Influence of Nonlinear Irregular Waves on the

Fatigue Loads of an Offshore Wind Turbine

Michiel B. van der Meulen1, Turaj Ashuri2, Gerard J.W. van Bussel3

and David P. Molenaar1

1 Offshore Center of Competence, Siemens, the Netherlands2 Department of Aerospace Engineering, University of Michigan, Ann Arbor, USA3 Faculty of Aerospace Engineering, Delft University of Technology, the Netherlands

E-mail: michiel van.der.meulen@siemens.com

Abstract. In order to make offshore wind power a cost effective solution that can competewith the traditional fossil energy sources, cost reductions on the expensive offshore supportstructures are required. One way to achieve this, is to reduce the uncertainty in wave loadcalculations by using a more advanced model for wave kinematics. As offshore wind turbinesare generally sited in shallow water, nonlinear effects which results in steeper waves with highervelocities and accelerations are common. Whereas extreme waves are modeled with higher-ordernonlinear regular wave models, fatigue loads are calculated from kinematics obtained by a low-fidelity linear irregular wave model. In this paper, a second-order wave model that is employedto simulate the dynamic response due to nonlinear irregular waves on a full set of IEC-standardload cases. This method is computationally efficient, which is particularly useful for designoptimization studies. It is shown that by using this method for a 25 m deep site in the GermanBight, equivalent fatigue loads increase by 7.5 % compared to the traditional linear wave model.The effect of nonlinear waves on fatigue is most prevalent in the foundation and tower partsnear the sea surface. Furthermore, it is found that the increase in fatigue damage accumulationis most prevalent in wind-wave misaligned load cases, in which aerodynamic damping is absent.

1. IntroductionGoing offshore and upscaling are the current technological trends in wind energy. Upscalingposes several fundamental design challenges mainly on the rotor [1], and going offshore makesin general the design more expensive than onshore wind turbines [2]. Cost reduction of thesupport structures has a fundamental importance, since it acts as a bottleneck to the realizationof offshore wind farms that can compete with traditional energy sources [3].

Currently, offshore wind farms are typically sited in coastal areas with water depths aroundor less than 30 m [4, 5]. For these water depths, the monopile foundation type is by far the mostpopular. Due to the limited water depth, nonlinear effects cause the waves to become moresharp-crested while the troughs are flattened. Besides that, the magnitude of particle velocitiesand accelerations below the waves are higher, which leads to higher hydrodynamic loads on thewind turbine support structure.

In the kinematic wave models used in offshore engineering, irregular waves are usuallyapproximated with classical linear (Airy) wave theory [6]. Since this method only describes wavekinematics up to the mean sea level, Wheeler stretching [7] is often applied to redistribute the

velocity and acceleration profiles up to the actual sea surface, but other extension and stretchingmethods exist and are sometimes used instead [8]. This traditional approach, based on deepwater experience from the oil and gas industry is accurate enough when the wave amplitude issmall with respect to water depth, but in shallow water kinematics magnitudes are likely to beunderestimated.

In order to be able to account for strong nonlinear effects in the highest waves that may occurduring the lifetime of an offshore structure, a nonlinear regular wave model such as the analytical5th-order Stokes method or a numerical Fourier approximation based on stream function theory iscommonly employed [9, 10]. This separate deterministic extreme wave is inserted in a stochasticwave record, such that in a aero-servo-elastic simulation of an extreme wave in the time-domain,the turbine dynamics due to an irregular sea state are included in the simulation.

Whereas in extreme wave events nonlinear effects are thus accounted for, a nonlinear modelfor irregular waves is uncommon in engineering practice. To compensate for the lack ofaccuracy of linear wave theory in shallow water, a safety factor is usually applied to obviatean underestimation of wave loads. Using a more accurate irregular wave model, the amountof uncertainty in fatigue load estimation could be lowered, which may lay the ground for adiscussion on the safety factor that is used. Altough nonlinear irregular waves are expectedto result in higher wave loads, the possible safety factor reduction might still result in a lowerstructural mass and hence reduced cost.

When it comes to modeling nonlinear irregular waves, there are several options. The second-order nonlinear irregular wave model developed by Sharma and Dean [11] is the one that hasreceived the most attention in similar research topics. In this model, which is a second-orderperturbation expansion of linear wave theory, the contribution of sum- and difference frequenciesare added to the first-order solution from linear wave theory. The Hybrid wave model byZhang [12] is a similar model with the addition of nonlinear effects due to short waves riding ontop of larger waves, which are empirically mixed with the results of second-order theory. Fullynonlinear waves can be modeled with more advanced Boussinesq models [13], but since these arevery computationally expensive, using this type of model is outside the scope of this research.Since the second-order wave model is mentioned as an alternative for linear wave theory inDNV Recommended Practice (DNV-RP-C205) [14], this is the model that will be used for thisresearch.

To this date, knowledge about the influence of nonlinear irregular waves on offshore windturbine fatigue loads is fairly limited. Several studies provided evidence that fatigue loadsdue to nonlinear irregular waves will increase [15, 16, 17]. A recent load extrapolation study byAgarwal [18] showed that long-term loads increase with approximately 10% when using nonlinearwaves. It must be remarked that this study was limited to two governing environmental statesonly, in which the significant wave height was high with respect to the water depth.

In general however, a large part of the wind turbine fatigue life is consumed during operationin mild conditions, in which nonlinear effects are much weaker. Therefore it may very well bethe case that the overall fatigue load increase is less than the mentioned 10%. However, suchan investigation of the influence of nonlinear waves on fatigue while taking into account a fullset of design load cases (IEC Standards 61400-3) [19], is not known by the author. This papertherefore aims at analyzing the influence of nonlinear irregular waves on response behavior andfatigue load for a complete set of load cases with a realistic design configuration, rather thaninspecting a single environmental state or using a simplified structural model. Additionally,since the method employs a frequency domain formulation, it is computationally very efficient.This trick together with some other useful techniques developed by the authors [20, 21] enablesdesign optimization studies where several thousand of function evaluations per optimizationiteration are required.

This paper is structured as follows: First, the methods required to generate time-series of

hydrodynamic loads on offshore wind turbines are presented. This comprises the procedure togenerate a random wave record, the theoretical formulation of linear and second-order wavetheory and the Morison equation to predict hydrodynamic loads from wave kinematics time-series. We then present the results of the second-order wave model simulations. The qualitativeeffect of the second-order contributions on a typical wave record and the spectral representationof the sea surface elevation and in-line bending moment are shown. The result of using nonlinearwave loads as input in the dynamic response simulation, which is carried out by the Siemenssoftware package BHawC in the time-domain, is then presented. In these simulations, a Siemenswind turbine from a wind farm in the German Bight, supported by a monopile foundation,is used. Next, we present the results of the calculated equivalent fatigue load and make acomparison with the outcome of simulations performed with the wave loads from the linearwave model. Finally, the paper is concluded by an analysis of the observations, conclusions andrecommendations for follow-up research.

2. MethodologyIn order to simulate the dynamic response due to the hydrodynamic forces acting on a monopilesupport structure, time-series of the hydrodynamic load for a number of vertical coordinates hasto be realized. Regardless of the type of model that is used for wave kinematics, the empiricalMorison equation is currently considered to be the most appropriate tool to create these time-series. In the Morison equation, the drag and inertia forces due to horizontal fluid particlevelocities and accelerations, U and U respectively, are added together to estimate the totalin-line wave force:

fMorison = fD + fI =1

2CdD |U |U + CmpiD

2

4U (1)

where and D denote the water density and structural diameter, respectively. The magnitude ofthe force components furthermore strongly depends on a proper selection of appropriate valuesfor the added inertia and drag force coefficients, Cm and Cd. In this formulation, structuralmotion is ignored, but the Morison equation can easily be extended to use the relative wavekinematics instead, see for example [18]. To account for diffraction effects, the MacCamy-Fuchscorrection can be applied as a low-pass filter on the acceleration terms [22]. Since this is acorrection to account for linear diffraction only, in this paper the MacCamy-Fuchs correction isassumed to be valid on first-order acceleration terms only.

In many occasions, a current due to tidal motion is to be taken into account. Since theinteraction between currents and wave kinematics is hard to predict and thus to model, a simpleassumption is often made to simply vectorially add a steady current velocity profile to thewave kinematics. Additionally, the Doppler shift that appears in the observed wave frequencyobserved from a stationary frame of reference, can be taken into account [23].

2.1. Simulation of linear and nonlinear irregular wavesThe common approach to simulate a random wave field is to take a high number of frequencycomponents, and use linear superposition to create an irregular wave record. This superpositionof random waves and the application of linear wave theory to predict wave kinematics is describedin many textbooks, for example [24, 25]. According to linear wave theory, the first-order surfaceelevation can be expressed as follows:

(1)(t) =

Nm=1

am cos(mt kmx m) (2)

Randomness in the simulations is obtained by uniformly distributing the phase angles mbetween 0 and 2pi. Besides that, the amplitudes am follow from the Rayleigh distributed

amplitude variances, which has an expected value that is obtained from the wave spectrum:E(a2m) = 2S(m). The wave number km is related to the angular frequency m through thelinear dispersion relation, 2m = gkm tanh(kmd), where d is the water depth below mean sealevel.

To ensure randomness, a high number of frequencies, typically more than 200, is desirable [25].The maximum or cut-off frequency of the simulations, if not bounded by the Nyquist samplingcriterion, can be assumed four times the peak frequency of the wave spectrum [26]. Using thiscut-off frequency, a good compromise between frequency discretization density and simulationbandwidth is achieved.

According to classical linear wave theory, the 1st-order velocity potential that corresponds tothe surface elevation given in Eq. 2 reads:

(1)(z, t) =

Nm=1

bmcosh km(z + d)

cosh kmdsin(mt kmx m) (3)

where bm are the amplitude coefficients given by:

bm =amg

m(4)

In the reference system used in these expressions, x is positive in the propagation directionof the waves. For a monopile support structure, x can simply be assumed zero. The verticalcoordinate z is positive upward and is zero at mean sea level. By deriving the velocity potential,expressions for the first-order horizontal velocity and acceleration can be obtained :

U (1)(z, t) =(1)

x, U (1)(z, t) =

t

((1)

x

)(5)

For the second-order nonlinear irregular wave model by Sharma and Dean [11], similarexpressions can be derived for the sea surface elevation and kinematics. The second-orderaccurate sea surface elevation, being a perturbation expansion of the first-order formulation,reads:

(2)(t) = (1) + (2) (6)

Here, (2) = (2+)+(2) is the second-order perturbation, which comprises the difference-and sum frequency corrections given by:

(2)(t) =

Nm=1

Nn=1

[aman

{Bmn cos(m n) +B+mn cos(m + n)

}](7)

The expressions for the transfer functions of the 2nd-order amplitude, Bmn and B+mn, are lengthyand are therefore given in the Appendix section. Furthermore, m and n are short notationsof the argument in the cosine in Eq. 2:

m = mt kmx m (8)The second-order difference- and sum velocity potential that corresponds to the surface elevationperturbations from Eq. 7 read:

(2)(z, t) =1

4

Nm=1

Nn=1

[bmbn

cosh kmn(z + d)cosh kmnd

Dmn(m n) sin(m n)

](9)

In a similar fashion as for the first-order kinematics, second-order perturbation contributionscan be obtained by deriving the above velocity potential.

2.2. Frequency-domain formulationThe linear and second-order nonlinear irregular wave models are formulated in the time-domain.As a large number of waves is to be superimposed, especially the double-summations in thesecond-order model will be time consuming. To avoid the numerical inefficiency of performingsummations in the time-domain, a better approach is to carry out the calculations in thefrequency-domain and subsequently use the Inverse Fast Fourier Transform (IFFT) to realize atime series.

This approach based on the Discrete Fourier Transform requires that the frequencycomponents are equally spaced, such that m = m, where m = 1, 2, . . . , N . For each

frequency component m, the Fourier coefficients for the first-order surface elevation, X(1),m(),

are then calculated as follows:

X(1),m(m) = am exp(im) (10)The IFFT can then be employed to create a time-series from the Fourier coefficientscorresponding to the frequency samples m:

(1)(tp) =