mooring response of a floating offshore wind turbine in ... waves are used for the sake of...

Journal of Ocean and Wind Energy (ISSN 2310-3604) http://www.isope.org/publicationsCopyright © by The International Society of Offshore and Polar EngineersVol. 2, No. 1, February 2015, pp. 37–44

Mooring Response of a Floating Offshore Wind Turbine inNonlinear Irregular Waves

Christof WehmeyerRambøll Offshore Wind

Esbjerg, Denmark

Jørgen Hvenekær RasmussenDONG Energy

Fredericia, Denmark

The current work focuses on the mooring loads obtained by a numerical model of a floating offshore wind turbine (FOWT)foundation, where the substructure is a three-legged Tension Leg Platform (TLP) and the influences of a flexible topside areconsidered. Representatively, the behaviour of one tendon is investigated. The FOWT is represented by a hybrid model thatcombines linear inertial excitation forces, obtained by potential flow, with nonlinear hydrodynamic viscous drag forces. Thelatter are obtained from the relative velocities between the linear and nonlinear sea state kinematics and the floater. Thenonlinear sea state surface elevation and kinematics are generated by a second-order wave model including all sub- andsuper-harmonics, with and without an embedded Stream-function wave. The numerical responses of the bow tendon arecompared to respective observations from model tests. The maximum bow tendon loads are overpredicted by the numericalmodel by 34% and 32% for the nonlinear and linear sea state generation, respectively. The embedded wave approach resultsin an overprediction of 37% to 23%, depending on the different crest front steepness values of the single wave, which isembedded in the nonlinear irregular background sea state.

INTRODUCTION

The focus of location choices for the possible deployment offloating offshore wind turbine (FOWT) foundations has shiftedlately. Until recently, the industry anticipated that the majority ofthe market will be in deep water areas outside of Europe, in waterdepths > 100 meters. However, a number of research and prototypeprojects have been initiated to realize installation in the deeperareas of the European continental shelf sea, i.e., in intermediatewater depths ≤ 60 meters. A recent example is the UK CrownEstate’s demonstrator project, which targets a commercially viableFOWT solution by 2020, i.e., a Tension Leg Platform (TLP) designin approximately 60 meters water depth and with significant waveheights of more than 10 meters. At such water depths, a linear waveassumption insufficiently describes realistic wave shapes, wavekinematics, and thereby wave force in the design conditions, whichdifferentiates FOWTs from earlier floater applications. Additionally,FOWTs are commonly composed of slender parts at the freesurface and voluminous submerged parts. The submerged parts areprone to excitation by diffractive wave forces in large and smallsea states. However, slender surface piercing parts are excited byhydrodynamic viscous drag in large sea states. Another importantaspect is that traditionally floating structures have been consideredrigid; the influence of their structural dynamics on the globalsystem behaviour was negligible. However, the high flexibility ofFOWT topsides including the rotor nacelle assembly (RNA) andtower changes the global system’s response, especially for TLPs(Matha, 2009). Therefore, the current work attempts to assess ifa key response of a FOWT located in intermediate water depths

Received August 21, 2014; revised manuscript received by the editorsOctober 16, 2014. The original version was submitted directly to theJournal.

KEY WORDS: Tension leg platform, second-order sea state, embeddedStream-function wave, floating wind turbine, mooring loads.

can be reasonably well represented by a hybrid model, whichconsiders a dynamically sensitive topside and combines linearradiation damping, linear inertial excitation, and quadratic viscousexcitation from linear and nonlinear waves.

The second-order inertial effects caused by the sum and differencefrequency interaction of the wave on FOWT TLP structures, e.g., asdiscussed in Bachynski and Moan (2013) and Roald et al. (2013),are not included in the current stage. The challenge is that thefirst-order hydrodynamic coefficients can be obtained solely fromthe structure’s geometry, i.e., the first-order wave forces are derivedbased on the assumption of a stationary floating body; however, thesecond-order wave forces are derived based on the assumptionthat the body is allowed to move and respond to the first-orderwave forces. This implies that the first-order body motions areneeded for the computation of the second-order fluid forces. For aFOWT TLP in particular, the rigid body assumption considered inthe potential flow theory does not hold. This was first highlightedby Matha (2009) and later supported by physical model tests andrespective numerical representations of natural frequencies fordifferent topside flexibilities in Wehmeyer et al. (2013). Furtherwork on this is ongoing; however, the current stage of the numericalmodel includes only higher-order viscous terms.

Recently, several efforts have been undertaken to investigatethe effect of higher-order wave impact on offshore wind turbines.Agarwal and Manuel (2010) investigated the effect of a second-orderirregular wave model on a bottom fixed structure, i.e., a monopile.Wheeler stretching was applied to determine the kinematics up tothe free surface. Unsurprisingly, they showed that the second-ordermodel predicted higher loads than a first-order model. Morison’sapproach was applied to determine the wave excitation force on a 6-meter monopile in 20 meters water depth, which supported a 5MWreference turbine. An increase of 2% and 18% of the ten-minutemaximum overturning moment at the mud line was found forsignificant wave heights of Hm0 = 505 meters and Hm0 = 705 meters,

38 Mooring Response of a Floating Offshore Wind Turbine in Nonlinear Irregular Waves

respectively. On the basis of this information, it can be concludedthat the maximum fore-aft tower bending moment was due to theviscous drag. The maximum values occurred simultaneously withthe wave crest passing the structure. It is known that second-orderirregular wave models predict more realistic crest height distributions(Forristall, 2000), which means higher individual wave crests andconsequently more realistic and higher viscous contributions abovethe still water level, especially for large sea states. It is thereforeassumed that accurate viscous force contributions are an importantpart of the incident wave excitation for the investigated FOWT andshould be considered accordingly.

METHODS

The DNV-OS-J103 guideline (2013) recommends two approachesfor modelling the ultimate limit state (ULS) design conditions inshallow water or in conditions where waves become steep:

(A) Second-order irregular waves(B) Embedding a Stream-function wave into an irregular Airy

wave modelThe overall purpose of the current work is to obtain insight

into the applicability of options (A) and (B) in terms of FOWTbehaviour in ULS conditions. The current work, however, modifiesoption B and embeds the Stream-function wave into a second-ordernonlinear irregular wave in order to better match the overall crestdistribution, as outlined above. The general purpose of embedding aparticular wave is to control the largest wave in a design process, inwhich a defined maximum wave needs to be considered. First-orderirregular waves are used for the sake of comparison.

While it is known that both approaches, strictly speaking, violatethe linear theory assumption, it is widely accepted that the linearequation of motion for floating bodies can be extended by mildlynonlinear terms and can be practically applied to a wide rangeof engineering challenges. In the current work the applicabilityis evaluated, comparing numerical results to measurements of akey parameter, i.e., the loads experienced by the bow mooring ofthe FOWT TLP, obtained by a physical model testing campaign(Wehmeyer et al., 2013). The numerical response quality of option(A) is evaluated by spectral comparisons and by the probabilityof exceedance. The numerical response quality of option (B) isevaluated by the running average results of a consecutive numberof short sea states, in which a Stream-function wave is embedded.

While the wind loads on the tower and the wind loads on theblades during fault are important in the design of offshore windturbines, the physical model and the numerical model neglect thewind loads. The primary focus of the current work is to describeand understand the responses due to the hydrodynamic excitation.The topside inertia of the mass of the tower and the RNA is basedon the well-known National Renewable Energy Laboratory (NREL)5MW turbine (Jonkman et al., 2009).

Numerical Model

The numerical model is an integrated five degrees of freedom(5DOF) model of a rigid body floating substructure connected to asingle linear elastic Bernoulli beam element, which represents theflexibility of the wind turbine tower and includes a lumped masson top that represents the RNA. Figure 1 shows the numericalrepresentation of the investigated system, which illustrates theincident wave direction and the consequent definition of the bowtendon (cf. Wehmeyer et al., 2014). TB is the tower bottom, i.e.,the connection point between the floater and the idealised windturbine, COG is the centre of gravity of the substructure, and COis the origin of the mooring attachment points. The small letter q

Fig. 1 Left: Numerical representation of investigated FOWT TLP,showing incident wave direction with blue arrow. Right: 3D viewof FOWT TLP, defining the bow tendon and the incident wavedirection.

indicates the degrees of freedom by its indices. The shaded areaindicates the flexible topside.

The assumed rigid substructure and the flexible topside arecoupled through their shared degrees of freedom. The idealizedstructure has lateral symmetry in the x-z plane, which consequentlymeans that surge, heave, and pitch are decoupled from sway, roll,and yaw, and hence a 2D system is assumed to be sufficient. Thestructural masses, the tower’s elasticity, and the structural dampingare considered according to the physical model. The hydrodynamiccoefficients for the added mass, hydrostatic stiffness, linear diffrac-tive wave excitation, and linear radiation damping for the rigid bodysubstructure are determined by use of the commercial potentialtheory solver, ANSYS AQWA (2013). The model includes thefirst-order inertial excitation forces (Froude-Krylov and diffraction)and the viscous hydrodynamic drag excitation force (the drag partof Morison’s equation), applying the relative velocities between thestructure and the wave particle velocities up to the free surface.The radiation damping force is approximated by a state-spacerepresentation. The respective A, B, C, and D matrices of the fluidmemory effect and the added mass at infinity are determined bya direct frequency domain identification procedure that appliesthe MSS FDI MATLAB toolbox (Perez and Fossen, 2009). Thenumerical model is consequently a time domain solver of theequation of motion, as given by Eq. 1:

4Mtower +Mij +mh�

ij 5xi (1)

+

∫ t

0C rad

ij 4t − �5xi4�5d�

+ 4Khydij +K tower

ij 5xi

Journal of Ocean and Wind Energy, Vol. 2, No. 1, February 2015, pp. 37–44 39

+ FMooringij +C tower

ij xi

= F Exc_inertiali + F

Exc_viscous dragi

The left-hand side describes the body motion by:• 4Mtower +Mij +mh�

ij 5= Combined mass matrix, consisting ofthe topside mass matrix, the substructure mass matrix, and theconstant infinite frequency added mass

•∫ t

0 Cradij 4t− �5xi4�5d� = Fluid memory effect or retardation

function• 4K

hydij +K tower

ij 5= Combined stiffness matrix, consisting of thehydrostatic restoring matrix and the stiffness of the topside

• FMooringij = Station keeping restoring force vector.

• C towerij = Damping matrix of the beam element representing the

towerThe right-hand side describes the excitation force by:• F Exc_inertial

i + FExc_viscous dragi = Wave excitation force vectors

consisting of the inertial Froude-Krylov and diffraction force andthe relative viscous drag part of Morison’s equation

The equation of motion is solved at the interface point betweenthe floater and the tower in the initial position of the structureby applying the ODE45 routine in MATLAB (2011). Individualtransformation matrices are applied in order to transform all forcesfrom their respective point of attack to the interface point. Thenumerical model is further described in Wehmeyer et al. (2014),where the response model is validated by a decay test and theapplied drag coefficients, which agree well with commonly usedvalues for cylindrical and angular structures, are reported. For thesake of orientation, the natural frequencies of the structure appliedin the current work are given in model scale in Table 1.

It needs to be noted that the original TLP design was made for100 meters water depth, which was realized in the physical modeltest by a deep section located in the middle of the test basin. Inorder to avoid wave refraction, the deep section was covered bya plate with holes leaving just enough space for the tendons tomove freely. The resulting prototype water depth was 56 meters,which resulted in nonlinear storm event waves. This circumstanceenabled the current investigation of the nonlinear wave impact onthe FOWT TLP.

Irregular Sea State

The second-order transfer functions for the generation of thenonlinear irregular waves are based on the complete theory bySchäffer (1996). The derivation and validation of the velocitypotentials and respective expressions for the wave kinematics areshown in the Appendix. Hu and Zhao (1993) presented an empiricallimit up to which the second-order wave model is valid, as givenby Eq. 2:

��

�P

< 0002 (2)

where �� is the standard deviation of the surface elevation and �P isthe wave length of the peak period wave. The numerical realizationof the numerical linear and nonlinear sea states uses Hm0 and TPvalues according to the mean of the measured sea state parameters(Wehmeyer et al., 2013), which resulted in ��/�P = 00011.

Surge [Hz] Heave [Hz] Pitch [Hz]

0.37 2.49 1.92

Table 1 Global natural frequencies of the investigated structurefor a water depth of around 100 meters

Fig. 2 Normalized wave crest distribution from observation, first-and second-order irregular sea state realizations, and respectivedistribution models

An extensive description of the physical model tests can befound in Wehmeyer et al. (2013). As stated before, the numericalwater depth is taken according to the physical wave generationwater depth. Three JONSWAP spectra with constant Hm0 and TPbut with varying � values of 1.25, 2.50, and 4.48 are used tosimulate the numerical sea states in accordance with the physicalwave generation. The wave elevation and kinematics for linear andnonlinear realizations are computed by using a random phase andrandomized frequency approach. 26 frequencies are used and 20 timeseries of 60 seconds each are computed and composed into a seastate, thus representing three hours in a model scale of 1:80. Theprocedure is repeated three times for each of the three � values, thusresulting in nine linear sea states and nine nonlinear sea states. Thismay not be entirely statistically stable; however, the total numberof sea states is limited by time restrictions. In Fig. 2, a wave crestdistribution from the tank test is compared to one of the first-orderwave realizations and to one of the second-order wave realizationsand is compared against their respective distribution models. Byvisual inspection, the second-order distribution compares better toobservations and the distribution given by Forristall (2000) than thelinear distribution. The wave particle velocities are stretched to theinstantaneous water surface by Wheeler stretching. A sensitivityanalysis of the effect of different stretching approaches is notcarried out in the current work.

Embedded Stream-Function Wave

It is common practice in the offshore wind industry to generatestorm time series with a shorter duration than three hours. In orderto ensure that the design wave and its nonlinearity are covered, thehighest wave of the random sea state is replaced by the nonlineardesign wave (see Fig. 3). The single design wave is modeled by aStream-function wave (Fenton, 1988) and subsequently embeddedinto a reduced irregular time series, typically around 300 seconds.

The transition between the random sea and the Stream-functionwave is defined by a sine-cosine transition. In the current work,1.5 times the wavelength of the design wave is embedded in theirregular wave train in order to ensure a representation of the wavetroughs. 0.25 times the wavelength of the total embedded waveis chosen as the overlap, which defines the transition region. Asmooth transition between the background sea state and the Stream-function wave is ensured by a multiplication with sin405 to sin4�5and cos405 to cos4�5 of the respective elevation and kinematics in


Fig. 3 Example of a Stream-function wave embedded into a second-order irregular sea state. �0 is the first-order irregular sea state,�0 &�21− &�21+ is the second-order sea state composed of thefirst-order contribution and the difference and sum frequencycontributions. The red stars mark all crests and troughs, and thecoloured circles mark the crest and trough of the highest second-order random wave, which is replaced by the Stream-functionwave (SF) shown as a black line. The cyan and red lines mark thetransition between the original second-order background sea stateand the embedded Stream-function wave. The resulting wave timeseries is shown by a green dashed line.

the transition region. A sensitivity analysis of the overlap lengthis outside of the scope of the current work. It is assumed that itis more appropriate to embed the Stream-function wave in thesecond-order irregular wave train because, as argued above, adesign sea state should be represented by a higher-order model.This ensures more accurately the influence of the irregular sea stateon the maximum wave. Figure 3 shows an example of a generatedwave elevation where a Stream-function wave is embedded. Thenonlinear background sea state is shown in blue. For the sake ofcompleteness, the linear wave elevation is shown in magenta. Thehighest wave in the irregular sea state is defined by the circles, i.e.,the trough and crest. The crest defines the point where the Stream-function wave crest should be positioned during embedment. TheStream-function wave is shown in black. Red and cyan describe thetransitions. The final wave time series including the embedded waveis shown by the green dashed line. The Stream-function wave heightand period are defined by the respective highest wave measured inthe observation, with Hmax/Hm0 values given in Table 2.

Three irregular wave runs have been conducted in the physicalmodel test, one for each chosen peak enhancement factor 4�5,which resulted in three distinct maximum wave heights. In additionto the three-hour irregular sea states described above, twentynumerical sea states are generated for each � value, but now withthe respective maximum wave embedded. Each nonlinear irregularsea state with the embedded wave has a length of 270 seconds (inprototype scale). These three times twenty incident sea states resultin three times twenty maximum values of the bow tendon load,

� 1.25 2.5 4.48

Hmax/Hm0 1086 1.73 1063

Table 2 Hmax/Hm0 ratios from observations

Fig. 4 Normalized results and upscaled probability density functionof normalized bow tendon loads from second-order sea statesincluding an embedded Stream-function wave. Loads are normalizedby the maximum observed bow tendon load. The circle indicatesan area equal to 10−4.

which are used to calculate a running average in order to find aconvergence value (see Fig. 4).

RESULTS

The numerically obtained bow tendon loads serve as an indicatorof the numerical model performance. For each of the three � values,which have been used according to the physical sea state generation,the bow tendon load results are shown in Fig. 5 by a normalizedtime domain representation in the upper row and by a frequencydomain representation in the lower row. The observed tendon loadsignal is given in black. The numerical tendon load signal dueto a first-order incident sea state realization is given in magenta.The numerical tendon load signal due to a second-order incidentsea state realization is given in blue. No Stream-function wave isembedded at that stage, which means that the highest wave in eachnumerical sea state is random, as is the highest bow tendon load.

The time domain shows the probability of exceedance (POE)of peak tendon load ratios within one period, i.e., the maximumnumerically predicted load within one wave period is normalized bythe overall maximum observed load. The physical wave realizationsresulted in a marginal scatter of Hm0 and Tp values. For thenumerical realizations, constant Hm0 and Tp values (i.e., the meanof the measured values) are applied in order to assess individualsteepness effects at a later stage, which is not part of the scopeof the current work. The differences between the numerical andphysical sea states are considered by normalizing the tendon loadsby the square of the sea state steepness ratios. This means thatthe numerical loads increase where the numerical steepness isbelow the measured steepness and vice versa. The steepness ratiosare calculated based on the � value for independent steepnessapproximations for irregular sea states given by Eq. 3 (Forristall,2000):

S1 =2�g

Hm0

T 21

(3)

where T 21 is the square of the mean wave period in the spectrum.


Fig. 5 Upper row from left to right: Probability of exceedance of normalized bow tendon loads for � = 1025, � = 2050, and � = 4048, givenfor observations, first-order incident wave realization, and second-order incident wave realization. Loads are normalized by the maximumobserved bow tendon load and scaled to match the observed steepness. Lower row from left to right: Power spectra of bow tendon loads for� = 1025, � = 2050, and � = 4048, given for observations, first-order incident wave realization, and second-order incident wave realization.

It can be seen in Fig. 5 that the � value specific maximummeasured tendon load decreases with increasing � value. This isin line with the expectation that higher � values cause less steepwaves, which result in fewer overturning moments around the stillwater line. A nonzero value can be observed for a POE of 1, whichis due to the pretension. Between 90% and 95% of the tendon loadsignals are predicted satisfactorily by the numerical model. Theremaining 5% to 10% are overpredicted by the numerical model,where the mean ratio of the nine first-order load predictions isfound to be 1.32 and the mean ratio of the nine second-order loadpredictions is found to be 1.34 (see Table 3).

The frequency domain results of the bow tendon load (see thelower row in Fig. 5) show a satisfactory match in the range between0.5 Hz and 1.25 Hz in model scale, and the different spectralshapes due to the different � values are obvious. Pitch and heavenatural frequencies are easily identifiable; however, a distinct surgenatural frequency is neither seen in the physical model nor in thenumerical model. A significant lack of energy is observed between1.3 Hz and 2 Hz, i.e., close to the pitch natural frequency, whereasan excess of energy can be observed between 2 Hz and 2.7 Hz, i.e.,close to the heave natural frequency.

The maximum bow tendon loads obtained by the incidentembedded Stream-function wave show an expected scatter from the

� → 1.25 2.5 4.48 Mean Mean - SF Runs

Mean First Order 1.37 1.30 1.30 1.32 1.29Mean Second Order 1.33 1.32 1.37 1.34

Table 3 Mean values of maximum bow tendon load ratios foreach � over three runs; total mean value over all � and mean valuefrom Stream-function runs.

� → 1.25 2.5 4.48

Crest Front Steepness Values 0.233 0.218 0.184of Embedded Wave

Mean Value of Load Ratios 1.37 1.28 1.23

Table 4 Crest front steepness values of embedded waves comparedto converged average value of load ratios resulting from embeddedStream-function wave, for three � values

twenty � values for specific realizations. The embedded wavesmatch the maximum measured waves and show different crest frontsteepness values (Myrhaug and Kjeldsen, 1986). Being a directresponse to the maximum wave, the obtained load ratios, againnormalized by the overall maximum measured load, follow thetrend of the crest front steepness values (see Table 4). The totalmean value of the Stream-function runs, averaged over all � values,is 1.29 (see Table 3). No steepness scaling is applied for this case.The running average converges to the mean values presented inTable 4 after approximately 18 runs (see Fig. 5); however, a smalldeviation of the last value can still be observed, which indicatesthe need for more runs. The load ratios from all three � valueruns are used to fit a normal distribution and to determine thearea resembling 98% coverage, i.e., 10−4. The remaining 2% isindicated by the encircled red area (see Fig. 4).

DISCUSSION

A shortcoming of the embedded wave approach is that the mea-surements were actually made in front of the structure. Therefore,the measured waves do not exactly resemble the ones impacting onthe structure, which makes it difficult to define the exact wave that


is responsible for the highest measured impact. The most interestingload ratio value is therefore the one from the lowest � value, wherethe ratio between the measured Hm0 and measured Hmax is veryclose to current design requirements (IEC, 2009). In that case, themean load ratio converges towards 1.37 and is equal to the highestaverage ratio of the irregular sea state runs. For the remaining� values, the mean load ratios follow the crest front steepnessvalues, which is expected (see Table 4). The approach outlined byMyrhaug and Kjeldsen (1986) to determine the crest front steepnesswas originally intended to describe wave crest asymmetries aroundthe horizontal and vertical axes. A Stream-function wave is sym-metrical around the vertical axis; however, the reason for applyingthis approach was the strong asymmetry of a Stream-function wavearound the horizontal axis. Hence, the applied steepness descriptionconsiders the crest height, which is believed to be an importantparameter considering a structure response that is sensible to pitchmotions resulting from hydrodynamic drag up to the free surface.

A Gaussian distribution is fitted through the maximum bowtendon load ratio scatter originating from the embedded Stream-function waves. The ratio between the 98th percentile, defined byan area of 10−4 in the probability density function (PDF), and themeasured value generally determines the safety level. In this case,a ratio below 1 would be required (see Fig. 4), which highlightsthe degree of conservatism of the current approach.

The overall mean load ratio values, which result from the first-and second-order incident wave realizations, follow the expectedtrend, in which the second-order responses are slightly higher (seeTable 3). However, unexpected high values for � values of 1.25and 4.48 for the first- and second-order incident wave realizations,respectively, are found. This highlights the need for a higher numberof runs in order to obtain a higher degree of statistical stability.

The influences of the nonlinear incident wave realizations canbe seen only for the highest 5% to 10% of the bow tendon loadevents. This is expected, as the higher crests of the second-orderrealizations result in higher overturning moments. The spectralenergy contents of the numerical bow tendon loads do not showany distinct differences upon application of the linear and nonlinearincident wave realizations. This is not surprising, as the higher-orderwave model mainly influences the highest waves; its influence onthe spectral energy is marginal. The measured spectral energy peakaround 1.0 Hz (see the middle plot in the lower row in Fig. 5) isassumed to be due to a yaw motion, which cannot be captured bythe 2D numerical model. However, most of the observed differencesare in the high frequency region, and it is consequently believedthat the missing second-order sum frequency inertial forces causethe difference. This is in line with the findings of other authors andunderlines the importance of further model improvements, i.e., theinclusion of those higher-order inertial forces.

CONCLUSIONS

Not surprisingly, the higher-order irregular wave generationis well established, and good agreement can be seen betweenthe distributions of the numerical results and measurements. Theembedment of a Stream-function wave is a common industrypractice. It provides measures to control the highest wave in anotherwise random sea state, which is very beneficial for engineeringapplications. The embedment into a second-order background seastate is new, but results from the fact that a second-order sea stateis more realistic, considering the enhanced higher-order terms forlarge sea states. To the authors’ knowledge, it is the first time thatan embedded Stream-function wave is used as an incident wavefor a FOWT. The approach is certainly controversial due to theviolation of linear theory. However, the intention is not so much to

accurately depict the observed responses; rather, the intention ismuch more to assess how a somewhat simple numerical hybridmodel reacts if exposed to nonlinear wave elevations and kinematics.In the short-term future, the fundamentally new challenge ofinstalling dynamically sensitive FOWTs in limited water depths willrequire clarification of the applicability of time-efficient solutionsto respective design tasks. The current work intends to assess thepossibility of this approach and the degree of conservatism that thisapproach contains. The irregular incident wave runs overpredict themeasured maximum loads by 32% on average, where the nonlinearirregular wave runs yield more conservative load ratios. Fromthis, it can be concluded that the linear irregular incident wave issufficiently conservative even though the nonlinear incident wavemodel better resembles the surface elevation (and probably thewave kinematics). The embedded wave approach results in slightlylower overall averaged maximum loads, i.e., an overprediction of29%. Therefore, based on the current status, it is concluded that theembedded wave approach provides a controlled and time-efficientengineering tool for the investigated FOWT. However, optimizationsare required in order to reduce the degree of conservatism.

ACKNOWLEDGEMENTS

The Danish Ministry of Higher Education and Science has partlyfunded this work through the industrial Ph.D. program. The GlostenAssociates provided an early-stage PelaStar design, which was thegenesis of the TLP subject. Their contribution to the project isgratefully acknowledged. An invaluable contribution to the projectwas the access to the wave laboratory of Aalborg University andthe support provided by Aalborg University’s staff. The supportfrom the co-author is highly appreciated.

REFERENCES

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APPENDIX

The theoretical background for the computation of a nonlineartime series is given in Schäffer (1994) and later on in Skourup andSterndorff (2002). However, the final equations are not given andare therefore presented below as they have been implemented.

Surface Elevation

It is assumed that the nonlinear surface elevation can be expressedas a summation of the first- and second-order components. Hence,the surface elevation is expressed as:

�415+425= �415

+�42+5+�42−5

The superscripts indicate the well-known first-order solution (1)and the second-order solution 42+5 and 42−5. The two latterexpressions thereby indicate that the second-order solution is asuperposition of a difference frequency term (i.e., sub-harmonics inthe following superscript “−”) and a sum frequency term (i.e.,super-harmonics in the following superscript “+”).

The first-order wave elevation is given in complex notation by:

�4154x1 t5=

N∑

n=1

0054Anei4�nt−knx55

with

An = anei4�n5

an =√

2S4�n5ã�

whereAn = complex amplitude of the nth wavean = frequency-respective linear wave amplitude, determined from

a spectrumS4�n5= spectral energy density�n = random phase angle

�n = wave frequencykn = wave numberi = imaginary unitc.c. = complex conjugate

The second-order wave elevation for the sub- and super-harmonicsis given in complex notation by:

�42∓54x1 t5=

N∑

n=1

N∑

m=1

0054G∓

nmAnA∓

mei4�n∓�m5

+ c0c05

with

A∓

m = complex amplitude of the mth wave

4�n ∓ �m5= 4�n ∓�m5t − 4kn ∓ kmn5x

and the following coefficients:

G∓

nm =�mn

g

{

4�n ∓�m5H∓

nm

D∓nm

−L∓

nm

}

H∓

nm = 4�n ∓�m5

(

∓�n�m −g2knkm�n�m

)

+�3

n ∓�3m

2−

g2

2

(

k2n

�n

∓k2m

�m

)

D∓

nm = g4kn ∓ km5 tanh4kn ∓ km5h− 4�n ∓�m52

L∓

nm =12

{

g2knkm�n�m

±�n�m − 4�2n +�2

m5

}

Kinematics

The kinematics and pressures are well-known derivatives of thevelocity potential. Following the approach for the surface elevation,it is assumed that, based on a spectral wave description, all relevantwave parameters can be expressed as a summation of the first- andsecond-order terms of a velocity potential � and its derivativesaccording to:

�415+425= �415

+�42+5+�42−5

The first-order velocity potential is given in complex notation by:

�4154x1 z1 t5=

N∑

n=1

005(

igAn

�n

cosh4kn4z+h55

cosh4knh5ei4�nt−knx5 + c.c.

)

Similar to the surface elevation, the second-order velocity potentialneeds to have two contributions. These are obtained by a summationof two first-order components and the difference of the two first-order components as follows:

�42∓54x1 z1 t5

=

N∑

n=1

N∑

m=1

0054i�mnF∓

nmAnA∓

m

cosh44kn ∓ km54z+h55

cosh44kn ∓ km5h5

× ei4�n∓�m5+ c.c.5

with

�mn =

121 n=m

11 n 6=m

The coefficient F is given by:

F ∓

nm =H∓

nm/D∓

nm


For the sub-harmonics, it is seen that D∓nm tends to zero and hence

would cause F ∓nm to tend to infinity. It is assumed to be appropriate

to step back to the Laplace equation and boundary conditions forthis particular case, to solve these under the assumption that thereis no oscillation, and hence to obtain a solution that involves acurrent and a change in sea level. Both contributions are includedin the calculations according to Stokes’ second-order theory.

Particle velocities and accelerations to the first order can beobtained as derivatives of the velocity potential as follows:

¡�

¡x

415

= u415

=

N∑

n=1

005(

gknAn

�n

cosh4kn4z+h55


)

¡�

¡z

415

=w415

=

N∑

n=1

005(

igknAn

�n

sinh4kn4z+h55


)

¡u

¡t

415

= u415

=

N∑

n=1

005(

igknAn

cosh4kn4z+h55


)

¡w

¡t

415

= w415

=

N∑

n=1

005(

−gknAn

sinh4kn4z+h55


)

Wave-induced excess pressure to the first order is given by:

p415

�= −

¡�415

¡t

with

¡�

¡t

415

=

N∑

n=1

005(

−gAn

cosh4kn4z+h55


)

Particle velocities and accelerations to the second order can be

derived accordingly as:

¡�

¡x

42+5

= u42+5=

N∑

n=1

N∑

m=1

005(

�mnF+

nmAnA+

m4kn + km5

×cosh44kn + km54z+h55

cosh44kn + km5h5ei4�n+�n5 + c.c.

)

¡�

¡x

42−5

= u42−5=

N∑

n=1

N∑

m=1

005(

�mnF−

nmAnA−

m4kn − km5

×cosh44kn − km54z+h55

cosh44kn − km5h5ei4�n−�n5 + c.c.

)

¡�

¡z

42+5

=w42+5=

N∑

n=1

N∑

m=1

005(

i�mnF+

nmAnA+

m4kn + km5

×sinh44kn + km54z+h55


)

¡�

¡z

42−5

=w42−5=

N∑

n=1

N∑

m=1

005(

i�mnF−

nmAnA−

m4kn − km5

×sinh44kn − km54z+h55


)

¡u

¡t

42+5

= u42+5=

N∑

n=1

N∑

m=1

005(

i�mnF+

nmAnA+

m4kn + km54�n +�m5



)

¡u

¡t

42−5

= u42−5=

N∑

n=1

N∑

m=1

005(

i�mnF−

nmAnA−

m4kn − km54�n −�m5



)

¡w

¡t

42+5

= w42+5=

N∑

n=1

N∑

m=1

005(

−�mnF+

nmAnA+

m4kn + km54�n +�m5



)

¡w

¡t

42−5

= w42−5=

N∑

n=1

N∑

m=1

005(

−�mnF−

nmAnA−

m4kn − km54�n −�m5


cosh44kn + km5h5ei4�n−�n5 + c.c.

)

mooring response of a floating offshore wind turbine in ... waves are used for the sake of...

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