Design of a Wind Turbine Pitch Controller for Loads and Fatigue Reduction

Mate Jelavi1, Nedjeljko Peri1, Ivan Petrovi1, Stjepan Car2, Miroslav Maeri3

1 Faculty of Electrical Engineering and Computing, University of Zagreb, Croatia 2 Konar Electrical Engineering Institute, Zagreb, Croatia

3 Konar Power Plant and Electric Traction Engineering, Zagreb, Croatia

Contact: mate.jelavic@fer.hr Phone: +385 1 6129 713 Fax: +385 1 6129 809

1. Summary

Wind turbines are constantly growing in size and power output level. This growth results in significant increase of structural loads and fatigue that have become limiting factors in wind turbine design. It introduces a need to control and limit structural loads and fatigue during wind turbine operation. This can be done by careful control system design. In this paper we focus on variable speed pitch controlled wind turbine that is today an industrial standard. A design procedure for three types of pitch controller is described and their performance is tested by simulations. Like first classic PID controller is described that aims only at achieving satisfactory rotor speed control. This classic pitch controller is then augmented and reduction of tower oscillations is used as additional demand during controller design. Two controller concepts used for this purpose are compared SISO and full state feedback controller. Both controllers are designed by pole placement method that assures closed loop system to behave in desired manner. Desired closed loop system behavior is carefully chosen in such a way that controlled system behaves as if its structural damping has increased. It results in reduced tower motion what leads to loads and fatigue reduction. All controllers described in this paper are linear so an adaptation algorithm is needed to account for nonlinear nature of wind turbine. For this purpose we propose the use of Takagi-Sugeno fuzzy model that calculates process model output as a combination of "local" linear models identified in chosen operating points.

2. Introduction

Over last few decades wind turbines have been constantly growing in size and power output level. This growth has resulted in significant increase of structural loads and fatigue that have become limiting factors in wind turbine design. It introduces a need to control and limit structural loads and fatigue during wind turbine operation what can be done by means of sophisticated control system. The main task of wind turbine control system is to obtain continuous power production under all operating conditions determined

by various wind speeds. As turbine's power is directly proportional to its speed, power control can be done by controlling turbine's speed. In the classic control systems this was the only task that turbine controller was designed to achieve. Wind turbine's characteristics are nonlinearly dependant on wind speed [1] what results in two very different operation regions each of them placing specific demands upon wind turbine control system. During weak winds power contained in the wind is lower than the rated power output of wind turbine generator. Therefore, the main task of the control system in this region is to maximize wind turbine power output by maximizing wind energy capture. This can be done by controlling rotor speed to values that guarantee maximal wind energy capture for each wind speed. Since modern wind turbines are connected to grid using AC-DC-AC frequency converters, generator frequency is decoupled from grid frequency what enables variable speed operation. On the other hand, during strong winds power of the wind is greater than the rated power output of wind turbine generator. Therefore, the wind energy conversion has to be constrained in this region to assure generator operation without overloading. Very efficient method for constraining wind energy conversion, which has become an industrial standard is pitching the rotor blades around their longitudinal axis what deteriorates their aerodynamic efficiency and therefore only a part of wind energy is used for driving the generator.

The borderline between two operation regions is the lowest wind speed at which turbine generator reaches its rated power output. This wind speed is termed rated wind speed [1]. Hence the wind turbine operation regions are known as below and above rated regions.

In modern wind turbines rotor speed control remains the priority but reduction of structural loads and fatigue appears as additional demand for the controller. So previously described classic control system has to be augmented to fulfill these additional demands.

In this paper we address the problem of tower oscillations that is becoming more and more pronounced as wind turbine towers grow in height. Firstly we describe the classic approach for controller design in above rated operation region and analyze influence of its actions on tower oscillations. Then two methods for reducing tower oscillations are described and analyzed by simulations. All described

controllers are linear so a method for adapting their parameters to accommodate for the system's nonlinearity is proposed at the end.

The paper is organized as follows. Section 3 gives a general description of the wind turbine control system. Section 4 introduces mathematical model of the wind turbine that is used for controller design. Section 5 describes the design of PID controller which is nowadays still a standard solution for wind turbine control. In section 6 and 7 two alternative control concepts are proposed and their performances are compared with the performance of PID controller. In section 8 a method for adapting controller's parameters based on Takagi-Sugeno fuzzy model is proposed. Conclusions are given in section 9.

3. Wind turbine control system

The principle scheme of wind turbine speed control system is shown in fig. 1. As it can be seen in this figure turbine speed can be influenced and thus controlled by two means by generator electromagnetic torque Mg which opposes rotor driving torque Mr and by pitch angle which alters the wind energy conversion.

Fig. 1: Principle scheme of wind turbine control

system For this reason turbine speed control system

consists of two control loops: torque control loop and pitch control loop. Those control loops operate simultaneously but depending on operation region one of them is dominant. In the below rated operation region the torque control loop is used to control turbine speed to values that will result in maximal wind power capture. This control loop is not in the scope of this paper. Details on its specifics can be found in e.g. [2]. In the above rated region this control loop just holds generator torque at its rated value.

The pitch control loop is used for setting the adequate pitch angle that will keep turbine speed at its reference value under all operating conditions determined by various winds. Below rated wind speed this loop sets pitch angle to value that assures maximal wind power capture which is usually around 0o. In this paper we assume that all blades have the same pitch angle what is known as "collective pitch". Controller in this loop, although used to control turbine speed, is commonly termed pitch controller. Blade positioning is mostly done using electrical servo drives that rotate blades by means of gearboxes and slewing rings. Position control of servo drives is usually achieved using frequency converters. This

control loop design is rather simple and is not in the scope of the paper. For simulation analysis it is modeled as second order system:

( ) ( )( )2

2 22n

SDref n n

sG s

s s s

= = + +

. (1)

In this paper we use n = 6.28 rad/s and 2 / 2 = what is fairly good approximation of real pitch positioning system.

4. Wind turbine model

The first step in control system design is construction of a suitable process model for wind turbine in scope. Wind energy conversion process is highly nonlinear and difficult for mathematical description. It can be described quite well using combined blade element and momentum theory [1]. However this approach yields implicit mathematical expressions that can only be solved iteratively. This form, although very common in simulation tools, is not suitable for controller design. So in this paper we use another somewhat simplified mathematical model that is usual in the literature dealing with controller design. Here we describe it briefly, while details on it can be found in [3] and [4].

Wind power or the power of air that moves at speed wv over the area swept by turbine rotor with radius R is given by [1]:

2 312w air w

P R v = , (2)

where air is density of air. Power contained in wind given by expression (2)

can never be completely transformed into wind turbine power and afterwards into electrical power. The amount of wind power that is converted into turbine power Pr can be described by means of performance coefficient CP [1]:

r w PP P C= . (3) The theoretical maximum for CP is determined by

the Betz' law [1] and equals 16/27. In practice wind turbines don't reach this limit but approach the value of 0.5 at best.

CP is not a constant parameter but its value is dependant on wind speed wv , rotor speed and blade pitch angle . Wind speed and rotor speed are usually bound together introducing parameter that is called tip speed ratio which represents the ratio between blade tip speed and wind speed [1]:

w

Rv = . (4)

Typical dependence of performance coefficient upon tip speed ratio with pitch angle used as a parameter is shown in fig. 2.

Fig. 2: Performance coefficient as a function of tip speed ratio

Aerodynamic torque Mr that drives wind turbine

rotor and thus generator is given by:

( )2 3 ,12

air w Prr

R v CPM

= =. (5)

Rearrangement of expression (5) yields:

( )3 2 ,12

air w Pr

R v CM

= . (6)

A quotient of performance coefficient and tip speed

ratio forms a new dimensionless parameter that is known as torque coefficient CQ [1]:

( ) ( ),, PQ CC = . (7) Having aerodynamic torque calculated according to

(6) rotor speed can easily be found using principle equation of motion:

t r g lJ M M M = , (8)

where Mg is generator electromagnetic torque, Jt is total moment of inertia of rotor and generator while Ml is loss torque. Loss torque Ml, caused mostly by friction is rather small and will be neglected here.

In this paper we consider wind turbine with generator that is directly coupled with turbine rotor. This turbine setting known as direct drive system uses synchronous multipole generator that rotates at small speed of turbine rotor. Since rotor and generator speeds are the same no distinction between them is made throughout the paper. Because there is no

gearbox between rotor and generator their moments of inertia can just be summed together in order to calculate total moment of inertia Jt. The coupling of rotor to the generator in direct drive solutions is very stiff and it can be considered as rigid thus removing any torsional oscillations what simplifies the control system design.

Before going further an important issue has to be addressed. Namely, expressions (4), (5) and (6) in this form would be valid only for structure with rigid tower and blades. In real situation the absolute wind speed wv in mentioned expressions has to be replaced by wind speed that is "seen" by rotor blades. This wind speed seen by the rotor is the resultant of three factors: absolute wind speed wv , speed of the tower movement perpendicular to wind speed (i.e. tower nodding speed) tx and speed of blade movement perpendicular to wind speed (i.e. speed of blade flapwise movement). Influence of tower nodding on wind turbine control is much more pronounced than influence of blade flapwise movement. Therefore in this paper we focus only on tower nodding considering rotor blades as rigid.

Tower nodding originates from the fact that wind turbine tower is very lightly damped structure due to its great height (more than 100 meters in modern wind turbines) and need for moderate mass. To model the wind turbine tower precisely we would have to use model with distributed parameters and to describe it in terms of mass and stiffness distribution. Such a model wouldn't be very suitable for controller design so it has to be substituted by model with concentrated parameters. This can be done using modal analysis that is very common tool in wind turbine analysis [1], [3]. It describes a complex oscillatory structure as a composition of several simple oscillatory systems each of them being described by means of mass, stiffness and damping. By this representation complex tower oscillations are seen as a sum of many simple oscillations characterized by their modal frequencies which are one of the most important structural properties of wind turbine. It has been shown in practice [5] that fairly good modeling of wind turbine tower nodding can be achieved using two modal frequencies (two modes). Since we are here primarily interested in building model suitable for controller design we use only the first modal frequency. The justification for this lies in the fact that for the turbine in scope second modal frequency is more than 6 times greater than the first modal frequency and therefore falls out of the controller frequency bandwidth. By using only one modal frequency tower dynamics can be described as:

t t tMx Dx Cx F+ + = , (9) where M, D, and C are modal mass damping and stiffness respectively and F is the generalized force that is originated by wind and that causes wind turbine tower oscillations. Tower modal properties in expression (9) are related to first tower modal frequency 0t as follows [4]:

( )02

0

2 ,

,t t

t

D M

C M

= =

(10)

where

t is structural damping. For steel structure structural damping is mostly set to 0.005 [4]. Modal mass M can be calculated as [1]:

( ) ( )20

th

M m h h dh= , (11) where ht is the height of the tower, m(h) is the mass distribution along the tower and ( )h is the tower's first mode shape. Note that actual distribution of mass along tower has to be modified in order to include mass of the rotor and the nacelle which is assumed to be concentrated at the tower top. Driving force F is mostly the rotor thrust force Ft caused by wind. It can be shown [4] that thrust force, similar to aerodynamic torque, depends upon wind speed, rotor speed and pitch angle. So similarly to (6) it can be expressed as [4]:

( )2 21 ,2t air w t

F R v C = , (12) where Ct is so called thrust coefficient. Expressions (6), (8), (9) and (12) form the simplified nonlinear model of wind turbine that is used in the following sections for controller design. Model is summarized below taking into account the fact that wind speed seen by the rotor is a sum of wind speed and tower nodding speed:

( ) ( )( ) ( )

23

22

,

1 , ,21 , ,2

.

t r g l

r air Q w t

t air t w t

t t t t

J M M M

M R C v x

F R C v x

F Mx Dx Cx

= =

= = + +

. (13)

Torque and thrust coefficients Cq and Ct are usually provided by wind turbine blade manufacturers or can be calculated using professional simulation tools.

5. PID Controller

The PID controllers are still by far the most used controllers for wind turbine speed and power control. This is due to their simplicity and rather high robustness. The small number of parameters makes possible for designer to quickly arrive at satisfactory, although in many cases suboptimal, system behavior. As stated before wind turbine dynamics change in nonlinear fashion with change in wind speed. To control the wind turbine with linear controllers gain

scheduling has to be used [3]. Although it seems straightforward to use wind speed as scheduling criterion this is not an appropriate solution since the wind speed is not measured fast and accurately enough. Therefore measured pitch angle is usually used as scheduling variable.

For the design of a PID controller using analytical methods process model (13) has to be linearised around chosen operating point. After linearization of expressions (13) and transition to Laplace domain simple algebraic manipulations yield transfer functions that are needed for controller design. Those transfer functions are:

( ) ( )( )s

G ss

= (14)

and

( ) ( )( )w ws

G sv s=

. (15)

These transfer functions are of the third order. A good insight in system properties of the wind turbine can be gained if we examine frequency characteristics of transfer function (14) shown in fig. 3.

Fig. 3: Frequency characteristics of G It can be observed that frequency characteristics of transfer function (14) at tower modal frequency exhibits magnitude and phase drop. Similar phenomena are present in frequency characteristics of (15) as well. This fact makes the pitch controller design very difficult. Physical explanation for observed effects becomes clear from the following analysis. Change in wind speed causes change in rotor speed what requires controller action and pitching of the blades in order to regulate the rotor speed to its rated value. Pitching the rotor blades, besides the aerodynamic torque, alters the thrust force significantly. Thrust force, according to (13), causes change in wind turbine tower top speed and thus the wind speed seen by the rotor is changed. This alters the aerodynamic conversion and in this way a feedback is formed. For this reason wind turbine can easily be driven into oscillatory behavior if the pitch controller is not designed properly.

To prevent the pitch controller from driving the wind turbine into oscillatory behavior it must be assured that system frequency bandwidth is below the first tower modal frequency. Moreover, sufficiently small magnitude is required at the first modal frequency. As it can be seen from fig. 3. the first modal frequency of the turbine in scope is 3 rad/s so a bandwidth of 1 rad/s was chosen. PID controller was designed to assure phase margin of around 60o what gives satisfactory behavior of the system.

To fully explore the system behavior with chosen controller simulation tool GH Bladed was used. GH Bladed is professional simulation package designed for wind turbine simulations and load calculations [5]. It relies upon very complex mathematical model based on combined blade element and momentum theory [1]. Structural properties of the wind turbine are modeled in detail and inertial and gravitational loads are taken into account along with aerodynamic ones. Extensive testing showed that simulation results obtained in Bladed are in accordance with measurements taken on actual wind turbines what was recognized by major standardization and certification institutions (e.g. Germanischer Llyod). To model the structural properties of explored turbine many modes are used [5]. Tower nodding is modeled with two modes as well as tower naying (tower side-side motion). Rotor blades' motion in flapwise direction is modeled with 6 modal frequencies while blades' motion in edgewise direction (displacement of the rotor blades in the plane of rotation) is modeled with 5 modal frequencies. Wind shear and tower shadow are included in the model as well. PID controller designed based on linearised model (14) and (15) was implemented in C and included as external discrete time controller. In that way we could use Bladed for controller testing. In the following figures behavior of the system when PID controller is used for rotor speed control is shown for one representative operating point ( wv = 15 m/s). Wind that was used for simulation observed positive and negative stepwise change shown in fig. 4. Responses of rotor speed, pitch angle and tower top displacement are shown in figs. 5. 6. and 7. respectively. From these figures it can be seen that rotor speed is well regulated and quickly compensated for the influences of wind speed changes. The pitch control actions are moderate without any oscillations. Similar results were obtained for all operating points throughout wind turbine operating range.

0 10 20 30 40 50 60 7012

13

14

15

16

17

18

t [s]

Win

d sp

eed

[m/s

]

Fig. 4: Wind speed used for simulation in Bladed

0 10 20 30 40 50 60 7022

22.5

23

23.5

24

24.5

25

t [s]

Rot

or s

peed

[rpm

]

Fig. 5: Response of rotor speed of the system controlled with PID controller

0 10 20 30 40 50 60 707

8

9

10

11

12

13

14

15

16

t [s]

Pitc

h an

gle

[deg

]

Fig. 6: Response of pitch angle of the system controlled with PID controller

0 10 20 30 40 50 60 70-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

t [s]

Tow

er to

p di

spla

cem

ent [

m]

Fig. 7: Response of tower top displacement of the system controlled with PID controller

While rotor speed and pitch angle behavior is satisfactory, attention should be paid to the tower top oscillations shown in fig. 7. It can be observed that tower top experiences lightly damped oscillatory behavior. These lasting oscillations, although small in magnitude, contribute to material fatigue and can lead to structure premature failure. So in the following sections two methods for controller design that aim at reduction of tower top oscillations are proposed.

6. SISO pole placement controller

In this section the pitch controller with SISO structure is designed using well known pole placement method [6]. As design objective in this method the desired behavior of the closed loop system has to be chosen. The pitch controller GPC is then designed to assure that closed loop system behaves in the chosen manner. Transfer functions (14) and (15) form the linearised wind turbine model that can be described with the principle scheme shown in fig. 8.

Fig 8: Principle scheme of the linearised wind turbine

model From the principle scheme given in fig. 8 closed loop transfer function with respect to wind speed change can be derived:

( )_( )( )

( ) 1 ( ) ( )w

CL ww PC SDCL

G ssGv s G s G s G s

= = +. (16)

The pitch controller GPC has to assure that closed loop transfer function (16) is equal to the chosen model transfer function Gm:

( ) !_ ( )CL w mG s G s= . (17) From (16) and (17) it follows that pitch controller has a form of :

( )( ) ( )1( )

( ) ( )w m

PCSD m

G s G sG sG s G s G s

= . (18) It should be noted that the above expression often needs to be modified to assure that the controller is causal. Details on this design method can be found in [6]. The crucial step in controller design using described method is the choice of model transfer function Gm. It is important to choose model transfer that is achievable for the system in scope and that assures its satisfactory behavior. One possibility that is very common in the literature is the use of standard forms such as Butterworth or binomial form. However, these forms can be rather difficult to relate to system physical properties. So another approach is used in this paper.

As previously said our primary goal is the reduction of tower oscillations. Tower oscillations can be reduced if the tower modal damping increases what would require change in tower structural parameters such as mass and stiffness distributions. Our goal is to achieve similar increase of tower damping by means of pitch controller actions without change in tower structural parameters. In that sense as a controller design objective we set a desired increase of tower modal damping. In other words tower modal damping D in the last expression in (13) is replaced with desired modal damping D' thus forming system model with new set of parameters. This model is linearised and transfer functions (14) and (15) are calculated. Using calculated transfer functions PID controller is designed following the guidelines described in section 5. Having the controller designed it is possible to calculate closed loop transfer function (16). This closed loop transfer function, obtained using system with increased damping and PID controller is then regarded as desired model transfer functions Gm for the real system. In other words our goal is to design a controller that assures that real system behaves as its tower damping has increased. The system response with SISO pole placement controller was simulated in Bladed using previously described full featured model. Wind used for simulation was the same as in section 4. Simulation results are given in the figs. 9-11. From these figures it can be seen that pole placement controller achieves almost the same regulation of rotor speed as PID controller does.

0 10 20 30 40 50 60 7022

22.5

23

23.5

24

24.5

25

t [s]

Rot

or s

peed

[rpm

]

Fig. 9: Response of rotor speed of the system

controlled with SISO pole placement controller

0 10 20 30 40 50 60 707

8

9

10

11

12

13

14

15

16

t [s]

Pitc

h an

gle

[deg

]

Fig. 10: Response of pitch angle of the system controlled with SISO pole placement controller

0 10 20 30 40 50 60 70-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

t [s]

Tow

er to

p di

spla

cem

ent [

m]

Fig. 11: Response of tower top displacement of the system controlled with SISO pole placement

controller This was to be expected since the PID controller is the "core" of the pole placement controller design. At the same time tower oscillations are more damped. Since the tower structure remains the same this damping is achieved by higher pitch control activity than in the case of PID controller. The question that naturally appears is the limit to which extend the tower oscillations can be damped in this way. Answer to this question can be obtained analyzing fig. 12. This figure shows comparison of Bode plots of controllers designed with different values of desired tower damping D'.

Fig. 12: Bode plots of SISO pole placement controllers designed to achieve different values of tower damping

D' From fig. 12 it becomes clear that increased tower damping results in high pass controller behavior. This has a consequence of increased pitch activity which in turn results in additional oscillations and finally cancels out the advantages of proposed design methodology. So a tradeoff between desired increase in tower damping and pitch activity has to be made.

7. Full state feedback controller

The SISO pole placement controller described in section 6 has shown some promising results but its ability to damp the tower oscillations is limited. Key cause for this is the fact that it doesn't use information about actual tower oscillations but only rotor speed feedback. Damping of tower oscillations by addition of tower top speed feedback to PID controller is proposed in [1]. Tower top speed measurement can be obtained from tower top acceleration measured by accelerometers that are nowadays almost a standard part of wind turbine control system. In our approach we use slightly different methodology. Instead of extended PID controller we use full state feedback controller designed using pole placement method. Desired closed loop behavior is chosen in the same way as for described pole placement controller. For this purpose process model (13) has to be rewritten in the state space form:

,.

x A x B uy C x D u= + = +

(19)

State variables used for system description and control are rotor speed , rotor acceleration , tower top speed tx and tower top acceleration tx , while system inputs are wind speed wv , pitch angle and generator torque Mg:

,w

tg

t

vx u

xM

x

= =

. (20)

Rotor speed and tower acceleration are measured

variables while other two states are derived from them. Using Ackermann's formula [6] vector of feedback gains for selected states can be calculated. System with such a controller was tested in Bladed using the same wind stepwise change as in sections 5 and 6. Simulation results are shown in the figs. 13-15. From these figures it can be seen that full state feedback controller maintains good rotor speed regulation while at the same time achieving better damping of the tower oscillations and practically removing the oscillatory movement of the tower. The pitch activity in this case increases considerably so a tradeoff between tower oscillations' damping and pitch activity is necessary. To examine the behavior of three described controllers in more realistic conditions the system behavior in 3D turbulent wind field was simulated in Bladed. Further information about simulations of 3D turbulent wind fields can be found in e.g. [7]. Let's just mention here that turbulent wind field was generated in Bladed using Kaimal spectrum recommended by the international standards for wind turbine design [8].

0 10 20 30 40 50 60 7022

22.5

23

23.5

24

24.5

25

t [s]

Rot

or s

peed

[rpm

]

Fig. 13: Response of rotor speed of the system controlled with full state feedback controller

0 10 20 30 40 50 60 707

8

9

10

11

12

13

14

15

16

t [s]

Pitc

h an

gle

[deg

]

Fig. 14: Response of pitch angle of the system controlled with full state feedback controller

0 10 20 30 40 50 60 70-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

t [s]

Tow

er to

p di

spla

cem

ent [

m]

Fig. 15: Response of tower top displacement of the system controlled with full state feedback controller

To gain valid information about system behavior in

3D turbulent conditions it is necessary for the simulations to last for at least 10 minutes what generates lots of data. Therefore, simulation results are not presented here due to limited space. Simulations in 3D turbulent wind field have shown that SISO pole placement controller in turbulent conditions achieves only modest improvement of tower oscillations when compared to the PID controller. The reason for this is the mentioned fact that SISO controller doesn't use

information about actual tower oscillations. On the other hand full state feedback controller maintains shown ability to reduce tower oscillations even under turbulent winds. The cost of that is increased pitch activity what must be taken into account to avoid pitch system excessive wear. This can be very efficiently done using LQR algorithm but that exceeds the scope of this paper.

8. Takagi-Sugeno process model

Controllers presented in sections 5-7 are linear. At the other hand wind turbine is highly nonlinear system as it was shown in section 4. To get the feeling of this high nonlinearity we simulated system response upon stepwise change of wind speed and pitch angle in various operating points from above rated operation region. The controller was only used to bring the system into steady state and then it was disabled. In this way it was possible to perform open loop tests ant to change wind speed and pitch angle independently one from another. Figure 16 shows open loop response upon 1m/s stepwise wind speed change in three representative operating points - 11 m/s (rated wind speed), 15 m/s and 25 m/s (cut out wind speed). It can be observed that turbine rotor speed response to the same wind speed step change is almost 10 times faster at cut-out wind speed than it is at the rated wind speed while the magnitude of turbine rotor speed change is around 7 times smaller.

0 50 100 150 200 25024

26

28

30

32

t [s]

Turb

ine

roto

r spe

ed [r

pm]

vw = 11 m/svw = 15 m/s

vw = 30 m/s

Fig. 16. Response of turbine rotor speed to wind

speed step change

Equivalent open loop experiments done for pitch angle showed that system dynamics change more than 10 times through above rated operation region. The observed situation shows that it would be impossible to control the system with a linear controller with fixed parameters in the entire operation region. However the use of linear controllers is very appealing for their simplicity and comprehensive design theory. Also, as it was shown in sections 5-7, linear controllers' performance can be very satisfactory around particular operating point for which they were designed. So the classic solution to tackle the wind turbine nonlinearity is the use of linear controllers with parameter scheduling [3]. In this classic approach, which is widely used in practice, operating region is divided into many operating points and at each operating point a linear controller is designed. During wind turbine operation as operating point changes controller chooses from predesigned set of parameters based upon certain scheduling variable. The scheduling variable is mostly measured pitch

angle since the wind speed is not measured fast and reliably enough.

The problem with described classic approach is the fact that the controller is designed based upon process model that is completely valid only in several operating points characterized by wind speed, pitch angle and rotor speed. In this way controller parameters are often not optimal for actual operating point and system behavior can be different from the one demanded during controller design.

To assure that demanded process behavior is obtained under all possible operating conditions on-line controller synthesis based on the on-line identified process model should be used. This approach, however, brings along many drawbacks that are related with necessity of persistent excitation and identification algorithm convergence [9]. Besides this wind speed, which is at the same time the main driving force of the system and the main source of disturbance, is not measured fast and reliably enough to be used for the process model identification.

So another approach is proposed in this paper that tries to overcome the drawbacks of two aforementioned methods. The idea is to perform the on-line controller synthesis to assure better adaptation to actual operating point, but the process model is not identified on-line using system identification methods. Instead a Takagi-Sugeno fuzzy model is used. Takagi-Sugeno fuzzy model that is described in [10] can be described with number of fuzzy rules Ri that have a form of:

( )1 1 2 2 nx nx: IF ( ) is ( ) is ( ) is i i i iR x k F x k F x k F 0 1 1 nm nm ( 1) ( ) (k), 0,1, ,nr,

i i i iTHEN y k p p m k p m i+ = + + + =" " (21) where nr is number of fuzzy rules, x1xny are

fuzzy variables from the conditional part of the rule with corresponding fuzzy sets

1i i

nxF F . As it can be seen from expression (21) the right side of the fuzzy rule doesn't contain fuzzy sets but the linear regression with p0pnm being linear model parameters and m1mnm being previous values of process model inputs and outputs (elements of regression vector). This is due to the fact that Takagi-Sugeno model uses singleton fuzzy sets in the output part of fuzzy rule which are in fact sharp numerical values. The output of the Takagi-Sugeno process model is calculated as:

( )( )

nr

1nr

1

( ) ( 1)( 1)

( )

i i

i

i

i

k y ky k

k

=

=

++ =

x

x

(22)

where i is the membership function that describes fuzzy set Fi. Expressions (21) and (22) show that the Takagi-Sugeno fuzzy model in fact produces a linear model output that is calculated as combination of many "local" linear models. This characteristic makes Takagi-Sugeno model very appealing for use in wind turbine modeling and control. The local linear models needed on the right side of fuzzy rules (21) are wind turbine linear models obtained by

linearization of wind turbine model (13) that were already used for design of previously described controllers. Having those models calculated the crucial step that remains is the choice of conditional variables x1xnx and building a base of fuzzy rules that will describe the actual process dynamics. To demonstrate the possibilities of proposed methodology in this paper we use very simple approach where only pitch angle is used as conditional variable. Process model (13) is linearized in 18 operating points from above rated operation region that are determined by wind speed and corresponding pitch angle. So a Takagi-Sugeno fuzzy model is described with 18 fuzzy rules that have a form of:

( ) ( ) ( ) ( ) ( )IF is "Around i" THEN 1i i iw wy k G z v k G z k

+ = +

, (23)

where ( )iwG z and ( )iG z are discrete time

transfer functions obtained by discretisation of (14) and (15) and i is the pitch angle that determines the operating point around which process model was linearised to obtain i-th linear process model. To describe "Around i" fuzzy sets a triangular membership function were used. The question that naturally appears is the choice of membership functions' parameters that is critical for the process description. The manual tuning of the membership functions' parameters would be very demanding and it would have to be done more-less based on trial and error procedure since the process dynamics away from the operating points is not known. To overcome this problem we used a training algorithm that tuned the membership functions' parameters based on measured error between nonlinear process model output and Takagi-Sugeno approximation. Nonlinear process model and Takagi-Sugeno models were excited with the same input signals and the output error was calculated. Then the membership functions' parameters were changed in the way that guaranteed reduction of this error. The described procedure was repeated for several iterations until the error fell below certain chosen value. It should be pointed out here that linear model that is obtained using Takagi-Sugeno model can only model perturbation i.e. change of output variable (rotor speed) away from its steady state values when excited with perturbations of input signals (wind speed and pitch angle). This poses a serious problem since wind speed and pitch angle have different steady state values at different operation points what complicates the algorithm. At the other hand our primary goal is not to approximate the system output with Takagi-Sugeno model but to model its dynamics as a base for controller design. Therefore we didn't use absolute values of inputs and outputs but their deviations. In this way we eliminate the steady state value and focus just on the dynamics that is of prior interest here. The illustration of performance of the proposed methodology is shown in fig. 17 where a comparison of nonlinear process model and Takagi-Sugeno model is shown. The fig. 17 shows the deviation of process model output when wind speed ramps from rated wind speed to cut-out wind speed and back at rate of 0.5 m/s2 what drives the system through all 18 operating points. For comparison fig. 17 also shows the output of the linear

model whose parameters are simply scheduled and taken to be the parameters for the nearest operating point determined by measured pitch angle. From this figure it can be seen that Takagi-Sugeno model can assure better approximation of the actual process dynamics than linear model whose parameters are changed based on actual operating point. This leads to conclusion that controller designed based on Takagi-Sugeno model could be more capable to cope with nonlinear process dynamics than controller that uses classic parameter scheduling. This simple example was intended just to illustrate the possibility of proposed methodology. Much better performance is expected if more process variables are used in conditional part of fuzzy rules.

0 5 10 15 20 25 30 35 40 45 50-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

t [s]

Rot

or s

peed

dev

iatio

n [ra

d/s

2 ]

Nonlinear modelSet of linear modelsTakagi-Sugeno

Fig. 17. Comparison of Takagi-Sugeno model and

linear model with parameter scheduling

9. Conclusion

The control system for variable speed pitch controlled wind turbine is presented. The wind turbine system is highly nonlinear and its parameters change significantly with change of wind speed. Furthermore wind turbine mechanical structure is very flexible and can easily be driven into oscillatory behavior. All this makes the controller design a very demanding task. In this paper three methods for wind turbine pitch controller design are compared. It is shown that classic PID controller can assure good rotor speed regulation but tower oscillations are very pronounced. To reduce these undesired oscillations two alternative control structures are investigated: SISO pole placement controller and full state feedback controller. The design objective for these controllers, besides good rotor speed regulation, was the increase of tower damping. Both controllers have shown that owing to increased pitch control activity it becomes possible to damp the tower oscillations. To test the controllers' performances in more realistic conditions simulation under 3D turbulent wind field were conducted in Bladed. It was observed that under such conditions only slight reduction of tower oscillations is achieved by SISO pole placement controller. On the other hand full state feedback controller has shown that is capable of reducing the tower oscillations even under turbulent conditions.

The achieved damping of tower oscillations leads to fatigue reduction what enables production of lighter and less expensive wind turbines. All considered controllers are linear so a method for adapting their parameters to account for nonlinear nature of wind turbine is needed. An on-line controller synthesis based on Takagi-.Sugeno fuzzy process model is proposed in the paper that seems promising for this purpose.

Acknowledgements

This work was financially supported by Konar Electrical Engineering Institute and the Ministry of Science Education and Sports of the Republic of Croatia.

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