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Disturbance Accommodating LQR Method Based Pitch Control Strategy for

Wind Turbines

Jianlin Li1

,Hongyan Xu2

,Lei Zhang1

, Zhuying1

, Shuju,Hu1

1Institute Electrical Engineering of Chinese Academy of Science, Beijing, 100190,China

2Hebei polytechnic university,Tangshan 063009,China

ljl@mail.iee.ac.cn

Abstract

A disturbance accommodating Linear Quadratic

Regulator (LQR) method was applied in pitch control

system to achieve good performance. The disturbance can

be estimate by designing state estimator, and a feedback

was added into the input to eliminate disturbance effect.

The feed back matrix was calculated in accordance toLQR theory. A wind turbine dynamic modal was set up,

and simulation of the control system was preformed based

on Matlab7.1/simulink. The simulation results show that

the controller ensure pitch control actuator little fatigue,

and has smaller overshoot. The proposed method is has

better performance and easy to realize.

1. Introduction

Since the 1990s, the wind energy industry has been

growing rapidly. The wind power generation technology

had developed from stall-controlled to variable speed

pitch regulated. And wind turbine has demanded betterperformance of controller [1-3].

With the increasing of capacity of wind turbines, pitch-

control technique of large wind turbine has become a key

technique of wind energy. Pitch-control can not only

output power steadily, but also make wind turbine have

better starting and braking performance. Additionally,

using optimized control algorithm can lower load and

torque ripple of wind turbine, extending the life of wind

turbine. At present, in China most wind turbine is

controlled by PID algorithm, which cannot have a satisfy

effect. Abroad researchers have proposed many advanced

control theory and strategy about pitch-control. Senjyu, T

et al had applied GPC control method to pitch-control [1].

This is wind speed predict model based on average wind

speed and standard deviation, having pitch controlled

according to predicted wind speed.

This paper analyzed the process of pitch-control, built a

wind turbine dynamic model and studied a LQR optimal

control algorithm based on disturbance correction.

Adopting this kind of LQR algorithm to perform pitch-

control can not only optimize output power, but also

decrease variable propeller pitch mechanism wear. At last,

this dynamic model was simulated in Matlab 7.1/simulink.

2 Simulation of wind turbineThe equivalent model of wind turbine is shown in Fig.

1

genJ

rotJ

rTshaftT

genT

gen

dK

dC

frotK

fgenK

Fig .1 Wind Turbine model

The aerodynamic torque gained by blade from wind

energy [5]:

2

2

),(21 V

CRT Pr

= (1)

in which,

is the density of airKg/m3,R is the

radius of rotor (m),V is the wind speedm/s,

is the

pitch angle degree , is tip speed ratio

VR /= , is the rotor speed, pC

is power conversion

coefficient, which indicates wind turbines efficiency of

converting wind energy to usable mechanism power. pC

is function of tip speed ratio and blade pitch angle .

pC can be written as [5, 6]

ie

iP

C

5.22

)54.0116

(22.0),(

=

(2)

in which i

satisfies

13

035.0

08.0

11

+

+

=

i Although wind turbine is a nonlinear model, at some

Second International Symposium on Intelligent Information Technology Application

978-0-7695-3497-8/08 $25.00 2008 IEEE

DOI 10.1109/IITA.2008.247

766

Second International Symposium on Intelligent Information Technology Application

978-0-7695-3497-8/08 $25.00 2008 IEEE

DOI 10.1109/IITA.2008.247

766

Second International Symposium on Intelligent Information Technology Application

978-0-7695-3497-8/08 $25.00 2008 IEEE

DOI 10.1109/IITA.2008.247

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point near by it can be treated as linear model. Linearizing

torque Tr at point),,( 000 V nearby :

+++= VVTT rr ),,( 000 (3)

In which, 0=

, 0VVV =

, 0 =

)0,0,0()0,0,0()0,0,0(,,

=

=

=V

r

V

r

V

r TT

V

T

Let state variable q1 and q2 are blade angle and rotor

angle respectively (calculated in low speed shaft. Tshaft is

the reaction torque on the shaft. Then:

)()( 2121 qqCqqKT ddshaft += (4)

)()( 2121 qqCqqKT ddshaft += (5)

=rotfshaftrrot

KTTqJ 1 (6)

gengenfgenshaftgenKTTqJ = 2

(7)

Above, dK

is elastic coefficient of propeller shaft, dC

is damping coefficient on propeller shaft, rotJ and genJ

are rotation inera of low speed side and generator (

calculated in low speed side), rotfK

rotfK are friction

coefficient of low speed side and high speed side

respectively. 0shaftT

is counter torque at working point

),,( 000 V . The speed acceleration is 0, so

00

000 ),,( += rotfshaftr KTVT (8)Then:

=rotfshaftrrot

KTTqJ 1 (9)

Let

23

212

11

)(

qx

qqKx

qx

d

=

=

=

Then:

VxCxxKCxJ drotfdrot +++= 3211 )( (10)

)( 312 xxKx d = (11)

According to the torque equation of generation

gengenfddgenTxKCxxCxJ ++= 3213 )(

(12)

In state equation form

+=

++=

uxy

uuxx D

DC

BA

(13)

where

=

gen

fd

gengen

d

dd

rot

d

rotrot

fd

J

KC

JJ

C

KKJ

C

JJ

KC

gen

rot

1

0

1)(

A

=

gen

rot

J

J

10

00

0

B

=

0

0

rotJ

[ ]100=C

0D =

input genTu = ,

disturbance quantityVuD =

At present pitch actuator has hydraulic and electric two

forms. For simplicity, pitch actuator can be simplified to a

first-order inertia model, no matter it is hydraulic or

electric actuator. The pitch actuator transmission functionis:

1

1)(

+=

ssAct

(14)

3 Pitch control strategy based on LQRAfter connected to the grid, wind turbine can work in

two modes: one mode is when wind speed is slower thatrated wind speed, another is when faster. When wind

speed is slow, wind turbine output power is smaller than

rated power. So the pitch angle is set to 0 and wind

turbine runs in optimal tip speed by controlling generator

speed, in order to absorb as much wind energy as possible.

While wind speed is faster than rated speed, the outputpower will excess rated power. Because the electrical and

mechanical limitation of wind turbine, the rotator speed

and output power cannot excess rated value. So, when

output power is larger than rated power, pitch angleshould be increased to smaller wind energy utilization

efficiency. When output power is smaller than rated

power, pitch angle will be decreased to maintain theoutput power at about rated power nearby.

Nowadays variance speed pitch-control wind turbine

always has its electromagnetic torque given value constant,

maintaining output power by regulating generator speed.

The most common method is adopting PI control to

regulate generator speed. This method is simple and easilyapplied in engineering. However, PI control may have

overshoot problems, which makes pitch actuator

complicated and easily fatigued.LQR is linear quadrics regular, whose control object is

linear system given by state space form in modern control

theory. And its object function is object states andquadrics function which controls input. LQR optimalcontrol is designing state feedback controller G. In order

to minimize the quadrics object function J , and also G is

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decided only by weight matrix Q and R, the selection of Q

and R is very important. LQR theory is a relatively maturetheory in modern control theory. It provided an efficient

analysis method for multi-variable feedback system.

Object function J included state variable and input

variable, which requires state variable and input variableto be small. In the pitch-control system, input value is the

error of pitch angle. Because of large inertia of blade,

rapid pitch-control would damage pitch regulatedmechanism and aggravate the friction of pitch-controlshaft. So, having some limitation to input energy will be

reasonable. Additionally, choosing torque variation as

state variable can suppress torque ripple as much as

possible in LQR optimal control. Then the life of windturbine can be extanded.

Set linear length-determined systems state function as :

=

++=

xy

uuxx D

C

BA

(15)

Object function

[ ] +=ft

t

TTdttRututQxtxJ

0)()()()(

2

1

(16)

where Q is positive semidefinite matrixR is positive

definite matrixQ and R are weighted matrix for state

variable and input variable respectively. x(t) is n-

dimension state variable u(t) is m-dimension input

variable. According to control theory, in order tominimize object function, optimal control is:

)()( tGxtu = PBRG

T1=

Where P is Riccati function

01 =+ QPBPBRPAPA TT (17)

Positive definite symmetric solution. The LQR controldiagram is shown as Fig. 2. In engineering application,

state variable cannot be measured usually. So it needs to

design a state observer to estimate state variable value.Fig. 2(b) is the diagram used in actual application.

x

Fig. 2 LQR control theory diagramBecause there is a disturbance variable ud in wind

turbine model, only using LQR control cannot regulate

generator speed very well. And the disturbance fromdisturbance variable should be minimized as much as

possible. Disturbance Accommodating Control (DAC) is a

good method to solve this problem. DAC control method

was proposed by Johnson (1976), DAC control is a

reconstructed disturbance model method based on stateobserver. The disturbance variable is reconstructed and is

part of state feedback, can decrease or neutralize the

disturbance effect. This paper adopted LQR method with

DAC, which means that through LQR optimal controlhaving a optimal feedback matrix G, then using DAC

method to estimate disturbance variable and eliminatingthe disturbance from disturbance variable. DAC diagram

is shown as Fig. 3Using state observer to estimate statevariable and disturbance variable, disturbance can be

eliminated.

x

Dz

uinput

Fig. 3 disturbance correction control diagram

Presume the disturbance variable has forms as below:

==

=

DDDD

DD

zztztz

tzu

0)0();()(

)(

F

(18)

z0D is unknowpresume and F is already known.

According to DAC control theory, state feedback should

contain the feedback of disturbance:

)()()( tztxtu DDGG += (19)

Replace u(t) in the state function with the up function,

we have:)()()()()( tztxtx DBGBGA D +++= (20)

To elimilate the disturbance, it requires

0BGD =+ then it can be considered as a system

without disturbance. If system cannot

satisfy 0BGD =+ , then choosing GD to make

BGD + minimum.Because state variable x(t) and zD(t) cannot be

measured directly, designing state observer is needed to

predict state variable and disturbance variable. Wind

turbines state observers math model:

==

+++=

0)0();(

))()(()()()(

xtxy

tytytututxx d

C

KBA x

(21)Disturbance observer:

=

+=

)(

))()(()(

tzu

tytytzz

DD

DD

KF D

(22)Designing appropriate Kx and KD can let:

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0))()(()(

0))()(()(

==

==

tztzimlteiml

txtximlteiml

DDt

Dt

tx

t

(23)

Disturbance state function can be written as:

)()()( tete CKA= (24)

Where,[ ]TTDTx eete =)(

=

F0

AA

[ ]0CC =

=

D

x

K

KK

According to the formula above, errors expression can

be solved:

)0()( )( eete tCKA= (25)

If system)( CA

is measurable, then)( CKA

can

have any poles configuration, letting)(te

damping to 0

rapidly. Feedback control principal became:)()()( tztxtu DDGG += (26)

Simulation result

To verify the control performance of LQR algorithm

based on disturbance correlation, a numeric simulation

was performanced on Matlab 7.1/simulink. The wind

turbine model parameter is : rated power 650kW, rotor

diameter 43m, gear box transmission ratio 43.16, rotor

rated speed 42 rpm. LQR algorithm based on disturbance

correction and PI regulation method were simulated.

Choosing work point atsmV /170 =

rpm420 =

53.130 =

in LQR algorithm and linearizing at this point.Then wind turbines state function is function (13), where:

=

0624.01056.11056.1

1069.201069.2

10108.310108.3198.0

54

77

56

A

=

0

0

105.7 3

B

choosing 1=R

=

5000

01010

001

12Q

From matrix A, B, Q and R, state feedback matrix:

[ ]1.3289-101.69052.2219 -8=K In the simulation, wind speed stepped from 17m/s to

18m/s at t=0 moment. In PI regulation, Kp=8, KI=1.5,

simulation result is shown as Fig. 4.

0 20 40 60 80

1810

1815

1820

1825

1830

1835

1840

t/s

generato

rspeed(rpm)

LQR

PI

(a) Generator rotating speed

0 20 40 60 803.4

3.45

3.5

3.55

3.6

3.65

3.7

3.75

3.8x 10

4

t/s

drive-traintorsionals

pringforce(N)

LQR

PI

(b) Elastic force on drive link

0 20 40 60 8013

14

15

16

17

18

t/s

pitch

angle(

)

LQR

PI

(c)Pitch angle

Fig .4 simulation waveform of LQR algorithm base on disturbance andPI control

From the simulation we can tell, PI regulation method

has a lager overshoot, while LQR algorithm has a muchsmaller one. In Fig. 4(b), LQR algorithm can decrease the

elastic force on drive link. In Fig.4(c), after adopting LQR

algorithm, the overshoot can be very small, which can

reduce the action of pitch actuator. While PI regulation

has a larger overshoot, pitch angle fluctuated for a

moment, which is harmful for pitch actuator.

ConclusionSo as to enhance pitch control performance of large

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wind turbine, this paper constructed wind turbines

dynamics model, giving a LQR pitch-control algorithm

based on disturbance correction according to LQR control

theory. This method can reduce pitch actuators

movement efficiently and has good control performance

on generator rotate speed. The simulation results showed

that this method has good dynamics performance and it issimple and effective. So it has the great potential to be

applied into engineering.

AcknowledgementIt is a project supported by China Postdoctoral ScienceFoundation (No. 20060390092).References[1] Senjyu, T.; Sakamoto, R.; Urasaki, N.; Funabashi, T.

Output power leveling of wind turbine Generator for alloperating regions by pitch angle control [J]. Energy

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Output Power Leveling of Wind Turbine Generator for AllOperating Regions by Pitch Angle Control[C]. Power

Engineering Society General Meeting, 2005. IEEE, 2005, 1,pp. 45-52.

[3] Ye Hangzhi. Control technology of wind turbine [M]

Beijing: Mechanics Industry Press2002

[4] Fu Wangbao, Zhao Dongli, Pan Lei. Cutting-in Control ofthe VSCF Wind-power Generator Based on Auto-

disturbance Rejection Controller [J]. Proceedings of the

CSEE,2006, 26(3), pp.13-18.

[5] Rajib Datta V. T. Ranganathan. Variable-Speed WindPower Generation Using Doubly Fed Wound RotorInduction MachineA Comparison With AlternativeSchemes[J]. IEEE Trans. on Energy Conversion, 2002, 17,

pp. 414-421.

[6] Kanellos, F.D.; Hatziargyriou, N.D. A new control scheme

for variable speed wind turbines using neural networks[C].Power Engineering Society Winter Meeting, 2002. IEEE,

2002, 1: 260-365.[7] Schinas N. A., Vovos N. A., Giannakopoulos G. B. An

Autonomous System Supplied Only by a Pitch-Controlled

Variable-Speed Wind Turbine. IEEE Trans. on EnergyConversion, 2006, 21(1), pp. 1-7

[8] H.S.Ko , K.Y. Lee, H.C. Kim. An intelligent based LQRcontroller design to power system stabilization[J]. ElectricPower Systems Research. vol. 71. 2004, pp. 19

[9] Chao Hu, Max Qinghu Meng, Peter Xiaoping Liu ,Observer Based LQR Control of Shaping Process ofAutomobile Belt. Proceedings of the 5th World Congresson Intelligent Control. June 2004, China, pp. 3310-3314.

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