Download Doubly-fed induction generator (DFIG) as a hybrid of asynchronous

Electric Power Systems Research 76 (2005) 33–37
Doubly-fed induction generator (DFIG) as a
hybrid of asynchronous and synchronous machines
Lianwei Jiao a , Boon-Teck Ooi b,∗ , Géza Joós b , Fengquan Zhou b
b
a Department of Electrical Engineering, Tsinghua University, Beijing, PR China
Department of Electrical & Computer Engineering, McGill University, Montreal, P.Q., Canada H3A 2A7
Received 1 November 2004; received in revised form 12 February 2005; accepted 1 April 2005
Available online 11 July 2005
Abstract
With increasing concern over greenhouse gas (GHG) emission, more and more wind turbines are connected to electric power systems.
Since many of these new wind farms employ doubly-fed induction generators (DFIG), power engineers find it is necessary to understand the
characteristics of the DFIG better. This paper presents the DFIG as a synchronous/asynchronous hybrid. Like the synchronous generator, the
real power characteristics are shown to depend on the rotor voltage magnitude |VR | and angle δ. Like the induction machine, the real power
characteristics also depend on the slip s.
© 2005 Elsevier B.V. All rights reserved.
1. Introduction
As it was known very early that varying the rotor resistance
allowed a range of speed control in the induction motor, external resistances were introduced through the rotor slip rings
of wound rotor induction machines [1–5]. The next advance
in rotor-side control was slip power recovery. This uses a
diode bridge to rectify the slip power to dc power which is
then inverted to the ac supply at ωS , the angular frequency of
the 50 or 60 Hz standard. Before long, the diode bridge was
replaced by the thyristor rectifier which has phase-control.
Fig. 1 shows the modern version of the rotor controls based
on two IGBT Voltage-source converters (VSCs) connected
back-to-back. As the two pulse-width modulated (PWM)
IGBT VSCs offer bi-directional, independent real power P
and reactive power Q control, there is active interest in developing the system of Fig. 1 for highly sophisticated control
such as vector control [6–8]. One of the advantages of controlling at the slip-ring terminals comes from the lower kV A
ratings of the VSCs. The reduction is by a factor of smax , the
maximum slip used in the application.
∗
Corresponding author. Tel.: +1 514 398 7133; fax: +1 514 398 4470.
E-mail address: [email protected] (B.-T. Ooi).
0378-7796/$ – see front matter © 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.epsr.2005.04.007
The recent interest of research in wound rotor induction
machines comes from the fact that the aforesaid advantages
make them ideally suited as variable speed wind-turbine generators. The rotor speed does not have to be constant and can
vary with the wind velocity even though the stator is generating power at 50 or 60 Hz. Furthermore, the reactive power Q
can be controlled to provide voltage support, a very important factor as many wind farms are remotely situated and the
short circuit ratio is quite low at the point of connection to
the grid.
Much of recent research on doubly-fed induction generator (DFIG) [9,10] is towards sensorless vector control.
The authors themselves have in [11] realized the conceptual
design of a decoupled P–Q control which acquires maximum
power from the wind. In designing sensorless vector control,
it has not been necessary to know how the DFIG operates in
principle. This is because one can proceed by applying the
standard dynamic equations of the induction machine in the
γ–δ frame. As such, there is a gap in understanding the real
power (or torque) characteristics of the DFIG in terms of the
familiar torque versus speed curves of the induction motor.
In filling the gaps in knowledge, this paper presents the
DFIG as a hybrid of a synchronous machine and an induction
machine. Section 2 presents PS the real power of the DFIG
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L. Jiao et al. / Electric Power Systems Research 76 (2005) 33–37
1} accounts for the electrical power which is converted to
mechanical power Te ωm . Thus, the electromechanical torque
is calculated as
(|Ir |2 Rr + Re(VR Ir∗ )) 1s − 1
(1)
Te
ωm
Fig. 1. Wind-turbine driven doubly-fed induction generator (DFIG).
at the stator terminal (which is equivalent to the torque Te
since Te = PS /ωS ) as a two-dimensional function of the rotor
voltage angle δ and rotor speed ωm . The classical equivalent
circuit of the induction machine forms the framework of the
analysis and performance prediction. Section 3 decomposes
the real power output into four components by using the principle of superposition. It should be noted that Ref. [12] has
also applied the principle of superposition but this paper has
gone a step further by showing that the decomposition yields
a synchronous/asynchronous hybrid.
2. Doubly-fed induction machine—real power
characteristics
Fig. 2 shows the well-known equivalent circuit of the
induction machine. The conventional motor direction of stator current IS and rotor current Ir is adopted. On the stator
side, RS and jωs LSl are the resistance and leakage reactance.
On the rotor side, Rr and jωs Lrl are the resistance and leakage reactance of the rotor winding. The mutual reactance
is jωs M. When the rotor rotates at angular velocity of ωm
electrical radian/s, the rotor resistance Rr is modified as Rr /s
where s = (1 − ωm /ωS ) is the slip. The rotor-side VSC of Fig. 1
injects balanced three-phase voltages (vRa , vRb , vRc ) at slip
frequency ωr = sωs , voltage magnitude VR and voltage angle
δ. Because Fig. 2 is based on the stator-side frequency ωs , the
VSC voltage phasor VR = VR ∠δ representing (vRa , vRb , vRc )
has also to be divided by the slip s resulting in the equivalent
rotor voltage VR /s∠δ.
The real power on the rotor side are |Ir |2 Rr /s and
Re{(VR /s)Ir∗ }. Of this sum, |Ir |2 Rr is the ohmic loss in the
rotor and Re(VR Ir∗ ) is the real power supplied to the slip ring
terminals. The remainder {|Ir |2 Rr /s + Re{(VR )Ir∗ }/{(1/s) −
Fig. 2. Equivalent circuit of induction machine.
As it is well known in induction machine theory, the airgap power is Pairgap = Te ωs . Neglecting stator and rotor ohmic
losses, which are assumed to be small, the following approximate formulae are used in this paper:
stator-side real power PS ≈ Pairgap = Te ωs
(2)
rotor-side real power PR ≈ PS − Te ωm ≈ sPS
(3)
Asynchronous operation as a motor or as a generator
at rotor speed, ωm , is possible in the DFIG because the
power electronic converter across the slip rings supplies
the balanced wound rotor windings with slip frequency
ωr = (ωs − ωm ) currents which produce a rotor magnetic flux
ΦR which rotates at (ωm + ωs − ωm ) in synchronism with the
speed ωs of the stator flux ΦS . The voltage-source converter
has the degrees of freedom to control the voltage magnitude
|VR |, its frequency ωr and its phase angle δ.
The strategy of magnitude control adopted in this paper
is VR = sVRS + VRsy . This is because looking at the rotor in
its original rotor frequency ωr = sωs , the rotor impedance is
Rr + jsωs Lrl . As jsωs Lrl dominates over Rr , the voltage component sVRS is chosen to ensure that the rotor current magnitude |Ir | remains substantially constant over the operating
range. At synchronous speed ωm = ωs , the voltage component
sVRS is zero because s = 0. Therefore, the voltage component
VRsy needs to be added to ensure that the DFIG has threephase rotor currents of the form [IR cos δ, IR cos(δ − 120◦ ),
IR cos(δ − 240◦ )]. The angle δ is controlled by the voltagesource converter. (In contrast, the angle δ is attached to the
field winding in the synchronous machine and δ can only be
changed by shifting the rotor shaft position with respect to
the stator flux ΦS ).
The values of VS , VRS , VRsy and the parameters of the
DFIG used in the graphical examples are listed in Appendix
A. They have taken from a 1.5 MW DFIG [13].
2.1. DFIG as synchronous machine
Fig. 3(a) plots the family of PS versus δ curves of the DFIG
of Appendix A for positive slips s = 0.3, 0.2, and 0.1. The
curves of Fig. 3(b) are for negative slips s = −0.1, −0.2 and
−0.3. PS is computed by applying (1) and (2). As the motor
convention has been chosen, positive and negative values are
for motoring and generating powers, respectively.
Because they have the familiar sin(δ) dependency, the
curves of Fig. 3 resemble those of the synchronous machine,
except for the “dc offsets” with respect to the x-axis. In order
to grasp the synchronous machine characteristics, Fig. 2 can
be simplified to Fig. 4 which considers the large mutual
L. Jiao et al. / Electric Power Systems Research 76 (2005) 33–37
Fig. 3. PS vs. δ curves of a DFIG.
Fig. 5. PS vs. ωm .
reactance jωs M to be virtually an open circuit. In the approximation in Fig. 4, the impedance Z(s) = R + jX where
Z(s) = RS +
Rr
+ j (ωs LSl + ωs Lrl )
s
(4)
Considering the single loop with impedance Z(s) = R + jX
between voltages VS ∠0 and VR /s∠δ, it is a simple matter to
compute IS . The stator power is based on the formula PS =
Real(VS IS∗ ) which becomes:
X VSR sin δ − R Vs − VSR cos δ
Ps = −
(5)
Vs
R2 + X 2
When R = 0 in (5), one has the familiar synchronous generator
power equation Ps = −(Vs (VR /s)sin δ)/X. When the slip s is
small, R becomes large and the cos δ term in (5) accounts for
phase shifts in Fig. 3. The term RVs2 /(R2 + X2 ) accounts for
the “dc offsets”. The positive and negative values of the slip s
in (5) explain the differences between the families in Fig. 3(a
and b).
2.2. DFIG as asynchronous machine
Fig. 5 show the PS versus ωm curves for δ = 0◦ , −30◦ ,
−60◦ and −90◦ in (a) and δ = 30◦ , 60◦ and 90◦ in (b). One
sees certain resemblances of the well-known torque versus
speed curves of induction machines. From Figs. 3 and 5, one
can construct a 2D graph of PS plotted against δ and ωm axes.
Fig. 4. Approximation of Fig. 2.
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3. Decomposition by superposition principle
For a more methodical derivation of the synchronous/
asynchronous characteristics of the DFIG, one applies the
superposition principle and views the torque production as
the summation of the effects of the stator excitation, the rotor
excitation and the inter-action of the stator/rotor excitations,
one at a time:
3.1. Stator-side excitation
The stator excitation is VS ∠0 and the rotor excitation is
VR = 0. The rotor current is Ir = Vs /Z(s).
3.2. Rotor-side excitation
The stator excitation is VS = 0. The rotor excitation is
VR = (VR /s)∠δ. The rotor current is Ir = −(VR /s∠δ)/Z(s).
3.3. Simultaneous stator and rotor excitations
When the stator excitation is Vs = Vs ∠0 and the rotor excitation is VR = (VR /s)∠δ, the rotor current is Ir = Ir + Ir .
Of the rotor-side voltage (Ir Rr + VR )(1/s), one deducts two
voltage terms: Ir Rr , the voltage drop across the rotor resistance, and VR = (VR /s)∠δ, the voltage output of the VSC of
the rotor-side voltage. The voltage, which is left after the
deduction, is associated with electromechanical energy conversion:
1
Em = (Ir Rr + VR )
−1
(6)
s
ωm
Em = (Ir Rr + VR )
(7)
sωs
3.3.1. DFIG torque
Multiplying both sides by Ir∗ , where * is the complex conjugate operator, the electrical power is converted to mechanical power through the equation:
ωm
∗
∗
Te ωm = Re(Em Ir ) = Re
(8)
(Ir Rr + VR )Ir
sωs
L. Jiao et al. / Electric Power Systems Research 76 (2005) 33–37
36
where Te is the electromechanical torque. Substituting Ir =
Ir + Ir and dividing both sides by ωm
Te =
=
Re[(Ir + Ir )Rr + VR ](I ∗r + I ∗r )
sωs
VR jδ
V
Vs − s e
Vs − sR e−jδ
jδ
Rr + V R e
Re
Z(s)
Z(s)∗
sωs
(9)
The electromechanical torque in (9) can be factored into
four terms:
Te = TS + TR + Tsr cos cos δ + Tsr sin sin δ
where
TS =
VS2
Rr
s
(10)
ωs |Z(s)|2
2
VR
RS
TR = −
s
ωs |Z(s)|2
V Rr − R
S
S
VR
s
Tsr cos = −
s
ωs |Z(s)|2
VR VS (Xrl + Xsl )
Tsr sin = −
s
ωs |Z(s)|2
(11)
(12)
(13)
(14)
3.3.2. Hybrid characteristics
The dependence on the rotor speed ωm (or the slip s) in TS ,
TR , Tsr cos and Tsr sin comes from the terms VR = sVRS + VRsy
and Rr /s. Based on (11)–(14), the torque components TS , TR ,
Tsr sin and Tsr cos are displayed graphically in Fig. 6 as functions of rotor speed ωm in per unit.
3.3.2.1. Induction machine—stator-side excitation. TS in
Fig. 6 is the familiar torque speed curve of the induction
machine with the rotor terminals short circuited. The formula
in (11) is that of the induction machine. For s > 0, which is
the motoring regime, TS > 0. The motoring torque TS lifts the
PS versus δ curves in Fig. 3(a). Conversely, for s < 0 which
is the generating regime, TS < 0. In Fig. 3(b), the generating
counter-torque TS lowers the PS versus δ curves.
Fig. 6. Torque components TS , TR , Tsr sin and Tsr cos .
3.3.2.2. Induction machine–rotor-side excitation. TR is
from rotor-side excitation. The stator terminals are short circuited and the slip ring terminals are excited by the VSC
applying a voltage of magnitude VR = sVRS + VRsy and angular frequency ωr = sωs . Since this form of induction machine
operation is unusual, its graphical form in Fig. 6 is unfamiliar.
Its formula in (12) is similar to (11), with (VR /s)2 replacing
(VS )2 and Rr /s replacing RS .
3.3.2.3. Synchronous machine. Tsr cos and Tsr sin in Fig. 6 are
from the interaction between the stator excitation VS and the
rotor excitation VR = sVRS + VRsy . The terms Tsr cos and Tsr sin
can be expressed as:
VR VS Z (s) sin ξ
(15)
Tsr cos = −
s
ωs |Z(s)|2
VR VS Z (s) cos ξ
Tsr sin = −
(16)
s
ωs |Z(s)|2
where
sin ξ =
cos ξ =
Rr
s
− RS
(17)
|Z (s)|
(Xrl + Xsl )
|Z (s)|
(18)
and
2
Z (s) = (Xsl + Xrl )2 +
Rr
− RS
s
2
(19)
Substituting (13) and (14) in Tsr cos cos δ + Tsrsin sin δ and
multiplying this torque component by ωs , its stator power
component ωs (Tsr cos cosδ + Tsr sin δ) can be expressed as:
VR
VS sin(δ + ξ)
Z (s)
s
Psyn-DFIG = −
(20)
|Z(s)|
|Z(s)|
Because Z(s) defined in (4) differs from Z (s) of (19) by
the sign in RS , the term |Z (s)|/|Z(s)| in (20) does not cancel. Otherwise, Psyn-DFIG in (20) is very close to the power
of a synchronous generator of internal impedance Z(s) and
internal voltage VR /s transferred the stator bus voltage of
magnitude VS . The angle (δ + ξ) corresponds to the power
angle.
3.3.3. PS versus δ curves
Since PS = Te ωs and ωs = 1 pu, the δ dependence of the
PS versus δ curves of Fig. 3 is principally determined by
Tsr cos cos δ and Tsr sin sin δ. TS and TR are constant offsets at
each slip. Fig. 7(a and b) shows the decomposition of the PS
versus δ curves taken from Fig. 3 for two cases: s = +0.2 and
−0.2. As mentioned the induction machine torque component
TS is positive for positive slip in Fig. 7(a) and negative for
negative slip in Fig. 7(b).
L. Jiao et al. / Electric Power Systems Research 76 (2005) 33–37
37
Appendix A. Parameters of the DFIG
The parameters of the DFIG used in the graphical results
are taken from Ref. [13]:
RS = 0.012 , Lsl = 0.204 mH, Rr = 0.021 , Lrl = 0.175 mH,
M = 13.5 mH.
Rated stator side voltage: 690 V, rated capacity: 1.5 MW,
pole number: 4, VRS = 690 V, VRsy = 20 V.
References
Fig. 7. Decomposition of PS vs. δ curves.
3.3.4. PS versus ωm curves
Since PS ≈ Te ωs , the PS versus ωm curves in Fig. 5 can be
understood in terms of TS , TR , Tsr cos and Tsr sin in Fig. 6.
4. Conclusions
Using the principle of superposition, it is shown that the
DFIG can be viewed as a synchronous/asynchronous hybrid.
At any speed, the DFIG can operate as a motor or as a
generator by changing δ, the angle of the voltage injected
at the rotor terminals. In this respect, the DFIG resembles
the synchronous machine. The DFIG retains the induction
machine characteristics which are slip dependent. Thus, it has
a component which yields the well-known motoring power
characteristic for operation at positive slip and generating
power characteristic for negative slip. There is also a lesser
known component which comes from the induction machine
excited from the rotor terminals.
Acknowledgments
The late Dr. Lianwei Jiao thanked Tsinghua University,
Beijing, China for leave of absence to pursue research in
McGill University, Canada. Financial support from a Strategic Grant of the Natural Science and Engineering Council of Canada (NSERC) on “Integrating Distributed Energy
Sources into the Power Grid” is acknowledged.
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