stable operation of the self-
excited induction generator, the machine has to operate
in the region of magnetic saturation. Therefore, iron loss
must be included in any accurate analysis. For small
induction machines, the current associated with iron loss
is of almost the same per-unit value as the magnetizing
current [16]. Neglecting the iron loss will cause large
errors in the analysis. Few works have been reported
which include iron loss in the induction motor model in
the D-Q axes [17-19].
This paper presents a novel analysis for the dynamics of
the self-excited induction generator driven by a variable
speed prime mover taking iron loss into account and
establishes the error introduced if iron loss is neglected.
Iron loss is represented as resistance Rm in the standard
D-Q axis equivalent circuits.
Over all this paper presents a model which takes into
account the actual non-linear variation of magnetising
inductance Lm and Rm as functions of voltage and
predicts the dynamics of the self-excitation process in
the time domain. The stability behavior of the SEIG can
be studied using this model which so far has not been
possible. The instantaneous voltage is predicted
accurately by using the dynamic model, which is useful
in studying the behavior of the SEIG connected to an
inverter which may in turn connect to a load or to an
active system such as an utility grid or an automotive
starter/generator system [20].
2. SELF-EXCITED INDUCTION GENERATOR
DYNAMIC MODEL INCLUDING Rm
To model the SEIG effectively, the parameters of the
machine should be measured accurately. The parameters
used in the SEIG can be obtained by conducting tests on
the induction generator when it is used as a motor. The
traditional tests used to determine the parameters are the
open circuit (no load) test and the short circuit (locked
rotor) test. The induction machine used as the SEIG in
this investigation is a three-phase wound rotor induction
motor with specification: 415V, 7.8A, 3.6kW, 50Hz,
and 4 poles. The parameters given in the d-q equivalent
circuit shown in Fig.1 are obtained by conducting
parameter determination tests on the above mentioned
induction machine. As it is a wound rotor induction
machine there is no variation of rotor parameters with
speed. The parameters obtained from the test at rated
values of voltage and frequency are Lls=Llr=11.4mH,
Lm=178mH, Rm=1600Ω, Rs=1.7Ω, Rr=2.7Ω.
For motoring application these parameters can be used
directly. However, for self-excited induction generator
application the variation of Lm and Rm with voltage
should be taken into consideration to find the correct
voltage build up. With the correct parameters the
dynamic currents, output power and induced
electromagnetic torque can be predicted accurately
during loading.
Equation (1) is derived from the equivalent circuit given
in Fig. 1. This equation represents all dynamics of the
induction generator taking into account the initial
conditions for the self-excitation process.
In Equation (1) Kd and Kq are constants, which represent
the initial induced voltages along the d-axis and q-axis
respectively due to remnant magnetic flux in the core,
Vcqo and Vcdo initial voltages on the capacitors and
p=d/dt.
Rs
Lls
ids
V cd
λds
C
Rr + -ωr λ_qr
Llr
imd
Lm λ
Rm
idr
dr
(a)
Rs
Lls
iqs
V cq
C
λqs
Llr
imq
Lm
Rm
λqr
Rr +ωr λdr_
iqr
(b)
Fig. 1 D-Q model of SEIG at no load a) d-axis b) q-axis.
From Equation (1) it is given that
0=[Z][I]+[Vo]
(2)
then the self-excitation currents are obtained from
Equation (2) in the normal way, ie
[I]=-[Z]-1[Vo]
(3)
from which the stator self-excitation current along the qaxis is given by:
U
(4)
id = 8
7
6
5
Ap + Bp + Dp + Ep + Fp4 + Gp3 + Hp2 + Jp + M
Here,
U is a function of the machine parameters, capacitance
C and initial conditions.



RmLm 
Rm Lm p
1
0
0
Rs +  Lls +

 p+
Rm + Lm p 
pC
Rm + Lm p




 i  V 

RmLm 
Rm Lm p
1
0 
  qs   cqo 
0
Rs +  Lls +
0
 p+
0 
Rm + Lm p 
pC
Rm + Lm p
 ids  Vcdo 

 =
+

0 


ωr RmLm p
RmLm p
RmLm 
RmLm p   iqr  Kq 
Rr +  Llr +
 
 p - ωr  Llr +
    
Rm + Lm p
Rm + Lm p
Rm + Lm p 
Rm + Lm p   idr  Kd 
0 









Rm Lm p
RLp
RL
ωr RmLm p
ωr  Llr + m m  Rr +  Llr + m m  p

Rm + Lm p
Rm + Lm p
Rm + Lm p 
Rm + Lm p  



(1)
A, B, D, E, F, G, H, J, and M are functions of the
machine parameters.
Lm (H)
If one of the roots of the denominator in the expression
for id is with a positive real root then there is selfexcitation. A positive real root has a growing transient
response until saturation of the magnetising inductance
is reached. Hence to find the roots, the denominator is
set to zero as
Ap8 + Bp7 + Dp6 + Ep5 + Fp4 + Gp3 + Hp2 + Jp + M = 0
When there is self-excitation, out of the eight roots at
least one of them will be with positive real root. During
the initiation of self-excitation, as the generated voltage
is close to zero the values of Rm and Lm should be
selected corresponding to a phase voltage close to zero.
0.25
3. CHARACTERISTICS OF Lm and Rm
Lm starts from a small value then increases to reach its
peak value and finally starts to drop [5]. The
characteristic of Lm is helpful for the stability of
generated voltage and to determine the minimum
generated voltage with out loss of self-excitation. The
value of Rm exhibits variation but over all it increases
with generated voltage [15]. When the per-unit value of
Rm is almost the same as Xm, neglecting Rm will give rise
to an error in the analysis.
The magnetising inductance, Lm, used in this
investigation set up is given by a fourth order curve fit
as follows:
Lm=-1.56×10-11Vph4+2.44×10-8Vph3-1.19×10-3Vph2
+1.42×10-3Vph+0.245
(4)
The resistance Rm representing the iron loss for the
machine given here is large and the result does not have
significant difference to the case when Rm is ignored.
The Rm value is taken for an induction machine similar
to that reported by Grantham [16]. The variation in Rm is
modeled by the following curve fit
Rm=3Vph+50
(5)
where Vph is the phase rms voltage across Rm in parallel
with Lm. Vph can be calculated by subtracting the voltage
drop in the stator impedance from the terminal voltage
winding for impedance determination, i.e, while the
induction machine is motoring. For generator
application Vph is the sum of the voltage drop in the
stator impedance and the voltage at the terminals of the
stator.
B
0.3
A
0.2
C
0.15
Experimental
0.1
Fourth order curve fit
0.05
0
0
50
100
150
Vph (V)
200
250
300
Fig. 3 Variation of magnetizing inductance with phase
voltage.
The curve in Fig. 3 can be explained as follows.
Between point A and B is the unstable region. If the
SEIG starts to generate in this region, a small decrease
in speed will cause a decrease in voltage and this will
bring a decrease in Lm, which in turn decreases the
voltage, and finally the voltage will collapse to zero. If
the speed increases slowly and sometimes with zero
acceleration so that the operating point remains in the
region between A and B, there will not be any selfexcitation even at high speed. When the increase of
wind speed has this characteristic then there is a
possibility that self-excitation will not occur. To avoid
this problem the capacitors should be connected when
the speed reaches its set point because voltage build up
requires a transient phenomena in the region between A
and B.
Between point B and C is a stable operating region.
When the speed decreases voltage will decrease and Lm
increases to have a new steady state operating point at
lower voltage.
4. SIMULATION OF DYNAMIC
SELF-EXCITATION
The simulation was carried out using SIMNON [21] by
rearranging equation (1) of the SEIG and solving the 2nd
order differential equation given by
p 2 I = Ao pI + A1 I + Bo pV + B1V .
Where
(6)
  Rs Rm Rm 
Rm
0
0
+
−  +

Lls
  Lls Lm Lls 

R R
R 
R

0
0
− s + m + m 
− m
Lls

 Lls Lm Lls 
Ao = 


Rm
R
R R

− r + m + m 
ωr
0
 −L
Llr Lm Llr 
lr



R R
R
R
− m
− ωr −  r + m + m
0

Llr
 Llr Lm Llr

 Rm Rs
− L L
 m ls

 0

A1 = 
 0


 − ω r Rm
 Llr

0
ω r Rm
Llr
0
delayed. The error in the steady state developed voltage,
shown in Fig. 5, is very small such that it can not be
seen in the given scale. From the simulation it was
discovered that as the magnitude of Rm increases and
become much greater than Xm there is no significant
difference in the analyzed results whether Rm is included
or neglected.
Speed (rpm)
1500
Including Rm




0
0


 Rm Rm  
Rm Rr
ωr 
−
+

Lm Llr
 Llr Lm  

R
R 
R R

− ωr  m + m  − m r

Lm Llr
 Llr Lm  r

0
R R
− m s
Lm Lls












 
0
1000
Neglecting Rm
500
0
0
50
100
150
200
250
300
Capacitance (µF)
 Rm
− L L
 m ls

 0
Bo = 

 0


 0

 1
− L
 ls

 0
B1 = 

 0


 0

iqs 
 
ids
I = 
iqr 
 
idr 
Vcq =
0
−
0
Rm
Lm Lls
0
−
0
1
Lls
0
0
Rm
Lm Llr
0
0
0
0
−
0
−
1
Llr
0




0 


0 

Rm 
−

Lm Llr 
Fig.4 Values of capacitance and speed for self-excitation
at no load.
0

0 


0 


0 

1 
− 
Llr 
Fig.5 RMS phase voltage during self-excitation.
5. DYNAMICS OF SEIG DURING LOADING
When a load is connected across the capacitors in Fig.1
power will flow to the load. The current flowing to the
load, iL, is given by
(7)
i L = i s − iC
Vcq 
 
Vcd
V = 
 Kq 
 
 K d 
1
iqs dt + Vcq
C∫
t =0
, Vcd =
where is is total stator current and iC
capacitor current.
1
ids dt + Vcd
C∫
t =0
As can be seen in Fig. 4 when the magnitude of Rm is
almost the same as Xm the onset of self excitation is
is the total
The induced electromagnetic torque or the mechanical
torque required to drive the induction generator is given
by [22]:
3
(8)
T =
P λ × I
e
2
p
m
r
Analyzing equation (8), the dynamic induced
electromagnetic torque with Rm included can expressed
as
Te =
R
3
PP R m ( iqs idr − ids iqr ) − m Te
2
Lm
(9)
The results given here are the simulation for the
induction generator when it is driven at constant speed
(1500rpm) and variable capacitance and resistive load
are connected at the stator terminals. The variations of
capacitance and resistance are given in Fig. 6.
Fig. 8 The dynamic electromagnetic torque
Fig. 6 Variation of connected capacitor and resistor.
Fig. 9 Dynamic currents in the load, capacitor and stator
of the SEIG
Fig. 7 The dynamic rms generated voltage
The dynamic variation of voltage current, power and
torque with variation in load and capacitance are shown
in Figs. 7 to Fig. 10. At no load the effect of Rm is
insignificant. However, when the induction generator is
loaded neglecting Rm will result in an error. When Rm is
included, which depicts the actual situation, the
generated voltage, currents, and output power are lower
than that for Rm neglected. However, due to additional
losses the electromagnetic torque with Rm included is
higher than without Rm.
Fig. 10 The dynamic output power.
6. CONCLUSION
This paper has described a novel way of including Rm in
the dynamic analysis and simulation of the SEIG in the
D-Q axes model. To neglect the iron loss, first it has to
be shown that its effect is negligible. This paper
provides the tool to reach such a decision. When Rm is
included, which depicts the actual situation, the
generated voltage, currents, and output power are lower
than that for Rm neglected. However, due to additional
losses the electromagnetic torque with Rm included is
higher than without Rm. The paper additionally paves the
way for the more general dynamic analysis of induction
machines and their high performance drives with the
effects of iron loss included in an easily understood
model for the first time.
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