Engineering, 2013, 5, 89-95
doi:10.4236/eng.2013.51b016 Published Online January 2013 (http://www.SciRP.org/journ al/eng)
A Gain Scheduled Method for Speed Control of Wind
Driven Doubly Fed Induction Generator
Wei Wang, Kang-Zhi Liu, Tadanao Zanma
Dept. of Electrical and Electronics Engineering, Chiba University, Chiba, Japan
Received 2013
Abstract
This paper proposes a gain sc heduled c ontrol method for a doubl y fed inducti on generator dr iven by a wind turbine. The
purpose is to design a variable speed control system so as to extract the maximum power in the region below the rated
wind speed. Gain scheduled control appro ach is applied in order to achieve high perfor mance over a wide range of wind
speed. A double loop configuration is adopted. In the inner loop, the rotor speed is used as the scheduling parameter,
while a function of wind and rotor speed is used as the scheduling parameter in the o uter loop. I t is verified in simula-
tions that a high tracking perfo rmance has been achieved.
Keywords: Doubly fed induction generator, Gain scheduled control method, speed control, current control
1. Introdu ct ion
With rapid development of modern industries, fossil fu-
els are being exhausted and environment is being de-
stroyed seriously. For instance, burning of fossil fuels
generates much waste carbon dioxide and causes global
warming. As a solution to shortage of fossil fuels and
environmental p rob le ms, muc h at te nt io n ha s b een p ai d to
the wind energy utilization because the wind energy is
inexhaustible and has no emission of carbon dioxide and
radioactive waste.
However, the wind energy is heavi ly in fluenced by
weather and varies irregularly. Since the power captured
by the wind turbine is proportional to the swept area and
the cube of the wind speed, the utilization efficiency of
wind power system becomes more and more important.
In wind farms, doubly fed induction generator (DFIG)
based doubly fed system and permanent magnet genera-
tor (PMSG) based direct-drive system are generally in-
stalled. Since DFIG is advantageous in lower cost and
low power loss caused by power electronics device, it is
widely used. As shown in Fig 1, DFIG system is differ-
ent from conventional wind po wer system in that its sta-
tor is directly connected to the grid and the rotor is con-
nected to the grid thr ough a back-to-back converter [1].
However, it is hard for the conventional linear control
method to achieve high performance of DFIG syste ms in
the case of large wind speed variation because of the
high no nli ne ar ity of wind power. So in recent years, non-
Fig 1: Block diagram of DFIG s ystem.
linear control methods have been studied in order to im-
prove the performance of DFIG. For example, [7] ap-
plied the sliding-mode control to the direct active and
reactive power regulation of DFIG. In [8] the exact li-
nearization method is applied to the transient stability
control of DFIG in face of fault. [9] proposed a combina-
tion of PI control and state feedback nonlinear control so
as to improve the dynamic behavior after clearing the
fault.
In this paper, a gain scheduled method is proposed
aiming at high performance in variable speed control
which is indispensable in the maximum power point
tracking. To this end, the nonlinear model of DFIG sys-
tem will be transformed equivalently as an LPV model
first. Then, a gain scheduled control method for the rota-
tional speed control is proposed. Effectiveness of the
method is verified by simulations. T his method is d iffer-
grid-side converter
R
L
rotor-side converter
DFIG
gearbox
wind turbine
grid 3
rotor-side
controller
grid-side
controller

W. WANG ET AL.
90
ent from conventional variable speed control in that no
approximation is made in model transformation.
2. Operation modes of DFIG Sys tem
In general, a DFIG system has two operation modes
which are described briefly below. Operation mode 1 is
in the wind speed region between the cut-in wind speed
and the rated wind speed. In this region, the pitch angle
is fixed to 0, and the rotational speed of wind turbine is
controlled in order to get better conversion efficiency [1].
Operatio n mode 2 is in the wind speed region between
the rated wind speed and the cut-out wind speed. The
control objective is to maintain the rated power by con-
trolling the pitch angle as well as the rotational speed of
wind turb ine [2].
This paper deals with the rotational speed control of
DFIG system in op e ration mode 1.
3. Aerodynamic Characteristics
Tip speed ratio, which is us ed to evaluate the per for-
mance of the wind turbine, is defined as
R
V
ω
λ
⋅
=
（1）
where ω is the rotational speed of wind turbine, R is the
turbine radius, and V is the wind speed.
The power coefficient Cp(λ) r epresents the power con-
versio n effic iency o f a wind turbine . In the case of oper-
ation mode 1, the power and torque coefficients vary
only with the tip speed ratio. The power coefficient is
approximated as the following equation [3]:
21
116
()0.5176 (5)0.0068
11
0.035
i
p
i
i
Ce
λ
λλ
λ
λλ
−
=⋅−⋅ +



= −



（2）
The relationship between the power coefficient Cp(λ)
and torque coefficient Cq(λ) is
()
() p
q
C
C
λ
λλ
=

（3）
It can be seen from equation (2) that the maximum
power coefficient is 0.48 and the optimal tip speed ratio
is 8.10, at which the wind turbine can capture the wind
energy with maximum efficiency. It also can be seen
from equation (2) and equation (3) that the maximum
torque coefficient is 0.0647 which is achieved at the tip
speed ratio of 6.76. So the optimal rotational speed for a
given wind speed is
8.10
opt V
R
ω =⋅ 
（4）
According to equation (4), the optimal rotational speed
may be computed by measurement of wind speed. The
maximum power point tracking may be implemented by
setting the optimal rotational speed as the speed com-
mand of the wind turbine. In addition, there also exist
other methods which search the optimal rotational speed
by means of search algorithms. In this paper, the first
method is adopted in simulation because the focus her e is
on the rotational speed control.
4. Model of Wind Turbine
The mechanical power of wind turbine is given by
23
1()
2
mp
PRC V ρπ λ = ⋅⋅
（5）
where ρ is the air density. The maximum power for a
given wind speed is
3
m optp
P KV
−
= ⋅
（6）
where Kp is determined by equation (5) with λopt= 8.10
substituted into Cp(λ).
Moreover, the aerodynamic torque is given b y
32
1()
2
mq
TRC V ρπ λ = ⋅⋅

（7）
5. Dynamic Model of DFIG System
5.1. DFIG Model
The electric circuit configuration of DFIG is shown in
Fig 2, where uds and uqs denote the stator voltages, udr and
uqr denote the rotor voltages, ids and iqs denote the stator
currents, idr and iqr denote the rotor voltages, ψds and ψqs
denote the stator fluxes and ψdr and ψqr denote the rotor
fluxes in d-q frame. In modeling the stator and rotor of
DFIG, the motor convention is used.
Fig. 2: Circuit configuration of DFIG.
The values of physical parameters for DFIG studied
here are illustrated in table 1.
Table 1. The v alues of physical parameter s f or DFIG.
Stator resistance Rs Ω0.087
Rotor resistance Rr 0.225Ω
Stator inductance Ls 84.59mH
ubr
θ
re
i
br
stator A
i
ar
ω
re
i
cr
u
ar
u
cr
i
bs
ias
i
cs
u
as
u
bs
u
cs
stator A
rotor stator

W. WANG ET AL.
91
Rotor inductance Lr 85.71mH
Mutual inductance Lm Hm83
Rated power 35kW
Rated voltages of stator and rotor V380
The equivalent circuit of DFIG in the d-q reference
frame which rotates with sync hronous speed ω1= 2πf ( f :
power supply frequency) is used to set up the model eq-
uations. The voltage equations of DFIG with constant
coefficients in the d-q reference frame are
1
1
dss dsdsqs
qss qsdsqs
drr drdrsqr
qrr qrsdrqr
u Ri
u Ri
u Ri
u Ri
ψ ωψ
ωψ ψ
ψ ωψ
ωψ ψ
= ⋅+−⋅

= ⋅+⋅+

=⋅+− ⋅

=⋅+ ⋅+





（8）
The flux li nkage equat ions with co nsta nt coe ffici ents i n
the d-q reference frame are
dss dsm dr
qss qsm qr
drm dsr dr
qrm qsr qr
Li Li
Li Li
Li Li
Li Li
ψ
ψ
ψ
ψ
= ⋅+⋅

= ⋅+⋅

= ⋅+⋅

= ⋅+⋅


（9）
The relationship between the slip frequency ωs and slip
s is defined as
11sp rm
sn ωω ωω =⋅=− ⋅
（10）
where np and ωrm denote the number of pole pairs and the
generator mechanical angular speed, respectively.
The equations about the electromagnetic torque ( driving
torque) and the reactive power at the stator terminal are
()
epmqs drdsqr
qs dsds qs
TnL iiii
Qui ui
=⋅ ⋅−⋅


=⋅− ⋅



（11）
It is assumed that the stator of DFIG is connected to
the constant-voltage constant frequency power supply
system. To realize decoupling control, d axis is aligned
with the gr id vo ltage vector, which me a ns
,0
ds sqs
uv u= =
（12）
after the commenc eme nt of electricity generation [4],
where vs is the mag nitude o f the grid voltage vec t or.
Since the voltage drop across the stator resistance is
suf ficiently low, it is neglected and the relationship
between t he st ator vol t age and flux is approximated as
11
,0
sdsqsds qs
v ψωψωψ ψ =−⋅=⋅ +



（13）
It is assumed the stator flux has reached the steady
state at the starting time of ele c tric ity generation, i.e .
1
() 0,()/
ds ciqs cis
t tv ψψω == −
（14）
So after that, there holds
1
0, /
dsqs s
v ψψω == −


（15）
By substituting eq uation (15) back into (9), the equa-
tions of the stator and ro tor currents become
1
,
m ms
ds drqsqr
s ss
L Lv
iiii
L LL ω
=−⋅ =−⋅−


（16）
Further, substitution of equation (16) into (11) yields
the following electromagnetic torque and the reactive
power [4]:
1
1
()
p ms
e dr
s
ms
s qr
ss
nLv
Ti
L
Lv
Qv i
LL
ω
ω
=−⋅



=⋅ ⋅+





（17）
It is easy to se e that the electromagnetic tor que is pro-
portional to idr and the reactive power at the stator ter-
minal is a linear function of iqr . Therefore, the electro-
magnetic torque Te is controlled by idr and the reactive
power Q is controlle d by iqr. Hence, it is quite natural to
use a two-stage approach:
1. Control ω and Q by using idr and iqr respectively.
2. Design current feedback loops to track the current
commands computed in stage 1.
5.2. Drive-train Model
The sche matic diagra m of drive-train syste m is s hown i n
Fig 3, where Tl is the torsional torque at low-speed shaft
and Tg is the driving torque at high-speed sha ft. Si nc e t he
stiffness coefficient Kr and Kg are sufficiently low, it is
neglected.
Fig. 3: Drive-t rai n dy na mic s.
The values of physical parameters for drive-train sys-
tem are illustrated in table 2.
Table 2. The valu e s of physi cal parameter s f or dri ve-train
system.
Iner tia of wi nd turbine Jr 63 kg·m2
Inertia of DFIG Jg
4.97kg·m2
Damping coef ficient of wind turbine Br 3.2Nms/rad
Damping coef ficient of DFIG Bg 0.8N ms / rad
Br
Jg
ω
rm
K
r
Tm Tl
J
r
Tg Te
Kg
Bg
ω

W. WANG ET AL.
92
Regarding the drive-train as a two-inertia system, the dynamic equations ar e obtained as
Fig. 4: Total c ontrol s tructure.
ml rr
gegrmg rm
TTJ B
TTJ B
ωω
ωω
− =⋅+⋅


+= ⋅+ ⋅





（18）
Neglecting the power loss in gearbox, the following equ-
ation is established according to the law of conservation
of ene rgy.
lg rm
TT ωω ⋅= ⋅
（19）
The gearbox ratio is defined by
rm l
g
g
T
nT
ω
ω
= =
（20）
Substi tutio n of equation (20) into (18) leads to the sim-
plified model o f drive train s ystem [5].
22
,
m ge
rgg rgg
T nTJB
J JnJBB nB
ωω +⋅ =⋅+⋅


=+⋅= +⋅




（21）
6. Controller Design for DFIG System
As shown in Fig.4, a double loop control configuration
is adopted. In the inner loop, the current controller aims
at high accuracy tracking of the reference rotor current,
while in the outer loop, the rotational speed controller
aims at capturing the wind energy with maximum effi-
ciency and generates the reference rotor current. The
controllers of these two loops are designed in this sec-
tion.
6.1. LPV Model for DFIG
Based on equatio n (8) and (9), an LPV model equivalent
to the nonlinear model of DFIG is described as
()
:
ds ds
qsqs ds dr
p rmsr
drdr qs qr
qr qr
rc
ds
dr qs
p
dr qr
qr
ii
i iuu
dA BB
i iuu
dt
ii
Gi
ii
Ci
i
i
ω
 
  
 
=⋅+⋅+⋅
 
  
 
 
 


 
= ⋅
 
 



（22）
where Bs and Br are constant matrices, while Ap(ωrm) is
2
1
2
1
1
1
()
()
mp mp
s rm
rm rm
ssrsr s
mp mp
s rm
rm rm
srs ssr
p rm
mp p
sm r
rm rm
sr rr
mp p
sm r
rm rm
r srr
Ln Ln
R RL
LLLLL L
Ln Ln
R RL
LLL LLL
ALn n
RL R
LL LL
Ln n
RL R
L LLL
ωω ω
σσ σσ
ωω ω
σσσ σ
ω
ω ωω
σ σ σσ
ω ωω
σσ σσ
σ

− +⋅⋅



 −+⋅−− ⋅

=

− ⋅−−⋅



⋅−+ ⋅−


=
2
1
m
sr
L
LL
−
and is affine in ωrm .
The size of power converter is not related to the total
generator power but to the selected speed variation range.
Typically a range of ±40% around the synchronous
speed is used [1].
For f=50Hz, np=3 and ng=10, the speed of DFIG takes
value in ωrm=20π~140π/3 (rad/s) and the speed of wind
turb ine takes value in ω=2π~14π/3 (rad/s), respectively.
R
L
optimal speed
calculation reactive power measurement
SVPWM bus voltage
controller
speed
controller reactive power controller
rotor current
controller
Q
ref
Q
3
grid
u
r
ir
gearbox
grid-side converter rotor-side converter
DFIG
ω
ω
ref
wind turbine
V

W. WANG ET AL.
93
6.2. Gain Scheduled Controller Design for Rotor
Current Control Loop
The generalized feedback system used for controller de-
sign is shown in Fig.5 in which Krc(s) is the gain sche-
duled controller.
The contro lled output is selected as the tracking error of
rotor current (edr , eqr) and rotor voltage (udr , uqr), while
the stator vo ltage is treated as a d isturbance.
The closed-loop system should achieve a high tracking
performance in the low frequency band since the rotor
reference current is majorly a low frequency signal, so
we select the weight functions Wsd(s) and Wsq(s) as low-
pass filters. Meanwhile the weight functions Wud(s) and
Wuq(s) are chosen as hi gh -pass filters.
-
-
＋
＋
Fig. 5: Generalized plant f o r rotor current control design.
Weighting functions are selected as follows through
trial and err or
100
()() 5.5 5
0.0001
() () 1000
sd sq
ud uq
Ws Wsss
Ws Wss
= =

+

= =
+


（23）
The state-space equation of generalized plant can be
written as
12
1 1112
2 21
()
rm
xAx BdBu
zCxD d D u
yCxDd
ω =⋅+⋅+⋅

= ⋅+⋅+⋅

= ⋅+⋅


（24）
where the disturbance vector d, the cont ro l i np ut vector u,
the controlled output vector z and the measured output
vector y are a s s hown in Fig.5 .
An output feedback control law which is affine in ωrm is
considered:
01 01
01 01
( )( )
( )()
KK rmKKK rmK
K rmKKKrmK
xAAx BBy
uCC xDD y
ωω
ωω
=+⋅ ⋅++⋅ ⋅

=+⋅ ⋅++⋅⋅


（25）
where xK is the state vector o f the contro lle r .
An H∞ method is used in the design, i.e. we design an
output feedback control system whose L2 induced-gain
from d to z is minimized. The design specification is re-
duced to LMI s at the maximum and minimum values of
ωrm in it s operating ra nge, and solved numerically [6].
The bode plots of rotor current controller for the maxi-
mum (blue lines) and minimum (red lines) values of ωrm
are shown in Fig.6. It can be seen from this figure that
the mag nitud es of frequenc y r esp onse va r y wi th ωrm sub-
stantially.
10
-2
10
0
10
2
10
4
10
6
10
8
-80
-60
-40
-20
0
20
40
60
80
Magnitude (dB)
Frequency (rad/sec)
10
-2
10
0
10
2
10
4
10
6
10
8
-80
-60
-40
-20
0
20
40
60
80
Magnitude (dB)
Frequency (rad/sec)
10
-2
10
0
10
2
10
4
10
6
10
8
-80
-60
-40
-20
0
20
40
60
80
Magnitude (dB)
Frequency (rad/sec)
10
-2
10
0
10
2
10
4
10
6
10
8
-80
-60
-40
-20
0
20
40
60
80
Magnitude (dB)
Frequency (rad/sec)
edr-udr(wrm-max)
edr-udr(wrm-min)
eqr-udr(wrm-max)
eqr-udr(wrm-min)
edr-uqr(wrm-max)
edr-uqr(wrm-min)
eqr-uqr(wrm-max)
eqr-uqr(wrm-min)
Fig. 6: Bode gain plot s of rotor current controller.
6.3. LPV Model for Drive-train System
Substitution of equatio n (7) into (21) leads to an LPV
model of drive train system in the operating range from
the cut-in wind speed to the rated wind speed.
:
dr
s
BM pKi
PJ
y
ωω
ω

=− +⋅⋅+ ⋅



=


（26）
where
3
1
,
2
gp ms
s
n nLv
R
MK
J JL
ρπ
ω
= =−⋅

The scheduling parameter p is defined as
2
()
q
V
pC
λω = ⋅

（27）
It is assumed that the wind speed is measured. So p can
be computed on-line a nd takes value in p=0~1.03.
6.4. Gain Scheduled controller Design for
Rotational Speed Control Loop
The generalized feedback system shown in Fig.7 is used
for rotational speed controller design.
-
+
Fig. 7: Generalized plant f o r rotational spe ed
controller design.
Wud
W
uq
G
rc
K
rc
d z
Wsd
Wsq
W
s
Wu
P
K
z

W. WANG ET AL.
94
Similarly to the rotor current control design, the con-
trolled o utput is selected as the trackin g error of rotation-
al speed ωref-ω and the d component of rotor current idr.
The weighting functio ns ar e selected as follows through
trial and err or
0.1 1
() 2.85 0.001
12.4 31
() 3 800
s
u
s
Ws s
s
Ws s
+
=

+
+
=
+


（28）
The gain scheduled controller has the form below
01 01
01 01
( )()
( )()
KKKK KK
KKK KK
xApA xBpBy
u CpCxDpDy
=+⋅⋅ ++⋅⋅

=+⋅⋅++⋅⋅



（29）
The numerical design is similar to that of the rotor cur-
rent loop [6 ].
7. Simulation Results
In simulation s, an integrator controller
0.5
Q
Ks =
（30）
is used as the reactive power controller. The reactive
power command is set as
100var 0350
250var350700
550var 7001000
ref
st s
Qst s
st s
− ≤<


= −≤<

− ≤≤

（31）
The command of rotational speed is computed by equa-
tion (4) in which the wind spe ed is gi ven in Fig.8 .
Simulations results are shown in Fig. 8~Fig. 13 in the
case where the wind speed input is a rapidly changing
random signal.
0100 200 300 400 500 600 700 800 900 1000
6.500
7.000
7.500
8.000
8.500
9.000
9.500
Wind speed
(m/s)
0100 200 300 400 500 600 700 800 900 1000
0.472
0.474
0.476
0.478
0.480
0.482
Power coefficient
time (s)
Wind speed
Maximum power coefficientReal value
Fig. 8: Wind s peed (to p) and pow e r coefficient (bottom).
As shown in Fig.8, the power coefficient is almost not
influenced by the irregular variation of the wind speed
and has been maintained close to its maximum value.
0100200300400 500600700 8009001000
8.50
9.50
10.5
11.5
12.5
13.5
Rotational speed
of wind turbine (rad/s)
0100200300400 500600700 8009001000
-0.80
-0.50
-0.20
0.10
0.40
time (s)
Error of
rotational speed (rad/s)
Error of rotational speed
Reference valueReal value
Fig. 9: Rotational s peed of wind turbine (to p)
and tracking error of rotational speed (bottom).
0100200 300 400 500600 700800 9001000
10.0
15.0
20.0
25.0
30.0
Wind turbine power
(KW)
0100200 300 400 500600 700800 9001000
0.00
0.60
0.12
0.18
0.24
0.30
time (s)
Error of
wind turbine power (KW)
Reference valueReal value
Error of wind turbine power
Fig. 10: Wind tur bine power (top)
and tr acking erro r of wind tur bine power (bottom).
It can be confirmed from Fig.9 and Fig.10 that a high
tracking performance of rotational speed and the max-
imization of wind energy capture have been achieved.
0100 200 300 400 500 600 700 800 9001000
-0.6
-0.4
-0.2
0.0
0.2
0.4
Reactive power
(Kvar)
0100 200 300 400 500 600 700 800 9001000
-0.6
-0.4
-0.2
0.0
0.2
time (s)
Error of
reactive power (Kvar)
Error of reactive power
Reference valueReal value
Fig.11: Reactive power (top)
and trac king error of react ive power (bottom).

W. WANG ET AL.
95
The r espo nse of the reactive po wer is sho wn in F ig. 11 , in
which it ca n b e seen that the reactive po wer has achieved
a high a ccur ac y t racking of the co mmand .
0100 200 300 400 500 600 700 800 9001000
0.00
5.00
10.0
15.0
20.0
d axis
rotor current (A)
0100 200 300 400 500 600 700 800 9001000
-26.5
-26.0
-25.5
-25.0
-24.5
time (s)
q axis
rotor current (A)
d axis rotor current
q axis rotor current
Fig.12: d axis rotor cu rrent (to p)
and q axi s rotor cu r r ent (bottom).
As shown in Fig.1 2, idr varies with the wind speed while
iqr varies with the command of reactive power. The high
frequency component in the rotor currents is suppressed
efficientl y.
0100 200 300 400 500 600 700 800 900 1000
-200.0
-130.0
-60.00
10.00
80.00
150.0
d axis
rotor voltage (V)
0100 200 300 400 500 600 700 800 900 1000
-12.00
-9.000
-6.000
-3.000
0.000
time (s)
q axis
rotor voltage (V)
d axis rotor voltage
q axis rotor voltage
Fig.13: d axis rotor voltage (top)
and q axi s rotor voltage (bottom).
Finally, Fig.13 shows that the high frequency compo-
nent in the rotor voltages is suppressed efficie ntly and
the variation is in a reaso nable r ange.
8. Conclusion
This paper has proposed a gain scheduled control method
for a doubly fed induction generator driven by a wind
turbine. This method is based on equivalent LPV model-
ing of the nonl inear DF IG system and H∞ op timization. It
is confirmed by simulations that a quite high precision
tracking control of rotor speed as well as reactive power
is achieved by the proposed method.
As a future work, we plan to deal with the controller
design for DFIG systems operating in mode 2 in or der to
mai nt ain the rated power.
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