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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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