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Contract No. DE-AC36-08GO28308

Effects of Second-Order 

Hydrodynamic Forces on 

Floating Offshore Wind 

Turbines

T. Duarte and A. JNA. Sarmento 

Instituto Superior Técnico

J. Jonkman

National Renewable Energy Laboratory

Presented at AIAA SciTech 2014

National Harbor, Maryland

January 13‒17, 2014

Conference Paper

NREL/CP-5000-60966

April 2014
NOTICE

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Effects of Second-Order Hydrodynamic Forces on Floating 

Offshore Wind Turbines

Tiago M. Duarte* and António JNA. Sarmento†

Instituto Superior Técnico, Lisboa, 1049-001 Portugal

and

Jason Jonkman‡

National Renewable Energy Laboratory (NREL), Golden, Colorado, 80401

Relative to first-order, second-order wave-excitation loads are known to cause significant 

motions and additional loads in offshore oil and gas platforms. The design of floating 

offshore wind turbines was partially inherited from the offshore oil and gas industry. 

Floating offshore wind concepts have been studied with powerful aero-hydro-servo-elastic 

tools; however, most of the existing work on floating offshore wind turbines has neglected 

the contribution of second-order wave-excitation loads. As a result, this paper presents a 

computationally efficient methodology to consider these loads within FAST, a wind turbine 

computer-aided engineering tool developed by the National Renewable Energy Laboratory. 

The method implemented was verified against the commercial OrcaFlex tool, with good 

agreement, and low computational time. A reference floating offshore wind turbine was 

studied under several wind and wave load conditions, including the effects of second-order 

slow-drift and sum-frequency loads. Preliminary results revealed that these loads excite the 

turbine’s natural frequencies, namely the surge and pitch natural frequencies. 

Nomenclature 

Ak = complex wave amplitude

Fex(1)

 = first-order wave-excitation loads

Fex(2)

 = second-order wave-excitation loads

Xi(ωk) = first-order excitation transfer function

Xi

+(ωk, ωl) = sum-frequency quadratic transfer function (QTF)

Xi

−(ωk, ωl) = difference-frequency quadratic transfer function (QTF)

ωk = wave frequency

I. Introduction

EEP water offshore wind has the potential to be an almost endless source of renewable energy. Several floating 

platforms are currently under development to support wind turbines, based on the knowledge developed in the 

offshore oil and gas industry. However, floating offshore wind turbines (FOWTs) are not only subject to the same 

wind loads as land-based turbines, but also current and wave loads. Most of the existing state-of-the-art tools for 

simulating the coupled aero-hydro-servo-elastic dynamics of FOWTs consider only the first-order wave-excitation 

loads. However, the second-order wave-excitation forces are known to produce significant slow-drift motions and 

 

*

Ph.D. Student, MIT Portugal Program, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal, non-member of AIAA.

†

Associated Professor, Mechanical Engineering Department, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal, nonmember of AIAA.

‡

Senior Engineer, National Wind Technology Center at NREL, 15013 Denver West Parkway, Golden, CO, USA,

AIAA Professional Member.

D
2 

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ringing effects in offshore oil and gas platforms. Because little work has been conducted on this subject to date, it is 

still not clear how these loads affect FOWTs. 

The second-order effects, proportional to the square of the wave amplitudes, are derived based on the interaction 

of a pair of regular waves, with frequencies ωkand ωl. This interaction generates low-frequency loads with a 

frequency of |ωk − ωl|, known as difference-frequency loads. In the same way, it also creates loads at ωk + ωl, 

known as sum-frequency loads. The difference-frequency loads are known to excite the slow-drift motions in 

slacked moored structures, and are responsible for the mean-drift loads generated by the waves. These loads are 

particularly important for soft-moored platforms (e.g., catenary), which usually have very high natural periods 

(100−150 s in surge). The sum-frequency loads are typically important for stiffer structures, like fixed-bottom 

monopole or jacket foundations or tension-leg floating platforms (TLPs), in which they are known to cause ringing 

effects in these type of structures. For FOWTs, the range of frequencies of the sum-frequency force may also excite 

the first tower-bending modes. 

Figure 1 presents the preliminary results of the first- and second-order wave-excitation loads for the Offshore 

Code Comparison Collaboration (OC3)-Hywind spar buoy FOWT1

, under an irregular sea-state modeled by a 

JONSWAP spectrum, with a significant wave height of 6 m and a peak-spectral wave period of 10 s. The vertical 

black lines represent the natural frequency values of the different modes of the floating turbine. As shown in the 

figure, the second-order loads are often neglected because they have a smaller magnitude than the first-order waveexcitation loads. In addition, the platform natural frequencies tend to be far from the peak of the first-order 

excitation loads to avoid significant resonance. (The natural frequencies of a FOWT are designed to be either below 

the wave-excitation frequencies (lower than 0.05 Hz) or above the wave-excitation frequencies (above 0.25 Hz). In 

the OC3-Hywind FOWT, the first tower fore-aft and side-to-side bending modes are around 0.45 Hz and the first 

blade-bending modes are around 0.65 Hz.) This shows that the difference-frequency loads, while small, may excite 

the platform motions, while the sum-frequency loads may excite the turbine degrees of freedom (DOFs). 

Figure 1. First- and second-order excitation loads and turbine natural frequencies (vertical black lines), for 

the OC3-Hywind spar buoy under a 6-m, 10-s JONSWAP wave spectrum.

Some studies have already highlighted the importance of second-order loads for FOWTs in low wind conditions. 

Furthermore, results from scaled wind-wave-tank tests presented in Robertson et al.

 2 indicated the importance of the 

second-order loads for these types of systems. Significant differences were found between the experimental results 

and FAST simulations outside the peak wave-energy range with only first-order hydrodynamic loads. The 

contribution of the low-frequency loads was clearly identified for all three floating platforms tested. Moreover, some 

differences between the experimental results and the first-order predictions were found for higher frequencies, which 

indicate the influence of sum-frequency loads, especially for the TLP and tower-base bending loads. Lucas3
3 

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compared the influence of second-order hydrodynamic loads on the OC3-Hywind spar-type floater and on the 

semisubmersible platform WindFloat. The six rigid-body DOFs of the floating platform were compared for regular 

and irregular waves, and the comparison showed that the spar is mainly dominated by the first-order component, 

while the second-order component is important mainly for the surge and pitch DOFs. For the semisubmersible

platform, the second-order loads dominated the response, or at least they were on the same order of magnitude as the 

first-order loads. 

This paper presents the methodology used to implement second-order hydrodynamic loads in the time-domain

within the wind turbine aero-hydro-servo-elastic tool FAST4

. It makes use of quadratic transfer functions (QTFs), 

which are determined by a frequency-domain panel code (e.g., WAMIT) preprocess step. This method is described 

in Section II. Section IV includes results from a time-domain analysis for a reference FOWT under load cases 

presented in Section III. 

II. Calculation of Second-Order Forces

This section presents the method used to compute the time-domain second-order wave-excitation loads, which are 

based on the outputs derived with a panel code. WAMIT 6.1S was used to derive the required QTFs for the 

examples presented in this paper. 

A. Time-Domain Implementation

In order to include the second-order hydrodynamic effects, the second-order loads can be linearly superimposed 

with the first-order loads. Consider the equation of motion of a rigid platform with six DOFs subject to 

hydrodynamic loads, as given by the Cummins equation: 

(M + A∞)q̈

tot + � K(t − τ)q̇tot(τ)dτ

t

0

+ Chydroqtot = Fex(1) + Fex(2) (1) 

where M is the mass matrix of the body, A∞ is the added-mass matrix, K is the retardation matrix, Chydro is the 

hydrostatic stiffness matrix, and q̈tot, q̇tot, and qtot are the body acceleration, velocity, and displacements. The firstand second-order wave-excitation loads are described by the terms Fex(1) and Fex(2)

. For a single regular incoming 

wave train, the first-order wave loads are computed using:

Fex(1)

i = Re(AXiejωt) , i = 1, 2, ... , 6 (2)

where A represents the complex incoming wave amplitude (characterizing the wave amplitude and phase), j is the 

imaginary number √−1, and Xi describes the i

th component of the first-order wave loads per unit amplitude. For 

long-crested (without directional spreading) irregular sea states, the total load can be seen as a superposition of the 

different wave-frequency components:

Fex

(1)

i = Re(∑ AkXi(ωk)ejωk N t k=1 ) , i = 1, 2, ... , 6, ωk = (k − 1)δω (3)

The normalized first-order wave-excitation force Xi(ωk) is a function of the platform shape, wave frequency, 

and wave direction. It is a complex-value vector that accounts for loads out-of-phase with the wave elevation. In 

FAST, Eq. (3) is implemented via a fast Fourier transform (FFT).

Similar to the first-order excitation, the second-order hydrodynamic load is also frequency dependent. The 

second-order loads for long-crested (without directional spreading) irregular sea states are divided into the 

contributions of the sum- and difference-frequencies between pairs of incoming waves. For each pair of incoming 

waves with amplitude Ak and Al and frequencies ωk and ωl, one can obtain two different QTFs, Xi

+(ωk, ωl) and 

Xi

−(ωk, ωl), corresponding to the i

th component of the sum- and difference-frequency loads. The second-order loads 

can therefore be obtained using the double Fourier transform: 

Fex(2)

i = Re�∑ ∑ �AkAlXi

+(ωk, ωl)e

j(ωk+ωl)t + AkAl

∗

Xi

−(ωk, ωl)ej(ωk−ωl)t

� N

l=1

N

k=1 � , i = 1, 2, ... , 6 (4)

where the asterisk (*) denotes the complex conjugate. 

1. Symmetry of the QTF Matrices

One should note the following symmetry relations of the QTF matrices:

Xi

+(ωk, ωl) = Xi+(ωl, ωk) and Xi−(ωk, ωl) = Xi−(ωl, ωk)∗ (5)
4 

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Taking for now just the difference-frequency term of Eq. (4), one can separate the double summation into three 

regions: l = k, l > k, and l < k. In addition, by introducing the variable μ− = k − l, the double summation 

becomes:

Fex(−)(2)

i = Re �∑ AkAk

∗

Xi

−(ωk, ωk) N

k=1 + ∑ ∑ Al+μ− Al

∗

Xi

−�ωl+μ− , ωl�e

j�ωμ N− −�t μ−

l=1

N−1 μ−=1

+ ∑ ∑ AlAl+μ− ∗Xi

−(ωl, ωl+μ− )e−j(ωμ N−μ −)t −

l=1

N−1 μ−=1

� , i = 1, 2, ... , 6 (6)

Using the relation in Eq. (5), the last two terms of the previous equation can be grouped and one obtains5

: 

Fex(−)(2)

i = Re�∑ AkAk

∗

Xi

−(ωk, ωk) N

k=1 + 2 ∑ Hμ−e N−1 j(ωμ−)t μ−=1 � , i = 1, 2, ... , 6 (7)

with Hμ− = ∑ Al+μ−Al

∗

Xi

−�ωl+μ−, ωl� N−μ−

l=1 , μ− = 1,2, . . . , N − 1 (8)

The first term in Eq. (7) corresponds to the mean-drift loads and equals the sum of the terms on the diagonal of 

Figure 2 (a). The second term represents the slow-drift load and corresponds to the sum of the terms with a circle 

over the blue lines. 

Figure 2. Symmetries of the difference-frequency QTF (left) and sum-frequency QTF (right) matrices, for 

N = 10. 

a) b)

Comparing Eq. (6) to the difference-frequency term of Eq. (4), it must be noted that the computational effort is 

significantly reduced, as one makes use of the symmetry and only takes the lower triangular part of the 

matrix Xi

−(ωk, ωl). However, the great advantage of this method is that instead of computing a double inverse 

Fourier transform (over ωk and ωl as in Eq. (4)), it is only required to perform a single inverse Fourier transform 

over ωμ−. Therefore, instead of having N inverse Fourier transforms each with N terms, only N ∗ (N + 1)/2 sums 

and a single inverse Fourier transform with N − 1 terms are employed. 

Extending this result to the sum-frequency QTF by introducing the variable μ+ = k + l, only a single Fourier 

transform over ωμ+ can be performed. This corresponds to summing all the terms along each blue line in Figure 2

(b). It should be noted that 0 ≤ ωk ≤ ωmax, and for the difference-frequency case 0 < ωμ− ≤ ωmax, but for the 

sum-frequency case 0 ≤ ωμ+ ≤ 2ωmax. Therefore, we obtain:

Fex(+)(2)

i = Re�∑ Hμ+e

j(ωμ+−1 2N )t

μ+=2 � , i = 1, 2, ... , 6 (9)

with: �

Hμ+ = ∑ AlAμ+−lXi

+(ωl, ωμ+−l) μ+−1

l=1 , μ+ = 2,3, . . . , N + 1

Hμ+ = ∑ AlAμ+−lXi

+(ωl, ωμ+−l) N

l=μ+−N , μ+ = N + 2, . . . ,2N (10)

1 2 3 4 5 6 7 8 9 10 1

2

3

4

5

6

7

8

9

10

k

l

1 2 3 4 5 6 7 8 9 10

9

8

7

6

5

4

3

2

1

0

k

μ-=k-l

QTF- Symmetry

Constant Difference-Frequency

μ1 2 3 4 5 6 7 8 9 10

1

2

3

4

5

6

7

8

9

10

k

l

2 3 4 5 6 7 8 9 10 11

11

12

13

14

15

16

17

18

19

20

μ+

μ+=k+l

Constant Sum-Frequency

QTF+ Symmetry

μ+
5 

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The approach simplifies the computation of the sum-frequency loads, using only N2 sums and a single inverse

Fourier transform with 2N − 1 terms, instead of a double inverse Fourier transform with N2 terms in Eq. (4). To 

make use of the symmetry of the QTF matrix, the matrix should be split into three regions: l = k, l > k, and l < k. 

Using the symmetry relations in Eq. (5) we obtain:

Fex(+)(2)

i = Re�∑ AkAkXi

+(ωk, ωk)ej(2ωk N )t k=1 + 2 ∑ Hμ+e

j(ωμ+−1 2N )t

μ+=2 � , i = 1, 2, ... , 6 (11)

with: �

Hμ+ = ∑ AlAμ+−lXi

+(ωl, ωμ+−l) �(μ+−1)/2�

l=1 , μ+ = 2,3, . . . , N + 1

Hμ+ = ∑ AlAμ+−lXi

+(ωl, ωμ+−l) �(μ+−1)/2�

l=μ+−N , μ+ = N + 2, . . . ,2N (12)

where ⌊x⌋ represents the floor function of x defined as:

⌊x⌋ = max{m ∈ Z|m ≤ x} (13)

where Z is the set of integers (positive, negative, and zero). Once again, the first term in Eq. (11) represents the 

inverse Fourier transform over the terms on the main diagonal of Figure 2 (b). The second term of Eq. (11) 

represents the inverse Fourier transform over the diagonal blue lines. The first interval of sums in Eq. (12) represents 

the sum over the round blue dots and the second interval represents the sum over the blue triangles. With this 

simplification, we further reduce the computational effort to ((N2 − N)⁄2) sums and two inverse Fourier 

transforms with N and 2N − 1 terms, respectively.

The total second-order load is therefore the contribution of the following terms:

Fex(2)

i = Re�∑ AkAk

∗

Xi

−(ωk, ωk) N

k=1 + 2 ∑ Hμ−ej(ωμ N−1 −)t μ−=1 + ∑ AkAkXi

+(ωk, ωk)ej(2ωk N )t k=1 +

2 ∑ Hμ+ e

j(ωμ+−1 2N )t

μ+=2 � , i = 1, 2, ... , 6

(14)

where:

Hμ− = ∑ Al+μ−Al

∗

Xi

−�ωl+μ−, ωl� N−μ−

l=1 , μ− = 1,2, . . . , N − 1

�

Hμ+ = ∑ AlAμ+−lXi

+(ωl, ωμ+−l) �(μ+−1)/2�

l=1 , μ+ = 2,3, . . . , N + 1

Hμ+ = ∑ AlAμ+−lXi

+(ωl, ωμ+−l) �(μ+−1)/2�

l=μ+−N , μ+ = N + 2, . . . ,2N

(15)

2. Mean-Drift Load

The second-order excitation load includes a mean different from zero, called the mean-drift load (constant term 

in Eqs. (6) and (14)). This term arises from the quadratic interactions of the first-order problem, and therefore can be 

computed without requiring the solution of the second-order potential. This can be proven by averaging the solution 

of the second-order excitation loads. The mean excitation load can be given in terms of a WAMIT-derived transfer 

function Xi

Drift: 

Fex(drift)

i = Re(∑ AkAk

∗

Xi

Drift(ωk) N

k=1 ) , i = 1, 2, ... , 6 (16)

3. Newman’s Approximation

Because of the computational expense in a frequency-domain panel code of determining the full QTF matrices 

and the double sum present in Eq. (4), many of the commercial time-domain codes use Newman’s approximation 

[6]. This approach is useful in simplifying the calculation of the slowly varying drift forces, especially for platforms 

with large natural periods, like the case of moored floating systems. Newman’s approximation estimates the offdiagonal terms of the difference-frequency QTF from the diagonal terms. This means that the second-order 

difference-frequency behavior can be estimated using only the mean-drift QTF derived from the first-order solution. 

Using the Newman’s approximation, the second-order slow-drift loads are estimated using the following 

equation6

: 

Fex(−)(2)

i =

�Re�∑ Ak�2Xi

−(ωk, ωk)ejωk N t k=1 ��

2

�

Xi

−(ωk,ωk)>0

− �Re�∑ Ak�−2Xi

−(ωk, ωk)ejωk N t k=1 ��

2

�

Xi

−(ωk,ωk)<0

, 

i = 1, 2, ... , 6

(17)
6 

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Standing et al7 provided a more accurate alternative to this equation, which should be considered in future work. 

The accuracy of Newman’s approximation is better if there is a small variation in the off-diagonal terms when 

compared with the mean diagonal. On theoretical grounds, the function Xi

− should be continuous in both ωk and ωl, 

so the near-diagonal term Xi

−(ωk, ωl) should be well-approximated by the mean of Xi−(ωk, ωk) and Xi−(ωl, ωl). 

Also, the important difference terms are those with low frequency, which arise from the near-diagonal terms in Xi

−. 

The far-from-diagonal values produce much higher frequencies that have little effect on a platform of sufficiently 

large inertia. It is also particularly suitable for narrow-banded spectrums, where it can be assumed that the only 

terms of importance are when ωk and ωl are close to each other, therefore the off-diagonal terms can be neglected 

because of the low magnitude of AkAl for large difference-frequencies.

On the other hand, it should be noted that the Newman approximation can be poor in shallow water, so the full 

QTF method is preferable in such cases. Additionally, if the vessel is subject to a significant spread of different 

wave directions, wave pairs with very different directions use far-from-diagonal QTF values but still give low 

frequency drift-load contributions if their periods are close to each other. So, the full QTF method is also preferable 

in cases with a spread of wave directions. There is no equivalent approximation for the sum-frequency loads.

B. Verification of the Implementation

The equations above were implemented in MatLab before being fully incorporated into FAST. The same 

irregular wave cases were run in both OrcaFlex8 and the developed code. The results were derived using the same 

QTF matrices for a reference TLP in both codes. The comparison is discussed in the following sections. All of the 

results refer to a sea state represented by a JONSWAP wave spectrum with a 6-m significant wave height and 10-s 

peak-spectral wave period. The comparisons shown in this section are aimed only at verifying the implementation 

and assessment of the equations’ computational efficiency.

1. Mean-Drift Load

The mean-drift load is constant in time and only depends on the diagonal terms of the difference-frequency QTF 

matrix. The results of the mean-drift load obtained with the methods presented in Section II. A were compared with 

the OrcaFlex results. The error between the codes is smaller than 0.2%, which might be caused by the different 

frequency-domain discretizations. 

Figure 3. Second-order drift surge force calculated with the Newman’s approximation and the full QTF. 

0 20 40 60 80 10 120 140 160 180 200 -20

-15

-10

-5

0

5

x 105

Time 

F2

Surge [N]

Newman`s App. 

Diff-QTF

OrcaFlex Newman

OrcaFlex - Diff-QTF
7 

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Figure 4. Sum-frequency surge force.

2. Difference-Frequency QTF

The difference-frequency loads were implemented using the single inverse Fourier transform method, described 

by Eqs. (7) and (8) and made use of a computationally efficient FFT. The JONSWAP wave spectrum was 

discretized with 40,960 equally spaced frequency steps, while OrcaFlex used 100 discrete frequency steps, spaced 

with an equal energy discretization. An example of the time-series obtained can be seen in Figure 3. 

The agreement for the Newman’s approximation method (blue lines) is very good. For the full-QTF method (red 

and purple lines), some high-frequency content was found in the OrcaFlex signal not present in the implemented 

code. This occurred because the frequency range considered was limited to 0 to 1.5 rad/s to save computational time. 

A sensitivity analysis showed that increasing this range would significantly reduce the differences shown. 

3. Sum-Frequency QTF

The sum-frequency loads were implemented using the approach described by Eqs. (11) and (12) and made use of 

computationally efficient FFTs. To verify the sum-frequency loads, the same approach that was used in the 

difference-frequency case was applied. The OrcaFlex outputs were compared to the implemented code using the 

same wave elevation in both codes. An example of the obtained time-series is presented in Figure 4. Overall, the

OrcaFlex outputs agreed well with the implemented code. 

0 20 40 60 80 100 120 140 160 180 200 -5

-4

-3

-2

-1

0

1

2

3

4

5

x 106

Time [s]

F 2 Surge [N]

MatLab

OrcaFlex
8 

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Figure 5 presents the computational time for the implemented methods. The results were obtained for 4096 s of 

the previously described spectrum and were used for computing the loads in the 6 DOF. While the MatLab results 

used 40,960 equally spaced frequency steps to describe the spectrum, OrcaFlex used a discretized spectrum with the 

equal energy method (only 100 frequencies). The OrcaFlex model that was created neglected all features except the

second-order loads to minimize the additional calculations not performed in the MatLab code and obtain a realistic 

comparison. These results show the efficiency of the computational method that was implemented, which was 

enhanced by using the inverse FFT routines. OrcaFlex performs the sum for each discrete frequency, which 

significantly reduces the number of discrete frequencies used but still requires higher computational times.

III. Case Study

This section presents a reference FOWT and the load cases studied in this paper. For the case study, the 

modeling tool FAST was coupled with the MatLab implementation. As the second-order loads in the time domain 

depend only on the incoming waves, and not on the motion of the platform, they can be precalculated before running 

the FAST simulation. The second-order load time-series are precalculated by MatLab, read into FAST, summed 

with the first-order hydrodynamic loads calculated within FAST, and coupled with the additional aero-servo-elastic 

features of FAST. The same wave spectrum and time-domain realization was used by both programs.

A. Reference Platform

The OC3-Hywind spar-type FOWT was chosen for this detailed analysis. The wind-wave-tank tests presented by 

Robertson et al2 pointed out that this platform might be affected by both difference- and sum-frequency forces. The 

proprieties of this platform are defined in1 and summarized in Figure 6. The difference- and sum-frequency QTFs

were derived using WAMIT 6.1S9

. 

Figure 5. CPU time for the different frequency methods implemented

0

50

100

150

200

250

Mean Drift

Newman,

s Approximation

Diff-QTF

Sum-QTF

OrcaFlex - Newman

OrcaFlex - Diff-QTF

OrcaFlex - Sum-QTF

CPU Time [s]
9 

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Main Properties of OC3 Hywind

Rated Power 5 MW

Configuration Upwind, Three Blades, Variable 

Speed

Rotor Diameter 126 m

Hub Height 90 m

Cut-in, Rated, Cut-Out Wind Speed 3 m/s, 11.4 m/s, 25 m/s

Platform Draft 120 m

Platform Diameter Above Taper 6.5 m

Platform Diameter Below Taper 9.4 m

Depth to Top of Taper 4 m

Depth to Bottom of Taper 12 m

Center of Mass (0,0,-78)

Figure 6. Main properties of the reference OC3-Hywind FOWT. 

B. Load Cases

A partial set of design load cases (DLCs) from those defined in the International Electrotechnical Commission 

(IEC) 61400-3 offshore wind turbine design standard were used in this work. In order to assess the influence of the 

second-order loads on the overall system, normal production load cases were selected. Load cases 1.1 and 1.2 were 

used to assess the ultimate and fatigue loads under normal production. The load case with extreme wave loads (1.6) 

provides conditions that intensify the second-order effects, resulting in higher extreme loads. (Note that the 50-yr 

extreme sea state was used for all wind speeds in place of a conditional severe sea state as defined by the IEC for 

this load case). Finally, load case 2.1 was used to study the platform loads with the turbine parked (the 

corresponding simulations did not include the transient fault condition). These load cases are summarized in Table 

1. 

Table 2 summarizes the met-ocean conditions used for a reference site in the North Sea. For more details on the 

met-ocean data, please consult Jonkman10. Each load case was run for wind speed bins of 2 m/s. For each wind 

speed bin, six 10-min simulations of 10 min each were run with different wave seeds and turbulent wind speed

fields. The wave height was selected for each wind speed according to the table. For each wind speed, two 

simulations were run with each peak-spectral wave period value presented in the table, totaling the six simulations 

per wind speed bin.

Table 1. Design Load Cases.

Turbine 

Condition

DLC Wind 

Condition

Wave 

Condition

Wind Wave 

Directionality

Sea 

Current

Type of 

Analysis

Power 

Production

1.1 NTM 

Vin<Vhub<Vout

NSS 

Hs=E[Hs| Vhub]

Aligned Neglected Ultimate

1.2 NTM 

Vin<Vhub<Vout

NSS 

Hs=E[Hs| Vhub]

Aligned Neglected Fatigue

1.6 NTM 

Vin<Vhub<Vout

SSS

Hs= Hs50

Aligned Neglected Ultimate

Parked 

Condition

2.1 NTM 

Vin<Vhub<Vout

NSS

Hs=E[Hs| Vhub]

Aligned Neglected Ultimate

NTM: Normal Turbulence Model

NSS: Normal Sea State

SSS: Severe Sea State

Vin: Cut-In Wind Speed

Vhub: Hub-Height Wind Speed

Vout: Cut-Out Wind Speed

Hs: Significant Wave Height

Tp: Peak-Spectral Wave Period
10 

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Table 2. Met-Ocean Conditions.

Wind Speed (m/s) Significant Wave Height (m) Peak-Spectral Wave Period(s)

4 1.586 (NSS) 8,97 12,67 16,43

6 1.758 (NSS) 7,98 12,67 17,37

8 1.837 (NSS) 7,98 12,67 17,37

10 2.194 (NSS) 9,15 13,38 17,60

12 2.402 (NSS) 9,15 13,38 17,60

14 2.988 (NSS) 8,68 11,97 15,25

16 3.373 (NSS) 10,09 13,38 16,66

18 3.673 (NSS) 10,09 13,38 16,66

20 4.441 (NSS) 11,26 14,08 16,90

22 4.720 (NSS) 11,03 13,38 15,72

24 5.521 (NSS) 12,67 15,49 18,30

Any 15.04 (SSS) 19.22

IV. Results

This section presents the results from the analysis of the OC3-Hywind FOWT. Spectral analysis results of the 

different load cases are presented in the next subsection, followed by the ultimate loads and fatigue results. Each 

load case was run without second-order loads, and with Newman’s approximation, difference-frequency full QTF, 

sum-frequency full QTF, and both difference- and sum-frequency QTFs together (the complete second-order

solution). Platform surge and pitch motion, fairlead tension of one of the upwind mooring lines, and tower-base 

bending moments are also analyzed in the following sections.

A. Spectral Analysis

1. No Wind Loads

Figure 7 shows the power spectral results with and without second-order loads for conditions without wind. 

These results correspond to the IEC load case 2.1 after the fault-induced shutdown. For this analysis, a JONSWAP 

spectrum with 3-m significant wave height and an 8-s peak-spectral wave period was chosen. The influence of the 

second-order loads is clearly visible for frequencies below the wave range, as predicted by Lucas3

. In most of the 

figures, the results with the difference-frequency QTF (red line) are below the results of the complete second-order 

solution (purple line). The first-order results (dark blue line) are below the sum-frequency results, especially below 

the wave region, where the sum-frequency loads are negligible.

The surge power spectral density shows a significant increase in surge motion below the first-order wave range, 

especially around the pitch and surge natural frequencies (0.035 Hz and 0.01 Hz, respectively). There is a significant 

increase of the pitch motion caused by the difference-frequency load, which is largely neglected when using 

Newman’s approximation. The fairlead mooring tension shows a similar behavior, with an increase around the pitch 

natural frequency (0.3 Hz). It should be noted that by using the quasi-static mooring module of FAST, we are 

neglecting the mooring line dynamics that would contribute to an increased response for the sum-frequency range. 

Finally, the tower-base moment shows an increase around the pitch natural frequency caused by the differencefrequency loads. No significant effect related to the sum-frequency loads was observed. This result is most likely 

because the large platform inertia is acting as a filter to these higher frequency loads.
11 

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Figure 7. Power spectral results obtained without wind loads. Top left: surge motion; top right: pitch motion;

lower left: fairlead tension; lower right: tower-base moment.

2. Normal Sea State with Wind Loads

The operating turbine generates a significant thrust force acting on the tower top, forcing the turbine to drift from 

its equilibrium position. This force is usually significantly higher than the mean-drift surge force caused by secondorder wave loads. Figure 8 presents the power spectral density of the platform motion and loads under the operating 

load case (1.1 / 1.2). The surge motion is now dominated by the wind thrust force, and no significant differences can 

be found for the case including second-order loads. However, the influence of the difference-frequency loads around 

the platform-pitch natural frequency is still visible in the pitch motion. Once again, Newman’s approximation 

underestimates the second-order excitation for this region, when compared to the full QTF approach. 

The loads acting on the mooring line follow the surge-motion results. No significant differences were found 

between the cases, as the wind loads were driving the surge motion and consequently the mooring line tension. For 

the tower-base bending moment, some additional loads were found around the pitch natural frequency of the spar 

buoy.

3. Extreme Wave Load Case

Load case 1.6 is dominated by an extreme sea state of 15.69-m significant wave height and 19.22-s peak-spectral 

wave period (see Figure 9). Despite the wind load acting on the rotor, the influence of the second-order loads is now 

visible for the platform surge motion, especially around the pitch natural frequency. This effect is also visible on the 

pitch motion, where once again Newman’s approximation tends to underestimate this effect. This difference also 

occurs for the mooring line tension and tower-base moments. Because of the larger periods, the sum-frequency loads 

now have a visible influence on the tower-base moment (although still very small), around the first tower-bending 

natural frequency.

0 0.05 0.1 0.15 0.2 0.25 0.3 10-3

10-2

10-1

100

Frequency [Hz]

Surge PSD [m2/Hz]

First-Order

Newman

Diff-Frequency

Sum-Frequency

Complete Second-Order

0 0.05 0.1 0.15 0.2 0.25 0.3 10-4

10-3

10-2

10-1

100

Frequency [Hz]

Pitch PSD [deg2/Hz]

First-Order

Newman

Diff-Frequency

Sum-Frequency

Complete Second-Order

0 0.05 0.1 0.15 0.2 0.25 0.3 10-3

10-2

10-1

100

101

102

103

104

Frequency [Hz]

Fair2Ten PSD [kN2/Hz]

First-Order

Newman

Diff-Frequency

Sum-Frequency

Complete Second-Order

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 105

106

107

108

109

Frequency [Hz]

Tower Base My PSD [kN2/Hz]

First-Order

Newman

Diff-Frequency

Sum-Frequency

Complete Second-Order
12 

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.

Figure 8. Power spectral results obtained for IEC load case 1.1 and 1.2. Top left: surge motion; top right: 

pitch motion; lower left: fairlead tension; lower right: tower-base moment.

0 0.05 0.1 0.15 0.2 0.25 0.3

10-2

10-1

100

101

Frequency [Hz]

Surge PSD [m2/Hz]

First-Order

Newman

Diff-Frequency

Sum-Frequency

Complete Second-Order

0 0.05 0.1 0.15 0.2 0.25 0.3

10-4

10-3

10-2

10-1

100

101

Frequency [Hz]

Pitch PSD [deg2/Hz]

First-Order

Newman

Diff-Frequency

Sum-Frequency

Complete Second-Order

0 0.05 0.1 0.15 0.2

100

101

102

103

Frequency [Hz]

Fair2Ten PSD [kN2/Hz]

First-Order

Newman

Diff-Frequency

Sum-Frequency

Complete Second-Order

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

103

104

105

106

107

108

109

Frequency [Hz]

Tower Base My PSD [kN2/Hz]

First-Order

Newman

Diff-Frequency

Sum-Frequency

Complete Second-Order
13 

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.

Figure 9. Power spectral results obtained for IEC load case 1.6. Top left: surge motion; top right: pitch 

motion; lower left: fairlead tension; lower right: tower-base moment.

B. Ultimate Loads

Figure 10 presents the ultimate loads found for the different load cases studied. No partial load factors were 

included in the results. The difference-frequency loads caused the biggest increase in maximum surge displacement

(under the extreme wave conditions). However, no significant changes were observed in the maximum fairlead 

tension. As for the surge motion, IEC load case 1.6 caused the highest differences in the extreme platform-pitch 

motion. The same result occured for the ultimate tower-base bending moment, in which the case including the sumfrequency loads showed the highest moment.

0 0.05 0.1 0.15 0.2

10-1

100

101

102

103

104

Frequency [Hz]

Surge PSD [m2/Hz]

First-Order

Newman

Diff-Frequency

Sum-Frequency

Complete Second-Order

0 0.05 0.1 0.15 0.2

10-2

10-1

100

101

102

103

Frequency [Hz]

Pitch PSD [deg2/Hz]

First-Order

Newman

Diff-Frequency

Sum-Frequency

Complete Second-Order

0 0.05 0.1 0.15 0.2

100

101

102

103

104

105

Frequency [Hz]

Fair2Ten PSD [kN2/Hz]

First-Order

Newman

Diff-Frequency

Sum-Frequency

Complete Second-Order

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

105

106

107

108

109

1010

1011

Frequency [Hz]

Tower Base My PSD [kN2/Hz]

First-Order

Newman

Diff-Frequency

Sum-Frequency

Complete Second-Order
14 

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.

Figure 10. Ultimate loads for the different load cases studied. Top left: surge motion; top right: pitch motion; 

lower left: fairlead tension; lower right: tower-base moment.

C. Fatigue Analysis

Fatigue analysis was performed for load case 1.2. using MLife11. The lifetime damage results were normalized 

by the case without second-order loads. A value higher than one indicates an increase in the lifetime damage, as 

presented in Figure 11. Very small changes were found between the fatigue life predictions with and without 

second-order loads. The biggest changes occured for the tower-base bending moment with the full differencefrequency QTF. However, these changes still represent less than a 1% increase in fatigue life. One possible reason 

for the lack of difference is the length of the simulation. According to the IEC, each load case for fixed-bottom 

offshore wind turbines should be run for 10 minutes with different random seeds. However, this simulation length 

seems to be too short to capture the effects of the slow-drift motion of FOWTs, especially in platforms with very 

large surge/sway natural frequencies. An investigation of the sensitivity to the simulation length should be 

performed in a follow-up study.

IEC 1.1 IEC 1.6 IEC 2.1 0

5

10

15

20

25

30

35

Load case

Maximum Surge Displacement (m)

Without 2nd Order

Newman

Diff. Freq.

Sum Freq.

Complete 2nd Order

IEC 1.1 IEC 1.6 IEC 2.1 0

2

4

6

8

10

12

Load case

Maximum Pitch Displacement (o)

Without 2nd Order

Newman

Diff. Freq.

Sum Freq.

Complete 2nd Order

IEC 1.1 IEC 1.6 IEC 2.1 0

500

1000

1500

Load case

Maximum FairLead Tension (kN)

Without 2nd Order

Newman

Diff. Freq.

Sum Freq.

Complete 2nd Order

IEC 1.1 IEC 1.6 IEC 2.1 0

0.5

1

1.5

2

2.5 x 105

Load case

Maximum Tower Base Mxy (kNm)

Without 2nd Order

Newman

Diff. Freq.

Sum Freq.

Complete 2nd Order
15 

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.

Figure 11. Fatigue life damage normalized by the case without second-order effects.

V. Conclusions and Future Work

A computationally efficient method to determine the second-order wave loads on FOWT in the time-domain was 

presented in this paper. The method was implemented in MatLab and verified against the commercial code 

OrcaFlex. The results indicated very good agreement with significant time savings. 

A preliminary analysis of the influence of these second-order wave loads on the OC3-Hywind floating spar-type 

wind turbine was also performed. The spectral analysis revealed a significant increase of the surge and pitch motion 

of the floating platform below the first-order wave-frequency region. Significant differences were found between 

Newman’s approximation and the full QTF method, especially close to the pitch natural frequency, but no 

significant contribution was found from the sum-frequency loads. The effects of second-order wave loads were 

significantly reduced for the load cases with the active turbine because of the dominance of wind loads. The ultimate 

load analysis revealed minor influences of the difference-frequency loads, which occurred particularly for the 

extreme wave load case. The fatigue analysis did not show any significant effect from the second-order wave loads. 

The MatLab routine will soon be converted to Fortran and implemented directly within FAST, which will enable 

further analysis of FOWTs with second-order hydrodynamic loads. The spreading of the sea state across different 

wave directions (not discussed in this paper) is also being considered in this implementation. The simulation length 

used in this analysis should be revisited in a future study, as evidence has shown that 10-min simulations are not 

long enough to fully resolve the effects of the slow-drift second-order loads. Also, the importance of second-order 

wave loads on other floating platforms will be assessed; second-order loads are likely to be more important in 

semisubmersibles and TLPs.

References

1 J. Jonkman, "Definition of the Floating System for Phase IV of OC3," NREL, Golden, Colorado, 2010.

2 A. N. Robertson, J. M. Jonkman, A. J. Groupee, A. J. Coulling, I. Prowell, J. Browning, M. Masciola and P. 

Molta, "Summary of Conclusions and Recomendations drawn from the DeepCWind scaled floating offshore wind 

system test campaign," in Proceedings of the ASME 2013 32nd International Conference on Ocean, Offshore and 

Arctic Engineering, Nantes, France, 2013. 

3 J. Lucas, "UpWind Project: Comparison of First and Second-Order Hydrodynamic Results for Floating Offshore 

Wind Structures," Garrad Hassan & Partners Ltd., Bristol, UK, 2011.

4 NREL, "NWTC Computer-Aided Engineering Tools (FAST by Jason Jonkman, Ph.D.).," 02 11 2012. [Online]. 

Available: http://wind.nrel.gov/designcodes/simulators/fast/. [Accessed 12 11 2012].

5 R. S. Langley, "On the time domain simulation of second order wave forces and induced responses," Applied 

Ocean Research, vol. 8, no. 3, 1986. 

6 J. N. Newman, "Second-order, Slowly-varying Forces on Vessels in Irregular Waves," in Proceedings of 

International Symposium on Dynamics of Marine Vehicles and Structures in Waves., London, 1974. 

YawBrMMyz TwrBsMMxy Fair1Ten Fair2Ten Fair3Ten

0.9

0.95

1

1.05

1.1

Newman m=3

Newman m=4

Newman m=5

Diff m=3

Diff m=4

Diff m=5

Sum m=3

Sum m=4

Sum m=5

Second m=3

Second m=4

Second m=5
16 

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.

7 R. G. Standing, W. J. Brending and D. Wilson, "Recent Developments in the Analysis of Wave Drift Forces, 

Low-Frequency Damping and Response," in Proceedings of the 79th Annual OTC, Houston, Texas, 1987. 

8 Orcina, "OrcaFlex Manual - Version 9.6," 2012.

9 L. A. Roald, "The Effect of Second-Order Hydrodynamics on a Floating Offshore Wind Turbine," 

Eidgenössische Technische Hochschule Zürich-Master Thesis, Zürich, 2012.

10 J. Jonkman, "Dynamic Modeling and Loads Analysis of an Offshore Floating Wind Turbine," NREL, Golden, 

Colorado, 2007.

11 NWTC, "NWTC Design Codes (MLife by Greg Hayman)," 2012. [Online]. Available: 

http://wind.nrel.gov/designcodes/postprocessors/MLife/. [Accessed 2013].