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Control for a Variable Speed Wind Turbine
equipped with a Permanent Magnet
Synchronous Generator (PMSG)
Johanna Salazar1, Fernando Tadeo1, Kritchai Witheephanich2, Martin Hayes2,
Cesar de Prada1
1 UVA, University of Valladolid, Spain
johanna@autom.uva.es, fernando@autom.uva.es, prada@autom.uva.es
2 UL, University of Limerick, Ireland
Martin.j.hayes@ul.ie, kritchai.witheephanich@ul.ie
Abstract In wind turbine systems, one of the operational problems is the
unpredictable factor of wind. Hence, quality of produced energy becomes an
important problem in wind turbine systems. Several control techniques have been
applied to improve the quality of power generated from wind turbines. Pitch
control is the most efficient and popular power control method, especially for
variable-speed wind turbines. In this paper, a traditional PI controller is developed.
Simulation was carried out considering realistic values.
1 Introduction
Nowadays, the interest towards wind energy is increasing. The advances in wind
turbine (WT) technology made necessary the design of efficient powerful control
system in order to improve wind turbine behaviour, namely to make them more
reliable. Hence, a good regulation of the electrical power will be the main
objective.
Variable speed wind turbine has higher energy yields and lower component
stress than fixed speed wind turbines. There are usually two controllers for this
variable-speed wind turbine, which are cross-coupled: below-rated power and
above-rated power. In below rated value, the speed controller continuously adjusts
the speed of the rotor to maintain the tip speed ratio constant at the level that gives
the maximum power coefficient. Generator torque provides the control input to
vary the rotor speed, and blade pitch angle is held constant. In above-rated value,
the main objective is to keep a constant power output. This is generally achieved
by keeping generator torque constant and varying blade pitch angle.
2 Johanna Salazar et al.
The purpose of the pitch angle control should be to regulate the aerodynamic
power in high-wind-speed region to keep safe operation. The pitch angle is almost
constant in other wind-speed regions. Hence, the objective of pitch control is to
maintain a rated rotational speed and output power. The highly nonlinear
aerodynamics of variable speed wind turbine makes it difficult.
Most control strategies are based on a linear model of wind turbine which can
be obtained by linearizing the nonlinear model at a specific operating point [1].
Adin [2] designed a proportional-integral (PI) controller based on the linearized
model of wind turbine. However, the controller does not provide a good
performance when the working point deviates from operating point [3], because
the numbers of pitch gains are usually the partial derivatives of the rotor
aerodynamic torque respect to blade pitch and these gains will change with
different wind speeds and pitch angles. Lescher [4] designed a gain scheduling
controller, which gains change with the wind speed or other parameters. The
author reported satisfactory simulation results, but the controller was not verified
on the nonlinear model. Ahmet Serdar [5] proposed a neural network controller
and the experiments shows a convincing performance, but it seems too
complicated to be implemented in real WTs.
In this paper, a proportional-integral (PI) controller for below-rated power and
a gain scheduling controller for above-rated power are proposed and verified on
the nonlinear model.
The paper is organized as follows: Section II describes the modelling of the
wind turbine. Section III wind turbine linearized model. Section IV conventional
PI controller based on a linear turbine model is analyzed. Section V simulations
show the performances of PI and gain scheduling controller.
2 Wind Turbine Model
The description and the modeling of a wind turbine with PMSG are described
throughout this section (Figure 1). The mechanical components of the Wind
Turbine System (wind turbine rotor and drive train) will be briefly presented.
Fig. 1. Wind Turbine Model
Control for a Variable Wind Turbine 3
2.1 Wind Turbine Model
The power in the wind is known to be proportional to the cube of the wind speed,
so it may be expressed as
3
w
ρAυ
2
1
w
P
=
(1)
where ρ is the air density, A is the area swept by the blades and υw is the wind
speed. However, a wind turbine can only extract a fraction of the power, which is
limited by the Betz limit (maximum 59%). This fraction is described by a power
coefficient, Cp, which is a function of the blade pitch angle β and the tip speed
ratio λ. Therefore the mechanical power of the wind turbine extracted from the
wind by the turbine is
( )
3
w
ρAυλβ,
p
C
2
1
M
P
=
(2)
where the tip speed ratio
λ
is defined as the ratio between the blade tip speed and
the wind speed
υ
w:
w
υ
R
T
λ
ω
=
(3)
ω
T is the rotational speed and R is the radius of the blades.
In this paper, the power coefficient is given by [6]
( )
0.00571λ
21
exp200.4β
230
0.71βλ,
p
C
+
−
−−=
σσ
(4)
where
1
1
3
0350
080
1
−
+
−
+
=
β
βλ
σ
.
.
(5)
Thus, any changes in the rotor or wind speeds induce changes in the tip speed
ratio, leading to power coefficient variation. In this way, the generated power is
affected. Figure 2 shows a typical CP -λ curve for a wind turbine that follows (4).
The wind turbine power coefficient is maximized (0.4522) for a tip-speed ratio of
λopt=6.96 when the blades pitch angle is β=3°.
4 Johanna Salazar et al.
Fig. 2. Power coefficient versus tip speed ratio
The hydraulic actuator can be modelled as a first order system as follows:
1)s
servo
(τ
1
ref
β
β
+
=
, (6)
2.2. Drive Train Model
The mechanical system of the wind turbine can be simply modeled with the one
mass model given by [7]:
Tr
F
e
T
M
T
td
T
ωd
total
J
ω
−−=
, (7)
where Jtotal=JT+Jg is the inertia constant of the whole drive train, with JT and Jg the
inertia constants of the turbine and the generator, respectively; ωT is the rotor
speed; Fr is the friction coefficient; Te is the generator electromagnetic torque and
the mechanical torque of the turbine TM is given by
( )
T
ω
3
w
ρAυβλ,
p
C
2
1
T
ω
M
P
M
T
==
(8)
2.3. Permanent Magnet Synchronous Generator Model
The rotor excitation of the Permanent Magnet Synchronous Generator (PMSG) is
assumed to be constant, so its electrical model in the synchronous reference frame
is given by [8,9]:
q
i
e
ω
s
L
d
i
s
R
d
u
td
d
id
s
L
+−=
(9)
f
ψ
e
ω
d
i
e
ω
s
L
q
i
s
R
q
u
td
q
id
s
L
+−−=
(10)
where subscripts d and q refer to the physical quantities that have been
transformed into the (d,q) synchronous rotating reference frame; Rs is the stator
resistance; Ls is the inductances of the stator; ud and uq are, respectively, the d and
q axis components of stator voltage; id and iq are, respectively, the d and q axis
components of stator current; ψf is the permanent magnetic flux and the electrical
rotating speed ωe is given by:
T
ω
p
n
e
ω
=
(11)
Control for a Variable Wind Turbine 5
where np is the number of pole pairs. The power equations are given by
( )
q
i
q
u
d
i
d
u
2
3
P
+=
(12)
( )
q
i
d
u
d
i
q
u
2
3
Q
−=
(13)
where P and Q are the output active and reactive powers, respectively. The
electromagnetic torque Te can be derived from
q
i
f
ψ
p
n
2
3
e
T
=
(14)
2.4. Current Controller Model
The current control scheme of the generator side converter is show in Figure 3.
This control is based on projections which transform a three phase time and speed
dependent system into a two co-ordinate (d and q co-ordinates) time invariant
system. These projections lead to a structure similar to that of a DC control that
makes easier AC control [10].
Fig. 3. Current controller Model
In order to design independent controllers for the two coordinates, the influences
of the q axis on the d axis component, and vice versa, must be eliminated, see
Figure 4. For this the decoupling voltages udref and uqref are given by [11]
q
i
e
ω
s
L
d
u
dref
u
−=
ˆ
(15)
f
ψ
e
ω
d
i
e
ω
s
L
q
u
qref
u
−+=
ˆ
(16)
6 Johanna Salazar et al.
Fig. 4. Decoupling between d and q axis
These decoupling voltages are added to the current controller outputs, resulting in
the control signal for the PWM-rectifier. In order to combine a fast response of the
controlled variable to a change of the set point with zero steady state deviation,
proportional integral (PI) current controllers are chosen. Control equations are
given by:
d
i
dref
i
td
1
xd
−=
(17)
q
i
qref
i
td
2
xd
−=
(18)
1
x
i1
K
d
Δi
p1
K
d
U
+=
ˆ
(19)
2
x
i2
K
q
Δi
p2
K
q
U
+=
ˆ
(20)
The required d-q components of the rectifier voltage vector are given by:
qs
i
e
ω
s
L
1
x
i1
K
d
Δi
p1
K
dref
u
−+=
(21)
f
ψ
e
ω
d
i
e
ω
s
L
2
x
i2
K
q
Δi
p2
K
qref
u
−++=
(22)
The stator current reference in d-axis idref is maintained at zero, for producing max-
imum torque, due to the non-saliency of the generator. The stator current refer-
ence in q-axis iqref is calculated from the reference torque Teref as follows.
eref
T
f
ψ
p
3n
2
qref
i
=
(23)
Considering dynamic response of power converter with Space-vector modulation
(SVM) is faster than the rest of the system. From equation 9, 10, 15 and 16 results
the transfer function of the stator winding (Figure 5)
Control for a Variable Wind Turbine 7
)(
ˆˆ s
Rs
s
L
q
u
q
i
d
u
d
i
+
==
1
(24)
Fig. 5. Current controller scheme considering d and q axis
Conventional design techniques are used to calculate the controller parameters
(Pole Placement). Choosing a damping factor ξ = 0,7448 and ωn= 134,2636, the
resulting controller is Kp= 9,6 and Ki= 1334,4.
3 Linearized Wind Turbine Model
The approach to design a traditional proportional-integral (PI) controller requires
that the non-linear turbine dynamics are linearized at a specified operating point.
The linearization of the pitch actuator equation (6) results in the following model
β∆β∆β∆τ
−=
refservo
(25)
whereas the linearization of the turbine equation (7) gives the following
differential equation
Tr
F
q
i
T
w
υ
T
J
fp
n
ω∆∆ω∆γ∆αβ∆ςω∆
ψ
−−++=
2
3
(26)
with the linearization coefficients are given by
OP
q
dλ
dc
w
υ
4
ρπR
2
1
γ
=
(27)
−
=
OP
q
q
dλ
dc
λc2
w
υ
3
ρπR
2
1
α
(28)
OP
dβ
q
dC
2
w
υ
3
ρπR
2
1
ς
=
(29)
Here,
w
υand
T
∆β∆ω∆
,
represent the deviations of
w
υand
T
βω
,
respec-
tively.
The linearization of the PMSG equations (9, 10) results in the following model
8 Johanna Salazar et al.
1
x
i
K
d
i
p
K
s
R
d
i
s
L
∆∆∆
++−=
(30)
2
3
2
x
i
K
q
i
p
K
s
R
eref
T
fp
n
p
K
q
i
s
L
∆∆∆
ψ
∆
++−=
(31)
d
i
x
∆
∆
−=
1
(32)
q
i
eref
T
fp
n
x
∆
∆
ψ
∆
−=
3
2
2
(33)
Using the linearization of the pitch actuator (25), the turbine (26) and the
generator (30, 33) equations give the following continuous-time linear system:
BuAxx
+=
(34)
DuCxy
+=
(35)
where
=
2
1
x
x
i
i
x
d
q
T
∆
∆
∆
∆
β∆
ω∆
and
=
w
υ
ref
eref
T
u
∆
β∆
∆
Considering A, B, C and D defined as follows:
Control for a Variable Wind Turbine 9
−−
−−
−−
−
−
−
=
000100
001000
0000
0000
0000
1
0
000
2
3
s
i
s
sp
s
i
s
sp
servo
T
fp
tt
r
L
K
L
RK
L
K
L
RK
J
n
JJ
F
A
τ
ψ
ςγ
(36)
=
00
3
2
000
000
00
3
2
0
1
0
00
fp
sfp
p
servo
T
n
Ln
K
J
B
ψ
ψ
τ
α
[ ]
[ ]
000
000001
=
=
D
C
4 Proposed Control Scheme
The strategy of the proposed control scheme is based on dividing the control
strategies into three distinct regions (Figure 6):
−Region I consists of low wind speeds and is below the rated turbine power,
so the turbine runs at the maximum efficiency to extract all power (In other
words, the turbine controls with optimization in mind);
−On the other hand, Region III consists of high wind speeds and is at the
rated turbine power. The turbine then operates to limit the generated power;
In between,
10 Johanna Salazar et al.
−Region II is a transition region, mainly concerned with keeping rotor torque
and noise low [12].
−Outside these regions (when the wind speed is either lower than cut-in
speed or higher than the cut-out speed), the turbine does not generate
power, and the pitch angle is usually set to 90°.
Low wind speed: As it has been mentioned, the main objective is to capture as
much power as possible from the wind. A variable speed regulation is used to
keep the turbine at it is most effective operation. Here the pitch angle is kept at the
lower possible value (around 3º), which leads cp to the most efficient operating
point. This mode can be used up to certain wind speed (for example, 9 m/s). The
basic scheme is shown in Figure 7.
Fig. 6. Operating regions of wind turbines
Fig. 7. Closed-loop system at low wind speeds
To ensure maximal energy yield, the reference speed is set such that the tip speed
ratio, λ is maintained at its optimal value, λopt according to the following equation:
R
wopt
Tref
υλ
ω
=
(32)
For this a good estimation of the wind speed is required: Wind measurements by
anemometers do not sense the same turbulence; moreover the wind turbine and the
gusts arrive with a time delay.
Middle Wind Speed: The generator is controlled to keep nominal rotational
speed with a pitch angle kept at a small constant value (generally around 3º). Here
the reference will be the nominal rotational speed of the generator. Usually this in-
Control for a Variable Wind Turbine 11
terval ends when nominal generator power is reached, so in this region turbine op-
erate below rated power. The basic scheme is shown in Figure 8.
Fig. 8. Closed-loop system at middle wind speed
The Transfer function in low and middle wind speed is given by
( )
r
Fs
T
J
I
Ks
p
K
s
Rs
s
L
I
K
p
sK
eref
T
T
+−+++
+
=
γ
∆
ω∆
2
(33)
Selection of the operating point is critical to ensure stability in the system which
depends on the derivative of Cq respect to λ. Considering this derivative, the
typical CP -λ curve can be divided into two regions: stable and unstable regions as
shown in Figure 9. The controller has to be designed to keep inside the stable
region.
Fig. 9. Unstable and stable regions over Cp- λ and Cq- λ curves
High wind speed: The main objective is to keep the rotational speed and
especially the generated power as close as possible to the nominal. The pitch angle
is regulated to give the rated effect (For the wind turbine used as example, this in -
terval starts at wind speeds around 11 m/s and finishes at 27m/s. The electric
torque reference is defined to produce the rated electric power (
rated
eeref
TT
=
).
The basic scheme is shown in Figure 10.
12 Johanna Salazar et al.
Fig. 10. Closed-loop system at high wind speeds
The Transfer function is given by
))(( r
Fs
T
Js
servoref
T
+−+
=
γτ ς
β∆
ω∆
1
(34)
The whole speed area is not able to operate using only one Proportional Integrator
(PI) controller. The controller can only work well in the neighbourhood of the
linearization point. When the turbine operating point deviates, the controller fails
to provide acceptable performance. In consequence, the speed controller is
implemented as a gain scheduling controller, where the controller gains are
scheduled depending on blade pitch angle. The transfer function (34) is used to
determine the desired proportional gain (see Figure 10). For the installation used
for demonstration, the scheduling is given in (34)-(36). The operating points used
during controller design are shown in Figure 11.
20 25 30 35 40 45 50
0.8
1
1.2
1.4
1.6
1.8
2
Pitch Angle [ºC ]
KP Value
20 25 30 35 40 45 50
0.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24
0.26
0.28
0.3
Pitch Angle [ºC]
KI Value
Fig. 10. Variations of the controller gains based on the pitch angle
( ) ( )
s
i
K
p
KController
β
β
+=
(35)
( )
115820280 ,,
+−=
ββ
p
K
(36)
( )
3542000480 ,,
+−=
ββ
i
K
(37)
Control for a Variable Wind Turbine 13
0 2 4 6 8 10 12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Tip Speed Ratio
Coefficient power
Fig. 11. Operating points chosen at zone III
5 Some Results
As already mentioned in the previous section the strategy of control is normally
divided into three distinct regions: low, middle and high speeds. These regions are
now studied separately.
Low Wind Speed: Some validation experiments are shown in Figure 12 for this
region: the upper plot shows the wind speeds applied. The second plot, the rota-
tional speed: it can be seen how it increases following the wind speed. The third
plot shows the tip speed ratio, which is regulated at its optimal point. The fourth
plot shows the blade pitch angle, that is kept constant. Finally, the fifth plot shows
the electrical torque and the last the generator output power.
0 2 4 6 8 10 12 14 16 18
x 10
4
2
4
6
8
10
Wind Speed [m/s]
0 2 4 6 8 10 12 14 16 18
x 10
4
-10
-5
0
5
10
15
Time [s]
Rotational Speed [rad/s]
14 Johanna Salazar et al.
0 2 4 6 8 10 12 14 16 18
x 10
4
0
5
10
15
20
Tip Speed Ratio
0 2 4 6 8 10 12 14 16 18
x 10
4
2
2.5
3
3.5
4
Time [s]
Pitch Angle [ºC]
0 2 4 6 8 10 12 14 16 18
x 10
4
-500
0
500
1000
Torque [Nm]
0 2 4 6 8 10 12 14 16 18
x 10
4
0
5000
10000
15000
Time[s]
Output Power [W]
Fig. 12. Responses at low wind speeds
Middle Wind Speed: In Figure 13, the first plot shows wind speeds. The second
plot shows the rotational speed; which keep constant at rated value. The third plot
shows the tip speed ratio which decrease as long as wind speed increases. The
fourth plot shows the blade pitch angle which is maintained constant. The fifth
plot shows the electrical torque and the last the generator output power.
0 0.5 1 1.5 2 2.5
x 10
5
9
9.5
10
10.5
11
11.5
Wind Speed [m/s]
0 0.5 1 1.5 2 2.5
x 10
5
0
5
10
15
Time [s]
Rotational Speed [rad/s]
Control for a Variable Wind Turbine 15
0 0.5 1 1.5 2 2.5
x 10
5
5.5
6
6.5
7
Tip Speed Ratio
0 0.5 1 1.5 2 2.5
x 10
5
2
2.5
3
3.5
4
Time [s]
Pitch Angle [ºC]
0 0.5 1 1.5 2 2.5
x 10
5
800
1000
1200
1400
1600
Torque [Nm]
0 0.5 1 1.5 2 2.5
x 10
5
1.4
1.6
1.8
2
2.2
2.4 x 10
4
Time [s]
Output Power [W]
Fig. 13. Responses at middle wind speeds
High wind speed: In Figure 14, the first plot shows the simulated wind speeds.
The second plot shows the obtained rotational speed, which is regulated to be con-
stant. The third plot shows the tip speed ratio, which decreases when wind speed
increases. The fourth plot depicts the blade pitch angle, which increases when
wind speed increases. The fifth and sixth plots show respectively the electrical
torque and the generator output power: it can be sent that they are maintained at
the rated value, as desired.
16 Johanna Salazar et al.
0 0.5 1 1.5 2
x 10
5
10
15
20
25
30
Wind Speed [m/s]
0 0.5 1 1.5 2
x 10
5
14.2
14.4
14.6
14.8
15
15.2
Time [s]
Rotational Speed [rad/s]
0 0.5 1 1.5 2
x 10
5
2
3
4
5
6
Tip Speed Ratio
0 0.5 1 1.5 2
x 10
5
10
20
30
40
50
Time [s]
Pitch Angle [ºC]
0 0.5 1 1.5 2
x 10
5
1100
1200
1300
1400
1500
1600
Torque [Nm]
0 0.5 1 1.5 2
x 10
5
2.2
2.25
2.3
2.35
2.4 x 10
4
Time [s]
Output Power [W]
Fig. 14. Reponses at high wind speeds
Realistic Values: As final test we show some results that use data of wind speeds
measured at the final location (Borj Cedria, Tunisia). Considering the designed
controller for each area (low, middle and high wind speeds) within the overall pro-
posed control scheme, shown in Figure 15, the results presented in Figure 17 were
obtained. The first and second plots show wind speed and yaw angle magnitude,
which have an influence in the final value of the wind speed, as can be seen in the
Control for a Variable Wind Turbine 17
third plot (Explanation of yaw angle will be given below). The fifth and sixth plots
show the rotational and electrical torques, respectively, with values depending on
rotational speed. Notice that both variables never exceed its rated value. The sev-
enth plot show the power coefficient, which depends on blade pitch angle and tip
speed ratio value. These variables are shown in the eighth and ninth plots, respect-
ively.
Fig. 15. Overview of the overall control scheme
Yaw angle: The wind turbine is said to have a yaw error when the rotor is not per -
pendicular to the wind. Considering θw as the wind direction and θt the yaw tur-
bine angle, then the yaw error angle is the difference between θw and θt. Yaw error
angle implies that a smaller wind energy share is going to be converted, as the
wind speed that the turbine is physically able to capture follows a complex func -
tion of this error. Using a cosine function as an approximation of the aerodynamic
effects of yaw error gives
)(
cos
tw
e
V
w
θθ
υ
−
=
(38)
Fig. 16. Explanation of yaw angle error
18 Johanna Salazar et al.
Fig. 16. Responses considering realistic values
Control for a Variable Wind Turbine 19
6 Conclusion
This work has presented the dynamic model for wind turbines equipped with Per-
manent Magnet Synchronous Generator (PMSG). This model was developed for
simulation and controller design. A control scheme is proposed, based on standard
PI controllers at three operating regions, with gain scheduling. Focusing on an in-
stallation in Borj Cedria, Tunisia, controllers are designed and validated by simu-
lation using realistic data obtained from the location. Some simulation results are
provided to show that the expected responses are obtained. The future work is to
design advanced MPC controllers.
Acknowlegments
This work has been funded by the European Commission within the Sixth Frame-
work Programme (FP6-2004-INCO-MPC-3) and by MiCInn (DPI2010-21589-
C05-05). We would like to thank the rest of the groups of the EU project “Open-
Gain” for feedback and comments.
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