An efficient frequency-domain model for quick load analysis of floating offshore wind turbines

Abstract

A model for Quick Load Analysis of Floating wind turbines (QuLAF) is presented and validated here. The model is a linear, frequency-domain, efficient tool with four planar degrees of freedom: floater surge, heave, pitch and first tower modal deflection. The model relies on state-of-the-art tools from which hydrodynamic, aerodynamic and mooring loads are extracted and cascaded into QuLAF. Hydrodynamic and aerodynamic loads are pre-computed in WAMIT and FAST, respectively, while the mooring system is linearized around the equilibrium position for each wind speed using MoorDyn. An approximate approach to viscous hydrodynamic damping is developed, and the aerodynamic damping is extracted from decay tests specific for each degree of freedom. Without any calibration, the model predicts the motions of the system in stochastic wind and waves with good accuracy when compared to FAST. The damage-equivalent bending moment at the tower base is estimated with errors between 0.2% and 11.3% for all the load cases considered. The largest errors are associated with the most severe wave climates for wave-only conditions and with turbine operation around rated wind speed for combined wind and waves. The computational speed of the model is between 1300 and 2700 times faster than real time.

Full text

Wind Energ. Sci., 3, 693–712, 2018
https://doi.org/10.5194/wes-3-693-2018
© Author(s) 2018. This work is distributed under
the Creative Commons Attribution 4.0 License.
An efficient frequency-domain model for quick load
analysis of floating offshore wind turbines
Antonio Pegalajar-Jurado, Michael Borg, and Henrik Bredmose
Department of Wind Energy, Technical University of Denmark, Nils Koppels Allé 403,
2800 Kongens Lyngby, Denmark
Correspondence: Antonio Pegalajar-Jurado (ampj@dtu.dk)
Received: 24 March 2018 – Discussion started: 9 April 2018
Revised: 15 August 2018 – Accepted: 2 September 2018 – Published: 16 October 2018
Abstract. A model for Quick Load Analysis of Floating wind turbines (QuLAF) is presented and validated here.
The model is a linear, frequency-domain, efficient tool with four planar degrees of freedom: floater surge, heave,
pitch and first tower modal deflection. The model relies on state-of-the-art tools from which hydrodynamic,
aerodynamic and mooring loads are extracted and cascaded into QuLAF. Hydrodynamic and aerodynamic loads
are pre-computed in WAMIT and FAST, respectively, while the mooring system is linearized around the equilib-
rium position for each wind speed using MoorDyn. An approximate approach to viscous hydrodynamic damping
is developed, and the aerodynamic damping is extracted from decay tests specific for each degree of freedom.
Without any calibration, the model predicts the motions of the system in stochastic wind and waves with good
accuracy when compared to FAST. The damage-equivalent bending moment at the tower base is estimated with
errors between 0.2 % and 11.3% for all the load cases considered. The largest errors are associated with the most
severe wave climates for wave-only conditions and with turbine operation around rated wind speed for combined
wind and waves. The computational speed of the model is between 1300 and 2700 times faster than real time.
1 Introduction: the need for an efficient,
frequency-domain tool
Offshore wind energy is a key contributor to a carbon-free
energy supply. Most of today’s offshore wind farms are
bottom-fixed, meaning their feasibility is limited to shallow
and intermediate water depths. On the other hand, the wind
resource in deep water represents an enormous potential that
can be unlocked with the deployment of floating wind farms.
An important step in making floating wind turbines econom-
ically feasible is the application of larger wind turbines and
the ability to design the floater to a minimum cost. The design
of a floating substructure for offshore wind deployment de-
pends on many design variables, and each possible combina-
tion is a potential design. In the design process, the candidate
designs need to be simulated in different environmental con-
ditions in order to assess the magnitude of the motions and
loads in the system. These simulations are typically carried
out with time-domain numerical tools, which allow a repre-
sentative modelling of the physical phenomena involved and
can simulate at about real-time CPU speed. However, this ap-
proach can be computationally expensive, especially if one
needs to evaluate different floater designs under several envi-
ronmental conditions. For an improved design process, faster
tools are needed to allow optimization in the initial design
stage, where the design space has to be thoroughly explored
and a broad overview of the system response is desirable.
A few studies of simplified design models for offshore
wind turbine floaters exist in the literature. Lupton (2014)
presented a frequency-domain numerical tool for the analy-
sis of the OC3-Hywind spar floating wind turbine (Jonkman,
2010), with eight degrees of freedom (DoFs): one normal
mode per blade, two tower fore-aft modes, and floater surge,
heave and pitch. The model included linear hydrodynamics
computed with a potential-flow panel code. The aerodynamic
forces were included through harmonic linearization, and the
mooring lines were represented by a stiffness matrix. The
frequency-domain code was benchmarked against an equiva-
lent Bladed (DNV-GL AS, 2016) model with Morison-based
Published by Copernicus Publications on behalf of the European Academy of Wind Energy e.V.

694 A. Pegalajar-Jurado et al.: An efficient frequency-domain model
hydrodynamics, and with a stiffness mooring matrix. Nei-
ther the frequency-domain model nor the Bladed model in-
cluded viscous drag. Results were shown for regular waves
and uniform, harmonic wind, and the frequency-domain code
was reported to be up to 37 times faster than the Bladed
model. In Lemmer et al. (2016), simplified time-domain
models of the OC3-Hywind spar (Jonkman, 2010) and OC4-
DeepCwind semi-submersible (Robertson et al., 2014) float-
ing wind turbines were introduced. The models had four
DoFs: floater surge and pitch, tower first fore-aft mode,
and rotor azimuthal position. Linear hydrodynamics from a
radiation-diffraction panel code were included in the time-
domain model through the Cummins equation (Cummins,
1962). Aerodynamics were computed by coupling the code
to AeroDyn. Quasi-static mooring forces were computed by
solving the catenary mooring equations at each time step. A
linearized version of the code was also presented. In the re-
sults, the linearized frequency-domain version was success-
fully benchmarked against the nonlinear time-domain ver-
sion, by comparing the linear transfer function from wave
height to tower-top displacement with its nonlinear equiva-
lent. The work of Wang et al. (2017) involved a frequency-
domain model of the DeepCwind semi-submersible (Robert-
son et al., 2014) with two rigid-body DoFs: floater surge and
pitch. Linear hydrodynamics, linearized drag and drift forces
were computed with the commercial software AQWA. The
aerodynamic loads were included through a linearized ver-
sion of the actuator point equation, where the aerodynamic
contribution was divided into a constant force and a damping
term – thus neglecting stochastic wind forcing. The moor-
ing loads were included through a stiffness matrix, obtained
from both quasi-static and dynamic mooring models. The
model was validated against DeepCwind test data in terms of
natural frequencies, response-amplitude operators and power
spectral density (PSD) plots of surge and pitch response, gen-
erally obtaining a good agreement. However, a frequency-
domain model for floating wind turbines able to incorporate
realistic aerodynamic loads is still needed.
For bottom-fixed offshore wind turbines, Schløer et al.
(2018) recently developed an efficient, frequency-domain
model named QuLA (Quick Load Analysis), considering the
DTU 10 MW Reference Wind Turbine (RWT; Bak et al.,
2013). The mono-pile foundation and the wind turbine tower
were defined as an Euler beam, and the first fore-aft modal
deflection of this beam was the only DoF. Inspired by the
work of van der Tempel (2006), the rotor and nacelle were
represented by a point mass at the tower top, and aerody-
namic loads and damping coefficients were pre-computed in
the time-domain aero-elastic tool Flex5 (Øye, 1996). Com-
pared to the work of van der Tempel (2006), the aerodynamic
damping in QuLA was considered as dependent on mean
wind speed. Hydrodynamic forcing was included through the
Morison equation (Morison et al., 1950), where the struc-
ture velocity and acceleration were neglected. The code was
validated against Flex5 in terms of time series, PSD, ex-
ceedance probability curves and fatigue damage-equivalent
load (DEL). The bending moment at the seabed was esti-
mated by QuLA within a 5 % error, and the code was reported
to be approximately 40 times faster than its Flex5 equivalent.
This study presents the extension of QuLA to floating off-
shore wind turbines. The resulting model, QuLAF (Quick
Load Analysis of Floating wind turbines), was first pre-
sented in Pegalajar-Jurado et al. (2016), with only two DoFs:
floater surge and tower first fore-aft bending mode. Here
we present an improved version of the model, a frequency-
domain code that captures the four dominant DoFs in the
in-plane global motion: floater surge, heave and pitch, and
tower first fore-aft modal deflection. The model, which is
here adapted to the DTU 10 MW RWT mounted on the
OO-Star Wind Floater Semi 10 MW (Yu et al., 2018) was
set up through cascading techniques. In the cascading pro-
cess, information is pre-computed or extracted from more
advanced models (parent models) to enhance the simplified
models (children models). In this case, the hydrodynamic
loads are extracted from the radiation-diffraction potential-
flow solver WAMIT (Lee and Newman, 2016). The aero-
dynamic loads and aerodynamic damping coefficients are
pre-computed in the numerical tool FAST v8 (Jonkman and
Jonkman, 2016), and the mooring module MoorDyn (Hall,
2017) is employed to extract a mooring stiffness matrix for
different operating positions. This way, the model includes
standard radiation-diffraction theory and realistic rotor loads
through pre-computed aero-elastic simulations. In the model,
the system response is obtained by solving the linear equa-
tions of motion (EoMs) in the frequency domain, leading to
a very efficient tool. While the radiation-diffraction results
allow for a full linear response evaluation for rigid structure
motion in waves, the ambition of this model is to extend them
with the flexible tower and realistic stochastic rotor loads,
thus going one step further than other simplified models in
the literature. The results from QuLAF are here benchmarked
against results from its time-domain, state-of-the-art (SoA)
parent model in terms of time series, PSD, exceedance prob-
ability and fatigue DEL. We assess the strengths and weak-
nesses of the cascading process by comparison with the orig-
inal time-domain model, and further develop techniques to
improve the accuracy of the simplified model with respect to
planar motion and tower-base loads. In this way, the potential
of the model as a reliable tool for pre-design and as a comple-
ment to SoA models is demonstrated. The idea is that once
the conceptual floater design is established with the efficient
pre-design model, more advanced SoA models can be used
for further design verification with a full design load basis
that includes extreme and transient events.
2 The case study
The floating wind turbine chosen for the present study is the
DTU 10 MW RWT (Bak et al., 2013) mounted on the OO-
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A. Pegalajar-Jurado et al.: An efficient frequency-domain model 695
Figure 1. The OO-Star Wind Floater Semi 10 MW concept (http:
//www.olavolsen.no; last access: 24 November 2017).
Star Wind Floater Semi 10 MW (Yu et al., 2018). The main
properties of the DTU 10 MW RWT are given in Table 1, and
further information can be found in Bak et al. (2013). The
basic DTU Wind Energy controller (Hansen and Henriksen,
2013) is utilized, and tuned to avoid the floater pitch instabil-
ity (commonly known as the “negative damping problem”)
reported in, for example, Larsen and Hanson (2007).
The floating substructure (see Fig. 1), developed by
Dr.techn. Olav Olsen AS (http://www.olavolsen.no; last ac-
cess: 24 November 2017), is a semi-submersible floater made
of post-tensioned concrete. It has a central column and three
outer columns mounted on a star-shaped pontoon with three
legs. Each outer column is connected to the seabed by a
catenary mooring line with a suspended clump weight. The
main properties of the floating substructure are collected in
Table 2, and further information can be found in Yu et al.
(2018).
3 Time vs. frequency domain: advantages and
disadvantages
Floating wind turbines can be considered harmonic oscilla-
tors with multiple coupled DoFs. To illustrate the strengths
and weaknesses of solving the relevant EoM in the time
or the frequency domain, a simple one-DoF mass-spring-
damper system is considered,
m¨
ξ(t)+b˙
ξ(t)+cξ (t)=F(t),(1)
where mis the system mass, bis the damping coefficient, c
is the restoring coefficient, ξ(t) is the system displacement
from its equilibrium position, and F(t) is a harmonic excita-
tion force. Equation (1) can be also written in complex nota-
tion, by expressing the excitation force at the frequency ωas
F(t)= <{ ˆ
F(ω)eiωt }, where <indicates the real part, ˆ
F(ω)
is the Fourier transform of F(t) and iis the imaginary unit.
If the initial transient part of the response is neglected, the
steady-state system response at the given frequency can also
be written as ξ(t)= <{ˆ
ξ(ω)eiωt }, leading to the EoM in the
frequency domain,
[−ω2m+iωb +c]ˆ
ξ(ω)=ˆ
F(ω),
⇒ˆ
ξ(ω)=ˆ
F(ω)
−ω2m+iωb +c≡H(ω)ˆ
F(ω).(2)
The frequency-domain response ˆ
ξ(ω) may be obtained by
simply multiplying the frequency-domain excitation force
ˆ
F(ω) by the transfer function H(ω). This can be done at
all frequencies and, due to the linearity, one can add the
results at each frequency to obtain the total solution. Thus,
once ˆ
ξ(ω) has been determined for all frequencies, the time-
domain response ξ(t) is obtained through an inverse Fourier
transform of ˆ
ξ(ω). If fast Fourier transform (FFT) and in-
verse fast Fourier transform (iFFT) are used, the solution can
be obtained very quickly.
Figure 2 shows the surge response ξ(t) of a one-DoF
model of the OC3-Hywind spar floating wind turbine
(Jonkman, 2010) subjected to stochastic hydrodynamic lin-
ear forcing. The response labelled as “Time domain” was ob-
tained by time-stepping of Eq. (1) with the classical fourth-
order Runge–Kutta method and initial conditions ξ(0) =0
and ˙
ξ(0) =0. The response labelled as “Frequency domain”
was computed by first obtaining the frequency-domain ex-
citation force ˆ
F(ω)=FFT(F(t)), calculating the frequency-
domain response using Eq. (2) and finally going back to
the time-domain response ξ(t)= <{iFFT(ˆ
ξ(ω))}. The sim-
ulation time step was 0.025 s and the total simulated time
was 5400 s, although only the first 1000 s are shown here.
The time-domain solution took 69.41 s to run, while the
frequency-domain solution was done in 0.03 s, or 2344 times
faster. The two responses diverge at the beginning, where
the time-domain solution is dominated by the transient re-
sponse to the initial conditions, which is not present in the
frequency-domain solution. However, after approximately
800 s (or six natural periods) and until the end of the sim-
ulation, the two solutions are practically identical, with er-
rors between 0.2 % and 0.5 %, likely due to the time and fre-
quency discretizations.
In addition to the gain in CPU speed, solving the EoM
in the frequency domain allows for an easier handling
of frequency-dependent properties, such as hydrodynamic
added mass and radiation damping. On the other hand, it has
also been shown that transient response due to initial con-
ditions is only captured by time-domain models. However,
as in the above example, the transient response due to ini-
tial conditions is an artifact of the time-domain formulation
and is often discarded in the analysis. Perhaps the most clear
disadvantage of frequency-domain models is that they can
only accommodate loads that depend linearly on the response
and its time derivatives, such as hydrodynamic added mass
loads or hydrostatic loads. They cannot directly accommo-
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696 A. Pegalajar-Jurado et al.: An efficient frequency-domain model
0 100 200 300 400 500 600 700 800 900 1000
Time [s]
-1
0
1
2
[m]
Time domain
Frequency domain
0 100 200 300 400 500 600 700 800 900 1000
Time [s]
-150
-100
-50
0
50
100
Error [%]
(a)
(b)
Figure 2. (a) Surge response of the OC3-Hywind floating wind turbine (Jonkman, 2010) to stochastic hydrodynamic linear forcing in a one-
DoF linear model, computed in both the time and frequency domains. (b) Relative error between the time- and frequency-domain solutions.
Only the first 1000 s are shown.
Table 1. Key figures for the DTU 10 MW RWT.
Rated power Rated wind speed Wind regime Rotor diameter Hub height
10 MW 11.4 m s−1IEC Class 1A 178.3 m 119 m
date loads that depend on the response in a nonlinear manner,
such as viscous drag from relative structural motion or cate-
nary mooring loads. In those cases, simplified or linearized
formulations have to be implemented instead.
4 The time-domain, state-of-the-art numerical model
In SoA models the nacelle, hub and floater are often con-
sidered rigid, whereas the tower and blades are flexible.
The floater motion typically has six DoFs: surge, sway,
heave, roll, pitch and yaw. Aerodynamics are normally
computed using unsteady blade element momentum theory
(Hansen, 2008). Hydrodynamics are typically represented by
radiation-diffraction theory (Newman, 1980), the Morison
equation or a combination of both. The mooring lines can
be modelled with either quasi-static or dynamic approaches.
In general, SoA models are more accurate than simplified
models, but they also have a higher CPU cost.
A SoA, time-domain numerical model of the OO-Star
Semi +DTU 10 MW floating wind turbine was used in this
study as a parent model to QuLAF. The SoA model was im-
plemented in FAST v8.16.00a-bjj (Jonkman and Jonkman,
2016) with active control and 15 DoFs for turbine and floater:
first and second flap-wise blade modal deflections; first edge-
wise blade modal deflection; drivetrain rotational flexibility;
drivetrain speed; first and second fore-aft and side-side tower
modal deflections; and floater surge, sway, heave, roll, pitch
and yaw. The turbulent wind fields were computed in Turb-
Sim, and the aerodynamic loads were modelled with Aero-
Dyn v14. The basic DTU Wind Energy controller was ap-
plied through a dynamic-link library (DLL). The mooring
loads, calculated by MoorDyn (Hall, 2017), included buoy-
ancy, mass inertia and hydrodynamic loads resulting from
the motion of the mooring lines in calm water. Hydrody-
namic loads on the floater were first computed in WAMIT
(Lee and Newman, 2016) and coupled to FAST through the
Cummins equation. Viscous effects were modelled internally
by the Morison drag term. Further details on the modelling
of floating wind turbines in FAST can be found in Jonkman
(2009), while a thorough description of the FAST model used
in this study is presented in Pegalajar-Jurado et al. (2018b)
and Pegalajar-Jurado et al. (2018a).
5 The frequency-domain, cascaded numerical
model
The simplest model for the dynamic analysis of floating wind
turbines would only have a few DoFs, typically rigid-body
motion of the floater in surge and pitch. Aerodynamic loads
would be represented by a point force at the rotor hub and de-
fined by an actuator point model. If the floating substructure
is slender compared to the incident waves, a strip-theory ap-
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A. Pegalajar-Jurado et al.: An efficient frequency-domain model 697
Table 2. Key figures for the OO-Star Wind Floater Semi 10 MW anchored at the selected site.
Water depth Mooring length Draught Freeboard Displaced volume Mass incl. ballast
130 m 703 m 22 m 11 m 23 509 m321 709 t
Figure 3. Sketch of the floating wind turbine as seen by the QuLAF model.
proach may be applied to compute the hydrodynamic loads
from the Morison equation. The forces exerted by the moor-
ing system can be included through a stiffness matrix in the
linear EoMs. Simplified, low-order models are very CPU ef-
ficient but their accuracy is often limited. In the following we
present a simplified model that combines elements extracted
from a SoA model into a very efficient tool, which aims at
getting close to the accuracy of the SoA model while still
retaining the CPU efficiency of low-order models.
QuLAF represents the floating wind turbine as two lumped
masses – floater and rotor-nacelle assembly – connected by a
flexible tower. The model captures four planar DoFs – floater
surge, heave, pitch and first tower fore-aft modal deflection
– and is thus applicable to aligned wind and wave situations.
The floating wind turbine is represented as depicted in Fig. 3.
The EoM is a matrix version of Eq. (2),
h−ω2(M+A(ω)) +iωB(ω)+Ciˆ
ξ(ω)=ˆ
F(ω),
⇒ˆ
ξ(ω)=H(ω)ˆ
F(ω),(3)
where Mis the structural mass and inertia matrix, A(ω) is
the frequency-dependent, hydrodynamic added mass and in-
ertia matrix, B(ω) is the frequency-dependent damping ma-
trix, and Cis the restoring matrix. The vector ˆ
ξ(ω) is the
dynamic response in the frequency domain for the four DoFs
and ˆ
F(ω) is the dynamic vector of excitation forces and mo-
ments in the frequency domain. The system transfer function
is given by H(ω). The different elements in Eq. (3) are de-
scribed in detail below.
5.1 Dynamic response vector
The dynamic response vector,
ˆ
ξ(ω)=



ˆ
ξ1(ω)
ˆ
ξ3(ω)
ˆ
ξ5(ω)
ˆα(ω)



,(4)
has one element for each DoF: floater surge, heave, pitch
and first tower fore-aft modal deflection. The sign conven-
tion is that shown in Fig. 3, with positive surge in the down-
wind direction, positive heave upwards, positive pitch (about
flotation point O) clockwise and positive tower deflection in
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698 A. Pegalajar-Jurado et al.: An efficient frequency-domain model
the downwind direction. The physical tower deflection at any
height zcan be obtained by multiplying the mode shape φ(z)
and the modal deflection α(t). The tower deflection at the hub
height hhub is therefore given by δ(t)=φhubα(t). If the ab-
solute nacelle displacement is sought, the contributions from
floater surge and pitch motions must be added to the tower
deflection, and the global response vector ˆ
ξglob(ω) is found
by introducing a transformation matrix Tglob,
ˆ
ξglob(ω)=



1 0 0 0
0 1 0 0
0 0 1 0
1 0 hhub φhub







ˆ
ξ1(ω)
ˆ
ξ3(ω)
ˆ
ξ5(ω)
ˆα(ω)



=Tglob ˆ
ξ(ω).
(5)
5.2 Dynamic load vector
The dynamic load vector,
ˆ
F(ω)=ˆ
Fhydro(ω)+ˆ
Faero(ω),(6)
contains hydrodynamic loads ˆ
Fhydro(ω) and aerodynamic
loads ˆ
Faero(ω). Hydrodynamic loads are extracted from the
solution to the diffraction problem, which provides a vec-
tor of wave excitation forces and moments in all six DoFs,
namely ˆ
X(ω). These excitation forces are normalized to
waves of unit amplitude, therefore the wave loads for a spe-
cific time series of free-surface elevation η(t) are obtained by
the product ˆ
X(ω)ˆη(ω). The vector of wave excitation forces
and moments is reduced to adapt it to the simplified model,
and a zero is added in the fourth element for the tower DoF,
ˆ
Fhydro(ω)=ˆ
X(ω)ˆη(ω)≡



ˆ
X1(ω)
ˆ
X3(ω)
ˆ
X5(ω)
0



ˆη(ω),(7)
where ˆη(ω) can be computed from an input time series η(t)
or from a theoretical wave spectrum. The only viscous effect
considered in the model is viscous damping (see Sect. 5.8.1),
but viscous forcing is neglected to keep the model computa-
tionally efficient. This simplification, however, is considered
reasonable because hydrodynamics for this floater are domi-
nated by inertia loads, and viscous forcing is expected to be
relevant mainly for severe sea states, which lie on the border
of the model’s applicability. The vector of aerodynamic loads
only contains the dynamic part of the wind loads and has the
format
ˆ
Faero(ω)=



ˆ
Faero,1(ω)
ˆ
Faero,3(ω)
ˆ
Faero,1(ω)hhub + ˆτaero(ω)
ˆ
Faero,1(ω)φhub + ˆτaero(ω)φz,hub



,(8)
where ˆ
Faero,1(ω) and ˆ
Faero,3(ω) represent the horizontal and
vertical components of the aerodynamic loads on the rotor,
respectively. The aerodynamic tilt torque on the rotor is given
by ˆτaero(ω). The fourth element of ˆ
Faero represents the ef-
fect of the aerodynamic loads on the tower modal deflection,
hence the mode shape deflection φhub and its slope φz,hub
evaluated at the hub are involved. The time-domain aerody-
namic loads for each mean wind speed Ware pre-computed
in the SoA model, as detailed in Sect. 5.8.2.
5.3 Structural mass and inertia matrix
The symmetric matrix of structural mass and inertia, ob-
tained by looking at the forces needed to produce unit ac-
celerations in the different DoFs, is defined as
M=















mtot 0mtotzCM
tot mrnφhub +
Nt
P
i=1eρiφi1zi
mtot 0 0
IO
tot mrnφhub hhub +ITT
rn φz,hub
+
Nt
P
i=1eρiφizi1zi
mrnφ2
hub +ITT
rn φ2
z,hub
+
Nt
P
i=1eρiφ2
i1zi















,(9)
where mtot is the total mass of the floating wind turbine,
mtot =mf+mrn +PNt
i=1eρi1zi, which includes the mass of
the floater mf, the rotor-nacelle mass mrn and the mass sum
of all the Ntelements that compose the flexible tower, each
with a mass per length eρiand a height 1zi. The total mass
inertia of the system about the yaxis at the flotation point
Ois given by IO
tot =IO
f+IO
rn +
Nt
P
i=1eρiz2
i1zi, including the
floater inertia IO
f, the rotor-nacelle inertia IO
rn and the inertia
of each of the tower elements, located at an absolute height
zi=(zt,i +ht), where zt ,i is the vertical position of the el-
ement iwith respect to the tower base, located at a height
ht. The centre of mass (CM) of the whole structure is located
at zCM
tot =(mfzCM
f+mrnhhub +PNt
i=1eρizi1zi)/mtot, with con-
tributions from the floater CM at zCM
f, the rotor-nacelle CM
at the hub height hhub and the CM of each of the tower el-
ements. The mode shape deflection of the tower evaluated
at a generic tower element iis φi, while φhub and φz,hub are
the mode shape deflection and its slope evaluated at the hub.
Finally, ITT
rn represents the mass inertia of the rotor-nacelle
assembly referred to the tower top. The tower structural prop-
erties and first mode shape are the same as the ones given as
an input to the SoA model.
5.4 Hydrodynamic added mass matrix and damping
matrix
The frequency-dependent, hydrodynamic added-mass and
radiation-damping matrices, A(ω) and Brad(ω), can be pre-
computed in a radiation-diffraction solver. Here, the same
WAMIT output files used for the SoA model are loaded into
QuLAF. However, the original 6 ×6 matrices are reduced by
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A. Pegalajar-Jurado et al.: An efficient frequency-domain model 699
removing the rows and columns corresponding to the DoFs
not included in the simplified model (sway, roll, yaw), and a
row and column of zeros is added for compatibility with the
tower DoF,
A(ω)=



a11(ω)a13 (ω)a15(ω) 0
a31(ω)a33 (ω)a35(ω) 0
a51(ω)a53 (ω)a55(ω) 0
0 0 0 0



,
Brad(ω)=



b11(ω)b13 (ω)b15(ω) 0
b31(ω)b33 (ω)b35(ω) 0
b51(ω)b53 (ω)b55(ω) 0
0 0 0 0



.(10)
The global damping matrix includes contributions from
the hydrodynamic radiation damping Brad(ω), the hydro-
dynamic viscous damping Bvis, the aerodynamic damping
Baero(ω) and the tower structural damping Bstruc:
B(ω)=Brad(ω)+Bvis +Baero (ω)+Bstruc.(11)
The hydrodynamic viscous damping matrix Bvis is ana-
lytically extracted from the Morison equation, as shown in
Section 5.8.1. The diagonal matrix of aerodynamic damping,
Baero(ω)=


baero,11(ω) 0 0 0
0 0 0
baero,55(ω) 0
baero,tow


,(12)
is extracted from the SoA model for each mean wind speed
W, as detailed in Section 5.8.2. The matrix of structural
damping only concerns the tower and is given by
Bstruc =


0 0 0 0
0 0 0
0 0
2ζstruc,tow√CtowMtow


,(13)
where the structural damping ratio for the first fore-aft tower
mode, ζstruc,tow, is directly taken from the input to the SoA
model, and Ctow and Mtow are the last diagonal elements of
the system restoring matrix Cand the mass inertia matrix M,
respectively.
5.5 Restoring matrix
The restoring matrix includes hydrostatic stiffness Chst,
structural stiffness Cstruc and mooring stiffness Cmoor:
C=Chst +Cstruc +Cmoor.(14)
The hydrostatic matrix should only include the contribu-
tions from the centre of buoyancy (CB) and waterplane area.
It is computed as part of the radiation-diffraction solution,
and is reduced following the same procedure as for the added
mass and radiation damping matrices. The symmetric matrix
of structural stiffness is given by
Cstruc =








0 0 0 0
0 0 0
−mtotgzCM
tot −mrngφhub −
Nt
P
i=1eρigφi1zi
Nt
P
i=1
EIiφ2
zz,i 1zi








,(15)
where gis the acceleration of gravity, and EIiand φzz,i are
the bending stiffness and the curvature of the mode shape
for the tower element i, respectively. The off-diagonal term
represents the negative restoring effect of the tower and rotor-
nacelle mass on the tower DoF when the floater pitches. The
mooring restoring matrix Cmoor is position-dependent and
therefore extracted from the SoA model for each mean wind
speed W, as detailed in Sect. 5.8.3. Although in this study
wind is the only effect considered to affect the mean posi-
tion of the floating wind turbine, other effects such as mean
drift forces and current can be taken into account in the SoA
model when linearizing the mooring system.
5.6 Static load and response
Static loads are related to the equilibrium of the structure. In
the model, the static part of the response, ξst, is added to the
dynamic part ˆ
ξ(ω) when it is converted from the frequency
to the time domain via iFFT. The static loads applied are
Fst =Faero,st +Fgrav +Fbuoy,(16)
which include the static part of the aerodynamic loads
Faero,st, the gravity loads Fgrav and the buoyancy loads
Fbuoy. The gravity load vector is given by
Fgrav =



0
−mtotg−Fmoor,z
mrngxCM
rn
mrngxCM
rn φz,hub



,(17)
where Fmoor,z is the vertical force exerted by the mooring
lines in equilibrium, and xCM
rn is the horizontal coordinate of
the rotor-nacelle CM.
The buoyancy load vector is
Fbuoy =



0
ρwgVf
−ρwgVfxCB
f
0



,(18)
where ρwis the water density, Vfis the volume displaced by
the floater and xCB
fis the horizontal coordinate of the floater
CB. With no wind and only linear wave forcing, the floating
wind turbine operates around its equilibrium position with
a stiffness matrix C0. If wind (or any other mean force) is
introduced, the floating wind turbine is moved to a new equi-
librium position, where the stiffness matrix is CW. The static
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700 A. Pegalajar-Jurado et al.: An efficient frequency-domain model
response ξst is therefore obtained from the static loads by
considering a mean stiffness matrix Cst,
Cst =C0+CW
2⇒Cstξst =Fst .(19)
This approximation is accurate to second order.
5.7 System natural frequencies
The vector of natural frequencies ω0is found by solving the
undamped eigenvalue problem given by
h−ω02(M+A(ω0)) +Ciˆ
ξ(ω0)=0,
⇒ω2
0ˆ
ξ(ω0)=(M+A(ω0))−1Cˆ
ξ(ω0).(20)
Since the matrix of added mass depends on frequency, the
eigenvalue problem is solved in a frequency loop. For each
frequency ω, the four possible natural frequencies are com-
puted. When one of the four possible frequencies obtained is
equal to the frequency of that particular iteration in the loop,
then a system natural frequency has been found. The sys-
tem natural frequencies computed in QuLAF are compared
to those obtained with the SoA model in Sect. 6.1.
5.8 Cascading techniques applied to the simplified
model
In Sect. 3 it was stated that one disadvantage of frequency-
domain models is their inability to directly capture loads that
depend on the response in a nonlinear way. Some relevant
examples are viscous drag, aerodynamic loads and catenary
mooring loads. This section gives a description of the cascad-
ing methods employed to incorporate such nonlinear loads
into the simplified model.
5.8.1 Hydrodynamic viscous loads
Viscous effects on submerged bodies depend nonlinearly on
the relative velocity between the wave particles and the struc-
ture, hence they can only be directly incorporated in time-
domain models. In the offshore community this is normally
done through the drag term of the Morison equation, which
provides the transversal drag force Fdon a cylindrical mem-
ber section of diameter Dand length dlas
Fd=1
2ρCDD|vf−vs|(vf−vs)dl, (21)
where ρis the fluid density, CDis a drag coefficient, and vf
and vsare the local fluid and structure velocities perpendic-
ular to the member axis. The equation can be also written
as
Fd=1
2ρCDDsgn (vf−vs) (vf−vs)2dl
=1
2ρCDDsgn (vf−vs)v2
f+v2
s−2vfvsdl, (22)
which shows that the drag effects can be separated into a pure
forcing term, a nonlinear damping term and a linear damp-
ing term. Since the hydrodynamics on the given floating sub-
structure are inertia-dominated and under the assumption of
small displacements around the equilibrium position, the two
first terms are neglected and only the linear damping term is
retained in the QuLAF model. Invoking further the assump-
tion of small displacements and velocities relative to the fluid
velocity, we have sgn(vf−vs)≈sgn(vf). With this assump-
tion, the linear damping term of the viscous force becomes
Fdl=1
2ρCDDsgn (vf−vs) (−2vfvs)dl
≈ −ρCDD|vf|vsdl. (23)
A symmetric viscous damping matrix Bvis is now derived
by applying Eq. (23) to the different DoFs. For the surge mo-
tion, integration over the submerged body gives the total vis-
cous force in the xdirection as
F1= −
0
Z
zmin
ρCDD|u|˙
ξ1dz, (24)
where zmin is the structure’s deepest submerged point, uis
the horizontal wave particle velocity and ˙
ξ1is the surge ve-
locity. The integral in Eq. (24) requires the estimation of drag
coefficients and the computation of wave kinematics at sev-
eral locations on the submerged structure, which can be in-
volved for complex geometries. These computations would
reduce the CPU efficiency relative to the radiation-diffraction
terms, so instead the local drag coefficient and wave veloc-
ity inside the integral are replaced by global, representative
values outside the integral, CDxand urep. Hereby the force
becomes
F1= −ρ˙
ξ1
0
Z
zmin
CDD|u|dz≈ −ρCDxurep ˙
ξ1
0
Z
zmin
Ddz
= −ρCDxAxurep ˙
ξ1≡ −b11 ˙
ξ1,(25)
where Axis the integral of the local diameter Dover depth,
or the floater’s area projected on the yz plane. This defines
the surge-surge element of the viscous damping matrix Bvis.
Further the b51 element of the matrix is obtained by consid-
eration of the moment from F1around the point of flotation,
τ1= −ρ˙
ξ1
0
Z
zmin
CDD|u|zdz≈ −ρCDxurep ˙
ξ1
0
Z
zmin
Dzdz
= −ρCDxSy ,Ax urep ˙
ξ1≡ −b51 ˙
ξ1,(26)
where Sy,Axis the first moment of area of Axabout the yaxis
(negative due to z≤0) and b51 is the surge-pitch element
of the viscous damping matrix. In a similar way, the heave-
heave and heave-pitch coefficients of Bvis are obtained by
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A. Pegalajar-Jurado et al.: An efficient frequency-domain model 701
applying Eq. (23) to the heave motion,
F3= −ρ˙
ξ3
xmax
Z
xmin
CDD|w|dx≈ −ρCD zwrep ˙
ξ3
xmax
Z
xmin
Ddx
= −ρCD zAzwrep ˙
ξ3≡ −b33 ˙
ξ3,(27)
τ3=ρ˙
ξ3
xmax
Z
xmin
CDD|w|xdx≈ρCD zwrep ˙
ξ3
xmax
Z
xmin
Dxdx
=ρCDzSy ,Azwrep ˙
ξ3≡ −b53 ˙
ξ3.(28)
Here ˙
ξ3is the heave velocity, wis the wave particle verti-
cal velocity, Azis the floater’s bottom area projected on the
xy plane and Sy,Azis the first moment of area of Azabout the
yaxis, which is zero for the present floating substructure due
to symmetry. Finally, by applying Eq. (23) to the pitch mo-
tion, the pitch-pitch element of the viscous damping matrix,
b55, is found. When the floater pitches with a velocity ˙
ξ5, an
arbitrary point on the floater with coordinates (x, z) moves
with a velocity (z˙
ξ5,−x˙
ξ5). The motion creates a moment
due to viscous effects given by
τ5= −ρ˙
ξ5
0
Z
zmin
CDD|u|z2dz−ρ˙
ξ5
xmax
Z
xmin
CDD|w|x2dx
≈ −ρ(CDxIy,Ax urep +CDzIy ,Azwrep)˙
ξ5≡ −b55 ˙
ξ5,(29)
where Iy,Ax and Iy ,Az are the second moments of area of Ax
and Azabout the yaxis, respectively. The complete symmet-
ric matrix of viscous damping is therefore
Bvis =




ρCDxAxurep 0ρ CDxSy,Ax urep 0
ρCDz Azwrep 0 0
ρCDxIy,Ax urep 0
+CDz Iy,Azwrep 0




.(30)
The global drag coefficients above have been chosen as
CDx=1 and CDz =2, given that the bottom slab of the
floater under consideration has sharp corners and is expected
to oppose a greater resistance to the flow than the smooth
vertical columns (see Fig. 1). To obtain the representative
velocity urep, the time- and depth-dependent horizontal wave
velocity at the floater’s centreline u(0,z, t ) is first averaged
over depth and then over time,
uavg(t)=1
|zmin|
0
Z
zmin
u(0,z, t)dz≡
1
|zmin|<iFFT ωˆη(ω)
k1−sinh(k(zmin +h))
sinh(kh),
⇒urep = |uavg|.(31)
Here kis the wave number for the angular frequency ω
and his the water depth. The representative velocity wrep is
chosen as the time average of the vertical wave velocity at
the centre of the bottom plate,
wavg(t)=w(0, zmin,t )⇒wrep = |wavg|.(32)
This simplification of the wave kinematics history, al-
though drastic, allows for the characterization of the viscous
damping for each sea state and avoids the need to compute
wave kinematics locally and integrate the drag loads.
5.8.2 Aerodynamic loads
Aerodynamic loads depend on the square of the relative wind
speed seen by the blades. The relative wind speed includes
contributions from the rotor speed, the blade deflection, the
tower deflection and the motion of the floater. The fact that
the aerodynamic thrust depends on the blade relative veloc-
ity produces the well-known aerodynamic damping. State-
of-the-art numerical models incorporate aerodynamic loads
based on relative velocity, because both the wind speed and
the blade structural velocity are known at each time step.
However, this cannot be done in a frequency-domain model.
In the approach implemented in QuLAF, the aerodynamic
loads considering the motion of the blades are simplified and
approximated by loads considering a fixed hub with rigid
blades and linear damping terms. The time series of fixed-
hub loads and the aerodynamic damping coefficients are ex-
tracted from the SoA model for each mean wind speed.
The aerodynamic loads are obtained at each wind speed
Wby a SoA simulation with turbulent wind and no waves,
where all DoFs except shaft rotation and blade pitch are dis-
abled and where the wind turbine controller is enabled. The
time series of fixed-hub, pure aerodynamic loads Faero,1(t),
Faero,3(t) and τaero(t) are extracted from the results and stored
in a data file which is loaded into the model. Hence, these
SoA simulations need to be as long as the maximum sim-
ulation time needed in the simplified model (5400 s in this
case).
For a given rotor, the work carried out by the aerodynamic
damping is a function of wind speed, rotational speed, turbu-
lence intensity, motion frequency and oscillation amplitude.
Here, we define an equivalent linear damping which deliv-
ers the same work over one oscillation cycle and can be ex-
tracted from a decay test. Schløer et al. (2018) used this prin-
ciple for the tower fore-aft mode of a bottom-fixed offshore
turbine and found that the damping was only slightly depen-
dent of the motion amplitude. We make a further simplifica-
tion and carry out the decay tests in steady wind. Since the
mass and stiffness of floater and tower only affect the aero-
dynamic damping through the motion frequency, we trans-
fer the damping coefficients bfrom the decay tests in FAST
to the QuLAF model. On the contrary, if the damping ra-
tio ζwas transferred, changes in mass or stiffness properties
would imply a change in the aerodynamic forcing, which is
not physically correct. With the transfer of damping coeffi-
cients b, recalculation of the decay tests is only necessary in
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702 A. Pegalajar-Jurado et al.: An efficient frequency-domain model
0 100 200 300 400 500
-20
0
20
x
hub
[m]
0 100 200 300 400 500 600
-2
0
2
x
hub
[m]
0 100 200 300 400 500 600
Time [s]
-0.02
0
0.02
x
hub
[m]
(a)
(b)
(c)
Figure 4. Example of time series of hub position and selected peaks for the extraction of aerodynamic damping. From top to bottom: surge,
pitch and clamped tower DoFs.
the event that the change of natural frequencies should affect
the damping values significantly. Here, the decay tests from
which aerodynamic damping ratios were extracted were car-
ried out at representative natural frequencies equal to those of
the present floater. These decay tests in calm water and with
the wind turbine controller active were carried out for each
DoF with all the other DoFs locked. This way, the floating
wind turbine was a one-DoF spring-mass-damper system in
each case, where the horizontal position of the hub xhub was
of interest. The decay tests were carried out as a step test in
steady wind where the wind speed goes from the minimum
to the maximum value with step changes every 600s. With
every step change of wind speed, the structure moves to a
new equilibrium position. If all sources of hydrodynamic and
structural damping are disabled, the aerodynamic damping is
the only one responsible for the decay of the hub motion, and
it can be extracted from the time series of xhub. The npeaks
extracted from the signal are used in pairs to estimate each
local logarithmic decrement diand, from it, a local damping
ratio ζi, which is then averaged to obtain the aerodynamic
damping ratio ζaero for the given DoF and W:
di=log xhub,i
xhub,i+1
,⇒ζi=di
q4π2+d2
i
,
⇒ζaero =1
n−1
n−1
X
1
ζi.(33)
Figure 4 shows examples of xhub(t) and selected peaks for
surge, pitch and clamped tower DoFs for a wind speed of
13 m s−1. The wind changed from 12 to 13 ms−1at t=0,
and the mean of the signals has been subtracted. For surge
and pitch, peaks within the first 40 s are neglected to allow
the unsteady aerodynamic effects to disappear. For the tower
DoF, however, the frequency is much higher and the signal
has died out by the time the aerodynamics are steady. For that
reason, the tower decay peaks are extracted after 300 s, and
a sudden impulse in wind speed is introduced at t=300 s to
excite the tower. This method was chosen since the standard
version of FAST does not allow an instantaneous force to be
applied.
In Fig. 5 the aerodynamic damping ratio is shown for all
DoFs as a function of W. It is observed that the aerodynamic
damping in surge is negative for wind speeds between 11.4
and 16 m s−1, due to the wind turbine controller. However,
in real environmental conditions with wind and waves, it has
been observed that the hydrodynamic damping contributes
to a positive global damping of the surge motion. This con-
troller effect is similar to the “negative damping problem”
reported in, for example, Larsen and Hanson (2007). The
negative aerodynamic damping in surge may be eliminated
if one tunes the controller natural frequency so it lies suffi-
ciently below the surge natural frequency of the floating wind
turbine, as it was done in Larsen and Hanson (2007) for the
floater pitch motion. This solution, however, would make the
controller too slow and would affect power production, thus
it was not adopted here because the global damping in surge
has been observed to be positive when all other damping con-
tributions are taken into account.
The damping ratio for the ith DoF and each wind speed,
ζaero,i (W), is next converted to a damping coefficient by
baero,i (W)=2ζaero,i (W)pCii (Mi i +Aii (ω)),(34)
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A. Pegalajar-Jurado et al.: An efficient frequency-domain model 703
4 6 8 10 12 14 16 18 20 22 24
W [m s- 1]
-5
0
5
10
15
20
25
30
ζ
aero
[%]
Surge
Pitch
Tower
Figure 5. Aerodynamic damping ratios for different DoFs as a function of wind speed.
where Cii ,Mi i and Aii (ω) are taken from the one-DoF os-
cillator in the corresponding decay test. The table of aero-
dynamic damping coefficients as a function of wind speed
baero(W) is stored in a data file, which is loaded into the
model. Since the aerodynamic damping coefficients are ex-
tracted from simulations with steady wind, but applied in the
model in simulations with turbulent wind, an averaging is ap-
plied to account for the variability in the wind speed in tur-
bulent conditions. Given the time series of wind speed at hub
height V(t), the probability density function (PDF) of a nor-
mal distribution given by N(V ,σV) is used to estimate the
probability of occurrence within V(t) of each discrete value
of W. Then the aerodynamic coefficient for the given turbu-
lent wind conditions and the ith DoF is
baero,i =
NW
X
j=1
PDF(Wj)baer o,i (Wj).(35)
5.8.3 Mooring loads
The equations that provide the loads on a catenary cable de-
pend nonlinearly on the fairlead position. In dynamic moor-
ing models the drag forces on the mooring cables are also
included; therefore, the mooring loads also depend on the
square of the relative velocity seen by the lines. These non-
linear effects can easily be captured by time-domain models,
but cannot be directly accommodated in a linear frequency-
domain model. In QuLAF, the mooring system is represented
by a linearized stiffness matrix for each wind speed, which
is extracted from the SoA model and where hydrodynamic
loads on the mooring lines are neglected. The dependence of
the mooring matrix on wind speed is necessary because dif-
ferent mean wind speeds generally produce different mean
thrust forces, which displace the floating wind turbine to dif-
ferent equilibrium states. The stiffness of the mooring system
is different at each equilibrium position because of the non-
linear force-displacement behaviour of the catenary mooring
lines.
For each wind speed a first SoA simulation is needed with
steady, uniform wind and no waves, where only the tower
fore-aft and floater surge, heave and pitch DoFs are enabled.
After some time the floating wind turbine settles at its equi-
librium position (ξeq,1,ξeq,3,ξeq,5), which is stored. These
simulations should be just long enough so that the equilib-
rium state is reached (600 s in this case). Then, a new short
SoA simulation with all DoFs disabled is run, where the
floater initial position is the equilibrium with a small posi-
tive perturbation in surge, (ξeq,1+1ξ1,ξeq,3, ξeq,5). This sim-
ulation should be just long enough for the mooring lines to
settle at rest (120 s in this case). The global mooring forces
in surge and heave and the global mooring moment in pitch
are stored, namely (Fξ1+
moor,1, F ξ1+
moor,3, τ ξ1+
moor,5). The process is
repeated now with a negative perturbation in surge (ξeq,1−
1ξ1,ξeq,3,ξeq,5), giving (Fξ1−
moor,1, F ξ1−
moor,3, τ ξ1−
moor,5). All this
information is enough to compute the first column of the
mooring matrix Cmoor for the wind speed W. Perturbations
in heave ±1ξ3and pitch ±1ξ5provide the necessary infor-
mation to compute the rest of the columns, and therefore the
full matrix:
Cmoor(W)=(36)
−











Fξ1+
moor,1−Fξ1−
moor,1
21ξ1
Fξ3+
moor,1−Fξ3−
moor,1
21ξ3
Fξ5+
moor,1−Fξ5−
moor,1
21ξ5
0
Fξ1+
moor,3−Fξ1−
moor,3
21ξ1
Fξ3+
moor,3−Fξ3−
moor,3
21ξ3
Fξ5+
moor,3−Fξ5−
moor,3
21ξ5
0
τξ1+
moor,5−τξ1−
moor,5
21ξ1
τξ3+
moor,5−τξ3−
moor,5
21ξ3
τξ5+
moor,5−τξ5−
moor,5
21ξ5
0
0 0 0 0











The first element of the mooring matrix Cmoor,11 is shown
as a function of wind speed in Fig. 6. It is observed that the
stiffness in surge reaches its maximum around rated wind
speed (11.4 m s−1), where the thrust is also maximum and
the floating wind turbine is the furthest from the equilibrium
position with no wind.
In the method applied here the linearization of the mooring
system has been done with the SoA model. However, in a real
design study where the mooring characteristics change, the
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704 A. Pegalajar-Jurado et al.: An efficient frequency-domain model
0 5 10 15 20 25
W [m s-1]
4
5
6
7
8
9
10
11
C
moor,11
[N m- 1]
× 10
4
Figure 6. Surge mooring stiffness as a function of wind speed.
above procedure can be made significantly faster by direct
static analysis of the nonlinear mooring reactions around the
floater equilibrium positions.
5.9 Estimation of extreme responses: a spectral
approach
Classical Monte Carlo analysis of response to stochastic
loads entails running a simulation, extracting the peaks from
the response time series, sorting them in ascending order and
assigning an exceedance probability to each peak based on
their position in the sorted list. Several simulations of the
same environmental conditions with different random seeds
provide a family of curves in the exceedance probability plot,
which can be used to estimate the expected response level for
a given exceedance probability. We note that the extracted
exceedance probability curves are based on the assumption
that the peaks are independent, which may not always be the
case. Yet, in this section the linear nature of the simplified
model will be further exploited to obtain an estimation of the
extreme responses to wave loads by solely using the wave
spectrum and the system transfer function, thus eliminating
the need for a response time series and the bias introduced
by a particular random seed. An extension of the method to
wind and wave forcing is further presented and discussed.
In a Gaussian, narrow-banded process, the peaks follow
a Rayleigh distribution. In linear stochastic sea states, the
free-surface elevation η(t) is a Gaussian random variable Rη
with zero mean. Thus, within the narrow-banded assump-
tion, which often applies to good approximation, the crest
heights follow a Rayleigh distribution (Longuet-Higgins,
1956) given by
P(Rη> η)=e−1
2η
ση2
,(37)
where the variance in η(t) is σ2
η, which can be obtained from
the integral of the wave spectrum,
σ2
η=
∞
Z
0
Sη(ω)dω . (38)
If we only consider linear wave forcing, the response is
also Gaussian for the linear system in Eq. (3). If the re-
sponse is also narrow-banded, its exceedance probability can
be found via the standard deviation of the response, which in
turn can be obtained by integration of the response spectrum.
From Eq. (3) we have
ˆ
ξ(ω)=H(ω)ˆ
X(ω)ˆη(ω),
⇒ˆ
ξglob(ω)=Tglob H(ω)ˆ
X(ω)ˆη(ω)≡TFη→ξ(ω)ˆη(ω),(39)
where TFη→ξ(ω) is a direct transfer function from surface
elevation to global response. The global response spectra
Sξ,g lob (ω) is related to the wave spectrum Sη(ω) in a simi-
lar way (Naess and Moan, 2013),
Sξ,glob (ω)=TFη→ξ(ω)Sη(ω)TF∗T
η→ξ(ω).(40)
Here ∗Tindicates the transpose and complex conjugate.
By virtue of Eq. (37), the exceedance probability of, for ex-
ample, the surge response ξ1is known from the variance in
the surge response σ2
ξ,1, which is given by
σ2
ξ,1=
∞
Z
0
Sξ,glob,11 (ω)dω . (41)
For nacelle acceleration we can write the response as a
function of the global nacelle displacement ξglob,4; therefore,
ˆ
¨
ξglob,4(ω)= −ω2ˆ
ξglob,4(ω),
⇒σ2
¨
ξ,4=
∞
Z
0
ω4Sξ,glob,44 (ω)dω . (42)
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A. Pegalajar-Jurado et al.: An efficient frequency-domain model 705
The turbulent part of the wind speed can also be consid-
ered a Gaussian random variable (Longuet-Higgins, 1956).
On the other hand, aerodynamic loads are not a linear func-
tion of wind speed. Therefore the response to wind loads
cannot be assumed to be Gaussian, and the approach shown
above is not valid. However, the method above can be applied
to cases with wind and wave forcing, bearing in mind that the
results may not be accurate since the necessary assumptions
are not fulfilled. If wind and wave forcing are considered,
Eq. (3) can be written as
ˆ
ξ(ω)=H(ω)ˆ
F(ω),
⇒ˆ
ξglob(ω)=Tglob H(ω)ˆ
F(ω)≡TFF→ξ(ω)ˆ
F(ω),(43)
where TFF→ξ(ω) is a direct transfer function from load to
global response. The global response spectra Sξ,glob (ω) is
now given by (Naess and Moan, 2013)
Sξ,g lob (ω)=TFF→ξ(ω)SF(ω)TF∗T
F→ξ(ω).(44)
Here SF(ω) is the spectra of the total loads (hydrodynamic
and aerodynamic),
SF(ω)=1
2dω ˆ
F(ω)ˆ
F∗T(ω).(45)
This method provides the exceedance probability of the
dynamic part of the response, therefore the static part should
be added after applying Eq. (37). Exceedance probability re-
sults from this method are compared in the next section to the
traditional way of peak extraction from response time series.
5.10 Integration of QuLAF in optimization loops
The main purpose of QuLAF is to provide a quick assess-
ment of loads, response and natural frequencies early in the
design phase, where several variations of the baseline design
are to be evaluated. The efficiency in the model is achieved
by (i) considering only a few DoFs; (ii) solving the linear
EoMs in the frequency domain; and (iii) pre-computing the
aerodynamic loads and aerodynamic damping coefficients.
The application of the model to an optimization loop can be
divided into two stages: a preparation stage, which needs to
be done only once for a given baseline floating wind turbine,
and a calculation stage, which can be repeated for each varia-
tion in the baseline design. After the optimal design has been
found through optimization, it should be verified by running
a complete load basis in a SoA model.
–Preparation stage. Once the wind turbine, the base-
line floater design and the design basis are defined, the
preparation stage entails the following.
a. Computation of time series of aerodynamic loads
at the shaft for the needed wind speeds and turbu-
lence random seeds, considering rigid blades and
fixed nacelle. The wind turbine controller should be
active and tuned according to the pitch frequency of
the baseline design.
b. Extraction of aerodynamic damping coefficients for
the needed wind speeds, by carrying out decay tests
in steady wind of the surge, pitch and clamped
tower DoFs.
c. Storage of the aerodynamic loads and damping co-
efficients in a database that can be reused for several
candidate designs.
–Calculation stage. The calculation stage is done for
each candidate design in the pre-design optimization
loop by following these steps.
a. Computation of the radiation-diffraction solution
in, for example, WAMIT.
b. Extraction of structural mass and stiffness proper-
ties.
c. For each wind speed, calculation of the equilibrium
position and linearization of the mooring system
around it.
d. Prediction of the natural frequencies, response and
loads for several environmental conditions using
QuLAF.
When compared to the same number of simulations in
the SoA model, the advantage of the simplified model re-
sides in the low computational cost of applying the calcu-
lation step 4 to several environmental conditions and differ-
ent variations in the baseline design. The extra work needed
to achieve the speed up comes from the aerodynamic pre-
computations in the preparation stage and from the lineariza-
tion of the mooring system (step 3) in each iteration of the
calculation stage. However, the aerodynamic loads need to
be extracted from the SoA model only once for a given wind
turbine, while the aerodynamic damping coefficients can also
be reused for different variations of the baseline floating wind
turbine, provided that the system natural frequencies do not
change significantly between different design iterations. The
linearization of the mooring system and the computation of
the radiation-diffraction solution may also be automated. Al-
ternatively, for slender, simpler geometries (such as spars),
a Morison approach may be implemented in QuLAF, thus
eliminating step 1 in the calculation stage. For the present
study, however, the radiation-diffraction solution was chosen
due to the shape and size of the floating substructure in con-
sideration, and a comparison to the Morison-based alterna-
tive has not been conducted.
6 Validation of the QuLAF model
We now compare and discuss the QuLAF and FAST re-
sponses to the same environmental conditions (see Table 4)
representative of the Gulf of Maine (Krieger et al., 2015).
The cases considered include five irregular sea states with
and without turbulent wind, with a single realization for each
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706 A. Pegalajar-Jurado et al.: An efficient frequency-domain model
Table 3. Natural frequencies and periods obtained in FAST and
QuLAF.
Surge Heave Pitch Tower
Natural frequency 0.0054 0.0478 0.0316 0.746
FAST (Hz)
Natural frequency 0.0054 0.0480 0.0320 0.682
QuLAF (Hz)
Error (%) 0.00 +0.42 +1.27 −8.58
Natural period 185.19 20.92 31.65 1.34
FAST (s)
Natural period 185.19 20.83 31.25 1.47
QuLAF (s)
Error (%) 0.00 −0.42 −1.25 +9.38
sea state. In all cases the total simulated time was 5400 s in
both models. The first 1800 s were neglected to discard initial
transient effects in the time-domain model. The free-surface
elevation of irregular sea states was computed in FAST from
a Pierson–Moskowitz spectrum, and the turbulent wind fields
in TurbSim from an IEC Kaimal spectrum. Since the turbu-
lent wind fields used in the SoA simulations are the same
employed for the pre-computation of aerodynamic loads, and
the free-surface elevation signal in the cascaded model is also
taken from the FAST simulation, a deterministic comparison
of time series is possible for all cases. In the plots shown in
this section (Figs. 7 and 9), the left-hand side shows a portion
of the time series of wind speed at hub height, free-surface
elevation, floater surge, heave and pitch, and nacelle accel-
eration; and the right-hand side shows the PSDs of the same
signals. The PSD signals were smoothened with a moving-
average filter of 20 points to ease the spectral comparison be-
tween models. The short blue vertical lines in the PSD plots
indicate the position of the system natural frequencies pre-
dicted by the simplified model (see Table 3). In addition, ex-
ceedance probability plots of the responses with both models
are shown (Figs. 8 and 10), based on peaks extracted from
the time series. The peaks were sorted and assigned an ex-
ceedance probability based on their position in the sorted
list. The exceedance probability of the extracted peaks is
compared to the one estimated with the method described
in Sect. 5.9, labelled as “Rayleigh”.
6.1 System identification
The system natural frequencies were calculated in QuLAF by
solving the eigenvalue problem in Eq. (20). In FAST, decay
simulations were carried out with all DoFs active, where an
initial displacement was introduced in each relevant DoF and
the system was left to decay. A PSD of the relevant response
revealed the natural frequency of each DoF. A comparison of
natural frequencies and periods found with the two models
is given in Table 3, where it is shown that all floater natu-
ral frequencies in the simplified model are within 1.3 % error
compared to the SoA model. On the other hand, the tower
frequency is 8.6 % below the one estimated in FAST. This
difference is due to the absence of flexible blades in the sim-
plified model, which are known to affect the coupled tower
natural frequency. With rigid blades, the SoA model predicts
a coupled tower natural frequency of 0.684 Hz, only 0.3 %
above the tower frequency in QuLAF.
The model presented here may be calibrated against other
numerical or physical models if needed, by introducing user-
defined additional restoring and damping matrices. For the
present study, however, no calibration against the SoA model
was applied, in order to keep the model calibration free and
assess its suitability for optimization loops.
6.2 Response to irregular waves
The response to irregular waves with Hs=6.14 m and Tp=
12.5 s (case “Waves 5” in Table 4) is shown in Fig. 7. On
the frequency side, all motions show response mainly at the
wave frequency range, and there is a very good agreement
between both models for surge and heave. In pitch – and con-
sequently in nacelle acceleration – the QuLAF model shows
a lower level of excitation at the wave frequency range when
compared to FAST. This deviation was traced to the absence
of viscous forcing in the simplified model, since the two
pitch responses are almost identical if viscous effects are dis-
abled in both models. As expected, the agreement is better
for milder sea states, where viscous forcing is less important.
In surge and pitch some energy is visible at the natural fre-
quencies, but only in the FAST model. Since the peaks lie out
of the wave spectrum and are not captured by QuLAF, they
could originate from nonlinear mooring effects or from the
drag loads, which are also nonlinear.
Figure 8 shows exceedance probability plots of the re-
sponse to irregular waves. The Rayleigh curves fit well to the
responses given by the simplified model, which is expected,
given that the free-surface elevation and the hydrodynamic
forcing are linear in the model, and the response can be con-
sidered narrow-banded. In the comparison between the two
models, the surge and heave peaks are very well estimated by
QuLAF. In nacelle acceleration and especially in pitch, how-
ever, the model underpredicts the response, with a difference
of about 30 % in pitch and about 8 % in nacelle acceleration
for the largest peak when compared to FAST. These obser-
vations in extreme response are consistent with the spectral
results of Fig. 7 discussed above.
6.3 Response to irregular waves and turbulent wind
The response to irregular waves with Hs=6.14 m and Tp=
12.5 s and turbulent wind at W=22 m s−1(case “Waves +
wind 5” in Table 4) is shown in Fig. 9. The surge motion is
dominated by the surge natural frequency, which is clearly
excited by the wind forcing. The linear model slightly un-
derpredicts this resonance of the wind forcing with the surge
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A. Pegalajar-Jurado et al.: An efficient frequency-domain model 707
3450 3500 3550 3600 3650 3700 3750
-1
0
1
Wind sp. [m s ]
Time domain
0 0.2 0.4 0.6 0.8 1
-1
0
1
PSD [m s Hz ]
2 -2
Frequency domain
FAST
QuLAF
3450 3500 3550 3600 3650 3700 3750
-5
0
5
Surf. elev. [m]
0 0.2 0.4 0.6 0.8 1
10
20
30
40
50
PSD [m Hz ]
2
3450 3500 3550 3600 3650 3700 3750
-5
0
5
Surge [m]
0 0.2 0.4 0.6 0.8 1
10
20
30
PSD [m Hz ]
2
3450 3500 3550 3600 3650 3700 3750
-2
0
2
Heave [m]
0 0.2 0.4 0.6 0.8 1
5
10
15
20
PSD [m Hz ]
2
3450 3500 3550 3600 3650 3700 3750
-2
0
2
Pitch [deg]
0 0.2 0.4 0.6 0.8 1
2
4
PSD [deg Hz ]
2
3450 3500 3550 3600 3650 3700 3750
Time [s]
-1
0
1
Nacelle acc. [m s
-2 ]
0 0.2 0.4 0.6 0.8 1
Frequency [Hz]
0.5
1
1.5
PSD [m s Hz ]
2 -4
-1
-1
-1
-1
-1
-1
-1
Figure 7. Response to irregular waves in the time and frequency domains.
natural frequency. Heave is dominated by the wave forcing,
and the response of both models agree well. In pitch, res-
onance with the natural frequency also exists in both mod-
els, although QuLAF predicts more energy at that frequency
than FAST. Both surge and pitch responses are resonant, thus
they are especially sensitive to the amount of damping. The
overprediction of pitch motion also leaves a footprint on the
PSD of nacelle acceleration, which shows energy at the pitch
natural frequency, the wave frequency range and the tower
natural frequency. The level of excitation of the tower mode
at 0.682 Hz, however, is slightly underpredicted by QuLAF,
likely due to an overestimation of the aerodynamic damping
on the tower DoF.
The associated exceedance probability plots are shown in
Fig. 10. In this case the Rayleigh curves generally do not
fit the responses predicted by the linear model, as the ex-
treme peaks are no longer Rayleigh-distributed. This is be-
cause the nonlinear nature of the wind loads makes the re-
sponse non-Gaussian, and in some cases broad-banded with
distinct frequency bands excited (e.g. the tower response can-
not be considered narrow-banded here). The best fit is seen
for heave, which is mainly excited by linear wave loads and
is also narrow-banded. When compared to FAST, however,
QuLAF shows a good agreement with errors in the largest
response peaks of approximately 8 % in surge, 12 % in pitch
and 4 % in nacelle acceleration.
6.4 Comparison of fatigue damage-equivalent loads
Table 4 shows a summary of fatigue DELs for a wider range
of environmental conditions. Each case is defined by the sig-
nificant wave height Hs, the wave peak period Tpand the
mean wind speed W. The fatigue damage-equivalent bend-
ing moment at the tower base estimated with the two models
is presented, as well as the error for the simplified model. Fi-
nally, the last column shows the ratio between the simulated
time and the CPU time in QuLAF, Trel. The cases labelled as
“5” correspond to the results discussed in the previous sec-
tion. The two DEL columns in Table 4 are also shown in
Fig. 11 as a bar plot.
For the cases with waves only, the model underpredicts the
DEL at the tower base with errors from 0.2 % to 11.3 % that
increase with the sea state, as observed in Fig. 11. The sig-
nificant wave height also increases with the sea state, as do
the associated nonlinear effects of position-dependent moor-
ing stiffness and viscous hydrodynamic forcing, which are
both included in FAST. QuLAF does not include viscous hy-
drodynamic forcing, and as a linear model, its accuracy is
bound to the assumptions of small displacements around the
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708 A. Pegalajar-Jurado et al.: An efficient frequency-domain model
0 0.5 1 1.5 2
Surge [m]
10-3
10-2
10-1
100
Exc. prob.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Heave [m]
10-3
10-2
10-1
100
Exc. prob.
FAST
QuLAF
Rayleigh
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Pitch [deg]
10-3
10-2
10-1
100
Exc. prob.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Nacelle acc. [m s ]
-2
10-3
10-2
10-1
100
Exc. prob.
Figure 8. Exceedance probability of the response to irregular waves.
Table 4. Summary of environmental conditions (Krieger et al., 2015) and DEL results obtained in FAST and QuLAF.
Case HsTpWDELFAST DELQuLAF Error Trel
(m) (s) (m s−1) (MNm) (MNm) (%) (–)
Waves 1 1.51 7.65 – 75.69 76.44 +1.00 2402
Waves 2 1.97 8.00 – 98.44 98.62 +0.19 2695
Waves 3 2.43 8.29 – 120.74 119.95 −0.65 2595
Waves 4 3.97 9.85 – 179.45 170.55 −4.96 2404
Waves 5 (Figs. 7, 8) 6.14 12.50 – 219.31 194.63 −11.25 2595
Waves +wind 1 1.51 7.65 6.0 167.13 158.74 −5.02 1354
Waves +wind 2 1.97 8.00 9.0 290.96 284.53 −2.21 1409
Waves +wind 3 2.43 8.29 11.4 375.12 349.37 −6.87 1400
Waves +wind 4 3.97 9.85 17.0 319.95 324.68 +1.48 1365
Waves +wind 5 (Figs. 9, 10) 6.14 12.50 22.0 339.01 348.77 +2.88 1408
equilibrium point. Hence, it is expected that the linear model
performs worse for the environmental conditions where non-
linear effects are not negligible. This observation is also con-
sistent with the discussion around Fig. 7, which corresponds
to the most severe sea state considered here.
For the cases with wind, the errors range from 1.5 % to
6.9 %, but the trend is not as clear. The predictions seem to
be worst for the environmental condition corresponding to
rated wind speed. Around rated speed the wind turbine oper-
ation switches between the partial- and the full-load regions,
which correspond to very distinct regimes of the generator
torque and blade pitch controller. The complexity of the dy-
namics involved in this transition zone is not well captured
by the simplified model. The vibration of the tower is also
more likely to be excited around rated wind speed, where the
thrust is maximum. As the coupled tower natural frequency
is different for the two models, this will also have an im-
pact on the resulting DEL. This effect has been quantified
for rated wind speed (”Waves +wind 3”), where the DEL
error becomes −5.6 % when the FAST simulation is carried
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A. Pegalajar-Jurado et al.: An efficient frequency-domain model 709
3450 3500 3550 3600 3650 3700 3750
0
20
40 Time domain
0 0.2 0.4 0.6 0.8 1
200
400
600
Frequency domain
FAST
QuLAF
3450 3500 3550 3600 3650 3700 3750
-5
0
5
0 0.2 0.4 0.6 0.8 1
10
20
30
40
50
3450 3500 3550 3600 3650 3700 3750
5
10
15
0 0.2 0.4 0.6 0.8 1
50
100
150
3450 3500 3550 3600 3650 3700 3750
-2
0
2
0 0.2 0.4 0.6 0.8 1
5
10
15
20
3450 3500 3550 3600 3650 3700 3750
0
2
4
0 0.2 0.4 0.6 0.8 1
20
40
60
80
100
120
3450 3500 3550 3600 3650 3700 3750
Time [s]
-2
0
2
0 0.2 0.4 0.6 0.8 1
Frequency [Hz]
0.5
1
1.5
PSD [m s Hz ]
2 -2
Surf. elev. [m] Wind sp. [m s ]
PSD [m Hz ]
2
Surge [m]
PSD [m Hz ]
2
Heave [m]
PSD [m Hz ]
2
Pitch [deg]
PSD [deg Hz ]
2
Nacelle acc. [m s-2 ]
PSD [m s Hz ]
2 -4
- 1
-1
-1
-1
-1
-1
-1
-
Figure 9. Response to irregular waves and turbulent wind in the time and frequency domains.
out with rigid blades, which indicates that the difference in
coupled tower frequency has some impact on the DEL er-
ror. In addition, the aerodynamic damping – which plays an
important role in the resonant response of the tower – is de-
pendent on the frequency at which the rotor moves in and out
of the wind. Since the aerodynamic damping on the tower is
extracted from a SoA simulation with fixed foundation and
rigid blades, it corresponds to a tower natural frequency of
0.51 Hz, different to the coupled tower frequency observed
when the floater DoFs are active (0.682 Hz in QuLAF, 0.746
in FAST). This difference in the frequencies at which the
aerodynamic damping is extracted and applied is likely to
lead to an overprediction of the aerodynamic damping, and
an underprediction of the tower vibration and the DEL. This
observation is consistent with the level of tower response at
the coupled tower frequency shown in Fig. 9. On the other
hand, the aerodynamic simplifications in the cascaded model
seem to work best for wind speeds above rated, likely due
to the thrust curve being flatter in this region. The last col-
umn of Table 4 shows that the ratio between simulated time
and CPU time is between 1300 and 2700 for a standard lap-
top with an Intel Core i5-5300U processor at 2.30 GHz and
16 GB of RAM. In other words, all the simulations in Table 4
together, 1.5 h long each, can be done in about half a minute.
7 Conclusions
A model for Quick Load Analysis of Floating wind tur-
bines, QuLAF, has been presented and validated. The model
is a linear, frequency-domain tool with four planar degrees
of freedom (DoFs): floater surge, heave, pitch and tower
modal deflection. The model relies on higher-fidelity tools
from which hydrodynamic, aerodynamic and mooring loads
are extracted and cascaded. Hydrodynamic and aerodynamic
loads are pre-computed in WAMIT and FAST, respectively,
while the mooring system is linearized around the equilib-
rium position for each wind speed using MoorDyn. A sim-
plified approach for viscous hydrodynamic damping was im-
plemented, and the decay-based extraction of aerodynamic
damping of Schløer et al. (2018) was extended to multi-
ple DoFs. Without introducing any calibration, a case study
with a semi-submersible 10 MW configuration showed that
the model is able to predict the motions of the system in
stochastic wind and waves with acceptable accuracy. The
damage-equivalent bending moment at the tower base is es-
timated with errors between 0.2 % and 11.3 % for all the
five load cases considered in this study, covering the opera-
tional wind speed range. The largest errors were observed for
the most severe wave climates in wave-only conditions and
for turbine operation around rated wind speed for combined
wind and wave conditions, due to three main limitations in
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710 A. Pegalajar-Jurado et al.: An efficient frequency-domain model
024681012
Surge [m]
10-3
10-2
10-1
100
Exc. prob.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Heave [m]
10-3
10-2
10-1
100
Exc. prob.
FAST
QuLAF
Rayleigh
012345
Pitch [deg]
10-3
10-2
10-1
100
Exc. prob.
0 0.2 0.4 0.6 0.8 1
10-4
10-3
10-2
10-1
100
Exc. prob.
Nacelle acc. [m s ]
-2
Figure 10. Exceedance probability of the response to irregular waves and turbulent wind.
12345
0
200
400
DEL [MNm]
Waves
FAST
QuLAF
12345
Environmental condition
0
200
400
DEL [MNm]
Waves + wind
FAST
QuLAF
(a)
(b)
Figure 11. Damage-equivalent bending moment at the tower base for different environmental conditions.
the model: (i) underprediction of hydrodynamic loads in se-
vere sea states due to the omission of viscous drag forcing;
(ii) difficulty to capture the complexity of aerodynamic loads
around rated wind speed, where the controller switches be-
tween the partial- and full-load regions; and (iii) errors in the
estimation of the tower response due to underprediction of
the coupled tower natural frequency and overprediction of
the aerodynamic damping on the tower. The computational
speed in QuLAF is between 1300 and 2700 times faster than
real time. Although not done in this study, introducing vis-
cous hydrodynamic forcing and calibration of the damping
against the SoA model would likely result in improved ac-
curacy, but at the expense of lower CPU efficiency and less
generality in the model formulation.
It has been shown that the model can be used as a tool
to explore the design space in the preliminary design stages
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A. Pegalajar-Jurado et al.: An efficient frequency-domain model 711
of a floating substructure for offshore wind. The model can
quickly give an estimate of the main natural frequencies, re-
sponse and loads for a wide range of environmental con-
ditions with aligned wind and waves, which makes it use-
ful for optimization loops. Although a better performance
may be achieved through calibration, a calibration-free ap-
proach was used here to emulate the reality of an optimiza-
tion loop, where calibration is not possible. In such a pro-
cess, once an optimized design has been found, a full aero-
hydro-servo-elastic model is still necessary to assess the per-
formance in a wider range of environmental conditions, in-
cluding nonlinearities, transient effects and real-time con-
trol. Since the model is directly extracted from such a SoA
model, this step can readily be taken. While the SoA model
should thus still be used in the design verification, the present
model provides an efficient and relatively accurate comple-
mentary tool for rational engineering design of offshore wind
turbine floaters. In addition, the QuLAF and FAST mod-
els presented in this study have been recently used in the
LIFES50+project for a broader analysis of different design-
driving load cases, including normal operation, extreme and
transient events (Madsen et al., 2018). Generally, the results
of the broader study and the conclusions drawn are aligned
with the ones presented here, as well as the limitations ob-
served in the simplified model when compared to its SoA
counterpart.
Given the model limitations observed in this study and in
Madsen et al. (2018), possible improvements of QuLAF may
involve (i) inclusion of viscous drag forcing, (ii) modelling
the effect of blade flexibility on the tower natural frequency,
(iii) improvement of the extraction of aerodynamic damping
from the SoA model, and (iv) extension of the model to out-
of-plane DoFs to make it applicable to cases with misaligned
wind and waves.
Code and data availability. The FAST model is publicly avail-
able, as detailed in Pegalajar-Jurado et al. (2018a) and Pegalajar-
Jurado et al. (2018b). The QuLAF source code is not public due to
possible commercialization in the future. The data used in figures
and tables can be obtained by contacting the first author.
Author contributions. APJ is the main responsible person for
developing the simplified model, carrying out the simulations,
analysing the results and preparing the manuscript. MB provided
input to the model development and to the first version of the
manuscript. HB developed the conceptual idea, provided input to
the model development and reviewed the paper in all its versions.
Competing interests. The authors declare that they have no con-
flict of interest.
Acknowledgements. This work was carried out as part of
the LIFES50+project (http://www.lifes50plus.eu; last access:
16 February 2018). The research leading to these results has
received funding from the European Union’s Horizon 2020
research and innovation programme under grant agreement
no. 640741. Also, the authors are grateful to Dr.techn. Olav Olsen
AS (http://www.olavolsen.no; last access: 24 November 2017) for
the permission to use their concept of the OO-Star Wind Floater
Semi 10 MW as the case study.
Edited by: Joachim Peinke
Reviewed by: Tor Anders Nygaard and one anonymous referee
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