STEADYSTATE
PERFORMANCE
AND
DYNAMIC STABILITY
OF
A
SELF :EXCITED
INDUCTION
GENERATOR
EEDING
AN
INDUCTION
MOTOR
SungChun
Kuo
Li
Wang
Department
of
Electncal
Engineenng
National
Cheng
Kung
University
Tainan,
Taiwan 70101,
R.
0
C.
Abstract: This paper presents a novel scheme on steadystate performance of
an
isolated threephase selfexcited induction generator
(SEIG)
upplying a loaded induction motor.
An
approach based on
dq
axis inductionmachine model is employed to derive steadystate equations
of
the studied SEIG. Eigenvalue analyses based
on
synchronous
reference
frame are employed
to
determine the critical operating conditions
and
dynamic stability
of
the
studied machines. The required minimum excitation capacitance
of
the
SEIG,
the
maximum torque of
the
induction motor load, the
combined
maximum
operating efficiency,
etc.
of
the studied system can
b:
easily investigated. Experimental results obtained from a
laboratory
I.
1
kW
induction machine
driven
by
a dc motor and a
300
W
induction motor
with
a
dc generator
as
its shaft
load
are
also performed to confirm the feasibility
of
the
proposed method. Keywords: selfexcited
induction
generator, steadystate performance, dynamic stability, induction motor load.
NOMENCLATURE
perphase resistance, inductance, and reactance
r
perphase magnetizing reactance and inductance magnetizing current voltage and current angular speeds
of
reference fiame and rotor perphase excitation capacitance differentiation operator with respect to time t induction generator and motor quantities daxis and qaxis quantities stator and rotor quantities magnetization quantities base quantities I. INTRODUCTION
It
is well
known
that an extemally driven induction machine can maintain self excitation when an appropriate value of a capacitor bank is appropriately connected across the twminals of the induction machine
[l].
Such induction machine is called a selfexcited induction generator (SEIG). The primary advantages of a SEIG over conventional synchronous generator are brushless construction with squirrelcage rotor, reduced size, without DC supply for excit;3tion, reduced maintenance cost, and better transient characteristics. In recent years,
SEIGs
have received increased attention and they have been widely employed as suitable isolated power sources in small hydroelectric and wind energy applications
[24].
According to the analyzed results
of
available references, the studies of an
SEIG
feeding static loads have been reported. Performances of
an
SEIG feeding a dynamic load such
as
an induction motor need to be further investigated. Shridhar, et al.
[ ]
reported a method to predict
the
steady state behavior of a
SEIG
feeding an induction motor. The perphase equivalent circuit model was obtained from
a
steadystate condition and
it
can not be used
for
analyzing
SEIG’s
transient characteristics. The author employed
a
novel eigenvalue sensitivity technique to analyze parallel operated SEIG feeding an induction motor load
[6]
In this paper, the authors propose an eigenavlue analysis based on synchronous reference frame to determine dynamic stability of a
SEIG
feeding a loaded induction motor. Experimental results obtained from a laboratory
1.1
kW
induction generator supplying
a
300
W
induction machine load are also employed to validate the proposed method.
11.
MACHINE
MODELS
Fig.
I
Oneline diagram
of
the
studied
SEE
feeding
an
IM
load
Fig.
1
shows the oneline diagram of a selfexcited induction generator
SEIG)
feeding
an
induction motor load. The excitation capacitor C provides the required reactive power for the SEIG. Fig.
2
shows the dq axis equivalent circuit model of a threephase symmetrical induction machine with arbitrary reference frame.
The
excitation capacitance
C
is connected to the induction machine’s stator terminals. The voltage equations of Fig.
2
can be written
as
below
[7].
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2000
IEEE
211
PMg
< g Nig 4g'PL.g
The dq axis equivalentcircuit model
of
the induction motor load
is
similar to Fig.
2
except that the currents in Fig.
2
are
in
the opposite direction. Connecting the induction motor load to the common bus, the voltage equations can be written
as
below.
c'
I
I
I
I+
I
+I
I I
I
Fig
2
Dq
xis
equivalent circuit
of
an
nduction
generator.
The voltagecurrent equations
of
the capacitor bank can be expressed
as
below.
3)
The shaft torque and the rotor speed of the
SEIG
are related by thr: following equation. where T,
is
mechanical input torque and
H,
is inertia constant
of
the studied
SEIG.
The prime mover
of
the
SEIG
is simulated by a shunt DC motor whose speedtorque characteristic is given by
5)
where
VDC,
RA,
,
and are the
DC
voltage, armature resistance, machine constant, and field flux
of
the DC motor, respectively. The noload speed of the prime
is
given by
vDC
go
Ka

The torque equation of the induction motor load is given
by
where
T,
is the load torque.
If
the qaxis
is
aligned with the stator terminal voltage phasor
by
setting vdp
0
and p(vdl)
=
0
in
1) 3),
the studied machine can be referred to
as
the synchronous reference frame. The system synchronous frequency is given by
I2O
10
7
70
\.
i
X:
experimental result
I
I
I
1
I
0.5
IO
1.5
2.0
25
30
Im
A)
Fig.
3
Magnetization curve between
X,
and
I,
of the studied
SEIG
Due to nonlinear behavior of the magnetizing reactance
X,
in the studied
SEIG,
previous authors intended to approximate the airgap voltage
V,
versus
X, by
a piecewise linear equation
[4]
or
to
fit the airgap flux linkage versus 1/M using a polynomial method
[SI.
The nonlinear characteristic relating
X,
SZ)
versus
I,
A)
shown in Fig.
3
is
determined by experimental tests. Such nonlinear relationship can be fitted
with
continuous function as:
X
=
a[arctan(pI,

)
+
91
1
I
9)
where magnetizing current
I,
is defined by:
I
The coefficient
a
s employed
to
make the estimated
X,
to
match the measured reactance with respect
to
the measured maximum
I,.
The coefficients
p
and
y
decide the
maximum
value and the initial value
of
the magnetization reactance, respectively. The coefficient
9
makes the flux linkages to be zero when I, is equal to zero. The maximum value and initial value
of
magnetization reactance can be derived from
(9)
by using the limit theorem. Combining
1) 7)
and replacing arbitrary angular speed
o
with synchronous angular speed
a,,
the complete system consists of
11
nonlinear differential equations. The major advantage of using synchronous reference frame is that all variables are presented
as
dc quantities. Under steady state
0780359356/00/ 10.00
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2000
IEEE
278
condition, the rates of change
of
all state variables in
1) 6)
are identically zero.
0,0.793 pu,
VI
=
0.726
pu,
111.
THE CONTINUATION
POWERFLOW
ANALYSIS
w
.
..
U =
0.793 pu.
V,
=
0.676 pu,
U =
0.793 pu, VI
=
0.704
pu,
To
express the above equations under steadystate powerflow analysis, the system equations with
a
new added parameter
h
can be written as below:
TL
I
,973
Nm
147.726ij 1111.227 190.419
kj
624.315 223.682kj 150.819 52.138kj 177.153
where [y] denotes the state vector
of
the studied system, and
h
=
0
represents the base load condition.
A
nontrivial solution
of
(11)
is called the equilibrium point
or
the stationary solution. The continuation powerflow method belongs to a general class of methods for solving nonlinear algebraic equations known as the pathfollowing methods. Such method has been utilized as a tool
for
analyzing power system steadystate voltage stability [9lo]. The continuation powerflow method uses
an
iterative process involving a predictor step and a corrector step through a known solution of
(1
1
to
obtain a series of solution branch. The stable and unstable equilibrium points of the studied system can then be effectively determined. Iluring the predictor step, a linear approximation is employed to estimate the next solution for a change in one of [y]
or
h
Taking the derivatives
of
both sides
of
1
I),
with the state variables corresponding to the initial solution, will lead to the following hear equations:
T,
=2
0
Nm
T,=
1
973
Nm
147.7152~ 1111.637 147.645k~
1111.290
190.844
kj
623.211 191.457
kj
621
337
228.691
kj
152.113 233.236Itj 153 961 49.832
j
177.153 48.384kj 183.934
Since the parameter
h
is
unknown,
an additional equation must be employed to solve
(12).
By setting one
of
the components of the tangent vector to be
+I
or
1.
This
component
is
referred to
as
the continuation parameter. Equation (12) becomes:
48.375

5.624
1.998
(13)
41.846
36.378
15.657 15.724 0.0
2
394
where
yk
s a row vector with all elements equal to zero except for the kth element being equal to 1. The continuation parameter
is
usually selected to be the state variable that has the greatest rate
of
change near the given solution. The sign
of
its slope determines the sign of the corrtsponding component
of
the tangent vector.
Once
the tangent vector is obtained, the prediction
for
the next solution
is
given by where s the step length and
it
should be properly chosen so that a solution exists under the specified continuous parameter.
0780359356/00/ 10.00
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2000
IEEE
In
the corrector procedure, the srcinal equation
1
1)
is augmented by one equation that specifies the state variable selected as the continuation parameter. The new augmented equation is written as below. where yk is a state variable selected as the continuation parameter and
q
equals the predicted value of
yk.
This set
of
equations can be solved using a slightly modified Newton Raphson powerflow method. The introduction of the additional equation specifying y, makes the Jacobian non singular at the critical operating point. The continuation powerflow analysis can be continued beyond the critical point and determine the unstable region where the dynamic responses can not be effectively simulated.
Table
I
Eigenvalues
rads)
of
the studied system under three different
IV
RESULTS AND
DISCUSSIONS
Three different variables, i.e., the load
torque
T,
of the
IM
load, the excitation capacitance
C
and noload speed
ago
of
the
SEIG,
an be selected as the continuation parameter
h
depicted
in
the previous section. Figs.
46
respectively illustrate the continuous critical points
of
the minimum excitation capacitance C,,,,
the
minimum
required reactive power
Qc,
and the maximum load torque
TLmax
ersus different values
of
axe
It is found from Fig.
4
that the higher the value of
ago,
he smaller the value
of
C,,,
is. The large the value
of
T, the larger the value
of
C
requires.
It
is seen from
Fig.
that the higher he value of
ago
he larger the value of Qc is. The large the value of
TL
he larger the value
of
Qc requires.
It
is
discovered from
Fig.
that the higher the value
of
a@,
the larger the value
of
TLma, s. The large the value
of
C,
the larger the value
of
Twill be. Table
1
also
lists
the associated eigenvalues (dynamic stability) of the studied system under three different equilibrium points. According to the eigenvalue results listed in the last row
of
Table
1,
the studied system
is
operated fiom
a
stable point (equilibrium point
1
through the critical operating point (equilibrium point
2)
to the unstable point (equilibrium point 3) when the terminal voltage
V,
is continuously decreased.
27
9
V.
CONCLUSIONS

0.5


04

a
U
=
0.3
11.2

0 1

3*0
.5
his
paper
has
presented a novel scheme on steadystate performance of an isolated threephase selfexcited induction generator
(SEIG)
supplying a loaded induction motor, Eigenvalue analyses based
on
synchronous reference frame
has
been employed to determine
the
critical operating conditions and dynamic stability
of
the studied machines. Experimental results obtained
from
a laboratory
1.1
kW
induction machine driven by a dc motor and a
300
W
induction
motor
with
a
dc generator as
its
shaft
load
are
also performed
to
confirm the feasibility
of
the proposed method.
LL
.
E
125
100
TL=1
Nm
I
I I
I
I
I
I
0 4
0 5
0 6 0 7 0 8
0 9
1 0
1.1
1 2
Speed
of
SEIG
P.u.)
Fig
4
?Characteristics f C,,,,,, versus different values of rotor speed of SEE and TL.
0 8
0.7
1
0.6
i
TL=1
Nm
TL O
Nm
0.0
1
4
0 5
0 6
0 7
0 8
0 9
1.0
1.1
1 2
Speed
of
SEIG
pu)
Fig. Characteristics of minimum reactive power versus various values of rotor speed of SEIG and T,
2.0
1.5 1
o
OS
/+
0.0
I
I
I
0.6
0.7
0.8
0.9
Speed
of
SEIG
pu)
Fig.
6
Characteristics of maximum load torque versus speed of
SEIG
and
C
VI.
REFERENCES
[l]
E. D. Basset and F. M. Potter, “Capacitive excitation of induction generators.”
Tram. Americon Institute Electrical Engineering,
vol.
54,
[2]
D.
B. Watson.
J.
Amlaga and T. Densem. “Controllable d. c. power supply from winddriven selfexcited induction machines,”
IEE
Proceedrngs.
vol.
126.
no.
12, 1979,
pp.
12451248. [3]
J.
B
Patton and D. Curtice, “Analysis of utility protection problems associated with small wind turbine interconnections,”
IEEE Truns.
Power
ApparatusondSystems,vol.
101,
no
10,
1982,
pp.
39573966. [4]
S.
S
Murthy,
0
P.
Malik and A.
K.
Tandon, *Analysis
of
self excited induction generators,”
IEE Proceedings,
Pt.
C.
vol.
129,
no.
6,
1982.
pp,
[ ]
L.
Shndhar, Bhim Singh,
C.
S.
ha and B. P.Singh, “Analysis of self excited induchon generator feeding induction motor,”
IEEE
Trans. EnergVConversron,
vol.
9,
no.
2,
1994.
pp.
390396.
[a]
L. Wang and C.
H.
Lee. “Dynamic analyses of parallel operated self excited induction generators feeding
an
induction motor load,” Paper
PE337EC0121997,
EEWPES
1997
Summer
Meeting
Berlin,
Germany,
1997.
[7]
P. C. Krause,
Analysis
ojElectric Mochineiy,
New York: McGrawHill
Book
Co.,
1987. [8]
0
Ojo
“Minimum airgap flux linkage requirement
for
selfexcitation in standalone induction generators,”
IEEE Trans. Energy Conversion.
vol.
10.
no.
3, 1995,
pp.
484492. [9]
V.
Aljarapu
and
C. Christy, “The continuation power flow:
A
tool for steady state voltage stability analysis,”
]E€€
Tmns
Power
Systems.
vol
7.
no.
1.
1992,
pp.
416423.
[IO]
P.
Kundur,
Power
system
Stabili4
and Control.
New York: McGraw Hill Book Company,
1993.
1935,
pp
540545. 260265
VII. BIOGRAPHIES
SungChun
Kuo
was
bom
on
July\9, 1957
in Tainan, Taiwan. He obtained his
M.
Sc. degree from Department
of
Electrical Engineering, National Cheng Kung University. He
is
now a Ph. D. candidate at the Department of Electrical Engineering, National Cheng Kung University, Tainan, Taiwan. He
is
currently pursuing his Ph. D degree at the Department
of
Electrical Engineering, National Cheng Kung University, Tainan, Taiwan.
His
interest includes AC electric machines and power electronics
Ll
Wang (S’87M‘88) Dr. Wang was
bom
in
Changhua, Taiwan, on December
20, 1963
He received a Ph.
D.
degree from Department
of
Electrical Engineering, National Taiwan University, Taipei, Taiwan,
in
June
1988.
Since August
1995,
he has been
a
professor at the Department
of
Electrical Engineering, National Cheng Kung University, Tainan. Taiwan.
At
present,
his
interests include the science research of power engineering
such
as
power systems, electric machinery, and power electronics.
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