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Periodic wind disturbance rejection using robust individual pitch control

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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.
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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
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.
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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
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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.
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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
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
4P 3P@tilt
M
and yaw
M
5P 6P@tilt
M
and yaw
M
6P 6P@avg
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)
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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
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..: ;.
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
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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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