Energy and Power En gi neering, 2011, 3, 393-400
doi:10.4236/epe.2011.34050 Published Online September 2011 (http://www.SciRP.org/journal/epe)
Steady State Analysis of a Doubly Fed Induction Generator
Ahmad M. Alkandari1, Soliman Abd-Elhady Soliman2, Mansour H. Abdel-Rahman3
1College of Technological Studies, Department of Electrical Technology Sheweikh, Kuwait, Kuwait
2Electrical Power and Machines Department, Misr University for
Science and Technology, Giza, Egypt
3Department of Electrical Engineering, Mansoura University, Mansoura, Egypt
Received January 25, 2011; revised February 28, 2011; accepted March 16, 2011
Abstract
In this paper, we present the steady state analysis of a doubly-fed induction generator (DFIG) adopted for
wind power generation. The three-phase induction machine connected to the network, to work as a generator
for wind farms, is excited on the rotor circuit by a slip-frequency current injected to the rotor, from an exciter
mounted on the same shaft of the machine. The resulting rotating magnetic field rotates at synchronous speed;
as such the generated power has a constant frequency independent of the shaft speed. Effects of the excita-
tion voltage magnitude and phase angle on the active and reactive power are studied, when the machine runs
at constant speed. It has been shown that by controlling the excitation voltage magnitude and phase angle
would control the mode of operation of the machine; motor mode or generator mode. Furthermore, the ef-
fects of the shaft speed on the active and reactive power at constant excitation voltage magnitude and con-
stant phase angle are also investigated.
Keywords: Asynchronous Operation, Doubly Feed Induction Generator (DFIG), Wind Power
1. Introduction
The growing demand of energy in industrialized world
and environmental problems determined some important
decision at political level that consider even more im-
portant to improve the percentage of energy produced by
renewable sources. It is well-known the Kyoto Protocol
and other important regulations that take care of envi-
ronmental problems. Also the European Parliament ap-
proved the document 2001/77/CE on the “promotion of
electrical energy produced by renewable sources” at-
tending that before 2020 the 22% of total consumed en-
ergy should be produced by renewable sources. In this
frame a lot of efforts are devoted to increase the effi-
ciency of the generation systems based on wind, sun,
hydro and biomass.
Wind power seems to be one of the most interesting
technologies, especially considering the developments in
the last decade. The electrical energy generation by wind
depends on different factors, in particular the wind speed
and the characteristics of the wind turbine generator [1].
The unpredictability of the wind power, that is variable
with time, determines fluctuating power outputs of wind
energy conversion systems. This variable power genera-
tion nature requires a careful analysis, design, and man-
agement of the generation system so that various ap-
proaches have been developed to study wind turbine
generato r be haviors.
There are different technical solutions that have been
set up for different cases, for wind turbines in a range
from less 1 kW to as large as 3 MW or more, to obtain
the maximum efficiency and reliability. Traditionally the
wind power generation has used fixed-speed induction
generators that represent a simple and robust solution,
then variable speed turbines have been considered be-
cause they give higher energy, allow an extended control
of both active and reactive power, and present less fluc-
tuation in output power. The main solution adopted to
realize wind generators can be divided, by electrical to-
pology point of view and depending on the power, in the
following categories [2]:
 Standard squirrel cage induction generator directly
connected to the grid;
 Wound-rotor induction generator with variable rotor
resistance;
 Doubly fed inductio n generator;
 Synchronous or induction generator with full-size
power converter.

A. M. ALKANDARI ET AL. 394
Both induction and synchronous generators can be
used for wind turbine systems. Induction generators can
be used in a fixed-speed system or a variable-speed sys-
tem, while synchronous generators are normally used in
power electronic interfac ed variable-speed systems. Mainly,
three types of induction generators are used in wind
power conversion systems: cage rotor, wound rotor with
slip control by changing rotor resistance, and DFIG. The
cage rotor induction machine can be directly connected
into an ac system and operates at a fixed speed or uses a
full-rated power electronic system to operate at variable
speed. The wound rotor generator with rotorresistance-
slip control is normally directly connected to an ac sys-
tem, but the slip control provides the ability of changing
the operation speed in a certain range. The DFIG pro-
vides a wide range of speed variation depending on the
size of power electronic converter systems. In this chap-
ter we first discuss the systems without power electronics
except the thyristor soft starter, and then discuss the
variable-speed wind turbine systems, including those
with partially rated power electronics and the full-scale
power electronic interfaced wind turbine systems.
In fixed-speed wind turbines, the generator is directly
connected to the mains source grid. The frequency of the
grid determines the rotational speed of the generator and
thus of the rotor. The generator speed depends on the
number of pole pairs and the frequency of the grid. The
scheme consists of a squirrel-cage induction generator
(SCIG), connected via a transformer to the grid. The
wind turbine systems using cage rotor induction genera-
tors almost operate at a fixed speed (variation of 1% -
2%). The power can be limited aerodynamically by stall
control, active stall control, or by pitch control. The ad-
vantage of wind turbin es with induction generators is the
simple and cheap construction. In addition, no synchro-
nization device is required. These systems are attractive
due to cost and reliability, but they are not fast enough
(within a few ms) to control the active power. There are
some other drawbacks also: the wind turbine has to oper-
ate at constant speed, it requires a stiff power grid to en-
able stable operation, and it may require a more expen-
sive mechanical construction in order to absorb high
mechanical stress since wind gusts may cause torque
pulsations in the drive train and the gearbox. Other dis-
advantages with the induction generators are high start-
ing currents and their demand for reactive power. They
need a reactive power compensator to reduce (almost
eliminate) the reactive power demand from the turbine
generators to the grid. It is usually done by continuously
switching capacitor banks following the production
variation (5 - 25 steps) [2].
In variable-speed systems the generator is normally
connected to the grid by a power electronic system. For
synchronous generators and for induction generators
without rotor windings, a full-rated power electronic
system is connected between the stator of the generator
and the grid, where the total power production must be
fed through the power electronic system. For induction
generators with rotor windings, the stator of the genera-
tor is connected to the grid directly. On ly the rotor of the
generator is conn ected through a power electronic system.
This gives the advantage that only a part of the power
production is fed through the power electronic converter.
This means the nominal power of the converter system
can be less than the nominal power of the wind turbine.
In general the nominal power of the converter may be
30% of the power rating of the wind turbine, enabling a
rotor speed variation in the range of 30% of the nominal
speed. By controlling the active power of the converter,
it is possible to vary the rotational speed of the generator
and thus of the rotor of the wind turb ines [2].
Doubly-fed induction machines can be operated as a
generator as well as a motor in both sub-synchronous and
super synchronous speeds, thus giving four possible op-
erating modes. Only the two generating modes at sub-
synchronous and super-synchronous speeds are of inter-
est for wind power generation.
In a DIFG the slip rings are making the electrical con-
nection to the rotor. If the generator is running su-
per-synchronously, electrical power is delivered to the
grid through both the rotor an d th e stator. If the generator
is running sub-synchronously, electrical power is deliv-
ered into the rotor from the grid. A speed variation of
±30% around synchronous speed can be obtained by the
use of a p ower converter of 30 % of nominal power. Fur-
thermore, it is possible to control both active (Pref) and
reactive power (Qref), which gives a better grid perform-
ance, and the power electronics enables the wind turb ine
to act as a more dynamic power source to the grid. The
DFIG system does not need either a soft starter or a reac-
tive power compensator. The system is naturally a little
bit more expensive compared to the classical systems
However, it is possible to save money on the safety mar-
gin of gear and reactiv e power compensation units, and it
is also possible to capture more energy from the wind.
[2]
In Reference [1] the case of variable speed dual- exc ited
synchronous machine is inv estigated. For its control pos-
sibilities, this machine is particularly suitable for vari-
able-speed constant frequency applications like wind
power generation systems. The mathematical model of
the wind turbine and the generator is presented a control
technique for output power maximization is proposed. It
has been demonstrated that using the proposed control
algorithm it is possible to achiev e a significant increment
on the power generated in comparison with that obtained

A. M. ALKANDARI ET AL. 395
using an unregulated induction machine. the stability of WT system. Applying a set of optimized
controller parameters, the stability can be further en-
hanced [7].
Reference [3] presents a dynamic model of an impor-
tant contemporary wind turbine concept namely a doubly
fed (wound rotor) induction generator with a voltage
source converter feeding the rotor. This wind turbine
concept is equipped with rotor speed, pitch angle and
terminal voltage controllers. It was shown that it is pos-
sible to develop a set of equations describing the behav-
ior of the wind turbine. Furthermore, controllers for the
rotor speed, the pitch ang le and th e terminal vo ltag e were
developed. The behavior of the system was investigated
using two measured wind sequences.
This paper presents the steady state analysis of DFIG,
where w e assume that the three phase induction machine
is connected to work as a generator for wind turbine
farms, the machine is excited in the rotor circuit by a slip
frequency current comes from an exciter moun ted on the
same shaft, with the main generator, so that the speed of
the resulting flux in the air gap is always synchronous
speed independent of the wind speed. Effects of the ex-
citation voltage as well as th e excitation vo ltage angle on
the total active and reactive power, when the machine
runs at constant speed are studied in this paper. Further-
more, effects of the shaft speed on the active and reactive
power at constant excitation voltage and phase angle are
also studied.
Reference [4] presents the modal analysis of a grid c on-
nected to a DFIG. The change in modal properties for
different system parameters, operating points, and grid str-
engths are computed and observed. The results offer a better
understanding of the DFIG intrinsic dynamics, which can
also be useful for control design and model justification. 2. Steady State Analysis of DFIG
Vector control is already applied to the DFIG control,
which makes the DFIG gain good performance in the
wind energy capturing operation. But in the two tradi-
tional vector control schemes, the stator magnetizing cur-
rent is considered invariant in order to simplify the rotor
current inner-loop controller. The two schemes are capa-
ble of performing very well when the grid is in normal
condition [5]. However, when the grid disturbance, such
as grid voltage dip or swell fault, occurs, the control per-
formance will be getting wo rse, the rotor over current will
occur, which seriously reduce the ride-through ability of
the DFIG wind energy generation system. Based on the
accurate model of the DFIG, the deficiency of the tradi-
tional vector control is deeply investigated. The improved
control schemes of the two typical traditional vector con-
trol schemes used in DFIG is proposed, and the simula-
tion study of the proposed and traditional control schemes
is carried out. The validity of the proposed modified
schemes to control the rotor current and to improve the
ride-through ability of the DFIG wind energy generation
system is proved in Reference [6] by the comparison study.
In the steady state analysis we assume a three-phase
slip-ring induction machine, where the rotor is excited
from a slip frequency exciter mounted on the same shaft
of the rotating machine, and the stator is connected to the
power grid having a constant voltage and constant fre-
quency. The excitation voltage is expressed in a phasor
form as f
V


. If the core resistance is neglected, then
the equivalent circuit in p.u is given in Figure 1.
The loop voltage equations in per unit are given by:

 
11

s
rm
VIR jXIjX (1)

 
2fsmr
VIjSXIRjSX   

1
(2)
Solving Equations (1) and (2) yields

 


2
2
sin cos
fmr fm
S
srs rms rrs
VRV XjSVXV X
IRRSX XSXjSRXRX


  (3)
while the rotor current is given by Equation (4).
Let us define the parameters a, b, c, d, e and g as
The growing integration of wind energy into power
networks will have a significant impact on power syste m
stability. With the development of Wind Turbine (WT)
techniques, the DFI becomes the dominant WT type used
in wind farms [6]. In this situation, DFIG should be
modeled properly in power system stability analysis. A
detailed model of the WT with DFIG and its associated
controllers is presented, based on which the small signal
stability model is derived. Small signal stability analysis
shows that the DFIG control can significantly improve







2
sin ;
cos ;
;
cossin ;
cos sin.
rfm
fm
srs rm
sr rs
fsf s
fs sfm
aVRVX
bSVXrVX
cRRSXXSX
dSRXRX
eVR VX
g
VXRV SXV






 



(5)
2
(cossin )(cossin)
()()
fs fsfssfm
r
srs rms rr s
VR VXjVXRVSXV
IRRSX XSXjSRXRX
 
 
 (4)

A. M. ALKANDARI ET AL. 396
V
R
S
j
x
s
JX
m
j
x
r
R
r
/S
I
S
I r
S
f
V


Figure 1. Steady state equivalent circuit of DFIG referred
to the stator.
ator and rotor currents can be written as:
Then the st
s
ajb
Icjd


r
ej g
Icjd


(6)
The active and reactive power of the stator can be ob-
tained as:
S
(7)
Substituting for
*
SVI PJQ
*
*
Real
Im
SSS
Ss
SS
PV
I
QVI




S
I
then the active and reactive
power of the stator is n by: give


22
S
Vacad
Pcd


22
S
Vb c ad
Qcd


(8)
Note that the stator active power is positive for m
toring operation and is negative for generating operation
an
r
o-
d vice versa for the reactive power.
The expressions for active and reactive power of the
rotor (exciter) are given as:
*
rf
SVI PJQ
*
*
Real
Im
rr
rfr
rfr
PV
I
QVI









 
(
Substituting
9)
r
I
into (9) we obtain
 
 
22
cos
ff
r
Ve gdVcg
P
22
si n
sin cos
ff
r
c ed
cd
Vecgd Vcged
Qcd





 

(10)
Note that, if the excitation voltage


f
Vth equals zero,
the machine performs in accordance with e basic oper-
at tio
of a doubly feed induction machine.
teristics we assume
d to an
uit of the exciter
is connected to the same source as such the output of the
ge is
applied to the machine rotor while the machine speed
, i.e.
varies from 1 to –1. The parameters of the machine
ch .6
p.
his is obviously correct, since the machines
de of 0.6, the machine
es the results ob tained for th e s tat or reactive
ing characteristics similar to a convennal induction
machine. Obviously, no power active or reactive is gen-
erated. Equation (10) describes the steady state operation
3. Steady State Characteristics
In studying the steady state charac
that the stator of the main machine is connecte
infinite bus source, and the primary circ
exciter is a source of a slip frequency and injects power
to the slip rings of the rotor of the main machine. As
such the current in the stator of the main machine has a
constant frequency independent of the shaft speed.
3.1. Effects of Excitation Voltage Magnitude
In this study we assume a constant excitation volta
changes from zero to double the synchronous speed
S
used in this study are given in Tables A.1 and A.2 [7].
Figure 2 gives the results obtained for the stator active
power when the slip is changed from –1 to 1, i.e. speed
anges to double the synchronous speed, while we
changed the excitation voltage magnitude from 0 to 0
u in step of 0.2 p.u. Examining these curves reveals the
following:
 At Vf =0, induction motor operation, the machine
delivers power to the network, via the stator windings,
at speed greater synchronous speed, slip greater than
–0.05. T
runs as an induction generator
 At excitation voltage magnitude of 0.2 and 0.4 p.u the
machine delivers power to the source at this speed
ranges.
 For excitation voltage magnitu
delivers power to the source at sub-synchronous
speed, speed less than synchronous speed.
Figure 3 giv
power versus the speed at a constant excitation voltage
Figure 2. Stator active power versus speed at constant exci-
tation voltage.

A. M. ALKANDARI ET AL.397
Figure 3. Stator reactive power versus the speed at constant
excitation voltage.
magnitude ranging from Vf = 0.0 p.u to Vf = 0.60 p.u.
Examining these curves reveals the following:
 At sub-synchronous the machine absorbs reactive
power from the network, while at super-synchronous
speed the machine delivers reactive power to the net-
work.
 The maximum absorbed or delivered reactive power
to the network occurs at the same speed.
 At the same speed, the reactive power increases as the
excitation voltage increases.
Figure 4 depicts the active power of the rotor at di
o to double the synchronous speed.
Examining this table reveals that:
d, no power is absorbed by the rotor (induc
tor reactive
ouble the synchronous
sp
f-
ferent excitation voltage magnitude when the machine
speed varies from zer
 The rotor is absorbed active power from the source at
all speeds regards the excitation voltage magnitude.
 The maximum power absorbed at different excitation
voltage magnitudes occurs at the same speed.
 When the excitation voltage is zero, the rotor is short
circuite -
tion motor operation).
Figure 5 shows the variation of the ro
power at different excitation voltage magnitudes, when
the speed varies from zero to d
eed. It can be noticed, from this figure, that:
Figure 4. Rotor active power versus the speed at different
excitation voltage.
Figure 5. Rotor reactive power versus speed at different
excitation voltage.
 At sub-synchronous speed the machine absorbs reac-
tive power from the source, while at super-synchro-
nous speed the machine delivers reactive power to the
network, (asynchronous capacitor) at different excita-
tion voltage magnitudes, and the maximum of these
reactive powers occurs at the same speed.
3.2. Effects of the Excitation Voltage Angle
In this section the characteristics of the machine is stud-
ied, where we assume that the machine run at su
angle between the excitation voltage
nd the source voltage, from –90˚ to 90˚ while we kept
-
per-synchronous speeds and we change the excitation
voltage angle, the
a
excitation voltage magnitude constant at Vf = 0.2 p.u.
Figure 6 shows the results obtained for the stator active
power. Examining this curve reveals that:
 At synchronous speed, s = 0.0, the machine behaves
as a synchronous machine, generator, and delivers
power to the network, (,,P

) characteristics. rs
oltage
angle and absorbs power from the network for nega-
o the network
ifferent
on voltage angle is zero.
 At super-synchronous speed the machine delive
power to the network for positive excitation v
tive excitation voltage angle.
 The maximum delivers powers occur at different ex-
citation voltage angles.
Figure 7 gives the same characteristics but the ma-
chines runs above synchronous speed. Examining this
curve reveals the following remarks:
 When the machine runs just speed 20% more than the
synchronous speed, it delivers power t
for the range of negative excitation voltage angle and
little bit positive excitation voltage angle.
 The maximum delivered power occurs at d
excitation voltage angle, negative angle, for every
super-synchronous speed.
 The machine delivers power to the network at this rang
of speeds, although the excitati

A. M. ALKANDARI ET AL. 398
-25
-20
-15
-10
Stator Power (p.u)
-5
0
5
-90 -80 -70 -60 -50 -40 -30 -20 -100102030405060708090
Excitation Angle(Degree)
10
S-0.0
S=0.1
S=0.15
S=0.2
Figure 6. Stator power versus excitation angle at constant
speed.
-10
-8
-6
-4
-2
0
2
4
6
8
-90-80-70-60-50-40-30-20-10010 20 30 40 50 60 70 80 90
Excitation Angle
Stator Power (p.u)
S=-0.10
S=-0.15
S=-0.2
Figure 7. The stator power versus the excitation angle when
the machine above synchronous spee d, Vf = 0.2 p.u.
Figures 8 and 9 give the excitation power for sub-
synchronous and super-synchronous speed operation, re-
spectively, and the excitation voltage magnitude is kept
constant at Vf = 0.2 p.u
Examining these curves reveals the following conclu-
sions:
 At synchronous speed, S = 0.0, the excitation power
is constant regardless the magnitude of the excitation
voltage angle.
 The excitation power having the same trend regard-
less the speed of the machine, sup or super synchro
negative excitation voltage angle and at sub-syn-
give the stator reactive power
va uns at sub-synchronous and super-synchronous
t :
haves as a non-salient pole syn chronous machin e and the
-
nous speed.
 The machine delivers power to the network at almost
chronous and super-synchronous speed, respectively.
 The maximum positive or maximum negative powers
occur at different excitation voltage angle.
Figures 10 and 11
riation with the excitation voltage angle, when the ma-
chine r
speeds, respectively, while the excitation voltage magni-
tude is kept constant at Vf = 0.20 p.u. Examining these
figures reveals tha
1) At synchronous speed, S = 0.0, the machine be-
-1
0
1
-90 -80 -70 -60-50 -40 -30 -20 -100102030405060708090
Excitation Angle(Degr e e )
2
3
4
n Power (p.u)
5
Excitatio
S=0.0
S=0.1
S=0.15
S=0.2
Figure 8. Excitation power versus excitation angle.
-1
-0.5
0
0.5
1
1.5
-90 -80-70-60 -50 -40-30 -20 -100102030405060708090
Excitation Angle (degree)
Exc itation P ower P 2 (p .u )
S=-0.10
S=-0.15
S=-0.20
Figure 9. Excitation power versus excitation angle when th
mac e hine runs above synchronous speed, Vf = 0.2 p.u.
-25
-20
-15
-10
-5
0
5
10
15
20
25
-90 -80 -70 -60 -50 -40 -30 -20 -100102030405060708090
Excitation Angle
QS (p.u)
S=0.0
S=0.1
S=0.15
S=0.2
Figure 10. Stator reactive power versus excitation angle at
Vf = 0.2 p.u.
excitation voltage angle becomes the rotor angle, i.e. the
power angle.
2) The machine absorbs and delivers reactive power to
the network at different excitation voltage angle (Note
that positive reactive power means absorption, inductive
reactive power, while negative reactive power means
delivering, capacitive reactive power).
3) The maximum stator reactive power either positive
or negative occurs at the same excitation voltage angle

A. M. ALKANDARI ET AL.399
0
2
4
6
8
10
12
-90 -80 -70 -60-50 -40 -30 -20 -100102030405060708090
Excitation Angle (degree)
Reactive Power Qs (p.u)
S=-0.10
S=-0.15
S=-0.20
Figure 11. Stator reactive power when the machine run
r every speed in the range.
s the excitation voltage angle when
ex
nous speed it de-
 At super-synchronous speeds the machine delivers
power to the network for positive excitation voltage
angle and absorbs power for negative excitation
voltage angles.
 At sup-synchronous speeds the machine delivers the
maximum power at the same excitation voltage angle
independent of the machine shaft speed.
Figure 13 depicts the variation of the total reactive
power versus the excitation voltage angle, when the ex-
citation voltage magnitude is kept constant at 0.2 p.u
onous speed, the
power to the source, when
s
above synchronous speed, Vf = 0.2 p.u.
fo Figure 12 shows the net active power delivered to the
network versu the
citation voltage magnitude is kept constant, while the
machine runs at super-synchronous speed. And the exci-
tation voltage magnitude is kept constant at 0.2 p.u. Ex-
amining this figure reveals the following remarks:
 When the machine runs at synchro
livers more power to the network than the other
speed.
,
n while the machine runs at sup synchronous speed. It ca
be noticed from this figure that:
When the machine runs at synchrmachine delivers reactive
the excitation voltage angle varies from zero to –90˚,
and absorbs the same magnitude of the reactive
power when the excitation voltage angle varies from
zero to 90˚. i.e. the curve is inversely symmetrical at
excitation voltage angle equals zero.
 At speeds rather than the synchronous speed, th e ma-
chine absorbs and delivers reactive power to the net-
work for different modes of excitation voltage angles.
 The maximum delivery of reactive power occurs at
the same excitation voltage angle independent of the
machine speed, except at synchronous speed.
4. Conclusions
In this paper the steady state analysis of DFIG is devel-
oped, we assume that the machine is excited on the rotor
-20
-15
-10
-5
0
5
-90-80-70-60-50-40-30-20-10010 20 30 4050 60 70 80 90
Excitation Angle (degree)
Active Power (p
10
.u)
S=0. 00
S=0. 10
S=0. 15
S=0. 20
Figure 12. Tatol active power (p.u) versus excitation angle.
-25
-20
-15
0
5
10
Total Reactive
15
20
25
Power (p.u)
S=0.00
S=0.10
S=0.15
S=0.20
-10
-5
-90 -80-70-60-50-40-30-20-10010 20 30 40 50 60 70 80 90
Excitation Angle (Degree)
Figure 13. Tatol reactive power versus excitation angle.
side by a slip-frequency current injected from an exciter
mounted on the same shaft of the machine. The resulting
rotating magnetic field rotates at synchronous speed.
Effects of the excitation voltage magnitude and angle on
both the active and reactive power when the machine
runs at constant speed are investigated. It has been
shown that controlling the excitation voltage magnitude
and phase angle controls the mode of operation of the
machine; motor or generator mode. Furthermore, effect
onstant excitation voltage magnitude and constant phase
angle are also investigated. It can be concluded that the
power angle stability of conventional synchronous ma-
chine has no meaning in this type of generators. Studying
the control of the machine terminal voltage is under
study and the results will appear in a forthcoming paper.
5. References
[1] L. Piegari, R. Rizzo and P. Tricoli, “High Efficiency Win d
Generators with Variable Speed Dual-Excited Synchro-
nous Machines,” International Conference on Clean Elec-
trical Power 2007, Capri, 21-23 May 2007, pp. 795-800.
[2] n
Wind Turbine,” Morgan & Claypool Publisher, San
s
of the shaft speed on the active and reactive power at
c
F. Blaabjerg and Z. Chen, “Power Electronics for Moder
Rafael, 2006.
[3] J. G. Slootweg, H. Polinder and W. L. Kling, “Dynamic
Modeling of a Wind Turbine with Doubly Fed Induction

A. M. ALKANDARI ET AL. 400
Generator,” IEEE Power Engineering Society Summer
Meeting, Vol. 1, 2001, pp. 644-649.
[4] F. Mei and B. Pal, “Modal Analysis of Grid-Connected
Doubly Fed Induction Generators,” IEEE Transactions
on Energy Conversion, Vol. 22, No. 3, 2007, pp. 728-736.
doi:10.1109/TEC.2006.881080
[5] Y. K. He, J. B. Hu and R. D. Zhao, “Modeling and Con-
trol of Wind-Turbine Used DFIG under Network Fault
Conditions,” Proceedings of the 8th International Con-
ference on Electrical Machines and Systems, 2005, Nan-
jing, Vol. 2, 29 September 2005, pp. 986-991
[6] F. Wu, X.-P. Zhang, K. Godfrey and P. Ju, “Modeling
and Control of Wind Turbine with Doubly Fed Induction
Generator,” 2006 IEEE PES Power Systems Conference
er Engineering, Depart-
and Exposition, Atlanta, 29 October-1 November 2006.
[7] A. Petersson, “Analysis, Modeling and Control of Dou-
bly-Fed Induction Generators for Wind Turbines” Ph.D.
Thesis, Division of Electric Pow
ment of Energy and Environment, Chalmers University
of Technology, Goteborg, 2005.
Nomenclature Appendix A.
,
s
r
I
I are the stator and rotor current, respectively.
and
s
r
RR are the stator and rotor resistance, respec-
tively.
&
s
r
X
X
ectively. are the total stator and rotor reactance, re- sp
s
sm
rr m
X
xX
X
xX


m
X
= Magnetizing reactance
&
s
r
x
x
spectivel are Leakage reactance of the stator and rotor y. re
s
is the machine slip =s
s
nn
n
,
s
is positive at
sub-s su-
per-synchron
the grid voltage a constant frequency
ynchronous speed S
nnand negative at
ous speed, S
nn respectively.
V is having
s
f
.
f
V is the excitation voltage magnitude, having con-
ncy stant freque
s
f
.

is the angle of the excitation voltage.
Table A.1. Parameters of the induction machines.
Stator resistance RS 0.0022  0.010 p.u
Stator leakage Reactance xS 0.1200 mH 0.180 p.u
Rotor resistance Rr 0.0018  0.009 p.u
Rotor leakage Reactance xr 0.05 mH 0.070 p.u
Magnetizing reactance Xm 2.9 mH 4.400 p.u
Table A.2. Base values.
Base Voltage (Phase-neutral)Vb 400 V
Base Current Ib 1900 A
Base Frequency ω
ance Zb =
b 314 r ad/s
Base Imped Vb/Ib 0.21 