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Version: Accepted Version
Article:
Lio, W.H., Jones, B. and Rossiter, J.A. (2019) Estimation and control of wind turbine tower
vibrations based on individual blade-pitch strategies. IEEE Transactions on Control
Systems Technology, 27 (4). pp. 1820-1828. ISSN 1063-6536
https://doi.org/10.1109/TCST.2018.2833064
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1
Estimation and control of wind turbine tower
vibrations based on individual blade-pitch strategies
Wai Hou Lio
1
, Bryn Ll. Jones
2
and J. Anthony Rossiter
2
.
Abstract—In this paper, we present a method to estimate the
tower fore-aft velocity based upon measurements from blade
load sensors. In addition, a tower dampening control strategy
is proposed, based upon an individual blade pitch control
architecture that employs this estimate. The observer design
presented in this paper exploits the Coleman transformations that
convert a time-varying turbine model into one that is linear and
time-invariant, greatly simplifying the observability analysis and
subsequent observer design. The proposed individual pitch-based
tower controller is decoupled from the rotor speed regulation loop
and hence does not interfere with the nominal turbine power reg-
ulation. Closed-loop results, obtained from high fidelity turbine
simulations, show close agreement between the tower estimates
and the actual tower velocity. Furthermore, the individual-pitch-
based tower controller achieves similar performance compared
to the collective-pitch-based approach but with negligible impact
upon the nominal turbine power output.
Index Terms—State estimation of dynamical systems, Kalman
filter, active damping control, wind energy.
I. INTRODUCTION
Large wind turbines experience uneven and intermittent
aerodynamic loads from the wind and such loads inevitably
contribute to fatigue damage upon the turbine structures. In
order to manage the competing demands of power capture and
load mitigation, most modern turbines employ a combination
of control systems based upon blade pitch actuation. Primary
amongst these is the use of collective pitch control (CPC) [1],
whereby the pitch angle of each blade is adjusted by an equal
amount in order to regulate the rotor speed in above rated
conditions. In addition, individual pitch control (IPC) and
tower damping control can be used to specifically attenuate
unsteady loads that play no part in power generation. The IPC
provides additional pitch demand signals to each blade in
order to balance the loads across the rotor plane, typically in
response to measurements of the flap-wise blade root bending
moments [2]–[4], whilst tower damping control provides a
further adjustment to the collective blade pitch angle in order
to reduce excessive tower vibrations, in response to tower fore-
aft velocity measurements [5]–[8]. Typically, and for reasons
of simplicity of implementation favoured by the industry, IPCs
and tower damping controllers are designed separately from
the CPC, and carefully in order to avoid cross-excitation [9]–
[12].
1
Wai Hou Lio is with Department of Wind Energy, Technical University
of Denmark, DK-4000 Roskilde, Denmark. (e-mail: wali@dtu.dk)
2
Bryn Ll. Jones and J. Anthony Rossiter are with Department of Automatic
Control and Systems Engineering, University of Sheffield, Sheffield, S1 3JD,
U.K (e-mail: b.l.jones@sheffield.ac.uk; j.a.rossiter@sheffield.ac.uk )
At present, most tower damping control strategies assume
a direct measurement of tower motion, typically from a
nacelle-mounted accelerometer (e.g. [4], [13]). However, the
turbine blades and tower are dynamically coupled and from
an estimator design perspective, such interactions may provide
an opportunity for the tower motion to be estimated based
upon the blade load measurements that are already available
to the IPC. If so, this indicates redundancy in the information
provided by the tower motion sensor that can either be
exploited in terms of a reduction in sensor count, or for fault
tolerant control purposes [14]–[16]. Moreover, typical tower
damping control strategies provide an additional blade pitch
signal collectively to all the blades in response to the tower
velocity [13], that is inevitably coupled with the rotor speed
regulation loop, thus, affecting the power output of the turbine.
On the other hand, well-designed IPCs are largely decoupled
from the CPC, thus there are potential benefits to designing
an IPC-based tower damping controller.
The contributions of this paper are thus twofold. Firstly,
a tower vibration observer design is proposed that estimates
the tower fore-aft velocity based solely upon standard blade-
load measurements. Secondly, an individual pitch-based tower
damping control strategy is presented that provides the blade
pitch command to each blade independently and with little
impact on the nominal turbine power regulation.
The remainder of this paper is as follows. Section II presents
the model of the blade and tower dynamics. In Section III,
a linear, time-invariant (LTI) model is derived that captures
the dynamics of the Coleman transform and establishes the
coupling between the blade load sensors and tower motion that
is key in establishing an observable system. The design of a
subsequent tower-top motion estimator and individual pitch-
based tower damping controller is described in Section IV.
In Section V, the performance of the proposed estimator and
controller are demonstrated in simulation upon a high-fidelity
and nonlinear wind turbine model.
Notation
Let R, C and Z denote the real and complex fields and set
of integers, respectively, j :=
√
−1 and let s ∈ C denote a
complex variable. The space R denotes the space of proper
real-rational transfer function matrices and ˙x represents the
time derivative of x. Let v
T
∈ R
1×n
v
denote the transpose
of a vector v ∈ R
n
v
and V
T
∈ R
n
y
×n
z
is the transpose of a
matrix V ∈ R
n
z
×n
y
. The identity matrix is denoted as I. Let
˜x denote the deviation of x from its equilibrium x
∗
.
2
Fig. 1: The perturbation on the free-stream stream-wise wind
speed ˜v
∞
i,l
on the shaded blade element at r
l
becomes the
apparent wind speed ˜v
i,l
after the effects of the tower fore-aft
velocity
˙
˜x
fa
and rotational velocity
˙
˜ϕ
fa
.
II. MODELLING
Typically, the dynamics of the blade flap-wise root-bending
moment and the tower-top fore-aft motion can be modelled as
second-order systems (e.g. [9], [3]), as follows:
¨
˜
M
i
(t) + 2ζ
b
ω
b
˙
˜
M
i
(t) + ω
2
b
˜
M
i
(t) = ω
2
b
˜
f
M
(
˜
θ
i
, ˜v
i
), (1a)
¨
˜x
fa
(t) + 2ζ
t
ω
t
˙
˜x
fa
(t) + ω
2
t
˜x
fa
(t) = ω
2
t
˜
f
x
(
˜
θ
col
, ˜v
col
), (1b)
where
˜
M
i
(t), ˜x
fa
(t) denote the deviations of the flap-wise
blade root bending moment of blade i ∈ {1, 2, 3} and tower
fore-aft displacement from an operating point, respectively.
The damping ratio of the blade and tower are ζ
b
, ζ
t
∈ R
and ω
b
, ω
t
∈ R are the respective natural frequencies of the
blade and tower. The nonlinear aerodynamic forcing functions
on the blade and tower are typically linearised around the
operating wind conditions to obtain the perturbation forces,
˜
f
M
(
˜
θ
i
, ˜v
i
) : R × R → R and
˜
f
x
(
˜
θ
col
, ˜v
col
) : R × R → R,
defined as follows:
˜
f
M
(
˜
θ
i
, ˜v
i
) =
df
M
dθ
θ
∗
,v
∗
˜
θ
i
(t) +
df
M
dv
θ
∗
,v
∗
˜v
i,
(t), (1c)
˜
f
x
(
˜
θ
col
, ˜v
col
) =
df
x
dθ
θ
∗
,v
∗
˜
θ
col
(t) +
df
x
dv
θ
∗
,v
∗
˜v
col
(t), (1d)
where
df
M
dθ
,
df
x
dθ
∈ R and
df
M
dv
,
df
x
dv
∈ R are the variations of
the forcing with respect to the pitch angle and apparent wind
speed. The deviations of the blade pitch angle and apparent
wind speed from their steady-states θ
∗
, v
∗
∈ R are
˜
θ
i
(t), ˜v
i
(t),
whilst
˜
θ
col
(t) :=
P
i
˜
θ
i
(t), ˜v
col
(t) =
P
i
˜v
i
(t) denote the
perturbations in collective pitch angle and the sum of the wind
speed effect on the rotor.
The wind turbine aerodynamic interactions of relevance
to this study are depicted in Figure 1. Owing to variable
blade geometry, the wind-induced forces are not uniformly
distributed on the blades and to model such forces, blade
element/momentum theory is adopted [13], where the blade
is discretised into small elements. Referring to Figure 1,
assuming the blade is rigid, the apparent stream-wise wind
speed perturbation ˜v
i,l
(t) experienced by blade i on span-
wise element l ∈ {1, . . . , L} ⊂ Z is dependent upon the
free-stream wind speed perturbation ˜v
∞
i,l
(t), deviations of the
fore-aft tower-top velocity
˙
˜x
fa
(t) and the tower-top rotational
velocity
˙
˜ϕ
fa
(t) from their equilibria, as follows:
˜v
i,l
(t) = ˜v
∞
i,l
(t) −
˙
˜x
fa
(t) +
˙
˜ϕ
fa
(t)r
l
sin
φ
i
(t)
, (2a)
where r
l
∈ R is the radial distance of the l-th blade
element. The azimuthal angle of each blade is defined
as [φ
1
(t), φ
2
(t), φ
3
(t)] : = [φ(t), φ(t) +
2π
3
, φ(t) +
4π
3
],
where φ(t) is the angle of the first blade from the horizontal
yaw axis with respect to the clockwise direction. This work
implicitly assumes the tower is a prismatic beam so that
the ratio between rotation and displacement is
2
3h
, where
h ∈ R is the height of the tower [3]. Thus, the fore-aft
rotational velocity of the tower-top can be approximated as
˙
˜ϕ
fa
(t) ≈
2
3h
˙
˜x
fa
(t). Since the focus of this work is on the
blade disturbance induced by the wind, the effect of the wind
perturbations upon the blade, ˜v
i
(t) in (1), can be approximated
by averaging the apparent wind speed perturbations ˜v
i,l
(t)
along the blade, as follows:
˜v
i
(t) ≈
1
L
X
l
˜v
i,l
(t),
= ˜v
∞,i
(t) −
˙
˜x
fa
(t) + k
ϕ
˙
˜x
fa
(t) sin
φ
i
(t)
. (2b)
Inspection of (1) and (2) indicates that coupling exists
between the dynamics of the blade flap-wise root-bending
moment and the tower, which is the key property that un-
derpins the subsequent work in this paper. By substituting (2)
into (1), the state-space representation of (1) can be formulated
as follows:
˙x(t) = A(t)x(t) + Bu(t) + B
d
d(t),
y(t) = Cx(t), (3)
where u(t) := [
˜
θ
1
(t),
˜
θ
2
(t),
˜
θ
3
(t)]
T
∈ R
n
u
and
y(t) := [
˜
M
1
(t),
˜
M
2
(t),
˜
M
3
(t)]
T
∈ R
n
y
are the control
inputs and measured outputs, respectively, whilst
d(t) := [˜v
∞,1
(t), ˜v
∞,2
(t), ˜v
∞,3
(t)]
T
∈ R
n
d
are the
wind disturbance inputs. The state vector is x(t) :=
[
˙
˜
M
1
(t),
˙
˜
M
2
(t),
˙
˜
M
3
(t),
˜
M
1
(t),
˜
M
2
(t),
˜
M
3
(t),
˙
˜x
fa
(t), ˜x
fa
(t)]
T
∈
R
n
x
. Notice that the system matrix A ∈ R
n
x
×n
x
is time-
dependent owing to the time-varying nature of the azimuth
angle.
III. TRANSFORMATION TO AN LTI SYSTEM AND
OBSERVABILITY ANALYSIS
For a linear time-varying (LTV) system (3), there exist
techniques for observability analysis and estimator design
(e.g. [17]). However, the problem of establishing the ob-
servability proof and synthesising an estimator for the LTV
system (3) can be greatly simplified by reformulating (3) as
an LTI system. As will now be shown, the key to achieving
this lies in the use of a coordinate transformation based upon
the Coleman Transform.
The Coleman Transform projects the blade loads in the
rotating frame of reference onto the fixed tilt and yaw turbine
axes. The typical Coleman transform T
cm
φ(t)
∈ R
3×3
is
3
defined as follows (e.g. [10] and references therein):
[
˜
M
col
(t),
˜
M
tilt
(t),
˜
M
yaw
(t)]
T
= T
cm
φ(t)
[
˜
M
1
(t),
˜
M
2
(t),
˜
M
3
(t)]
T
,
(4a)
T
cm
φ(t)
:=
2
3
1
2
1
2
1
2
sin (φ(t)) sin
φ(t) +
2π
3
sin
φ(t) +
4π
3
cos (φ(t)) cos
φ(t) +
2π
3
cos
φ(t) +
4π
3
,
(4b)
where
˜
M
col
(t),
˜
M
tilt
(t),
˜
M
yaw
(t) denote the perturbation on
the collective, tilt and yaw referred flap-wise blade root-
bending moments, respectively. The inverse Coleman trans-
form T
inv
cm
φ(t)
∈ R
3×3
is as follows:
[
˜
θ
1
(t),
˜
θ
2
(t),
˜
θ
3
(t)]
T
= T
inv
cm
φ(t)
[
˜
θ
col
(t),
˜
θ
tilt
(t),
˜
θ
yaw
(t)]
T
,
(4c)
T
inv
cm
φ(t)
:=
1 sin
φ(t)
cos
φ(t)
1 sin
φ(t) +
2π
3
cos
φ(t) +
2π
3
1 sin
φ(t) +
4π
3
cos
φ(t) +
4π
3
, (4d)
where
˜
θ
col
(t),
˜
θ
tilt
(t),
˜
θ
yaw
(t) represent the perturbations on
the collective pitch and the referred pitch signals upon the tilt
and yaw axis, respectively. The same also applies to the wind
speed ˜v
i
.
Clearly, the Coleman Transforms are time-dependent, and
hence their dynamics must be factored into any system model
that employs them. As shown in [10] from the perspective
of IPC design, models that arise from the misconceived
treatment of the Coleman Transforms as static projections give
rise to erroneous dynamics, leading to poor IPC performance.
Thus, this work presents the LTI reformulation of (3) with the
correct treatment of the Coleman Transforms in Theorem 3.1.
Theorem 3.1: Assuming a fixed rotor speed and Coleman
transformations (4), the linear time-varying system (3) can be
transformed into the following LTI form:
˙
ξ(t) = A
ξ
ξ(t) + B
ξ
u
cm
(t) + B
ξd
d
cm
(t),
y
cm
(t) = C
ξ
ξ(t), (5)
where y
cm
(t) = [
˜
M
col
(t),
˜
M
tilt
(t),
˜
M
yaw
(t)]
T
∈ R
n
y
,
u
cm
(t) = [
˜
θ
col
(t),
˜
θ
tilt
(t),
˜
θ
yaw
(t)]
T
∈ R
n
u
, d
cm
(t) =
[˜v
∞,col
(t), ˜v
∞,tilt
(t), ˜v
∞,yaw
(t)]
T
∈ R
n
d
are the referred
measurements of the flap-wise blade moments, pitch angle
signals and wind speeds upon the fixed reference frame, whilst
ξ(t) ∈ R
n
ξ
is the projection of the states associated with the
blade dynamics upon a non-rotating reference frame (19) and
the states of the tower dynamics (20).
Proof: See Appendix A.
Corollary 1: Assuming the model parameters obtained
from linearising the baseline turbine [18], the system (5) is
observable.
Proof: Trivial inspection of the rank of the system’s
observability matrix.
Hence, the tower motion states are observable from measure-
ments of the blade loads alone. This result lays the foundation
for the observer and controller designs of the next section.
Wind
Turbine
Coleman
Transform
T
cm
φ(t)
Inverse
Coleman
Transform
T
inv
cm
φ(t)
IPC
K
ipc
(s)
Observer
Γ
o
(s)
Tower
Controller
K
t
(s)
˜
M
col
(t)
˜
M
tilt
(t)
˜
M
tilt
(t)
˜
M
yaw
(t)
˜
M
yaw
(t)
˜
θ
tilt
(t)
+
˜
θ
tilt
(t)
˜
θ
yaw
(t)
˜
θ
yaw
(t)
˜
θ
col
(t)
G
cm
(s)
˜
M
1
(t)
˜
M
2
(t)
˜
M
3
(t)
˜
θ
1
(t)
˜
θ
2
(t)
˜
θ
3
(t)
˜
M
col
(t)
˜
M
tilt
(t)
˜
M
yaw
(t)
˜
θ
col
(t)
˜
θ
tilt
(t)
˜
θ
yaw
(t)
ˆ
˙x
fa
(t)
˜
θ
tilt
(t)
+
Proposed tower velocity estimation and control system
Fig. 2: Schematic of the proposed estimator and controller.
IV. DESIGN OF THE ESTIMATOR AND CONTROLLER
Figure 2 depicts the architecture of the proposed estima-
tion and control system, where the tower motion estima-
tor produces an estimate
ˆ
˙x
fa
(t) of the fore-aft velocity of
the tower-top based on Coleman-transformed blade moment
measurements
˜
M
col
(t),
˜
M
tilt
(t),
˜
M
yaw
(t) and pitch signals
˜
θ
col
(t),
˜
θ
tilt
(t),
˜
θ
yaw
(t). The individual pitch-based tower con-
troller subsequently employs this estimate to provide addi-
tional referred blade pitch signals upon the tilt axis
˜
θ
tilt
(t)
for attenuating the tower motion. Note that this architecture is
deliberately chosen so as to augment, rather than replace the
existing turbine controllers.
A. Estimator design
The system (5) is driven by the wind-induced disturbance,
which consists of slow-moving mean wind speeds and fast-
changing turbulence. We consider these wind speed distur-
bances as coloured noise. Given the known frequency spectra
of these wind speed disturbances, a linear wind model that
is driven by Gaussian white noise w(t) ∈ R
n
d
is defined as
follows:
˙
ξ
w
(t) = A
w
ξ
w
(t) + B
w
w(t), d
cm
(t) = C
w
ξ
w
(t), (6)
where the system matrices {A
w
, B
w
, C
w
} are determined by
fitting the spectra of the model output to the known spectra of
the wind speed disturbances. Combining the LTI system (5)
and the wind disturbance model (6), we yield the proposed
tower observer as follows:
˙x
a
(t) = A
a
x
a
(t) + B
a
u
cm
(t) + Le(t),
y
cm
(t) = C
a
x
a
(t), (7)
where x
a
(t) = [ξ(t), ξ
w
(t)]
T
∈ R
n
x
a
denotes the state of
the augmented system, whilst L ∈ R
n
x
a
×n
y
is a steady-state
Kalman filter gain and e(t) ∈ R
n
y
is the prediction error
between the plant and model output.
4
B. Estimation-based controller design
Typically, a tower controller provides an additional collec-
tive blade pitch signal on top of the CPC loop in response
to the tower fore-aft velocity, in order to dampen the fore-
aft structural mode. The excessive vibrations of the tower
are mainly concentrated around the resonant frequency of the
tower (0.32Hz in this work) [13]. However, the collective-
pitch-based approach might affect the rotor speed regulation
loop performance. Thus, this work proposes a novel tower
damping strategy using the existing Coleman transform-based
IPC architecture to decouple the CPC and IPC loops. The
proposed tower controller uses the referred pitch signal upon
the tilt axis in response to the tower-top velocity estimate,
as shown in Figure 2. The key challenge is to separate
the existing IPC loop and the tower damping control loop,
which is particularly important since the tower estimate is
also dependent upon the blade load measurements. To see
this, firstly consider the LTI system (5) in its transfer function
form:
y
cm
(s) = G
cm
(s)u
cm
(s). (8)
Secondly, consider the existing Coleman transform-based IPC
controller K
ipc
∈ R
2×2
, adopted from [11], [19]:
˜
θ
tilt
(s)
˜
θ
yaw
(s)
=
"
K
(1,1)
ipc
(s) K
(1,2)
ipc
(s)
K
(2,1)
ipc
(s) K
(2,2)
ipc
(s)
#
˜
M
tilt
(s)
˜
M
yaw
(s)
. (9)
Referring to Figure 2, together with the proposed tower
controller K
t
∈ R and the observer Γ
ob
∈ R
1×(n
u
+n
y
)
, the
pitch signal θ
tilt
on the tilt axis becomes:
˜
θ
tilt
(s) = K
(1,1)
ipc
(s)
˜
M
tilt
(s) + K
(1,2)
ipc
(s)
˜
M
yaw
(s) + K
t
(s)
ˆ
X
fa
(s),
(10)
where the estimate of the tower-top fore-aft velocity
ˆ
X
fa
∈ R
can be expressed as follows:
ˆ
X
fa
(s) = Γ
ob
(s)[u
cm
(s), y
cm
(s)]
T
, (11a)
Γ
ob
(s) := [Γ
(1,1)
ob
(s), Γ
(1,2)
ob
(s), Γ
(1,3)
ob
(s), ...
... Γ
(1,4)
ob
(s), Γ
(1,5)
ob
(s), Γ
(1,6)
ob
(s)]. (11b)
By substituting (11) into (10), the existing IPC K
ipc
in (9) is
inevitably coupled with the tower controller K
t
and becomes
K
m
ipc
∈ R
2×2
, where:
K
m
ipc
(s) =
"
K
m(1,1)
ipc
(s) K
m(1,2)
ipc
(s)
K
(2,1)
ipc
(s) K
(2,2)
ipc
(s)
#
, (12a)
K
m(1,1)
ipc
(s) =
I + K
t
(s)Γ
(1,2)
ob
(s)
K
(1,1)
ipc
(s) + Γ
(1,5)
ob
(s), (12b)
K
m(1,2)
ipc
(s) =
I + K
t
(s)Γ
(1,3)
ob
(s)
K
(1,2)
ipc
(s) + Γ
(1,6)
ob
(s), (12c)
Thus the observer introduces undesirable, but inevitable cou-
pling from the tower controller to the existing IPC. Nonethe-
less, the Coleman transform-based IPC typically targets the
static and 3p (thrice per revolution) non-rotating loads caused
by the blade (e.g. 0 and 0.6 Hz) [20], whilst tower loads occur
mainly at the tower resonant frequency (0.32Hz). Therefore,
with a view towards avoiding the undesired couplings, the
tower controller is designed as an inverse notch filter with
gain concentrated at the tower resonant frequency, away from
multiples of the blade rotational frequency:
K
t
(s) := K
p
s
2
+ 2D
1
ω
t
s + ω
2
t
s
2
+ 2D
2
ω
t
s + ω
2
t
, (13)
where K
p
= 0.03, D
1
= 10 and D
2
= 0.05.
To examine the coupling between the existing IPC and the
proposed tower controller, Figure 3 shows the closed-loop
0 0.5 1 1.5 2
Frequency f [Hz]
0
0.5
1
1.5
2
|S
(1,1)
(f)|, |S
m(1,1)
(f)|
0 0.5 1 1.5 2
Frequency f [Hz]
0
0.5
1
1.5
2
|S
(1,2)
(f)|, |S
m(1,2)
(f)|
0 0.5 1 1.5 2
Frequency f [Hz]
0
0.5
1
1.5
2
|S
(2,1)
(f)|, |S
m(2,1)
(f)|
0 0.5 1 1.5 2
Frequency f [Hz]
0
0.5
1
1.5
2
|S
(2,2)
(f)|, |S
m(2,2)
(f)|
Fig. 3: Magnitude Bode plots of the closed-loop sensitivity
functions of (I + G
cm
K
ipc
(s))
−1
(Solid blue line) and (I +
G
cm
K
m
ipc
(s))
−1
(Dashed red line).
sensitivity functions of the original IPC controller S(s) :=
(I + G
cm
K
ipc
(s))
−1
and the coupled controller structure
S
m
(s) := (I + G
cm
K
m
ipc
(s))
−1
. It is clear from the figure
that the disturbance gain of the coupled control structure
remains similar to the original IPC, which is also still largely
unaffected across all frequencies. In addition, the coupled
control structure and the existing controller possesses the same
robust stability margin (0.39), suggesting the proposed design
does not affect the robustness of the original IPC.
V. NUMERICAL RESULTS AND DISCUSSION
This section presents simulation results to demonstrate the
performance of the proposed estimator and estimation-based
controller for the tower fore-aft motion. The turbine model
employed in this work is the NREL 5MW turbine [18] and
the simulations are conducted on FAST [21]. This turbine
model is of much greater complexity than the linear model (7).
All degrees-of-freedom were enabled, including flap-wise and
edge-wise blade modes, in addition to the tower and shaft
dynamics.
A. Estimator Performance
The proposed observer (7) was compared with a typical
double-integrator Kalman-filter design based on measurements
from the tower fore-aft accelerometers (e.g. [13]), subse-
quently referred to as the baseline design. All measurements
were perturbed with additive white noise and simulations were
conducted under three time-varying wind field test cases: (i)
above-rated; (ii) below-rated and (iii) full operating wind
conditions.
Simulations in Figure 4 were conducted under a time-
varying wind field with a mean wind speed of 18 ms
−1
and a turbulence intensity of 5%, with the hub-height wind
speed shown in Figure 4a. It can be seen that in Figure 4b
good agreement was achieved between the proposed and
baseline design and actual tower velocity. Nonetheless, small
discrepancies for both methods are revealed by evaluating the
estimate error magnitude, auto-correlations and spectra, shown
in Figures 4c, 4d and 4e, respectively. A residual test [22] was
adopted, that suggests the estimate errors would be white noise
