NREL is a national laboratory of the U.S. Department of Energy

Office of Energy Efficiency & Renewable Energy

Operated by the Alliance for Sustainable Energy, LLC.

This report is available at no cost from the National Renewable Energy

Laboratory (NREL) at www.nrel.gov/publications.

Contract No. DE-AC36-08GO28308

The Effect of Second-Order 

Hydrodynamics on Floating 

Offshore Wind Turbines

Preprint

Line Roald and Ndaona Chokani

ETH Zurich

Jason Jonkman and Amy Robertson

National Renewable Energy Laboratory

Presented at DeepWind’2013 – 10th Deep Sea Offshore Wind 

R&D Conference

Trondheim, Norway

January 24 – 25, 2013

Conference Paper

NREL/CP-5000-58718

July 2013
NOTICE

The submitted manuscript has been offered by an employee of the Alliance for Sustainable Energy, LLC 

(Alliance), a contractor of the US Government under Contract No. DE-AC36-08GO28308. Accordingly, the US 

Government and Alliance retain a nonexclusive royalty-free license to publish or reproduce the published form of 

this contribution, or allow others to do so, for US Government purposes.

This report was prepared as an account of work sponsored by an agency of the United States government. 

Neither the United States government nor any agency thereof, nor any of their employees, makes any warranty, 

express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of 

any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately 

owned rights. Reference herein to any specific commercial product, process, or service by trade name, 

trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, 

or favoring by the United States government or any agency thereof. The views and opinions of authors 

expressed herein do not necessarily state or reflect those of the United States government or any agency thereof.

This report is available at no cost from the National Renewable Energy Laboratory (NREL) 

at www.nrel.gov/publications. 

Available electronically at http://www.osti.gov/bridge

Available for a processing fee to U.S. Department of Energy

and its contractors, in paper, from:

U.S. Department of Energy

Office of Scientific and Technical Information

P.O. Box 62

Oak Ridge, TN 37831-0062

phone: 865.576.8401

fax: 865.576.5728

email: mailto:reports@adonis.osti.gov

Available for sale to the public, in paper, from:

U.S. Department of Commerce

National Technical Information Service

5285 Port Royal Road

Springfield, VA 22161

phone: 800.553.6847

fax: 703.605.6900

email: orders@ntis.fedworld.gov

online ordering: http://www.ntis.gov/help/ordermethods.aspx

Cover Photos: (left to right) photo by Pat Corkery, NREL 16416, photo from SunEdison, NREL 17423, photo by Pat Corkery, NREL 

16560, photo by Dennis Schroeder, NREL 17613, photo by Dean Armstrong, NREL 17436, photo by Pat Corkery, NREL 17721.

 Printed on paper containing at least 50% wastepaper, including 10% post consumer waste. 
This report is available at no cost from the 1 

National Renewable Energy Laboratory (NREL) 

at www.nrel.gov/publications.

DeepWind'2013, 24-25 January, Trondheim, Norway 

The Effect of Second-Order Hydrodynamics on Floating Offshore 

Wind Turbines

Line Roalda*, Jason Jonkmanb, Amy Robertsonb, Ndaona Chokania

a

Laboratory of Energy Conversion, ETH Zürich, Sonneggstrasse 3, 8092 Zürich, Switzerland b

National Renewable Energy Laboratory (NREL), 15013 Denver West Parkway, Golden, CO 80401, United States of America

Abstract

The design of floating offshore wind turbines relies on the use of modeling tools that can simulate the entire coupled system 

behavior, including aerodynamics, hydrodynamics, structural elasticity, and the turbine control system. At present, most of these 

tools include only first-order hydrodynamic theory. However, observations of supposed second-order hydrodynamic responses in 

wave-tank tests suggest that second-order effects might be of critical importance to turbine design. In this paper, the second-order 

methodology used by the oil and gas industry has been modified to analyze the effect of second-order hydrodynamics on floating 

offshore wind turbines. The method combines the use of the frequency-domain simulation tool WAMIT and the time-domain 

simulation tool FAST. The proposed assessment method has been applied to two different floating wind concepts, a spar and a 

tension-leg platform (TLP), both supporting the National Renewable Energy Laboratory’s (NREL’s) 5-MW baseline wind 

turbine. Results showing the hydrodynamic forces and motion response of these systems are presented and compared to 

aerodynamic effects. The effect of the second-order hydrodynamics is relatively small for the spar configuration, and its motion 

response across the high- and low-frequency domain is largely determined by the turbine’s aerodynamics. On the other hand, the 

influence of second-order hydrodynamics is relatively large for the tension-leg-platform configuration, with the motion response 

in heave being dominated by sum-frequency effects.

Keywords: offshore floating wind turbine, second-order hydrodynamics, wave loads, spar buoy, tension-leg platform 

1. Introduction 

Designing, building, and maintaining wind farms offshore requires knowledge about both the wind turbines and 

the marine environment in which they are to function. Important tools in the process of finding an optimal design for 

a floating turbine are computer-aided engineering (CAE) tools that simulate the turbine nonlinearly in the time 

domain, including aerodynamics, hydrodynamics, structural elasticity and the turbine control system in the 

simulation. The tools that have been verified through the International Energy Agency (IEA) Wind Task 23 Offshore

Code Comparison Collaborative (OC3) [9] include (among others) FAST by the National Renewable Energy 

Laboratory (NREL), GH Bladed by GL Garrad Hassan, and HAWC2/SIMO and Riflex by DTU Wind 

Energy/MarinTek.

There are several approaches to the computation of hydrodynamic loading. Among the most suitable formulations 

for wind turbines are the Morison’s equation, an empirical formulation for inertia forces and viscous drag important 

to slender structures, and the radiation/diffraction approach, describing effects important for large-volume structures. 

The latter formulation is considered here. The radiation and diffraction approach incorporates wave reflection and 

scattering but ignores all viscous effects by assuming potential flow. Assuming small platform motions (relative to 

the waves) and a small wave slope, the radiation problem and the diffraction problem are expanded using a 

perturbation series with regards to the wave slope, and are split into first-order, second-order and higher-order parts. 

These parts can then be solved separately.

Often, only the first-order problem is solved (at least in a preliminary design) and all higher-order terms 

neglected, assuming that the higher-order forces will be at least an order of magnitude smaller. This significantly 

* Corresponding author. Tel.: +41 44 632 65 77; fax: +41 44 632 12 52. E-mail address: roald@eeh.ee.ethz.ch. 

Current Address: Power Systems Laboratory, ETH Zürich, Physikstrasse 3, 8092 Zürich, Switzerland
This report is available at no cost from the 2 

National Renewable Energy Laboratory (NREL) 

at www.nrel.gov/publications.

reduces the complexity of the problem, and the solution becomes much less computationally demanding, while it 

remains reasonably accurate in most cases. Due to the linearity of the first-order problem, the first-order forces and 

motions oscillate at the same frequency as the incident waves.

The second-order parts of the perturbation series form the second-order hydrodynamic problem, which is the 

topic of interest in this paper. The second-order problem addresses interactions between two harmonically oscillating 

components, resulting in forces and motions at the sum- and difference-frequencies of the incident waves. Offshore 

structures are normally designed to have their eigenfrequencies outside the excitation range of the incident waves, 

i.e. above or below 0.25–1.25 rad/s (periods of 5–25 s). The sum- and difference-frequency forces introduce 

excitation above and below the frequencies of the first-order forces and may potentially excite the eigenfrequencies 

of the structures. If the damping of the excited eigenmodes is sufficiently small, the result can be large, slow 

oscillations or problematic high-frequency vibrations.

The hydrodynamic modules of most floating wind CAE tools neglect radiation and diffraction forces beyond first 

order. This paper proposes an analysis methodology to quantify the second-order effects on offshore floating 

turbines based on the methodology used in the offshore oil and gas industry. However, there are a number of reasons 

why wind turbines are different from oil and gas installations. First, the dynamics of a wind turbine are significantly 

influenced by aerodynamic forces (treated in a very simple way for other offshore structures) and the properties of 

the control system. Second, the wind turbine does not behave like a rigid-body structure. Last but not the least, in 

comparison to oil and gas platforms, floating wind turbines are smaller volume structures for which viscous effects 

may be more important. The questions to be answered in this paper are therefore the following: Should second-order 

effects be included in floating wind simulation tools in the future, or do the assumptions that justify the disregard of 

higher-order effects for most traditional offshore structures also hold for wind turbines? How appropriate is the 

methodology commonly used in the offshore industry for analysis of offshore wind turbines?

There are very few previous studies applying second-order hydrodynamic theory to floating wind turbines. A 

report from the UpWind project [11] provides a summary of the theory of second-order hydrodynamics and some 

results for the first- and second-order hydrodynamic coefficients for the OC3-Hywind spar buoy and a 

semisubmersible. Agarwal [1] investigated second-order effects on a monopile structure in shallow water and used 

second-order wave kinematics in combination with Morison’s equation to compare linear and nonlinear effects. This 

approach is, however, limited to bottom-mounted slender cylinders. In the DeepCwind model tests performed at the 

MARIN wave basin in Wageningen, Holland, supposed second-order effects were observed, as reported in [3] and 

[4]. The significance of these effects inspired new interest in the loads and responses of floating wind turbines that 

are induced by second-order hydrodynamics. 

This paper introduces a general methodology for the assessment of second-order effects on floating turbines, 

accounting for system geometry, as well as linearized system properties described by the system mass, damping, and 

stiffness matrices. The proposed methodology is used to simulate the influence of second-order effects on two 

different floating wind turbine concepts, a spar and a tension-leg platform (TLP), both supporting the NREL 5-MW 

baseline wind turbine. Results for these systems are presented and analyzed in order to answer the questions posed 

above.

The remainder of this paper is organized as follows: Section 2 details the assessment methodology. Section 3 

describes the properties of the two studied wind turbine concepts, and the results of the assessment are presented in 

Section 4. Conclusion and an outlook to future work are found in Section 5.

2. Modeling Approach

The methodology described in this paper uses two simulation tools to simulate the second-order effects on

floating wind turbines, the time-domain tool FAST and the frequency-domain tool WAMIT. 

FAST is a wind turbine CAE tool developed by NREL. The tool is open source and publicly available [13]. 

FAST predicts the coupled dynamic response of an entire wind turbine system nonlinearly in the time domain, 

taking aerodynamics, structural elasticity, control system, and hydrodynamics into account.

WAMIT is a commercial 3D panel code designed to compute hydrodynamic loading from the radiation and 

diffraction problem in the frequency domain [16]. It is widely used in the offshore industry, and is capable of 

solving both the first- and second-order hydrodynamic problem for a rigid structure of arbitrary geometry.

An outline of the current approach for simulation of hydrodynamic forces on floating offshore wind turbine using 

these two tools is given below. The extension to the current analysis methodology needed to assess second-order 

forces and motion response is also presented.
This report is available at no cost from the 3 

National Renewable Energy Laboratory (NREL) 

at www.nrel.gov/publications.

2.1. Current Simulation Approach Including First-Order Hydrodynamics

Currently, the coupled response of floating offshore wind turbines is simulated in FAST with first-order 

hydrodynamic quantities determined in WAMIT or a similar program. The first step in such an approach is to 

calculate the hydrostatic restoring, the frequency-dependent hydrodynamic added mass and damping matrices from 

wave radiation, as well as the frequency- and direction-dependent hydrodynamic force coefficients from wave 

diffraction. This is done in WAMIT, based on the geometry of the submerged portion of the floating platform. In a 

second step, the hydrodynamic quantities are given as an input to the time-domain simulation in FAST, together 

with models of the wind turbine (including structural, aerodynamic, and control system properties), the floating 

platform, and the mooring system. For a specific simulation, FAST also needs input describing the environmental 

conditions, more specifically the wind speed and the sea state.

The main benefit of this approach is that FAST solves the equations of motions in the time domain. This is the 

key characteristic that allows the transient behavior and the nonlinear coupled dynamics of the platform, tower, and 

rotor to be accounted for in the simulation.

2.2. Simulation Approach to Include Second-Order Hydrodynamics

One important property of the first-order hydrodynamic forces is that they only depend on the geometry of the 

structure and not on the motion response to the incoming waves. It is thus possible to compute the first-order 

hydrodynamic quantities in WAMIT without solving any equations of motion. However, the forces arising from the 

second-order problem depend not only on the structure’s geometry, but also on the solution to the first-order 

equations of motion. Therefore, the first-order motion response must be given in the frequency domain before 

second-order forces can be computed. The solution is typically given in the form of response-amplitude operators 

(RAOs), which describe the motion response as a function of wave amplitude and wave frequency. The computation 

of the RAOs is an integral part of a second-order calculation in WAMIT, where the first-order equations of motions 

are solved in the frequency domain. Because FAST is currently not configured to make use of second-order force 

inputs, the second-order equations of motion are also solved by WAMIT in the frequency domain and output as 

second-order RAOs. The RAOs can be converted into a time-domain response by assuming a specific sea state with 

a given wave spectrum. Wind speeds are not needed by WAMIT, as aerodynamics cannot be accounted for. The 

complete approach is outlined in Figure 1.

Figure 1 Procedure to analyze second-order effects with WAMIT and FAST.
This report is available at no cost from the 4 

National Renewable Energy Laboratory (NREL) 

at www.nrel.gov/publications.

To solve the equations of motion, the system mass, damping, and stiffness matrices of the wind turbine must be 

imported to WAMIT. These matrices are computed by the linearization procedure in FAST, which needs the same 

hydrodynamic inputs as for the time-domain simulation described in the former section to calculate the system 

matrices and system eigenfrequencies correctly. The linearized system properties output from FAST include 

contributions from the turbine (mass/inertia, aerodynamic damping/stiffness, gyroscopics), the substructure 

(mass/inertia), and the mooring system (stiffness properties). Any hydrodynamic contributions in the system 

matrices are removed before the matrices are input to WAMIT to avoid counting these contributions twice.

The solution of the equations of motion in the frequency domain imposes some important limitations on the 

methodology. In the frequency domain, only steady-state, oscillatory forces can be taken into account and any nonlinear effects are ignored. This eliminates the possibility to compute transient behavior and to properly account for 

the influence of aerodynamics, control system actions, viscous drag, or other non-linear characteristics of the system 

(e.g. platform set-down for TLPs). Another important limitation to the calculation in WAMIT is that the structure is 

modeled as a rigid-body. The substructure could have been modeled as a flexible body using generalized modes, but 

this feature of WAMIT does not apply to any part of the structure outside of the water. Therefore, the turbine 

flexibility, which is of higher importance to the response of the structure, cannot be included in WAMIT. 

3. Properties of the Analyzed Wind Turbine Concepts

3.1. OC3-Hywind Spar

The first of the two analyzed floating wind turbine concepts is the OC3-Hywind spar. The spar buoy is a long, 

slender cylinder that relies on a low center of gravity for stability. The considered configuration is a modified 

version of the full-scale 2.3-MW floating wind turbine that is built and operated by Statoil close to the southwest 

Norwegian coast. The platform model is the same that was used in the OC3 project and is described in [6]. The 

OC3-Hywind platform is designed to carry the NREL 5-MW reference wind turbine that was developed to provide 

specifications for a model representative of a utility-scale multimegawatt wind turbine [8]. The tower of the 

reference turbine was slightly changed to fit on the floating platform, and the control system was adapted to 

accommodate the floating platform motion. The properties of the new tower and controller, as well as other 

properties specific to the floating system are described in [6].

3.1.1. WAMIT Model

The geometry of the OC3-Hywind substructure was modeled with quadrilateral panels. Because the spar has two 

planes of symmetry, only one quarter of the structure needs to be modeled, leading to shorter simulation time within 

WAMIT. A cosine-spaced mesh was used, giving a panel distribution with smaller panel size close to sharp edges or 

close to the free surface. This yields more accurate results for the same number of panels compared to equally sized 

panels [10]. The number of panels used to model the turbine and the free surface was chosen based on results from 

simulation convergence tests that were performed in a way similar to that described in [14]. Using the finest 

discretization as a benchmark, all first-order and difference-frequency results seem to have an error of at most 2%. 

For the sum-frequency results, the largest error is in the order of 6%. More information regarding the convergence 

tests and the results can be found in [15]. 

3.1.2. Derivation of System Matrices and System Eigenfrequencies in FAST

The FAST model of the OC3-Hywind is provided by NREL. This model was used for all simulations of the OC3-

Hywind in FAST. The mass, damping, and stiffness matrices were derived through the FAST linearization process 

with input from first-order hydrodynamic simulations in WAMIT. The linearized system parameters come from a 

case without any aerodynamic excitation.

The system eigenfrequencies are derived using the method outlined in [7]. WAMIT is not able to model turbine 

flexibility; therefore, any effects due to tower or blade flexibility will not be included when the equations of motion 

are solved. To see how the eigenfrequencies change, depending on whether the tower and blades are modelled as 

flexible or rigid, the eigenfrequencies are computed for both cases. For the OC3-Hywind, the difference is 

insignificant. The eigenfrequencies for OC3-Hywind are seen in Table 1 (left). It should be noted that the OC3-

Hywind spar is designed to have eigenfrequencies below the frequency range of the incident waves, i.e. below 0.25 

rad/s. The exception is the platform-yaw degree of freedom, which is not significantly influenced by wave excitation 

but rather by turbine effects like rotor gyroscopics. Due to the low eigenfrequencies, it can be expected that 
This report is available at no cost from the 5 

National Renewable Energy Laboratory (NREL) 

at www.nrel.gov/publications.

difference-frequency excitation from the waves might have an effect on the spar, while the effect of sum-frequency 

excitation will be negligible.

3.2. UMaine TLP

The second floating wind turbine concept analyzed here is a TLP developed by the University of Maine for use in 

the DeepCwind project. The TLP is a rather shallow structure that relies on the mooring system, the so-called 

tension legs, to remain stable. The turbine carried by the TLP is the same NREL 5-MW reference turbine that was 

used for the OC3-Hywind with slightly different tower and control system properties as described in [8]. Further 

information about the TLP system can be found in [3] and [4].

3.2.1. WAMIT Model

Due to the rather complicated shape, the geometry of the TLP was modeled using MultiSurf, a CAD program. As 

with the OC3-Hywind, cosine spacing was used to get a better trade-off between computational effort and accuracy, 

and the number of panels was chosen based on convergence tests [14]. Again, using the finest discretization as a 

benchmark, all first-order results seem to have converged to an error of less than 5%. For the sum- and differencefrequency results, the error is harder to quantify, as computational effort put a rather strict limit on the highest 

number of body panels that could be tested. Experience from the spar computations do, however, indicate that the 

error should be in the order of less than 10%. More information regarding the convergence tests and the results can 

be found in [15].

3.2.2. Derivation of System Matrices and System Eigenfrequencies

The FAST model used to simulate the TLP is the model that was used to develop the model-scale TLP wind 

turbine for the DeepCwind project. The system matrices and system eigenfrequencies were derived in the same way 

as for the OC3-Hywind, using FAST for linearization and the same assumptions. Also for the TLP, the system 

eigenfrequencies were derived for both rigid and flexible turbine tower and blades. The difference between the two 

cases is more significant than for the OC3-Hywind, as can be seen from the results presented in Table 1 (right). 

Table 1: Eigenvalues of OC3-Hywind Spar (left) and for UMaine TLP (right) with rigid and flexible wind turbine blades and tower. 

Degree of 

freedom

OC3-Hywind UMaine TLP

Eigenfrequency 

[rad/s]

Eigenfrequency 

(rigid) [rad/s]

Eigenfrequency 

(flexible) [rad/s]

Surge 0.051 0.156 0.156

Sway 0.051 0.156 0.156

Heave 0.204 5.975 5.948

Roll 0.215 3.388 2.005

Pitch

Yaw

0.215

0.761

3.392

0.374

2.021

0.374
This report is available at no cost from the 6 

National Renewable Energy Laboratory (NREL) 

at www.nrel.gov/publications.

Whereas the eigenfrequencies in surge, sway, heave and yaw remain almost the same, a significant shift of the 

pitch and roll frequencies from about 3.4 rad/s with a rigid tower to about 2.0 rad/s with a flexible tower is observed. 

This shift is due to a coupling between tower-bending and platform-pitch, an effect which has already been described 

for the NREL/Massachusetts Institute of Technology (MIT) TLP in [12]. Because it is not possible to model the 

flexible turbine tower within WAMIT, the results for the pitch and roll degrees of freedom may be inaccurate. It 

should also be noted that the eigenfrequencies of the UMaine TLP in heave, roll, and pitch are above the incident 

wave frequency range. For these degrees of freedom, the sum-frequency excitation are expected to be important. 

4. Results

In this section, the magnitudes of the first- and second-order forces and response are compared for both concepts. 

Excitation from second-order hydrodynamics and aerodynamics is compared for the OC3-Hywind spar. 

The first-order quantities were computed for incident wave frequencies in the range 0.005 to 5 rad/s. Due to the 

high computational effort needed to compute the second-order solution, the second-order quantities were computed 

for combinations of 21 different frequencies in the range from 0.26 to 1.5 rad/s. 

4.1. Comparison of First- and Second-Order Response in a Specific Sea State

The frequency-domain coefficients that are output from WAMIT are normalized such that they provide the force 

and motion response per wave amplitude. The first-order quantities are normalized by one incident wave amplitude, 

whereas the second-order quantities are normalized by pairs of incident wave amplitudes. Therefore, a direct 

comparison of first- and second-order quantities is only possible after a sea state with quantified wave amplitudes 

has been chosen. To get the force or motion experienced by the platform in the ocean, the coefficients are multiplied 

by the complex amplitude of the incident waves. The complex wave amplitude Am = ameiφm belonging to a wave 

with frequency ωm is specified by a magnitude am and a random phase φm. The wave amplitude am at a given

frequency ω is determined directly and uniquely from the wave spectrum S(ω). The wave phase φm is uniformly 

distributed between 0 and 2π and based on selection of a random wave seed. 

There are many pairs of waves that contribute to the overall second-order force (and motion) at a given frequency 

because there are many pairs of wave frequencies that share the same sum or difference frequency. The quantity of 

interest is, however, the total second-order force or motion response, which is given by the complex sum over all 

contributions at a given frequency. This summation is given by

�Ftotal

±(ωk

±)� = � � fmn±AmAn

ωm±ωn=ωk±

� = � � �fmn±�amanei(φmn±+φm+φn)

ωm±ωn=ωk±

� (1) 

Here fmn± = �fmn±�eiφmn±

is the complex second-order force coefficient of the wave pair with frequencies ωm and 

ωn. Because the incident wave phases φm and φn are random and depend on the chosen wave seed, the total force 

contribution differs between realizations. To get an impression of the range and variation of the total magnitudes, the 

summation is done for 15 different wave seeds. The result for the surge and heave degrees of freedom is shown in 

Figure 2 for the OC3-Hywind spar and in Figure 3 for the UMaine TLP. The wave spectrum in this study is a 

Pierson Moscowitz wave spectrum with Hs = 3.66 m and Tp = 9.7 s, which represents a moderate environmental 

condition with an operating turbine. 
This report is available at no cost from the 7 

National Renewable Energy Laboratory (NREL) 

at www.nrel.gov/publications.

For the OC3-Hywind spar, the second-order forces are very small compared to the first-order forces. However, 

where the difference-frequency forces coincide with the eigenfrequencies of the structure, some motion response is 

seen. Second-order difference-frequency response in heave is most significant compared to the first-order response. 

This is because the second-order excitation at the heave eigenfrequency is higher compared to the overall first-order 

excitation than for the surge or pitch directions. The difference-frequency response of the surge degree of freedom 

has two peaks. The lower one is response due to excitation at the surge eigenfrequency, and the upper one is due to 

excitation of the pitch eigenfrequency to which the surge response is coupled. Comparing the difference-frequency 

coefficients for the different wave pairs, the off-diagonal coefficients are found to be much larger than the 

coefficients along the diagonal. This is expected, as the deep penetration of the second-order diffraction potential 

means that it has a relatively large effect on the forces on deep draft structures compared to the forces arising from 

quadratic interactions of first-order quantities. 

The second-order forces are much higher in magnitude for the UMaine TLP than for the OC3-Hywind spar, and 

the motion response is therefore also much more significant. In heave, the sum-frequency forces are of the same or 

even higher magnitude as the first-order forces, meaning that the sum-frequency response actually dominates the 

overall heave motion response. Interestingly, this happens even without excitation of an eigenfrequency. For surge, 

the response due to difference-frequency excitation at the eigenfrequency dominates the overall response. Due to the 

coupling between pitch and surge, the same peak is seen in the plot for the pitch degree of freedom. One reason for 

this very large response might be that viscous drag is neglected. For such oscillations in surge, the viscous drag 

could be expected to be relatively high, especially since the TLP has relatively many slender parts. It should be 

noted that the pitch eigenfrequency (3.4 rad/s) is outside the range of sum-frequency excitation, and thus, there is no 

sum-frequency response in pitch. If it would be possible to model the flexibility of the turbine tower in WAMIT, the 

pitch eigenfrequency would shift to 2 rad/s, and we could expect to have significant sum-frequency response at this 

frequency as well.

Figure 2 First- and second-order forces (upper part) and motion responses (lower part) for the OC3-Hywind spar. 
This report is available at no cost from the 8 

National Renewable Energy Laboratory (NREL) 

at www.nrel.gov/publications.

4.2. Comparison of the Effects From Second-Order Hydrodynamics and Aerodynamics

A floating turbine experiences forces not only from hydrodynamics but also from aerodynamics. Aerodynamic 

loading on the rotor is known to produce slowly varying excitation in a frequency range similar to that of the 

difference-frequency hydrodynamic forces and with a magnitude that may be substantially higher. To create a case 

for comparison between aerodynamic and difference-frequency hydrodynamic responses, time series from FAST 

simulations (with aerodynamic and first-order hydrodynamic forces) and time series based on WAMIT first- and 

second-order RAOs are compared. The FAST time series were based on turbulent wind input data generated by 

TurbSim [5] and was run with rigid turbine tower and blades. The wave spectrum used for both the FAST time 

series and the time series based on WAMIT RAOs was a Pierson-Moskowitz spectrum generated by FAST.

4.2.1. Comparison of Mean-Drift Force and Mean Wind Turbine Thrust

The mean-drift force is a constant force that arises from interactions of oscillations with the same frequency. To 

compute the mean-drift force, all components with zero difference frequency are summed. This sum (given in Eq. 

(2)) does not need to account for random phases (as was done in the summation in Eq. (1)) because the random 

phases of two identical components cancel out. Fmm is the mean-drift force coefficient for the incident wave 

frequency ωm with magnitude fmm and phase φm. 

Fmean = �AmAm∗Fmm =

m

�amamei(φm−φm)fmmei(ωm−ωm) = �am2fmm

m m

(2) 

The mean-drift force and the mean turbine thrust are compared for 12 different environmental conditions identified 

in [15]. The mean-drift force was calculated as described in Eq. (2), and the mean wind turbine thrust was calculated 

from the FAST time series for each of the given environmental conditions. The results are shown in Figure 4 for the 

OC3-Hywind spar. For the cases where the turbine is operating, the mean-drift force is less than 1% of the rotor 

thrust. In more severe environmental conditions, where the turbine is idling and the rotor thrust is significantly 

lower, the mean-drift force amounts to about 10%–15% of the rotor thrust. However, this is still low enough to 

neglect the mean-drift force in general. The significance of the mean-drift force would likely be even smaller if the 

direct wind drag load on the tower, which has not been considered here, would be included in the simulation.

Figure 3 First- and second-order forces (upper part) and motion responses (lower part) for the UMaine TLP. 

The first-order force and motion response is represented by a single black line because it does not depend on the random phase 

and therefore does not vary between realizations. The single points (cyan for sum-frequency results and green for differencefrequency results) represent the total force or motion response at a certain sum or difference frequency for one realization of the 

sea state. The blue line (sum frequency) and the red line (difference frequency) represent the mean over all realizations.
This report is available at no cost from the 9 

National Renewable Energy Laboratory (NREL) 

at www.nrel.gov/publications.

4.2.2. Comparison of Motion Response Due to Excitation From Second-Order and Aerodynamic Forces

In this section, the importance of the difference-frequency forces on the motion response is compared to the 

importance of aerodynamics. The power-spectral densities (PSDs) of the motion response in surge, heave and pitch 

were computed for the same 12 environmental conditions as the mean-drift force. The results shown in Figure 5 are 

the PSDs of the surge response of the OC3-Hywind. The sea state is the same as in Section 4.1, with Hs = 3.66 m, Tp

= 9.7 s, and mean wind speed at hub height equal to 17.6 m/s.

The time series from FAST and the time series based on the frequency-coefficients from WAMIT give the same 

response in the incident wave frequency band (0.25–1.5 rad/s) in all modes of motion and for all sea states, 

confirming that the first-order wave excitation is the same for both systems, and that the first-order hydrodynamic 

excitation is dominating the motion response in this frequency range.

The main difference between the time-series from FAST, which contains excitation from aerodynamic forces, 

and the time-series based on frequency-domain coefficients from WAMIT, which contains excitation from secondorder hydrodynamic forces, is seen in the low-frequency domain. The response due to aerodynamic excitation is 

several orders of magnitude higher than the response due to second-order hydrodynamic effects and dominates the 

overall response of the turbine. This holds for all considered conditions and not only for the surge degree of freedom 

but also for heave and pitch.

From these results, it is concluded that in the case of the OC3-Hywind spar, aerodynamic excitation has a much 

more significant effect on the motion response than the excitation from difference-frequency hydrodynamic forces.

In the high-frequency domain above 1.5 rad/s, the response due to both wind and wave excitation is orders of 

magnitude smaller than at lower frequencies. High-frequency excitation seems to be of little importance to the spar 

motion response, and the sum-frequency wave excitation is even less important than high-frequency wind excitation. 

However, these results are based on a rigid turbine configuration, and could be different if structural dynamics were 

included. Also, no interactions between the second-order wave loading and high-frequency wind loading could be 

simulated in the current setup. 

Figure 4 Comparison of mean-drift force and mean-thrust force for the OC3-Hywind spar.
This report is available at no cost from the 10 

National Renewable Energy Laboratory (NREL) 

at www.nrel.gov/publications.

5. Conclusions

In this paper, a method to assess the effect of second-order hydrodynamics on floating offshore wind turbines has 

been presented. The proposed method relies mainly on simulation by the frequency-domain tool WAMIT and is 

based on the simulation methodology typically applied to more traditional offshore structures with the addition of 

inputs derived from the wind turbine CAE tool FAST. 

Using the proposed method, the importance of the second-order effects has been assessed for the OC3-Hywind 

spar and the UMaine TLP. More results than were presented here can be found in [15]. The second-order forces are 

very small for the OC3-Hywind spar, but where the forces excite eigenfrequencies, some second-order motion 

response is seen. For the UMaine TLP, the second-order forces are quite high, leading to a higher motion response. 

In heave, the sum-frequency effects dominate the overall motion response, even without exciting any 

eigenfrequencies. 

In a second step, the second-order effects have been compared to aerodynamic effects for the OC3-Hywind spar 

to assess their relative importance in more realistic environmental conditions. The mean-drift force was found to be 

insignificant compared to mean turbine thrust. A comparison between the motion response induced by aerodynamics 

and second-order hydrodynamics revealed that aerodynamics dominate the response of the turbine in the lowfrequency domain, and that the motion response due to difference-frequency excitation is negligible in comparison. 

A similar comparison of the relative importance of second-order effects and aerodynamics for the TLP has not yet 

been performed but is part of future work. As aerodynamics introduce less energy to the system at high frequencies, 

the sum-frequency effects of the TLP are likely to be relatively more important than the difference-frequency effects 

are for the OC3-Hywind spar. 

In the process of simulating the second-order effects of the OC3-Hywind spar and UMaine TLP, some limitations 

to the proposed method have been identified. First, the tower flexibility cannot be taken into account in WAMIT, 

leading to inaccuracies in the simulation of second-order quantities for structures where the tower flexibility couples 

to and influences the eigenfrequencies in pitch and roll (i.e. TLPs). One possible solution to this would be to derive 

RAOs based on FAST output and import these to WAMIT. However, this solution would require changes to the 

WAMIT source code. Another possibility might be to tune the FAST inputs to WAMIT, i.e. the stiffness matrix, to 

get the first-order RAOs predicted by WAMIT to match those derived from FAST. The second drawback is that 

viscous effects, which would likely damp some of the second-order motion response, are not accounted for when 

solving the equations of motion in WAMIT. One solution would be to linearize the viscous drag contribution and 

include this in the system damping matrix that is input to WAMIT. This is currently being tested. A more thorough 

solution would be to import second-order force coefficients from WAMIT to FAST in a way similar to the firstorder coefficients. This would require a change in FAST and is currently being worked on. It would allow 

simulation of the complete coupled response of the turbine, including first- and second-order hydrodynamics, as 

well as viscous drag and aerodynamics.

Figure 5 Surge response of the OC3-Hywind spar due to first-order hydrodynamic excitation combined with aerodynamic excitation 

(green) or second-order hydrodynamic excitation (blue). Sea state: Hs=3.66 m, Tp=9.7s, mean wind speed at hub height=17.6 m/s.
This report is available at no cost from the 11 

National Renewable Energy Laboratory (NREL) 

at www.nrel.gov/publications.

References

[1] Agarwal, P., Simulation Structural Reliability Offshore Wind Turbines, Ph.D. Dissertation, University of Texas at Austin, Austin, TX, 

August 2008. 

[2] de Ridder, E.J., Koop, A.H. and Doeveren, A.G., DeepCwind Floating Wind Turbine Model Tests, Report No. 24602-1-OB, MARIN, 

Wageningen, Netherlands, December 2011. 

[3] Goupee, A.J., Koo, B., Kimball, R.W. and Lambrakos, K.F., Experimental Comparison of Three Floating Wind Turbine Concepts, 

Proceedings of the 31st International Conference on Ocean, Offshore and Arctic Engineering, Rio de Janeiro, Brazil, June 10-15, 2012. 

[4] Goupee, A.J., Koo, B., Lambrakos, K.F. and Kimball, R.W., Offshore Wind Energy: Model Tests for Three Floating Wind Turbine 

Concepts, Proceedings of the Offshore Technology Conference held in Houston, Texas, USA, 30 April–3 May 2012.

[5] Jonkman, B., TurbSim User's Guide: Version 1.50, NREL/TP-500-46198, Golden, CO, National Renewable Energy Laboratory, September 

2009. 

[6] Jonkman, J., Definition of the Floating System for Phase IV of OC3, NREL/TP-500-41958, Golden, CO: National Renewable Energy 

Laboratory, November 2007.

[7] Jonkman, J. M. and Buhl, M. L., Jr., FAST User’s Guide, NREL/EL-500-38230, Golden, CO: National Renewable Energy Laboratory, 

August 2005. 

[8] Jonkman, J., Butterfield, S., Musial, W., and Scott, G., Definition of a 5-MW Reference Wind Turbine for Offshore System Development, 

NREL/TP-500-38060, Golden, CO: National Renewable Energy Laboratory, February 2009. 

[9] IEA Wind Task 23, Offshore Wind Technology and Deployment, “Subtask 2: The Offshore Code Comparison Collaboration, Final Report”, 

http://www.ieawind.org/task_23/23_FinalReports/Subtask2_Final_Report.pdf. 

[10] Lee, C.-H., Newman J.N., Kim, M.H., and Yue, D.K., The computation of second-order wave loads, OMAE Conf., Stavanger, Norway, 

1991. 

[11] Lucas, J., UpWind Project: Comparison of a First- and Second-Order Hydrodynamic Results for Floating Offshore Wind Structures, Garrad 

Hassan & Partners Ltd, Bristol, UK, January 2011. 

[12] Matha, D., Model Development and Loads Analysis of an Offshore Wind Turbine on a Tension-Leg Platform, with a Comparison to Other 

Floating Turbine Concepts, NREL/SR-500-45891, Golden, CO: National Renewable Energy Laboratory, February 2010. 

[13] National Wind Technology Center, URL: http://www.nrel.gov/wind/, Simulation code download URL: 

http://wind.nrel.gov/designcodes/simulators/ 

[14] Newman, J.N., Sensitivity of wave loads to the discretization of bodies, BOSS (Behaviour of Offshore Structures), ss. 50-64, London, 1992. 

[15] Roald, L.A., The Effect of Second-Order Hydrodynamics on a Floating Offshore Wind Turbine, Master Thesis, ETH Zürich, 2012. 

[16] Wamit, Inc. (2002). WAMIT User Manual, Version 6.4., WAMIT Inc.