Full expandable model of parallel self-excited
induction generators
*
F.A. Farret, B. Palle and M.G. Simoes
Abstract: Self-excited induction generators (SEIG) offer many advantages as variable-speed
generators in renewable energy systems. Small hydro and wind generating systems have constraints
on the size of individual machines, and several induction generators must be paralleled in order to
access fully the potential of the site. SEIGs connected in parallel may lose excitation momentarily
owing to large transient currents caused by differences in individual instantaneous voltages and
frequency. This phenomenon cannot be easily simulated using the conventional models because it
has such a fast transient nature. An innovative and automatic numerical solution for steady-state
and transient analysis of any number of SEIGs operating in parallel is presented. Experimental
results confirm the accuracy of the proposed model and open new possibilities for incorporating
advanced control to monitor and optimise a parallel installation of SEIGs. The proposed SEIG
model is applied to a two-turbine case, which can be extended to simulate a wind generating
system.
1
Introduction
There has been a huge increase in energy demand, during
the last few decades, which has accelerated the depletion of
world fossil fuel supplies. Environmental concerns and
international policies are supporting new interests and
developments for small-scale power generation. Therefore,
the study of self-excited induction generators has regained
importance, as they are particularly suitable for wind and
small hydro power plants [1, 2]. They have advantages over
conventional synchronous generators in their reduced
installation cost, lower maintenance requirements, absence
of power supply for excitation and natural protection
against system faults. The applications of self-excited
induction generators (SEIGs) are limited, based on
economic reasons depending on the combined costs of
machine, excitation capacitors and static switches. Typical
SEIG loads include electricity for rural residences [3],
heating, lighting and small induction motors. Typically,
generators rated 15 kVA are cost effective; but 100 kVA [4]
was found to be the upper limit where generator price is in
the crossover price curve.
A stand-alone SEIG is unlikely to supply the energy
demand of ordinarily growing loads for a long time. Thus,
multiple generators operating in parallel may be required to
harvest the maximum energy available at a site. Also, in the
last few years, the trend has changed from installing a few
wind turbines to planning large wind farm installations with
many induction generators connected electrically in parallel
[5, 6]. With increasing penetration of wind power into
power networks, an accurate dynamic model of the overall
wind farm system is required to analyse the interaction
between the wind farm and the power system. A system of
parallel-operated SEIGs, in a wind or small hydro power
plant, is subjected to various transient conditions, such as
initial self-excitation, load transient and generator/capacitor
switching. Transient interaction and resonant states may
also be reasons for concern, in small power plants, as they
may create unstable oscillations, cause mechanical elements
deterioration and trigger protection circuits. Therefore, it is
important to perform modelling and simulation analysis of
parallel-operated induction generators under transient
conditions. References [7–9] developed a wind farm model
that can be used in power system dynamic simulations with
steady-state representation of the generator, and do not deal
with the transient behaviour of SEIGs. Several papers [10–
14] have been published on stand-alone operation of selfexcited generators, but only a few references are available
on parallel operation of SEIGs. Steady-state analysis of
parallel-operated SEIGs has been discussed [15, 16], but
previous work related to transient analysis does not present
clear numerical modelling and experimental observations
[17, 18]. The current literature does not approach parallel
operation under transient analysis with limitations due to
approximated representation of the induction machine
models.
In this paper, an automatic procedure to build up a
matrix model for steady-state and transient analysisFvalid
for any number of self-excited induction generators
operating in parallel and supplying a common R–L
loadFis proposed [19]. Figure 1 shows n induction
G1
r IEE, 2005
IEE Proceedings online no. 20045057
doi:10.1049/ip-epa:20045057
Paper first received 12th June and in revised form 22nd October 2004
F.A. Farret is with the Federal University of Santa Maria, Rua Dr Bozano, 915
Apto. 203, Santa Maria – RS 97.015.003, Brazil
B. Palle and M.G. Sim*oes are with the Colorado School of Mines, 1610 Illinois
St., Golden, Colorado 80401-1887, USA
96
G2
......
Gn
......
C1
Fig. 1
load
C2
Cn
Induction machines supplying a common load
IEE Proc.-Electr. Power Appl., Vol. 152, No. 1, January 2005
machines ready to be connected in parallel according to
load requirements or when more energy becomes available.
Such a basic association of generators seems to cope with
the vast majority of practical cases since it is always
expected to have a new generator connected or disconnected
from an already existing parallel association. It should be
noted that each machine needs a self-excitation capacitance,
because each machine has its own magnetisation curve.
A computer algorithm is developed to simulate the
parallel operation of SEIGs. Simulated results for connection of two generators are compared with experimental
results on two laboratory machines. Individual variation in
the magnetising inductance is incorporated in the analysis.
A model is then developed for a wind turbine with an SEIG
as the AC generator. Parallel operation of two wind
turbines is simulated using real wind conditions. This model
allows us to reproduce several conditions encountered in
small power plants.
2
Model for parallel SEIGs
The model to be presented accords with the representation
of n SEIGs as shown in Fig. 1. A generalised state–space
equation for n generators operating in parallel is shown in
(1). Generator, excitation capacitance and load parameters
of the generation system are separated to obtain a matrix
representation of parallel generators in a wind or a hydro
power plant. Incoming generators can be incorporated into
the model by appending the generator matrix ‘G’ diagonally, as shown. This matrix representation allows us to
simulate the behavior of n generators in parallel during the
self-excitation process and in the steady state. It also allows
us to analyse the transient performance under varying load
conditions and generator switching. Eigenvalue and eigenvector analysis can also be performed using (1), as it is in the
form of the classical state–space equation. A detailed SEIG
model is developed in Section 3:
3 2
3
G1 0 L1
iG1
6i 7 6 0 G
L2 7
2
7
6 G2 7 6
6 . 7 6 .
.. 7
..
7
6 . 7 6 .
7 6
6
. 7
.
p 6 . 7 ¼6 .
7
6i 7 6
Gn Ln 7
Gn 7
7
6 .....
6 ..........................................
7 6
7
6
4 .....
CG CG CG CT 5
vL 5 4 ..........................................
iL
0
0
0
0
L
3
2
3
2
iG1
B1
2
3
6i 7
7 vG1
6 G2 7 6
B2
76
6 . 7 6
76 vG2 7
6 . 7 6
7
76
6 . 7 6
..
þ
7 6
76 . 7
6
.
74 .. 7
6i 7 6
5
7
6 Gn 7 6
Bn 5
7 4
6
4 vL 5
vGn
0
0
0
0
iL
....................................
2
Rr1
iqdr1
−
+
LIr1
Rs1
LIs1
ð1Þ
where
2
3
2
3
idsi
vdsi
6 iqsi 7
6
7
7; vGi ¼ 6 vqsi 7; vL ¼ vLd ; iL ¼ iLd
iGi ¼ 6
4 idri 5
4 vdri 5
vLq
iLq
iqri
vqri
3
Through (3) and Fig. 2, a classic matrix formulation [20]
using d–q axis modelling is used to represent the dynamics
of a conventional induction machine operating as a
generator. For an isolated generator, the parameters are
labelled according to Fig. 2. The representation includes the
self and mutual inductances as coefficients widely used in
machine theory. Using such a matrix representation, one
can obtain the instantaneous voltages and currents during
the self-excitation process, as well as during load variations.
At this point, it must be said that the traditional d–q axis
model is not convenient for the automatic building up of a
general model of parallel-operated induction generators,
because it does not isolate the machine parameters from the
self-excitation capacitor and load parameters. To isolate
those parameters, (4) has been formulated using eight firstorder differential equations that relate the stator and rotor
currents and voltages. The simultaneous solution of this
system of equations can be obtained using the Runge–
Kutta fourth-order integration method with automatic
adjustment of step. This gives the instantaneous values of d–
q axis voltages and currents for stator and rotor.
The following assumptions are made in this analysis:
(i) Core and mechanical losses in the machines are
neglected. (ii) All machine parameters, except the magnetizing inductance, are assumed to be constant. (iii) Stator
windings, self-excitation capacitors and the load are wye
connected. The variation of the magnetizing inductance is
the main factor in the dynamics of the voltage build up and
stabilization in SEIGs. When multiple SEIGs are operating
in parallel, the machine with the lowest saturation voltage
will control the voltage of the whole group. So, with
machines of different ratings, the larger machines will have
to be de-rated so as to stop them driving the smaller
machines. The relationship between magnetization inductance, Lm, and the magnetization current for each induction
machine was obtained experimentally. The non-linear
relationship between magnetising inductance and magnetising current for the generator G1 (see Section 5) used in the
experimental setup is shown below:
Lm ¼ 6:89 106 Im4 þ 1:38 104 Im3 1:22
103 Im2 þ 1:28 103 Im þ 4:62 102 H
excitation
capacitance
iqds1
jωr 1 qdr1
qdr1
Lm1
qds1
SEIG modelling
Rs 2
R
LIs 2
LIr 2
iqds1
qds2
C
+
−
Rr 2
jωr 2 qdr 2
Lm 2
ð2Þ
iqdr2
qdr 2
L
SEIG1
Fig. 2
load
SEIG2
Two self-excited induction generators in parallel represented in d–q model
IEE Proc.-Electr. Power Appl., Vol. 152, No. 1, January 2005
97
6v 7 6
6 qs 7 6
6 7 ¼6
4 vdr 5 6
4
vqr
Rs þ Ls p þ
RþLp
RCpþLCp2 þ1
0
Lm p
Lm p
or Lm
R r þ Lr p
Lm p
or Lr
0
32 3
ids
76 7
iqs 7
Lm p 7
76
7
76
4
or Lr 5 idr 5
iqr
Rr þ Lr p
0
3
ids
76 iqs 7
7
76
7
76
76 idr 7
Lm 0
0
0
7
76
76 i 7
0
Lm 0
0
76 qr 7
7
76
76 vLd 7
0
0
1=CK 0
6 ......7
............................................................7
7
6
0
0
0
1=CK 7
76 vLq 7
7
76
54 iLd 5
1=LK 0
R=LK 0
............................................................
......
iLq
0
1=LK 0
R=LK
3
Lr 0
Lm
0
0
Lr 0
Lm 7
7
72 3
Lm
0
Ls 0 7 vds
7
6 7
0
Lm
0
Ls 7
76 vqs 7
þ ......................................... 76 7g
ð4Þ
0
0
0
0 74 vdr 5
7
0
0
0
0 7
7 vqr
.........................................
7
0
0
0
0 5
0
0
0
0
................................................
0
0
0
0
0
0
................................................
0
0
Lr
32
which is in the form of classical state-space equation
p[x] ¼ [A][x]+[B][u], or:
2 3 2 32 3
G
iG
iG
ð5Þ
p4 vC 5 ¼ 4 C 54 vC 5 þ ½ B½vG L
iL
iL
where G, C and L refer, respectively, to the partition of
matrix [A] into matrices for the induction generator
parameters, the self-excitation capacitance and the load.
Vector [x] is the transposed matrix [iG vC iL], and the submatrixes [G], [C] and [L] are defined as:
2
98
......................
Rs Lr
6 o L2
6 r m
½G ¼K 6
4 Rs Lm
or Lm Ls
or Lm Lr Lr
Rr M
or L s L r
Rr Ls
1600
or L2m
100
0
ð3Þ
Lr
0
approximated
shaft speed
simulated
speed profile
1700
a
or Lm
0
RþLp
Rs þ Ls p þ RCpþLCp
2 þ1
Equation (3) can be expressed as a state variable matrix that
takes the following form:
3
2
2
ids
Rs Lr
or L2m Rr Lm
or Lm Lr
6 o L2
6 iqs 7
Rs Lr
or Lm Lr
Rr Lm
7
6 r m
6
7
6
6
6 Rs Lm
6 idr 7
o
L
L
R
L
or Ls Lr
r m s
r s
7
6
6
6 o L L R L
6i 7
or Ls Lr Rr Ls
r m s
s m
6
6 qr 7
p6
7 ¼Kf6
6 1=CK
6 vLd 7
0
0
0
7
6........................................................................
6
60
6v 7
1=CK
0
0
6
6 Lq 7
7
6
6
4........................................................................
4 iLd 5
0
0
0
0
iLq
1800
Rr Lm
Rs Lr
or Lm Lr
or Lm Ls Rr Ls
Rs Lm or Ls Lr
3
0
0 0
0
Lr
0 07
7
7
Lm 0
0 05
0
Lm 0 0
0
−100
b
100
0
− 100
0
1
2
3
4
5
6
7
8
9
c
time,s
Fig. 3
Self-excitation and load response of a stand alone SEIG
a Rotor speed profile
b Variation in phase voltage measured during self-excitation and load
switching on generator G1 (see Section 5)
c Variation in phase voltage simulated using MATLAB
½C¼ ½ CG
1=C 0
0 0 0 0 1=C 0
Cr ¼
0
1=C 0 0 0 0 0
1=C
0 0 0 0 1=L 0
R=L 0
½L ¼
0 0 0 0 0
1=L 0
R=L
.........
2
.........
3
rotor speed,
rpm
vds
phase voltage, V
2
and K ¼ 1=L2m Ls Lr
The excitation vector [u] ¼ [vG] of (5) is multiplied by the
excitation parameter matrix [B] shown below. Therefore,
[B][u] defines the voltages corresponding to the residual
magnetism in the machine core:
3
2
Lm
0
Lr 0
60
Lr 0
Lm 7
7
6
6 Lm
0
L
0 7
s
7
6
60
Lm
0
Ls 7
7
6
½B ¼ 6
0
0
0
0 7
7
6.........................................
60
0
0
0 7
7
6
4.........................................
0
0
0
0 5
0
0
0
0
Figure 3 shows the experimental and simulated plots of the
transient self-excitation process of a stand-alone SEIG.
Load is switched on at 7.6 s. The induction generator was
operated at 1780 rpm with a DC motor as prime mover. A
capacitor bank of 160 mF in star connection supplied about
700 VAR/phase of reactive power for the machine, and a
120 O–22.5 mH star load was connected after the generator
was completely excited. Variation of speed observed in the
laboratory when load was applied is incorporated in the
numerical simulation seen in Fig. 3a, which depicts
simulated and approximated shaft speed. Remanent
magnetism in the machine core is also taken into account
and is explained in Section 4. The complete simulation
process is detailed in the following sections.
Figure 4 shows an SEIG connected in star with an
excitation capacitor bank with neutrals isolated. The natural
consequence of this type of connection is third harmonics in
phase voltages. Simulation and experimental results do not
show any significant third-harmonic content as the machine
was run under light saturation. High currents can flow
through the capacitors when run under deep saturation.
Also, the excitation capacitor along with the machine
inductance acts as lowpass filter attenuating the thirdharmonic voltages present in the phase voltages. Figure 5
IEE Proc.-Electr. Power Appl., Vol. 152, No. 1, January 2005
of only a small source of applied voltage to the real
machine, for recovery of its active state during the
occurrence of fortuitous core de-excitation.
a
C
Van
4
Vcn
Vbn
C
C
b
Turbine model
Wind energy systems are usually composed of the turbine,
gearbox, generator and load. The wind turbine power is
given by:
c
Fig. 4
Pw ¼ rV 3 pr2 CP =2
SEIG connected in star with excitation capacitors
0
−4.7 dB at
180 Hz
attenuation, dB
−3
−5
ð6Þ
where
r ¼ air density
V ¼ wind speed
r ¼ propeller radius
CP ¼ wind turbine power coefficient (usually expressed as
a function of tip speed ratio l).
−10
Power coefficient is not constant, but varies with the wind
speed, rotational speed of the turbine and turbine blade
parameters. The torque generated by a turbine is given by:
−15
Tw ¼ rV 2 pr3 CT =2
where CT ¼ CP/l is the torque coefficient and the
electromagnetic torque generated by the induction generated is:
−20
−25
Tg ¼ 0:75PMðids iqr iqs idr Þ
1
Fig. 5
ð7Þ
100 180
1000
frequency, Hz
10
Frequency response of SEIG
Table 1: Generator ratings and parameters
Generator G1
Generator G2
P ¼ 1 HP
Rs ¼ 0.32 O
P ¼ 1 HP
Rs ¼ 0.39 O
1750 rpm, 60 Hz
Xls ¼ 0.8 O
1750 rpm, 60 Hz
Xls ¼ 0.93 O
V ¼ 120/220 V
Rr ¼ 0.41 O
V ¼ 120/220 V
Rr ¼ 0.44 O
I ¼ 11.5/5.5 A
Xlr ¼ 0.8 O
I ¼ 11.5/5.5 A
Xlr ¼ 0.93 O
The machine swing equation is given by:
dwr
P
2
¼ ðTw Tg Bwr Þ
ð9Þ
2J
P
dt
where P is the number of poles of the generator, M is the
magnetising inductance, and J and B are the overall system
inertia and viscous friction coefficient, respectively. Equation (9) represents the wind turbine as a single lumped
inertia; but, if required, (9) may be expanded to include the
masses of the generator, gearbox and blades with the
associated parameters. A matrix equation similar to (1) for
the rotor speeds of n SEIGs operating in parallel can be
obtained from (9). Using an iterative process, the instantaneous prime mover speeds can be calculated.
5
shows the frequency response of the SEIG for the
parameters shown in Table 1. The SEIG model presented
above is for the configuration in Fig. 4, and further work is
required to incorporate other configurations, although, in
practice, it has been observed that a cross-connection is not
desirable because of the floating neutral and voltage
imbalances.
At this point, it is interesting to note that, to begin the
self-excitation process of the induction generator, a certain
amount of residual magnetism must be present, that is, it is
a condition ‘sine qua non’. This effect also needs to be taken
into account in the numerical simulation of the selfexcitation process, without which it is not possible to start
the numerical integration process. At the beginning of the
integration process of (3) and (5), an impulse function was
used to represent the transient existence of the residual
magnetism that fades away after the first iterative step. Any
other representation of the way the residual magnetism
fades away may be acceptable. This observation is very
important in the dynamic understanding of the selfexcitation phenomenon, because it would justify the use
IEE Proc.-Electr. Power Appl., Vol. 152, No. 1, January 2005
ð8Þ
Simulation of parallel-operated SEIGs
Simulations have been developed in MATLAB. The routine
shown in Fig. 6 is used to predict the generated voltage
from two given self-excited induction generators operating
in parallel. This routine can be extended for any number of
generators operating in parallel. Remanent magnetism in
the machine is taken into account, without which it is not
possible for the generator to self-excite. As stated above, an
impulse function is used to represent the remanent magnetic
flux in the core. The main routine calls for the subroutine
shown in Fig. 6b, to solve the differential equations.
6
Results and discussion
The model proposed for parallel-operated SEIGs is
investigated in the laboratory using two identical induction
machines driven by two different DC motors. Figure 7
shows experimental magnetisation curves for each induction
generator. A fourth-order polynomial curve fitFdescribing
the non-linear relationship between airgap voltage and
magnetizing currentFwas obtained from experimental
data. The parameters in Table 1 were used for simulation.
In addition to the stand-alone operation of an SEIG
99
160
start
G2
140
120
read machine parameters,
magnetization
characteristics
G1
Vg , V
100
80
60
40
initialize residual magnetism; set itr = 0
20
0
0
simulate stand-alone operation of generator-1
using the sub-routine in fig. 4(b)
2
4
6
8
10
Im , A
Fig. 7
Magnetisation characteristics of generators G1 and G2
simulate stand-alone operation of generator-2
using the sub-routine in fig. 4(b)
rotor speed,
rpm
switch load when generator
is completely excited
1800
simulated
speed profile
approximated
shaft speed
1600
1400
a
close the paralleling switch at the chosen instant
100
a
phase voltage, v
0
simulate stand-alone operation of two generators
using the sub-routine in fig. 4(b)
−100
b
100
start
0
−100
Y
0
end
if itr < max_itr
1
2
3
c
4
5
6
time, s
N
Fig. 8
solve the state-space equation for
the d-q parameters using RungeKutta fourth order method
simulation of two wind turbines using real wind data
operating electrically in parallel.
calculate the magnetization current
and update mutual inductance
Y
if itr > 1
itr = itr + 1
N
remove the initial
impulse applied
for residual
magnetism
record instantaneous
generator phase
voltages
b
Fig. 6
a Main program to simulate parallel operation
b Sub-routine for numerical solution
supplying a medium load discussed in Section 3, four other
cases have been discussed in this paper:
voltage collapse in a stand-alone self-excited induction
generator due to a heavy load
transient phenomena of load and generator switching of
two SEIGs operating at different voltages levels
transient phenomena of load and generator switching of
two SEIGs operating at similar voltage levels
100
Voltage collapse in a stand-alone SEIG under heavy load
a Rotor speed profile
b Variation in phase voltage measured during collapse of G1
c Variation in phase voltage simulated
Figure 8 shows the experimental and simulated plots of
terminal voltage collapse of a stand-alone SEIG under
application of a heavy R–L load at time ¼ 2.6 s. Generator
G1 was self-excited with 160 mF capacitance bank, and a
45 O–45.5 mH star load was applied after the generator was
fully excited. Speed variations observed in the laboratory
were taken into account for simulation, as illustrated in
Fig. 8a. It took about 2.5 s for the voltage to collapse
completely. To avoid the collapse, an electronic protection
scheme can be used to detect the current levels. If the
collapse occurs, the machine has to be re-magnetised.
Figure 9 shows the transient process of load and
generator switching of two SEIGs operating in parallel.
Generator G1 was self-excited with 160 mF capacitance at
1800 rpm, and a 130 O–22.5 mH star-connected load was
applied at time ¼ 2 s. Generator G2 was previously selfexcited with 160 mF at 1800 rpm, and connected in parallel
at time ¼ 3.75 s. The sudden and brief collapse in voltage is
due to differences in phase voltage of the two generators, at
the paralleling instant. Full common voltage was recovered
at about time ¼ 6.0 s. The rotor speed variation of
generators G1 and G2, during parallel operation, was
carefully observed in the laboratory and was incorporated
in the simulation, as illustrated in Fig. 9a. At the precise
IEE Proc.-Electr. Power Appl., Vol. 152, No. 1, January 2005
rotor speed,
rpm
rotor speed,
rpm
1600
1200
800
simulated
speed profile
a
approximated
shaft speed
100
1200
0
phase voltage, v
b
100
0
−100
a
c
b
100
0
−100
c
100
100
0
−100
0
−100
0
1
2
3
4
5
6
7
8
9
0
d
1
2
3
4
5
d
time, s
time, s
a Rotor speed profile
b Variation in phase voltage of G1 measured. Load applied at
t ¼ 1.95 s, and paralleling switch closed at t ¼ 3.85 s
c Simulated variation in phase voltage of G1
d Simulated variation in phase voltage of G2
instant of parallel connection of the two generators, a heavy
dip in the overall speed of the machines was noticed.
Three moments should be pointed out in those graphs.
The first is related to the voltage reduction across generator
G1’s terminals when the R–L load was switched on. The
second is the partial voltage collapse at time ¼ 3.85 s, and its
recovery up to the parallel common voltage level at about
6.0 s. The third moment to observe is the reduced voltage of
generators G1 and G2 with respect to the no-load voltage
level of G2 alone representing an appreciable voltage
difference at the parallel connecting instant. These three
moments can be clearly and closely observed in both
theoretical and experimental setups. Synchronisation techniques could be included so as to switch the generators
when the phase voltages are in phase. The purpose of this
result was to demonstrate the possibility of representing the
worst-case scenario using the proposed model.
Figure 10 shows the experimental and simulated plots of
transient generator switching process when parallel connection was made between SEIGs operating at identical voltage
levels. Generator G1 was self-excited with 180 mF capacitance, and a 130 O–22.5 mH star load was applied at
time ¼ 1.2 s. Generator G2 was already self-excited with
160 mF and was connected in parallel at time ¼ 4.4 s. Full
common voltage was recovered at about time ¼ 5.2 s. As the
voltages were similar, there was no collapse as observed in
the previous case, and the voltage and speed dips were not
very pronounced.
Figure 11 shows the simulated plots of two 20 kW wind
turbines operating electrically in parallel. Figure 11a shows
the wind speed profile of the two turbines. The generator
begins to self-excite after the rotor has reached a certain
speed (1). As the wind speed increases, the generated voltage
also increases (2). The dip in voltage at (3) corresponds to
load switching. G2 is connected to G1 at (4) to share the
load, resulting in a slight increase in the G1 voltage.
Figure 12 shows the simulated rotor speed and torque
variations during load and generator switching process. The
torque surge experienced by the two machines, during the
switching of the second generator, is very high. However, it
settled down in a few milliseconds; therefore, the system is
able to sustain its excitation.
IEE Proc.-Electr. Power Appl., Vol. 152, No. 1, January 2005
Fig. 10
levels
6
7
8
9
Parallel connection of two SEIGs with similar voltage
a Rotor speed profile
b Variation in phase voltage of G1 measured. Load applied at t ¼ 1.2 s,
and paralleling switch is closed at t ¼ 4.6 s
c Simulated variation in phase voltage of G1
d Simulated variation in phase voltage of G2
wind speed,
m /s
Parallel connection of two SEIGs with different voltage
7
vw1
6
vw2
5
a
500
1
3
2
0
phase voltage, v
Fig. 9
levels
simulated
speed profile
100
0
−100
−100
phase voltage, v
approximated shaft
speed
1600
4
−500
b
500
0
−500
0
Fig. 11
5
10
15
20
25
c
time, s
30
35
40
45
50
Simulation of two wind turbines operating in parallel
a Wind speed profile
b Simulated phase voltage of G1. Load applied at t ¼ 36 s, and
paralleling switch closed at t ¼ 38 s
c Simulated phase voltage of G2
7
Conclusions
An innovative model to simulate the electromechanical
steady-state and transient performance of any number of
wind turbines with induction generators connected in
parallel has been presented. With this model, it is possible
to have an automatic computer process to generate the state
variable matrix representing the incorporation of any new
wind turbine to the previous parallel association. This
feature is guaranteed by the separate parameter representation of the machine model, the self-excitation bank of
capacitors and the load.
The approach presented enhances previous work based
only on electrical steady-state conditions that could not be
used for real analysis of the transient phenomenon that
occurs and may have adverse impact on system stability and
101
rotor speed,
rpm
9
2000
1500
1000
ωr 2
ωr 1
a
torque, N m
20
Tg1
Tw1
10
0
b
20
0
Fig. 12
Tg2
Tw 2
10
0
5
10
15
20
25
30
c
time, s
35
40
45
50
Simulation of two wind turbines operating in parallel
a Rotor speed profiles of two generators
b Turbine torque and electromagnetic torque of G1
c Turbine torque and electromagnetic torque of G2
protection. Previous work related to transient analysis does
not present clear numerical modelling and experimental
observations. Experimental results prove that the proposed
matrix partition is in agreement with the mathematical
modelling. In addition, the use of the matrix partition
proved to be a powerful numerical tool.
The main advantages of this approach are: (i) representation of the self-excited induction generator in the form of
classical state equations; (ii) separation of the machine
parameters from the self-excitation capacitors and load
parameters, to allow the automatic building up of the
matrix representation of SEIG parallel operation; (iii) an
increase of flexibility and simplicity in generalising a model
for n parallel-operated generators; (iv) inclusion of electromechanical steady-state and transient analysis of the
paralleling of induction generators; and (v) eigenvalues
(eigenvector analysis can be performed using the classical
state–space representation).
8
Acknowledgments
The authors sincerely thank the Coordination for Improvement of Advanced Education Personal (CAPES) and the
National Science Foundation (NSF) for their financial
support of this project.
102
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