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PROCEEDINGS OF SPIE

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Periodic wind disturbance rejection

using robust individual pitch control

Yuan Yuan, Xu Chen, Jiong Tang

Yuan Yuan, Xu Chen, Jiong Tang, "Periodic wind disturbance rejection using

robust individual pitch control," Proc. SPIE 10601, Smart Materials and

Nondestructive Evaluation for Energy Systems IV, 106010G (27 March 2018);

doi: 10.1117/12.2296771

Event: SPIE Smart Structures and Materials + Nondestructive Evaluation and

Health Monitoring, 2018, Denver, Colorado, United States

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Periodic Wind Disturbance Rejection using Robust Individual

Pitch Control

Yuan Yuan, Xu Chen, and J. Tang†

Department of Mechanical Engineering

University of Connecticut

Storrs, CT 06269, USA

ABSTRACT

A robust individual pitch control strategy is presented to deal with periodic load disturbances on wind

turbines under operating point variation. The asymmetric loads are mainly caused by the tower shadow and

wind shear effects. Multi-blade coordinate (MBC) transformation is utilized to model the turbine dynamics

under various operating points. The coupling dynamics of the multi-input multi-output (MIMO) system are

considered to reveal high harmonic frequency peak reduction. The stability and robustness performance of

the system under uncertainties are guaranteed by robust control design. The performance of the

synthesized controller is compared with a collective controller and a PI individual controller. The results

show significant load mitigation in periodic frequencies.

Keywords: periodic disturbance, robust control, individual pitch control, wind turbine.

1. INTRODUCTION

Wind energy is promising for renewable energy supply. The sizes of commercial wind turbines have

progressively increased in recent years to increase output power capacity. Meanwhile, the increased

mechanical loading on components subjected to aerodynamic forces may directly reduce the wind turbine

life-span. Collective pitch control is widely studied to regulate power and decrease symmetric loads [1, 2].

More recently, individual pitch control (IPC) is utilized to mitigate the asymmetric loading of the rotor

blades caused by the wind speed variations across the rotor plane [3]. The effectiveness of individual pitch

control for periodic load mitigation has been demonstrated by employing a LQG approach without

compromising energy capture [3]. The IPC strategy can also be realized through two decoupled single

input single output control loops, with the addition of feedforward control to remove the 3P component

from the input fixed frame load [4]. In another study, the IPC strategy is combined with preview-based

disturbance feedfoward approach to achieve load mitigation [5]. IPC implemented through robust

approach to mitigate loads have been presented by many researchers. A

H

∞

MISO controller is proposed

to improve both the performance of the closed-loop disturbance rejection and the tower fore-aft loads

which are deteriorated by the generator speed control [6]. A later study shows that the decoupled PI

controller is not sufficient because yaw and tilt modes are significantly coupled after MBC transformation

and cannot be neglected [7]. A multivariate

H

∞

approach is presented considering the coupling effects by

using a frequency dependent MIMO plant [8]. With the mixed sensitivity loop shaping approach, the

control efforts (actuator usage) can be also penalized when we try to achieve load mitigation. The

aforementioned investigations have shown that IPC is an effective method to mitigate asymmetric loads in

wind turbines. However, several control difficulties remain. The control design often relies on accurate

modeling of turbine dynamics while plant dynamics has significant parametric variation at different

operating points. The unmodeled dynamics of plant exist when the state of the system trajectories changes

from one equilibrium point to another because of the nonlinearities of the system. In addition, the

incoming wind effects are complex, and horizontal effects should not be the only factor considered.

This paper presents an individual robust pitch control method to reject periodic loads under

aforementioned model uncertainties. The robustness performance against operating point variation will be

taken into account. In particular, this paper proposes the structured singular value ()-synthesis approach

to attain robust stability and robust performance under model uncertainties due to operating-point variation.

† Corresponding author

Smart Materials and Nondestructive Evaluation for Energy Systems IV, edited by Theodoros E. Matikas, Proc.

of SPIE Vol. 10601, 106010G · © 2018 SPIE · CCC code: 0277-786X/18/$18 · doi: 10.1117/12.2296771

Proc. of SPIE Vol. 10601 106010G-1

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Meanwhile, the weighting functions are properly designed intending to reject periodic wind disturbances.

The modeling of the wind turbine is presented in Section 2. The robust individual pitch control strategy is

outlined in Section 3. The simulation and results are shown in Section 4.

2. MODELING OF THE WIND TURBINE

For wind turbines, aeroelastic simulators have been well developed to conduct numerical computation,

including FAST [9], BLADED [10], HAWC2 [11], and FLEX5(4) [12]. Here we adopt FAST to carry out

simulation. FAST can model 22 to 24 DOFs. In the model linearization process, we reduce the DOF

complexity to obtain a simplified model to represent the low frequency dynamics. Since our major

objective is to reduce the loads on blades, we only consider flapwise DOF and generator DOF in the linear

model. The tower and drive-train DOF are omitted. As we will focus on rejecting periodic loads on wind

turbines, individual pitch control is used to reduce the asymmetrical loads that are caused by the wind shear,

tower shadow and centrifugal forces. Therefore, the control inputs are the pitch angles of all three blades.

Correspondingly, we have to add more measurements in the output to maintain the observability of the

system. For the disturbance modeling, the perturbed horizontal hub-height wind speed is a common one to

represent the upcoming wind disturbance [13]. However, the wind shear effect has important effect on the

wind asymmetric output, and in the modeling process it is often neglected [13, 14]. Our contribution in this

research is that we will consider the horizontal and vertical wind shear effects in the modeling process to

attain a better output prediction and thus to facilitate more accurate controller design. The simplified model

from FAST is shown below,

(

)

(

)

(

)

(

)

() () () ()

1dd

dd

kkkk

kkk k

+= + +

=++

xAxBuBu

yCxDuDu (1)

where x is the state variable, y is the output, u is the control input, and d

u is the disturbance matrix. The

generator torque controller adopts the standard torque controller in [15].

As mentioned, a wind turbine is indeed a periodic system due to wind shear and tower shadow effects.

The dynamics of wind turbine rotor blades are generally expressed in rotating frames attached to the

individual blades. However, the responses of rotor dynamics relative to the nacelle and tower actually have

to be considered as an integral one. Multi-blade coordinates (MBC) can transform the dynamics of the

rotating frame to the non-rotating frame (consistent with the fixed tower frame) and coherently interconnect

the spinning rotor with the tower and nacelle. MBC is derived and first used in the helicopter system to

analyze the flap motion related stability [16]. The aforementioned LTI model is simple and often adopted

in collective pitch control strategy. Recent studies have found that multi-blade coordinates (MBC)

transformation can reduce the variations between linearizations obtained at different azimuths, and

therefore yields a better representation of the turbine dynamics [17]. The detailed transformation from the

rotational coordinate to the fixed coordinate can be found in [18].

The underlying transformation from the rotational coordinate to the fixed coordinate is defined as

()

()

1

2

3

1/2 1/2 1/2

224

,coscos cos

333

24

sin sin sin

33

avg

tilt

yaw

MM

MTMT

MM

θθ

θθπθπ

θθπθπ

⎡

⎤

⎢

⎥

⎢

⎥

⎡⎤⎡⎤

⎢

⎥

⎢⎥ ⎛⎞⎛⎞

⎢⎥

== ++

⎜⎟⎜⎟

⎢

⎥

⎢⎥⎢⎥ ⎝⎠⎝⎠

⎢

⎥

⎢⎥⎢⎥

⎣⎦

⎣⎦

⎢

⎥

⎛⎞⎛⎞

++

⎢

⎥

⎜⎟⎜⎟

⎝⎠⎝⎠

⎣

⎦

(2) where

θ

is t

h

blade moment which will induce the yaw motion of rotor. Here avg

M

is the symmetric moment and tilt

M

,

yaw

Mare the asymmetric moment. Equation (4) transforms moments in rotating coordinate to moments in

nonrotating coordinate. It is worth noting that we only take the asymmetric moments as the inputs to the

proposed controller.

Proc. of SPIE Vol. 10601 106010G-2

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The outputs of the controller are tilt

β

and yaw

β

that are both in the non-rotating coordinate. They can

be transformed back to the rotating coordinate by the inverse MBC transformation, denoted by 1

T

θ

−.

() ()

1

11

2

3

1cos sin

22

,1cos sin

33

44

1cos sin

33

avg

tilt

yaw

TT

θθ

θθ

ββ

ββ

θπ θπ

ββ

θ

πθπ

−−

⎡

⎤

⎢

⎥

⎢

⎥

⎡⎤

⎡⎤

⎢

⎥

⎢⎥ ⎛⎞⎛⎞

⎢⎥

==++

⎜⎟⎜⎟

⎢

⎥

⎢⎥

⎢⎥ ⎝⎠⎝⎠

⎢

⎥

⎢⎥

⎢⎥

⎣⎦ ⎣⎦

⎢

⎥

⎛⎞⎛⎞

++

⎢

⎥

⎜⎟⎜⎟

⎝⎠⎝⎠

⎣

⎦

(3)

The FAST code can calculate the linearized state-space model at a defined operating point in several

evenly distributed azimuth angles in one-revolution. Therefore, the system can be defined as a periodic

system. To facilitate controller design, this periodic state-space model is transformed to a LTI state-space

model by the aforementioned MBC transformation. The model for individual pitch control includes first

blade flapwise bending DOF (3 DOFs) and generator DOF. The measurements are generator speed and

flap bending moments at each blade root. Each blade pitches at different angle at the same time, which

depends on current azimuth in the rotor plane. Since the periodic loads are mainly from wind shear effects

on the rotor plane, we include horizontal and vertical wind shear in the disturbance modeling.

Equation (1) can be directly applied with MBC transformation. With a series of state-space model at

several azimuths, this representation is still a periodic model.

(

)

(

)

(

)

(

)

() () () ()

1

mbc mbc mbc mbc mbc mbc mbc

dd

mbc mbc mbc mbc mbc mbc mbc

dd

kkkk

kkkk

+= + +

=++

xAxBuBu

yCxDuDu (4)

where

,,

avg

tilt

yaw

d

avg avg

mbc mbc mbc

tilt tilt d d

yaw yaw d

u

y

yy u

yu

β

β

β

⎡

⎤

⎡⎤ ⎡⎤

⎢

⎥

⎢⎥ ⎢⎥

==

⎢

⎥

⎢⎥ ⎢⎥

⎢

⎥

⎢⎥ ⎢⎥

⎣⎦ ⎣⎦

⎢

⎥

⎣

⎦

=uu

(5)

An average state-space system is obtained from the complete set of linearizations at N azimuth angles by

computing

()

0

1N

mbc

i

i

N

θ

=

=∑

AA

(6)

The same average method can be applied to other state-space matrixes. As such, we can get a LTI model

of wind turbine. It will serve as the model for individual control design. Since the horizontal shear and

vertical shear have been included in the disturbance vector, the model will better serve the control design.

After MBC transformation, the intrinsic periodic characteristics have been handled. Other remaining

problems are modeling uncertainties caused by varying operating points. Here we will generate a

multiplicative uncertain system whose range of behavior includes all responses of sampled linearized

models at several operating points. We select 18 m/s as the operating point to formulate the nominal model

and select 14 m/s, 16 m/s, 20 m/s, and 22 m/s as the operating points to formulate the uncertain model.

3. INDIVIDUAL ROBUST CONTROL STRATEGY

Oftentimes, the collective controller is adopted to regulate the generator speed and power in high wind

speeds in case of the over-speed of rotor that may cause over-heat of rotor and generator. It can further

reduce the symmetric loads on the blade. Since the collective pitch loop is coupled with the tower loads,

the collective pitch control may increase the loads on the tower fore-aft or side-side loads. The reason is

the pitch angle signal may excide the resonance frequency of the first tower mode. From [3], we know that

in the frequency response of blade root moments, there are several peaks at the nP frequencies (P is per

revolution frequency of the rotor, 1, 2, 3,n=L). Indeed, the periodic disturbances on the loads come from

wind shear, tower shadow, and the centrifugal forces [19]. The wind turbine system is actually a periodic

Proc. of SPIE Vol. 10601 106010G-3

system that undergoes periodic disturbance excitation. The loads induced by the horizontal wind

disturbances are called symmetric loads that can be decreased by the collective pitch strategy. Meanwhile,

the unbalanced loads induced by the wind shear, tower shadow, and the centrifugal forces during the rotor

plane can be decreased by the individual pitch strategy. The pitch angle of each blade will be adjusted in

different values corresponding to the azimuth positions where the blade is at that sampling interval.

3.1 Baseline controllers

In this research, we compare the new controller with two other controllers. The first controller is the

collective pitch controller, i.e., GSPI in [15], which provides the basic comparison between the collective

pitch strategy and individual pitch strategy. The second controller is a PID individual control. Although

the tilt and yaw moments in the non-rotating coordinate after MBC transformation are coupled with each

other in reality, it is assumed that they are decoupled variables in order to design two separate SISO PID

controller to reduce the asymmetric loads.

The PID individual controllers that are employed to attenuate the tilt and yaw moments separately can

be expressed as

/

/

tilt tilt tilt tilt

yaw yaw yaw yaw

PI P I D

PI P I D

GKKsKs

GKKsKs

=+ +

=+ + (7)

where tilt

I

K

,

y

aw

I

K

are selected as 7

310

−

×, other gain values are selected as 0. The underlying reason that

PID can reduce the loads is that the integral part can reduce the low frequency response in the non-rotating

coordinate. Consequently, the individual controller can correspondingly reduce the 1P periodic loads in the

rotating coordinate.

3.2 Individual pitch controller

The augmented control block diagram of collective and individual control is shown in Figure 1. There

are two control loops in the turbine pitch system. One is the collective pitch loop regulating the generator

speed, which provides the collective signal. The other one is the individual pitch loop to provide small

modification that is summed with the collective pitch based on the blade azimuth angle in the rotor plane.

In the individual control loop, the blade root moments of each blade are transformed to tilt and yaw

moments in the non-rotating coordinate by MBC transformation. The individual pitch controller is a

multivariate

H

∞ controller. The tilt and yaw pitch angles which are the outputs of the individual controller

should be transformed back to the rotating coordinate by the inverse MBC transformation and then are

summed with the collective pitch signal. The signal provided by the individual loop is intended to mitigate

the asymmetric loads.

r

ω

ω

1

M

3

M

2

M

tilt

M

y

aw

M

tilt

β

y

aw

β

c

β

1

β

Δ

2

β

Δ

3

β

Δ

Figure 1. The augmented control block diagram of collective and individual control.

Proc. of SPIE Vol. 10601 106010G-4

We propose to use structured singular value ()-synthesis to develop a RSRP (robust stability and

robust performance) controller to address the structured uncertainties and load mitigation in wind turbines.

The parameters of nonlinear equation of motion are highly related with the operating wind speed.

Therefore, robustness against dynamic operating point variation should be maintained. The objective is to

find a controller minimizing the effect of parameter variation and disturbances. The

synthesis gives

better performance considering uncertainties compared to

H

∞

controller.

3.3 Weighting functions

The wind turbine structural loads exist at integer multiplies of the rotor frequency due to rotation

dynamics. After MBC transformation, the original 1P, 2P, 3P, ··· frequencies in the rotating frame are

changed to 0P, 3P, 6P, ··· frequencies [13]. The transformation relationship is shown in Table 1. It is

worth noting that, the 3P, 6P, … , 9P frequencies in the rotating coordinate cannot be counteracted since

avg

M

is neglected. Therefore, we concentrate on the reduction of low frequencies and 3P frequency

performances. The wind conditions are pre-specified as 14% turbulence intensity on top of the steady wind

condition.

The performance weights are incorporated with the plant model to form the generalized plant M. The

weighting matrices p

Wand u

W are 2×2 diagonal matrices. The optimization process minimizes the

infinity norm of the weighted closed-loop transfer function

()

()

-1

,

u

S= I+F M ΔK (i.e., the output

sensitivity function), and KS.

()

()

()

()

-1

-1

,

,

pu

uu

∞

⎡

⎤

⎢

⎥

⎢

⎥

⎣

⎦

WI+FMΔK

WK I+F M ΔK

(8)

S is the transfer function between w and z, and KS is the transfer function between w and u. The numerical

control design is carried on with MATLAB Robust Control toolbox.

Table 1. Transformation of system dynamics through MBC

Rotating coordinate Non-rotating coordinate

1P 0P @ tilt

M

and yaw

M

2P 3P@tilt

M

and yaw

M

3P 3P@avg

M

4P 3P@tilt

M

and yaw

M

5P 6P@tilt

M

and yaw

M

6P 6P@avg

M

7P 6P@tilt

M

and yaw

M

The closed-loop response characteristics can be shaped or tuned by the weighting functions, which are

defined as rational, stable, minimum-phase transfer functions. The disturbance w here can be assumed as a

combination of a low frequency signal and 3P sinusoidal signal (in the non-rotating coordinate), and

therefore it will be successfully rejected if the maximum signal value of S is made small over the frequency

bandwidth. The weighting matrices p

W are defined as 22pp

WI

×

=

W. The diagonal element p

W is a

combination of a low pass filter l

W and a second-order notch filter

3p

W. l

W has a high gain at low

frequencies to reject 0P frequency and

3p

W is an inverted notch filter in the 3P (0.6 Hz) frequency to reject

2P, 4P, … frequencies in rotating coordinate.

22

33

13

22

33

/2

,2

pp p p

lp

pp p p

sM s

WK W

se s

ω

αω ω

ω

βω ω

+++

==

+++

(9)

Proc. of SPIE Vol. 10601 106010G-5

3pl p

WWW

=

• (10)

(1)

(2)

Figure 2. Frequency response of weighting functions p

W(1) and u

W(2).

The weighting function u

W is selected to guarantee the actuator be functional in the proper bandwidth.

The diagonal element u

W is chosen as a high pass filter, which has a low gain below the actuator

bandwidth and has a high gain beyond the actuator bandwidth.

/M

uu

u

uu

s

Wes

ω

ω

+

=+ (11)

Figure 2 shows the bode diagram of diagonal element of weighting matrices p

W and u

W. The inversion

of weighting functions indicates the shape of the sensitivity function.

The shaping of multivariate transfer functions is based on the idea that a satisfactory definition of gain

for a matrix transfer function is given by the singular values of the transfer function [20]. The singular

values in the open-loop and closed-loop response from three component disturbances to tilt and yaw

moments show the disturbance rejection performance. Since the plant dynamics are uncertain, we obtain a

series of singular values in Figure 3. It can be observed that the low frequency response is lower in closed-

loop response compared with open-loop response. There is a deep notch in 3P frequency that is intended to

design in weighting functions. The magnitude around 3P is a little bit higher in closed-loop response

Proc. of SPIE Vol. 10601 106010G-6

..: ;.

r'. Ji

pbsu-loob

Clozsqloc

which can be explained from the Bode’s Integral Theorem [21]. With the Robust Control Toolbox, the

robust controller K achieved has a robust performance value of 0.1784. Therefore, we have realized the

RSRP design goal for the uncertain model sets under operating point variations.

Figure 3. Singular values comparison of uncertain systems between open-loop and closed-loop systems.

4. RESULTS AND DISCUSSION

To demonstrate the effectiveness of the proposed controller, simulations are carried out in

Matlab/Simulink environment. The parameters of wind turbine can be found in [15]. The turbulent wind

field is generated by TurbSim code at a series of wind fields for a 10-minute simulation. Blade flapwise

moments, blade edgewise moments and tower base moments are dominating loads on turbines. They will

be examined in this section. Here we study the fatigue damage equivalent load (DEL) which serves as an

important metric for comparing fatigue loads across the entire spectrum of turbulent wind files. The

equivalent damage is represented by a constant load and calculated by using MLife [14] based on the

rainflow counting algorithm.

Figure 4. PSD of flapwise moments under collective control, individual PID control and individual robust

control.

The controller is simulated in turbulent 18 m/s wind field. The turbulent intensity is 14%. Figure 4

illustrates the power spectral density (PSD) of flapwise moment of blade 1 root of the collective PID

controller, the individual PID controller and the individual robust controller. Both individual controllers

can decrease the peak magnitude at 1P (≈0.2Hz) frequency while the robust controller has more obvious

Proc. of SPIE Vol. 10601 106010G-7

decrease over wider frequency region around 0.2 Hz. It can be also observed that the PSD of the proposed

individual robust controller at the 2P (≈0.4 Hz) and 4P (≈0.8 Hz) frequency can be significantly decreased

while the individual PID controller has no effects at those frequencies. The reason is that the integral effect

of PID controller can only deal with low frequencies (0P in rotating coordinate, 1P in non-rotating

coordinate). The proposed controller can reveal better disturbance rejection because the dynamic coupling

effects of yaw mode and tilt mode are considered and the high harmonic frequencies can be taken care of

by weighting functions in robust controller.

5. CONCLUSION

In this research, we study robust individual pitch control to reject periodic loads under model

uncertainties. Since the nonlinear plant dynamics operate in wide operating point variation, an uncertain

model is formulated by Multi-blade coordinate (MBC) transformation to facilitate robust control design.

The multivariate individual controller can reveal high harmonic frequency peak reduction with taking

coupling dynamics into account. The robust structured singular values ()-synthesis approach is utilized to

guarantee the robust stability and robust performance under the model uncertainties due to the operating

point variation. The results from high-fidelity aeroelastic simulator present significant load mitigation

under periodic disturbances.

ACKNOWLEDGMENT

This research is supported by National Science Foundation under grant CMMI – 1300236.

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Proc. of SPIE Vol. 10601 106010G-9