Reactive Power Control in Doubly-Fed Induction
Generators for Wind Turbines
B. Rabelo*, W. Hofmann*, J.L. Silva**, R.G. Oliveira** and S.R. Silva**
*Dresden University of Technology/Dept. of Electrical Machines and Drives, Dresden, Germany
**Federal University Of Minas Gerais/Electric Engineering Research and Development Center, Belo Horizonte, Brazil
Abstract—High penetration level of wind power generation in
the interconnected electrical network and the increasing rated
power of the wind energy converter units imposes new technical
challenges to engineers. The first one is to guarantee grid stability
despite the stochastic nature of wind while the second one is to
limit the power losses since electrical machines efficiency do not
augment proportionally to the rated power increase in the MWclass. Attending to the new requirements of the power companies
on wind turbines concerning the contribution to network stability
through reactive power exchange with the mains supply, this
work proposes a reactive power control scheme for a doublyfed induction generator drive that minimises the system power
losses. This drive topology enables besides the control of active
power the independent control of reactive power with both
inverters. This degree of freedom is used in order to share the
reactive currents resulting on the reactive power required on the
network and increasing the efficiency. The steady-state power
losses are modeled and the optimal reactive power share working
points are found by an iterative procedure. The control structure
design is presented and discussed as well as implementation
issues. Experimental results are presented in order to prove the
theoretical assumptions.
I. I NTRODUCTION
Power generation and energy savings have become an issue
of major importance in the last decades. The shortening of
fossil energy sources, air pollution and global warming are
pushing the technological development of renewable energy
converters. Countries where this development first took place
like Germany and Denmark are nowadays market leader
manufacturer’s and present a high density on installed wind
turbines. Although less turbines are being erected compared
to earlier years installed power is still increasing due to repowering. Also the world wind energy market is booming. The
22,000 MW installed capacity in Germany until end of 2007
corresponds to 7% of the country total energy consumption.
In the northern and coast states, due to the favourable flat
topography and wind regime, the amount of installed power
in wind turbines performs more than 30% of the local energy
consumption [1]. Furthermore, today’s average wind generating unity power rates 2 MW and tends to increase, specially
for the off-shore application, where the benchmark is 5 MW.
The wind generation technology development, namely the
high penetration of wind power on the electrical network system and the increasing of the generating units rated power are
closely related to each other and bring new challenges to the
engineers. The considerable amount of wind energy generated
and delivered to the interconnected electrical network depends
978-1-4244-1668-4/08/$25.00 ©2008 IEEE
instantaneously on the site weather conditions. Although experience tells that wide-spread wind generation results in more
stable average generated powers [2], the dependency on the
actual wind conditions is still a risk factor for the balance
between energy offer and demand as a basic condition for
reliable operation of the electrical power system. On the other
hand the increased machine powers is accompanied by a nearly
proportional increase on the losses. Electrical machines in
the MW-class present very high efficiency values reaching
saturation with the state-of-the-art materials and construction
technology. Therefore only slight augmentation on efficiency
are to be expected with the progressing generator rated power.
In order to avoid network stability problems power companies launched guidelines for net connection of wind turbines
[3]. These grid codes established similar requirements on usual
generating plants to wind power converters, i.e. supporting
the network stability in normal operation and during faulty
conditions. During normal operation this is translated into the
capability of frequency regulation through active power control
and of voltage regulation through reactive power control. The
work presented in this paper is concentrated on this latter
theme and proposes a control strategy for the reactive power
flow in a doubly-fed induction generator (DFIG) drive aiming
the reduction of the electrical power losses.
The first attempt to reduce losses in a slip-ring induction
machine by sharing reactive power between generator stator
and rotor was found by the authors in [4]. Instead of a power
electronics converter a synchronous generator was connected
to the rotor terminals and a DC-machine was used as a prime
mover. Voltage and frequency were controlled by varying the
synchronous machine excitation and the DC-machine speed,
respectively. Later on, an expression for the optimal stator
reactive current that minimises the generator copper losses
was derived based on the induction machine model in [5].
In [6] this expression was extended to the rotor reactive
current and the converter losses were also taken into account
in order to find the system overall losses and a more general
optimal solution. A variant of the optimisation scheme using
the reactive currents instead of the powers was proposed in
[7] where experimental results were presented.
In this work the control structure for a DFIG drive with
reactive power production is described. The controllers design
is carried out and implementation issues are discussed. Experimental results in a 4 kW test bench are presented.
106
II. S YSTEM T OPOLOGY AND C ONTROL S TRUCTURE
The wound-rotor induction machine drive topologies are
well-known since many years as the sub-synchronous and
over-synchronous cascades. The reduced power electronics
converter rating to handle only part of the machine rated power
is one of its main advantages [8]. In modern DFIG drives the
variable voltage and frequency rotor circuit is connected to
the constant voltage and frequency grid by a back-to-back
converter composed of two voltage-source inverters linked
through a DC capacitor. The stator circuit is directly connected
to the mains supply. The rotor-side inverter (RSI) regulates the
slip power controlling the machine speed and torque while
the mains-side inverter (MSI) regulates the active power flow
between rotor and mains supply maintaining a constant DClink voltage. Output filters are used on the MSI output in order
to suppress inverter harmonics on the network. Additional
inductors are employed as filters between the RSI and the rotor
terminals because of the relative low leakage inductance. The
figure 1 shows the schematic diagram of a DFIG drive with
the power and current flow.
currents permitting the production of reactive power. The
choice of the system orientation to the net voltage allows the
regulation of reactive power independently from the active
power control. Furthermore, the voltage DC-link connecting
both rotor-side and mains-side inverters delivers in some extent
another degree of freedom, namely the production of reactive
power separately with both inverters.
The DFIG control is performed in a reference frame rotating
synchronously with the mains voltage vector uN = uN d +
juN q . The voltage phase displacement ϑ̂N is determined using
the phase-locked-loop (PLL) scheme described in [10]. If there
is a voltage deviation the PLL controller increases or diminishes the angular frequency ΔωN = ωN − ω̂N accelerating or
breaking the rotating coordinate system in order to track the
mains voltage vector until the angle deviation ΔϑN = ϑN −ϑ̂N
is reduced to a minimum. This implication is illustrated in
figure 2.
Wind Turbine
Pm
QS
3
uS
DFIG
PS
iS
iR
LR f
3
uR
PN
RSI
Mains
Supply
LN
Fig. 2.
The synchronous rotating coordinate system is oriented to
the mains voltage when uN = uN d and uN q = 0. According
to the voltage orientation the d-axis current is considered
the active and the q-axis the negative reactive component,
respectively. In this way the d-axis currents from mains and
rotor sides are responsible for respective the DC-link voltage
and speed/torque controls. The q-axis currents are available
in order to control the reactive power production on both
inverters.
On the rotor side the system is oriented to the slip angle
The mechanical rotor position ϑ required for the slip angle
ϑR = ϑN − ϑ computation is given by an encoder.
iN
DC-Link
uN
PDC
CDC
MSI
S
QN
Qn
3
Lf
Pn
in
uC
un
Voltage orientation
Cf
LC-Filter
III. S YSTEM DYNAMICAL E QUATIONS
Fig. 1.
DFIG drive topology
A. Generator Model
In generator mode active power flows from the network to
the rotor in sub-synchronous operation and from the rotor to
the network in super-synchronous operation. The converters
size is determined by the desired speed range. Usually a
±30% slip enables a suitable operating region for wind turbine
applications. At the same time, the use of bi-directional
switches, IGBTs and anti-parallel free-wheel diodes enable
the phase displacement between inverter output voltages and
The control structure developed is based on the induction
machine dynamic model whose voltage equations on the
synchronously rotating frame are
107
dΨ S
+ jωS Ψ S
dt
dΨ R
uR = RR
iR +
+ jωR Ψ R
dt
uS = RS iS +
(1)
(2)
where the flux linkages are given by the expressions
Ψ S = LS iS + Lm iR
(3)
Ψ R = LR iR + Lm iS ·
RS
LS iS
iR'
'
LR
Fig. 3.
uN q = 0 = Rf inq + Lf
RR'
dinq
+ ωN Lf ind + unq
dt
(10)
(11)
The DC-link can be modeled with simplicity neglecting the
parasitic elements. It consists of a T-circuit with the current
flowing into the capacitor given by the difference between the
input and output DC-currents
uR'
S
S
R
jSS
jRR
iDC (t) = iDCn (t) − iDCR (t)·
The DC-link voltage is then given by
1
uDC (t) =
iDC (t)dt + UDC0 ,
CDC
DFIG equivalent circuit in dq reference frame
The developed electromagnetic torque is given by
Te =
dind
− ωN Lf inq + und
dt
C. DC-Link Model
Lm
S
uN d = Rf ind + Lf
(4)
i
uS
is approximately the net voltage uC ∼
= uN the MSI output
current dynamics is described by the following equations
Lm
3
{Ψ S i∗
}·
PP
R
2
LS
(5)
Considering all the values referred to the stator side and letting the subscript for rotor values fall the current components
can be written as
Lm
i
(6)
iSd = −
LS Rd
Ψ
Lm
iSq = Sq −
i ·
(7)
LS
LS Rq
Substituting the stator current on the rotor flux linkage equation and developing yields
diRd
Lm
+ ωR σLR iRq + ωR
Ψ
(8)
dt
LS Sq
di
uRq = RR iRq + σLR Rq − ωR σLR iRd (9)
dt
These equations are used to design the rotor current controllers
as well as the feed-forward compensation of the cross-coupling
terms and of the stator flux.
uRd = RR iRd + σLR
(12)
(13)
where UDC0 is the DC-link initial voltage.
IV. C ONTROLLERS D ESIGN
The control strategy of the DFIG depicted in figure 5 is
based on a cascade structure with rapid inner current controllers similarly to the scheme presented in [11]. The current
controllers outputs are the reference voltages for the spacevector modulation (SVM) unities that generate the IGBTs
gate pulses for the inverters. The current reference values
are generated from outer voltage and speed/torque regulators.
The choice of voltage orientation presents some important
advantages like better stability range than flux oriented systems
[12] and direct decomposition of the currents in active and
reactive components.
A. Current Control
The current control design is based on equations 10 and
11. It is a well-known 2-dimensional cross-coupled control
problem. Applying the Laplace transform on both equations
and developing yields
ind (s) = [−und (s) + (uN d (s) + ωN Lf inq (s))] Gin (s); (14)
B. LC-Filter Model
Lf
Rf
in iN
LN
inq (s) = [−unq (s) − (ωN Lf ind (s))] Gin (s),
RN
where the terms inbetween brackets are the couplings and the
transfer function Gin is given by
iC
un
uC
uN
Gin (s) =
Cf
S
S
jNLN iN
Fig. 4.
1
K in
,
=
sLf + Rf
sTf + 1
LC-filter equivalent circuit
Consider voltage orientation and the inverter synchronised
with the mains supply voltage. If the voltage drop over the
mains impedance LN is neglected and the capacitor voltage
(16)
where Kin = R1f and Tf = Rff . It can be seen that the
LC-filter inductance and its resistance determine the system
dynamics. The transfer functions on both dq-channels are
identical as well as the controllers design. The inverter deadtime, signal conditioning, A/D-conversion and processing time
delays can be added up to a time constant TΣ of a few sample
times and modeled as a first order transfer function
1
Ginv (s) =
·
(17)
sTΣ + 1
L
jNLf in
(15)
108
Sq
iR Control
te Control
te *
te
-1
-
R
iRd*
Lm
L Sind
uRd*
-
iRd
RLR
ind
iRq
qS Control
qS*
qS
iRq*
iSq -
RLR
ind
im
in Control
qn Control
qn*
qn
uRq*
-
-
inq*
-1
-
inq
-
-
unq*
-
Fig. 6.
N Lif
Step response for the current control loop
nd
ind
uDC Control
uDC*
uDC
ind*
N Lif
nd
-
-
On the rotor side the current control design is carried out in a
similar way using equations 8 and 9. The controller parameters
are computed using the technical optimum method
und*
-
uNd
Fig. 5.
DFIG control structure
With the compensation of the coupling terms a simple linear
control problem arises. The technical optimum design method
can be employed enabling a relative fast control response
avoiding over-shoots as it is required to the inner current
control loop. The PI-controller parameters, the proportional
gain and the integral time, are found to be
KPin =
Lf
;
2TΣ
σLR
;
TIiR = TR ·
(21)
2TΣ
The response of the q-axis rotor current to a reference step
from −1 A to 1 A as well as the couplings with the d-axis
current is shown in figure 7.
KPiR =
0
ird (A)
ird*
Ŧ1
TIin = Tf ·
(18)
Ŧ1.5
Ŧ0.01
For the implementation the controllers were discretised
using the trapezoidal or Tustin rule given by the following
bilinear relation that maps the imaginary axis onto the unit
circle in the z-domain.
2 1 − z −1
s=
·
T 1 + z −1
b0 + b1 z −1
Y (z)
,
=
E(z)
a0 + a1 z −1
0.01
0.02
0.03
0.04
2
rq
(19)
1
0
irq
Ŧ1
irq*
Ŧ2
Ŧ0.01
0
0.01
Fig. 7.
(20)
where Y (z) = Z{y(t)} and E(z) = Z{e(t)}.
The step response for the closed loop current control on
the MSI-side for the continuous design Gc (s) and for the
digital design Gc (z) are presented in figure 6 together with
the measured values of the step response on the current inq .
The good correspondence between the theoretical and practical
values was achieved after modeling the system dead-time of
3.5 sample times and some tuning on the plant parameters that
were found to be slightly higher than the initial values.
0
i (A)
The continuous PI-controllers structure can be written as the
rational discrete form
GR (z) =
ird
Ŧ0.5
t (s)
0.02
0.03
0.04
Step response of iRq
B. Active Power Control
The active power is controlled indirectly through the electromagnetic torque on the rotor side and through the DC-link
voltage control on the mains side. A detailed description of
the design will be set aside in this paper and explained in
another publication. In this moment it is enough to mention
that the torque controller is tuned using the technical optimum
method in a similar way as it will be presented for the reactive
power controllers. The voltage controller is designed in order
109
to present good disturbance rejection characteristics using the
symmetrical optimum criteria [8], [9].
Although only active power can flow through the DC-link,
deviations from the constant value affect indirectly the reactive
power through deviations on the inverters output voltages.
Therefore, good dynamics of the DC-link voltage control turns
to an important issue that will be explored in detail in a future
work.
The figure 8 shows the response to a reference step as
well as the speed and rotor currents variation across the
synchronous speed.
T (Nm)
4
2
T
e
e
0
−2
Te*
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
C. Reactive Power Control
The MSI controls the reactive power to be controlled at the
NCP. Under voltage orientation it is translated by the equation
3
3
3
{uN i∗n } = (uN q ind − uN d inq ) = − uN d inq · (22)
2
2
2
Considering the mains voltage constant and the inner closed
loop current control, one has a linear control problem.
qn =
GP qn (s) =
5
Control
(rad/s)
160
m
ω
140
qn
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Plant
n
n
-1
-
.
GF q
n
-
3
2
qn
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Step response of Te
The approach to the reactive power controller design is
based on the common requirements on generation units, i.e.,
it should regulate the power avoiding over-shoots and oscillations. The controller parameters should be computed in such a
way that the control loop is well-damped. Therefore, technical
optimum criteria can be also applied here. The controller
parameters are determined considering the filter time constant
Tfqn as
(A)
KPqn =
165
160
U
155
150
0
(A)
5
DC
UDC*
0.05
0.1
0.15
0.2
0.25
0.3
ind
TIqn = Tfqn ·
(24)
3
3
{uS i∗S } = − uSd (iμ − iRq ),
2
2
(25)
Ψm
where the magnetising current is given by iμ = L
and can
m
be fed-forward compensated. In this way the linear control
problem can be solved using the technical optimum where the
controller parameters are
0
i
−5
0
Tfqn
;
2Kqn TΣ
The step response of the mains side reactive power control
can be observed in the figure 11. The power peaks with the
multiple of the fundamental frequency can be seen while the
averaged power value follows the designed dynamics.
Similar approach is carried out to the rotor side, i.e. to the
stator reactive power. Under the voltage orientation the stator
reactive power can be computed as follows
qS =
n
abc
Mains side reactive power control plant
−2
i
Rabc
Fig. 10.
0
A step response on the DC-link voltage as well as the MSI
output currents and the d-axis active current component are
shown in figure 9
(V)
inq
5
2
Fig. 8.
DC
inq*
4
−4
U
uNd
Gci
GR q
qn*
150
0
(23)
The computation of the reactive power involves the multiplication of measured voltages and currents. Due to drift and noise
these values can vary deteriorating the computed power and
influencing the control. Hence, the computed value is filtered
before being passed to the control. The block diagram of the
control system is depicted in figure 10.
170
130
qn
3
= Kqn = − ÛN d ·
inq
2
0.05
0.1
0.15
0.2
0.25
0.3
t (s)
K Pq =
S
Fig. 9.
Step response of UDC
TfqS
;
2KqS TΣ
TIqS = TfqS ·
(26)
TfqS is the filter time constant and the plant gain is given by
KqS = − 32 ÛSd .
110
The dynamic couplings with the active power are almost
imperceivable due to the well-damped reactive power control.
The decrease on the total active power caused by the increase
on the active power input at the MSI-side can be observed.
1.4
1.2
n
q /q *
1
V. R EACTIVE P OWER S HARE
The regulation of the voltage and/or power factor makes
more sense in the net connecting point (NCP) of a wind park
or in isolated systems where the single machine can be a
representative amount of the short-circuit power. Depending
on the system configuration and operating conditions these
higher level regulators deliver the reference reactive power that
is passed to the single machines. Locally the reactive power
can be produced by both RSI and MSI making possible the
reactive power share. This association can be seen in figure
13.
n
0.8
0.6
0.4
q simulated
n
0.2
q measured
n
0
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
t (sec)
Fig. 11.
Step response of qn
Wind Park
Active and Reactive Power
Management
In order to evaluate the dynamic response of both reactive
controllers in operation reference steps of qS∗ and qn∗ were
applied simultaneously. The recorded results of the active and
reactive powers on both sides as well as for the total power
on the NCP can be observed in figure 12.
Net Frequency Regulation
m
f N p N k
k 1
pN1
.
.
.
pNm
pNx
MPPT
me*
Local Reactive Power Control
q N Dividing
0
Net Voltage Regulation
PN Ŧ0.2
PS0
QN
QS0
Ŧ0.4
0
0.5
1
1.5
QS
PS Ŧ0.2
0.6
u N 0.4
0.2
2
0
PS0
0.8
k 1
0
0.5
1
1.5
2
Fig. 13.
0.8
Ŧ0.4
qC
iSq*
-
iRq*
-
i
qn Control
-
qn *
-
inq*
Frequency (active power) and voltage (reactive power) regulation
0.6
0.5
1
1.5
2
0.04
0
Qn
0.02
QS0
S0
0
qS
qNx
qn
0.5
1
1.5
The reactive power splitting factor
q
α= S
qN
2
0.05
P
Nk
1
QS0
0
Pn
m
q
qN1
.
.
.
qNm
qS Control
qS *
0
0.5
1
t (s)
Fig. 12.
1.5
2
0
Ŧ0.05
0
0.5
1
1.5
2
t (s)
∗ steps
Active and reactive power responses for qS∗ and qn
The power values are normalised by the generator rated
active power PS0 = 4000 W and rated reactive power QS0 =
3200 VAr. At the stator side the reference was varied from
2000 VAr to 3000 VAr at t = 0s and back at t = 1s. At the
same time instants the MSI reference value was stepped from
−100 VAr to 100 VAr and back. The steps are carried out
with the generator delivering pS = 1000 W.
A very important issue is to tune the reactive power controllers in order to have the same or very close dynamics. In
this way the total reactive power response will also present
a similar dynamic characteristic. Normally the generator-side
influence is bigger than the MSI-side but depending on the
capacitor contribution qC this latter can be also significant.
(27)
was defined in previous works [6] and can be found depending
on the operating conditions in order to reduce power losses.
In order to demonstrate the influence on the active power the
reference value α∗ is varied from 1.17 to 0.97 during 2 seconds
for a fixed operating point and compared to the measured one
given by 27, as shown in figure 14.
The implications of the reactive power share variation on
the power flow are depicted in figure 15. The left column
presents filtered instantaneous active powers normalised by
PS0 while the right column presents the respective filtered
reactive powers normalised by QS0 . The reactive powers
on the stator qS and at the MSI qn , shown in the down
right diagrams, follow their respective references given by the
relations 27 and
qn = qN (1 − α) − qC ·
(28)
In this way the total reactive power delivered at the NCP
qN remains constant as can be observed in the upper right
diagram. On the left column are the total generated active
111
couplings with the active power. Also the increasing on the
generated power was presented.
Future works should present the computation of the optimal
sharing factor and its range for the whole operating range. The
couplings between active and reactive power in the currents
and due to DC-link voltage variations will be further analysed
in order to compensation solutions to be found. Another power
control approaches should be studied like direct power control.
Finally, the impact of the reactive power control on lowvoltage ride through problem should be addressed.
1.2
q
α = qS
N
α
α∗
1.15
1.1
1.05
1
A PPENDIX A
PARAMETERS
0.95
0
Fig. 14.
1
2
3
4
t (s)
5
6
Rated Power
Stator Voltage
Stator Current
Power Factor
Mechanical Speed
Rotor Voltage
Rotor Current
tator Resistance
Rotor Resistance
Magnetising Inductance
Stator Inductance
Rotor Inductance
Moment of Inertia
Friction Coefficient
Filter Inductance
Filter Resistance
Filter Capacitance
Variation of reference and measured values of α
power pN at the NCP, the stator active power pS and the MSI
active power pn , respectively. It can be seen that the total
generated power as well as the stator power increase during
the reactive power share variation. The consumed active power
on the MSI reduces and passes through a minimum.
p
N
P
S0
−0.24
1
−0.26
q
N
Q 0.9
pS
PS0
0
2
4
6
2
4
6
R EFERENCES
−0.3
S0
0
2
4
6
0.03
0.8
0
2
4
6
4
6
0.2
pn
S0
0
q
S
Q 1
−0.28
−0.32
P
0.8
−0.26
−0.34
0.1
qn
QS0
0.02
0.01
4000 W
380 V
8.6 A
0.84
1440 rpm
160 V
15.5 A
1.5 Ω
0.9 Ω
139 mH
148 mH
141 mH
0.045kgm2
0.00727 N ms
8 mH
0.5 Ω
69 mF
S0
−0.28
−0.3
Pmec
US
IS
cos ϕS
nm
UR0
IR0
RS
RR
Lm
LS
LR
Jg
Bg
Lf
Rf
Cf
0
2
4
6
0
−0.1
0
t (s)
Fig. 15.
2
t(s)
Active and reactive powers during the variation of α
These results and other not shown here pointed out that there
exists optimal reactive power distribution values for different
operating conditions in order to reduce the power losses on
the whole system and hence increasing the efficiency.
VI. C ONCLUSION
A control structure and design for reactive power regulators
in DFIG drives was presented. The proposed scheme uses
well-known linear control techniques and can be implemented
easily in existing plants. The reactive power share between the
inverters enable the distribution of the reactive currents and
hence power losses reduction. Experimental results showed
the dynamics of the regulation of reactive power flow and its
[1] C. Ender, Wind Energy Use in Germany - Status 31.12.2007,
Dewi
Magazin NO. 32, pp.32-46, February, 2008
[2] S. Heier, Wind Energy Conversion Systems, 2nd ed. New York: John
Wiley & Sons, 1998.
[3] Grid Code High and Extra High Voltage, 2nd ed. E.On Netz GmbH,
2006.
[4] M.S. Vicatos and J.A. Tegopoulos. Steady state analysis of a doubly-fed
induction generator under synchronous operation. IEEE Transactions on
Energy Conversion, 1989. Vol.4, No.3,pp.495-501.
[5] Y. Tang and L. Xu, A Flexible Active and Reactive Power Control Strategy
for a Variable Speed Constant Frequency Generating System, IEEE Trans.
on Industry Applications, Vol. 42, No. 6, November/Dezember, 1993.
[6] B. Rabelo and W. Hofmann, Optimised Power Flow on Doubly -Fed
Induction Generators for Wind Power Plant, Proceedings of PEMC2000,
pp.275-282, Kosice, Slovakia, 2000.
[7] B. Rabelo and W. Hofmann, Loss reduction methods for doubly-fed
induction generator drives for wind turbines, Proceedings Power of
SPEEDAM, pp.1217 - 1222 , Taormina, Italy, 2006.
[8] W. Leonhard, Control of Electrical Drives, 3rd ed. Berlin, Germany:
Springer-Verlag, 2000.
[9] B. Rabelo and W. Hofmann, Optimal Active and Reactive Power Control
with Doubly-Fed Induction Generators in the MW-Class Wind-turbines,
Proceedings of PEDS, pp.53-58, Bali, Indonesia, 2001.
[10] V. Kraura and V. Blasko, Operation of a Phase Locked Loop System
under Distorted Utility Conditions, IEEE Trans. on Industry Applications
Vol. 33, no.1,pp58-63 1997.
[11] R. Pena and J.C. Clare and G. Asher, Doubly-Fed Induction Generator
using back-to-back PWM Converters and its application to VariableSpeed Wind Energy Generation, IEE Proced. Electr. Power Appl., No.3,
pp. 231-240, 1996.
[12] M. Heller, Die doppeltgespeiste Drehstrommaschine für drehzahlvariable Pumpspeicherwerke, PhD. Dissertation, Braunschweig, Germany,
1997.
112