Modern Mechanical Engineering, 2013, 3, 90-97
http://dx.doi.org/10.4236/mme.2013.32013 Published Online May 2013 (http://www.scirp.org/journal/mme)
Multidisciplinary Constrained Optimization of Power
Quality in Doubly Fed Wind Turbine Induction Generator
Seyed Javad Fattahi1, Abolghasem Zabihollah2
1Department of Mechanical Engineering, Ottawa University, Ottawa, Canada
2School of Science and Engineering, Sharif University of Technology, Inter’l Campus, Kish Island, Iran
Received October 16, 2012; revised March 21, 2013; accepted April 9, 2013
Copyright © 2013 Seyed Javad Fattahi, Abolghasem Zabihollah. This is an open access article distributed under the Creative Com-
mons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work
is properly cited.
ABSTRACT
Shape optimization of turbine blade to maximize the output power usually changes the power factor due to compensate
Repower in a wind turbine. This article presents a multidisciplinary optimization technique to maximize the output
power in Doubly Fed Induction Generator (DFIG) wind turbine. The most common parameters when operating the tur-
bine, namely, active power, reactive power and power factor, are considered as the problem constraints and the pitch
angle grid side variable frequency converter of the turbine blades are optimized to maximize the output power. Nu-
merical simulation has been illustrated to present the performance of the proposed design approach.
Keywords: Optimization; DFIG; WTG; Pitch Angle; RSC; GSC
1. Introduction
Wind Turbine machines are rapidly becoming an eco-
nomically viable source of renewable energy. Two key
elements of wind turbine, performance and availability,
serve optimization based on machine design and per-
formance. Wind turbine performance could be character-
ized by three basis parameters for power, torque and the
load of the tower of a wind turbine. The torque size is
important to design the mechanical elements like the
rotor, gear box and brake. The tower of the wind turbine
and foundation are designed to resist against the com-
pression, buckling and the axial loading. The power size
is important because it affects the transformed energy by
the rotor of wind turbine. Therefore, power quality is a
new challenge in wind turbine design [1-3]. Harmonics
generated by the interactive wind system were investi-
gated by Giraud and Salameh [4]. Boukhezzar et al. [5]
proposed a non-linear approach to control a variable-
speed turbine to maximize the power of turbine consid-
ering the generator torque. Muljadi and Butterfield [6]
developed a pitch control strategy to maximize the power
and minimize turbine loads for different wind speed sce-
narios. Chen et al. in [7] proposed the reactive power
optimization of the distribution network which contains
several wind farms by introducing the static VAR com-
pensators, and Bie et al. in [8] present a transition-opti-
mized approach based on adaptive load pattern classifi-
cation for DFIG based on the fluctuation of the output
active power and the reactive power capability and this
procedure was completed in [9]. Also in [10-12], differ-
ent multi-objective particle swarm optimization (MOPSO)
is proposed where MOPSO considers multi-objective simul-
taneously rather than a PSO but the output of MOPSO
consists of a group of non-dominated instead of PSO that
it has only one optimization solution, but these papers
focus on active and reactive power of DFIG and grid, on
the other hand, Kusiak and Zheng [13] introduced an
evolutionary computation approach for optimization of
power factor and power output of wind turbines. This
article contributes a multidisciplinary optimization for
doubly Fed Induction Generator (DFIG) wind turbine
subjected to active power, reactive power and power
factor mutually, through pitch angle and grid side vari-
able frequency converter. Numerical simulation has been
illustrated to present the performance of the proposed
design approach.
2. Wind Active Power
Theoretically, using Betz’s model, wind energy captured
by the rotor of a wind turbine can be expressed by:

2
1
0.5 π,
rp
PRC  3 v (1)
C
opyright © 2013 SciRes. MME

S. J. FATTAHI, A. ZABIHOLLAH 91
where ,R,
r
P

and 1 represent, respectively, the
wind energy captured by the rotor, the air density, the
rotor radius, and wind speed before passing the rotor.
The parameter p is the power coefficient that depends
on the blade pitch angle,
v
C

, and the tip-speed ratio,

,
and can be determined from the following relation as:
3
2
11
41
p
vv
Cvv
 

 
 
2
(2)
31
22
vv
v  (3)
2
rR
v

  (4)
where r

is the rotational speed of the rotor and 1
and 3 are wind speed in front and behind of turbine
blade respectively. The literature on optimization of the
power coefficient p is quite extensive. The maximum
value of the power coefficient p is obtained by the
derivation of (2) and optimal value of
v
v
C
C
21
vv
is equal to
0.593.
A wind turbine may operate when the tip speed ratio is
changing in large limits but a maximum power coeffi-
cient, p, could be obtained only for an optimal value
of
C

(tip speed ratio). It results that the maximum effi-
ciency in the wind energy conversion and the rotational
speed of the rotor wind turbine must be correlated with
the wind speed.
Wind turbines with capability to control the pitch an-
gle may have the power coefficient described by a func-
tion that depends on pitch angle and tip speed ratio:
 
5
123 46
,e
i
c
pi
CCCCC

where:
3
1 0.035
0.08 1
i



 (6)
The variables are given as: C1 = 0.5175, C2 = 116,
C3 = 0.4, C5 = 5, C6 = 21 [14]. Figure 1 shows the gener-
ated power with respect to different wind speed when
pitch angle equal to zero in which one may conclude that
the generated power has a large sensitivity to pitch angle.
The power coefficient Cp for the pitch angle β variation
from 0˚ to 20˚ is shown in Figure 2.
i
C
3. Sensitivity Analysis
Design optimization of wind turbine generators using
pitch angle does not guarantee that the power quality of
the output power to be desirable. Numerically, for rotor
speed of 54 rpm, zero pitch angle and the maximum
of 0.4296, the tip speed ratio is obtained as 7.9952.
p
C
It is worth noting that the variation of the rotor speed
is limited by the gear constraint and it is allowed to toler-
ate ±5%. As it is shown in Figure 3, the maximum out-
put power has a large sensitivity to set point of the
working rotor speed. Figure 4 shows the graph of the
working plane of a WTG for different rotor speed and
wind speed. It is shown that in desired wind speed
(around 13.5 - 14.5 m/s) the output power is changed
from 82% - 107.5% of nominal (0.95 MW) output power,
hence, for delivering a good power quality, multiple
variables, including pitch angle, rotor speed limitation or
generated reactive power need to be controlled simul-
taneously.
The power is given as:
C



   (5) 22
SPQ
2
(7)
0.2 0.4 0.60.811.2 1.4
0
0.2
0.4
0.6
0.8
1
1.2
S peed
Power
P ower/ wi n d velocity
V=8
V=10
V=12
V=14
V=16
V=18
V=20
Figure 1. Output power in different wind speed with respect to β = 0˚.

S. J. FATTAHI, A. ZABIHOLLAH
92
0246810 12 14 0
10
20
-0. 8
-0. 6
-0. 4
-0. 2
0
0. 2
0. 4
0. 6
Tip S peed Rat i o
P ower c oefficient in WTG in variabl e wind speed and fixed rotor speed=54 r.p.m
P i t ch A ngle (B et a)
P ower coeffic ient (Cp)
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
Figure 2. Power coefficient in WTG in variable wind speed and fixed rotor speed 54 r.p.m.
12 14 16 18 20
0
5
10
15
20
-5
0
5
10
15
20
x 10
5
W i nd S peed
Out put P ower in WTG in x (1)ariabl e wind s peed
P it ch A ngl e (Bet a)
Power
-2
0
2
4
6
8
10
12
14
16
18
x 10
5
Output Power in WTG in variable wind speed
Figure 3. Output power of WTG in when speed of wind and pitch angle varies.
where
P
is the stator active power measured in Watts
(W), S is the apparent power measured in volt-amperes
(VA), Q is the reactive power measured in reactive volt-
amperes. The Power factor is given by

cos
P
S ,
where is the phase angle between the current and the
voltage, measured in degrees.

3.1. Reactive Power Sensitivity
The reactive power flow from the grid to the wind plant
at the interconnecting point is simplified and given by
[15]:


22
32π
tg
QQ IXVfCQ 
where
g
Qis the positive reactive power, consumption, of
the turbine which is also negative when WTG is operat-
ing at a lagging power factor, X, denotes the equivalent
series reactance of cables, lines and transformers, C is the
equivalent shunt reactance of cables, and c
Q is the re-
active-power injected by any reactive power compensa-
tion system at the point of interaction. The second and
third terms of Equation (8) are not controllable and ap-
proximately, fixed in steady state, thus, control of t
Q
has to be de with on
g
Q and c
Q on. The wd tur-
bine is connected to the Doubly Fed Induction Genera-
tors (DFIG) through a mechanical shaft system, which
consists of a low-speed and a high-speed shaft which are
ly in
connected with a gearbox. The wound-rotor induction
c
(8)

S. J. FATTAHI, A. ZABIHOLLAH 93
1212.5 1313.5 1414.5 15 15.516
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
Wind Speed
Generated Power (MW)
Va riation of output power in nal rotor speed rang e ( Nr± 5%)omi n
54 rpm
54 rpm - 5%
54 rpm + 5%
Desired Wind
Speed Range
Figure 4. Generated pow er of WTG with constraint to rotor speed and wi nd speed.
achine in this configuration is fed from both stator and
Regulating the DFIG rotor speed for maximum
w
e DFIG stator output voltage fre-
qu
the DFIG reactive power
rators (DFIG),
th
is rotor current in d-axis,
s
L
V
inals of the
rm
a
m
rotor sides. The stator is directly connected to the grid
while the rotor is fed through a variable frequency con-
verter (VFC). In order to produce electrical active power
at constant voltage and frequency to the utility grid over
a wide operation range from sub-synchronous to super-
synchronous speed, the active power flow between the
rotor circuit and the grid must be controlled both in mag-
nitude and in direction. In DFIG the variable frequency
converter consists of two four-quadrant IGBT PWM con-
verters, rotor-side converter (RSC) and gridside con-
verter (GSC), connected back-to-back by a dc link ca-
pacitor. The crow-bar is used to short-circuit the RSC in
order to protect the RSC from over-current in the rotor
circuit during transient disturbances. The RSC control
scheme is expected to achieve the following objectives
[16].
1)
ind power capture
2) Maintaining th
ency constant
3) Controlling
Normally, for Doubly Fed Induction Gene
e reactive power consumption depends on the terminal
voltage and loading according to the following expres-
sions [16,17]:
m
s
qr
s
L
PV I
L
 (9)

2
3
2
s
mms msdr
g
s
LI II
QL
 
 (10)
qss qs
ms
sm
VRI
IL



dr
I
(11)
where, and are
stator leakage and velocity (rad/s) and is thag-
nitudee stator phase voltage i
powe controlled on the termachine
m
L
e m
m
hin
qs
n q-axis. The reactive of th
r can be
gror at the id side of the turbine transfoer wit the
WTG capability. The voltage variationsre much less
than the current variation, therefore, the current is the
main term that is required for compensation. Thus, by
using vector control with d-axis oriented stator flux vec-
tor in rotor side converter (Equations (10) and (11)), the
stator reactive power
g
Q can be controlled by regu-
lating dr
I
.
On the other hand, grid side converter is presented for
keeping DC-link voltage of capacitor constant, regardless
of the magnitude and dction of rotor power. Neglect-
ing the power lo
ire
sses in the converter, capacitor current
can be described as follow [12]:
3
4
D
Cdgc rDC
I
kI I (12)
where dgc
I
stands for the d-axis current flowing be-
tween grid and grid side converter, rDC
I
ac
is the rotor side
DC cu C is the DC-link cap
PWM modulation index of the grid
po
rrent, itance, and k is the
side converter. The
reactivewer flow into the grid from GSC can be ex-
pressed as:
3
2
c qgc
QVI (13)
where V is the magnitude of grid phase voltage (V) and
qgc
I
fore,
is q-axis current of grid side converter (A). There-
it is seen from Equations (11) and (12), by adjusting
and qgc
I
,
dgc
I
DC-link voltage and can be con-
Usi
c
trolled respectively.
ng Equations (10) and (13), the DFIG reactive po-
wer can be written as: (See Equation (14)).
Q

S. J. FATTAHI, A. ZABIHOLLAH
94
Tien
he wind turbines are controlled to
m
he power factor measures the efficcy of electric
power utilization. T
aintain a power factor of 1. Substituting Equations (9)
and (14) in Equation (8), leads to:

2
2
2
2
s
qr dr
m
SKII L










(15)
m
s
s
L
KV
L



However, the power factor of wind tu
to control, thus, its value is lower than
wind turbines and wind farms. Figure 5 shows the active
vs reactive power generation curve, in which one can
ob
rbines is difficult
1 for individual
serve that ideal point


SP occurs at “0.99 Pmax”.
On the other hand, it is known that the generated power
is highly sensitive to power coefficient and pitch angle
ratio. Therefore, to reach to a desired power factor in grid
several variables shouldn into consideration si-
multaneously, requiring a multidisciplinary optimization
technique.
4. The Optimization Problem
ions imposed by ambient temperature on the
be take
tronic device, is less restrictive than those
ntly, the generator
Safe operation
The limitat
power elec
fixed by the generator itself. Conseque
must be kept in a safe operation zone.
zone depends on the type of inverter and generator which
is considered as 95% of full power capacity. Optimiza-
tion of power quality needs solving two optimization
problems: 1) maximization of active power and 2) mini-
mization of reactive power, therefore, requiring a mul-
tidisciplinary optimization problem to be solved.
4.1. Statement of Optimization Problem
The optimization problems have been cast into a standard
format as: (See Equation (19)).
subject to:
05


2
0
rRv 14


Minimizing
1
1216m/sv


 
2
,
dr
QII ,,
2π
33
22
ms
smmsmsdr
qgc
s
V f
fxI Ia IVI
L
 

(20)
subject to:
0.951.05 Pu
180240 Volt
r
qss qs
dr
sm
qs
s
VRI I
L
V







, tip speed ratio



, In general, the pitch angle
wind speed


v and rotor speed

r
S are related-
straints of genrated active power and ,,
dr ms
con
e
I
IV and
f
are related constraints of reactive power.
. Statement of Multidisciplinary Opt4.2imization
The Multidisciplinary optimization problem
Problem
objective of
is defined as:
Maximize S through ;
22
SPQ (21)
The solution of this problem ha
fo
s been performed as the
llowing:
Solving the maximization problem described at i
without any optimizing procedure on reactive power “Q”,
or vice versa, means solving minimization problem de-
scribed at ii without any optimizing procedure on active
power “P”; both procedures lead to bad power quality
output from DFIG. Therefore, it needs to define a Mul-
tidisciplinary Optimization (MDO) control in two dif-
ferent domains: 1) MDO is based on active power opti-
mization by using the pitch angle, tip speed ratio, and
wind velocity and 2) reactive power optimization by us-
ing rotor speed, d-axis of rotor current, q-axis of stator
current and voltage simultaneously and priority concern.
The priority of solving optimization Problem (1) and
Problem (2) depends on the results obtained from each
section. This fact has been clearly presented in results
provided in Tables 1-3. These two problems have been
solved using Sequential Quadratic Programming (SQP)


2
33
22
sm msmsdr
ssm
s
drg cqgc
ss s
LI II
VL
QVIQQ V
LL L



 I (14)


53
10.035
0.08 1
2 3
12346 1
3
,,,
1 0.035
0.5 πe
0.08 1
r
c
PvS
RCCCCCv




 



























(19)

S. J. FATTAHI, A. ZABIHOLLAH 95
-1-0.500.51
0
0.5
1
1.5
Rective power (p.u)
Active power (p.u)
P/Q Ratio
Pmax
Ideal Point
O.P= 90% Pmax
Figure 5. Active vs. reactive power generation curve in DFIGWTG.
approach described in the following section. One may
note that due to complexity of the solution procedure and
as the (SQP) is a well-defined optimization technique in
many commercial package
nly the basis of the (SQP)
r optimization prob-
n
l
s, including MATLAB®, here,
is provided. However, to en- o
hance the accuracy and for faster convergence to optimal
results, the gradients of the objective and constraints func-
tions have been computed analytically and fed to the op-
timization toolbox of MATLAB®.
4.3. Sequential Quadratic Programming (SQP)
The optimization problem is formulated as Sequential
Quadratic Programming (SQP) which is a widely used
method for most complex nonlinea
lems owing to its robustness and high efficiency i
searching for the optimum point. Considering a genera
problem of minimizing the objective function


f
x, sub-
ject to constraint

x
, we can write the Lagrangian
function as
     
1
,
m
kiik
i
Lxfxgx 


 (22)
Then the quadraprogramming (QP) sub-problem
can be formulated bd on a quadratic approximation of
the Lagrangi
tic
ase
an function as:

T
T
1
min 2
n
dR 
Subject to

T
kk
dHdf qd  


H
(23)

 
T
0
0
ik ik
ik ik
gxd gx
xd x 


(24)
ere ,ik x
g
and i

are design variable vector, con-
strainns and Lagrange multipliers, respectively.
Table 1. Maximum output power and recommended prior-
ity of control in normal rotor speed (54 r.p.m).
Wind Speed Maxwer
W
Reactive Power Control
Priority
t functio
˚β (m/s) M
Po
0 12 0.53943 RSC, GSC, Pitch
0 13 0.74516 RSC, GSC, Pitch
0.0719630914 0.95
1.3919400215 0.95 Pitch, RSC, GSC
916626
RSC, Pitch, GSC
3.12210.95 Pitch, RSC, GSC
Table 2. Maximum output power and recommended prior-
ity of control in rotor sub-speed (54 × 0.95 r.p.m).
˚β Wind Speed
(m/s)
Max Power
MW
Reactive Power Control
Priority
0 12 0.57310 RSC, GSC, Pitch
0 13 0.76210 RSC, GSC, Pitch
0.00351158 140.95 RSC, Pitch, GSC
1.10133779 15 0.95 Pitch, RSC, GSC
1.83905968 16 0.95 Pitch, RSC, GSC
Table 3. Maximum output power and recommended prior-
ity of control in rotor super-speed (54 × 1.05 r.p.m).
˚β Wind Speed
(m/s)
Max Power
MW
Reactive Power Control
Priority
1.1525 12 0.54649 Pitch, RSC, GSC
0.8616 13
0.6
4. 39396085 16 0.95 Pitch, RSC, GSC
0.71390 Pitch, RSC, GSC
0 14 9435RSC, Pitch,GSC
1.76758958 15 0.95 Pitch, RSC, GSC

S. J. FATTAHI, A. ZABIHOLLAH
96
The matrix i
H
is a positive definite approximation of
the Hessian matrix of the Lagrangian function and can be
updated by usie
the suroblem i used m a new until
cone occur
To improve the chances of obtaining a minimum clos-
er tl minim, a trerrop-
proach used toger witP mle
heuthod iolves raly self
stts inhope one g
pe to tglobal um. i-
ni su
ng quasi-N
s then
wton meth
to for
ods. The solution to
iterationb-p
vergencs.
o globaumial-and-r or heuristic a
isethh the SQethod. A simp
ristic me
arting poin
nv
the
ndom
that
ecting a set o
of these startin
oints is closhe minimWhile global m
mum is not asred, the probability of obtaining better
minimum increases with the number of starting points.
Examining Equations (23) and (24), one can easily real-
ize that the gradients of objective and constraint func-
tions,
 
f
x and
 
ik
g
xare the most dominant
fac tor in convergence of the optimization procedure. As
it mentioned above, in order to reach the global opti-
mum the procedure should be repeated for different
starting points, requiring an efficient computations of the
function gradients. In the present work, the analytical
gradients of objective and constraint functions have been
derived and implemented in SQP method for all the op-
timization problems unless otherwise specified.
Solving the optimization problem described above,
leads to the maximum power output for normal, sub-
speed and-speed roes. Ta ble 1 provides the
maximum power output for normal speed (54 r.p.m), in
which one can conclude that the maximum power obtains
at wind speed of 16 m/s and 3.12
  Table 2 pro-
vides the maximum power output for rotor sub-speed (54
× 0.95 r.p.m), in which, once again, the maximum power
obtains at wind speed of 16 m/s and 1.84
  and the
priority in control is similar that of normal rotor speed.
For super-speed of rotor (54 × 1.05 r.p.m), the maximum
power achieves at 4.39
  as given in Tabl
supertor valu
onverter,
pitch an
ined fixed to
nominal speed (54 r.p.m)when the wind speed is varied
/srespectively. Fig-
e 3. It is
worth noting that to obtain the maximum power; the pri-
ority in control for normal, sub-speed and super-speed is
always recommended as: Pitch, RSC, GSC.
5. Modeling and Simulation
By simulating the DFIG that it’s connected directly to the
network through the stator, and controlled by its rotor
through a PWM direct torque control c based on
a vector control approach via stator flux. The simultane-
ous control methodology is implemented and simulated
on DFIG based ongle control, rotor-side con-
verter (RSC) control and grid-side converter (GSC) con-
trol. Figure 6 shows the regulated speed of GFIG ro-
tor,and it is shown that the rotor speed rema
in step down and up to 13 m/sand 15 m
ure 7 shows the rotor current regulated in DFIG by using
PWM converter on rotor side. Finally, the delivered cur-
rent (stator current) to grid side is presented in Figure 8,
in which one can see that after 0.2 second the delivered
power (current) to the grid is remained constant, and it’s
calculated RMSE value is 0.43% after steady state time
(after 500 ms) by concurrent using of RSC, GSC, Pitch
and PWM control together.
6. Conclusion
A multidisciplinary optimization technique has been
00.2 0.4 0.6 0.81
0
10
20
30
40
50
60
Tim e ( sec)
peed (Wind Sm/s) and R otor S peed ( rpm )
Wind S pee d
Rotor Speed
Figure 6. Rotor speed of DFIG in variable wind speed.
00.2 0.4 0.6 0.81
-150
-100
-50
0
50
100
150
Time
(
Second
)
Current (A)
Rotor Current
cur r ent in a pha se
cur r ent in b pha se
cu r rent in c pha se
Figure 7. Regulated rotor c urrent in DFIG in variable wind
speed.
00.20.4 0.6 0.81
-15 0
-10 0
-50
0
50
100
150
Time (Second)
Current (A)
Stator Current
cu r rent in a p hase
cu r rent in b p hase
cu r r en t in c phase
Figure 8. Stator current in DFIG in variable wind speed.

S. J. FATTAHI, A. ZABIHOLLAH
97
presented to maximize the power generated of Do
Fed Induction Generator (DFIG) wind turbine. The most
common constraints, namely, active power, reactive po-
wer, power factor, pitch angle and grid side variable
quency converter has been taken into consideration.
design and operation optimization problems have been
considered and solved simultaneously to determine the
optimal pitch angle and power control. It is observed
optimizing the pitch angle is not sufficient to maxi
the generated power. It may cause an undesired power
quality in grid, requiring a multi-disciplinary optimiza-
tion approach to obtain the optimum power quality.
ubly
fre-
The
that
mize
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