Quick assessment of semi-submersible floating offshore wind turbines under misaligned wind, wind-wave and swell-wave loading

Computer-intensive coupled time-domain simulations are used by both academia and industry to capture the nonlinear behavior of floating offshore wind turbines. To assist in pre-design and optimization, computationally efficient methods such as frequency-domain methods are required. However, state-of-the-art frequency-domain methods consider simplified cases which do not represent operational design cases of industry projects. This paper compares the effectiveness of a frequency-domain method based on Response Amplitudes of Floating Turbines (developed by the NREL) to time-domain simulations carried out in BHawC-OrcaFlex by Siemens Gamesa Renewable Energy. The load cases are part of IEC Design Load Case 1.2 (fatigue in normal power production) and are selected from an industry project. Reasonable agreement is found for the pitch/roll rigid body modes around rated wind speed. However, the mean response and dynamic response is overestimated by the frequency-domain method due to underestimation of the mooring stiffness and the underestimation of hydrodynamic damping. For the estimation of the fore-aft and side-to-side bending moment, main differences are identified due to the absence of the tower bending, excitation due to rotor rotation and second-order wave forcing in the frequency-domain method. The results show the need for extending the frequency-domain method with tower bending and rotor rotation as degrees of freedom.


Introduction
The dynamic response of a floating offshore wind turbine (FOWT) contains many nonlinearities.These include the geometric nonlinearity of the mooring system, the hydrodynamic viscous drag and the aeroelastic modelling of the turbine [1,2].To account for the resulting coupled dynamics, time-domain (TD) methods for the design process of FOWTs have been developed and verified [3].
These fully coupled nonlinear simulations may require extensive calculations when evaluated for the required Design Load Cases (DLCs) (presented in IEC 61400-3) [4].This can amount to about 20000 simulations with a processing speed of 1-3 times real time [5] and about 6-8 weeks of computation time for a single design iteration [6,7].This considerable simulation time is a constraint for research and leads to increased development costs if many iterations of the floater design of offshore wind turbines are performed.Therefore, research is performed to capture this non-linear behavior in a computationally efficient method.One of the ways to reduce the computation time for fatigue damage estimation is to use a surrogate model, such as neural networks [8].However, these methods use a black-box approach and therefore do not contribute to a better understanding of the physics in such a system.Furthermore, obtaining a data set to train and test the model would require running many fully-coupled simulations and would limit the application designs within the trained data set.
Another approach is to estimate the response of the FOWT in the frequency domain (FD).Using a FD-method requires linearisation of the non-linear loads and response characteristics around an operating point.Recent FD-methods are developed focusing on different specific applications.Each of these models uses linearisation techniques and degrees of freedom (DOF) relevant for the specific application.
Considerable simplifications are assumed in the method by Lupton [9].The hydrodynamic excitation and response are estimated using only the Morison equation.This limits the application to relatively small and slender structures, since diffraction and radiation effects occur for large structures [10].
In addition, there are developments for the QuLAF method (Quick Load Analysis of Floating wind turbines) [5,11], a 4 DOF model (floater surge, heave and pitching and the first fore-aft tower bending mode).This method only considers four DOFs and is therefore not suitable for considering misaligned wave and wind cases.One of the recommendations is to add out-of-plane DOF to include calculations for misaligned wind and waves [5].The required DOF for such an extended method have not been presented in the research.
The method presented by Hegseth and Bachynski [12] is developed for spar-type floaters.They propose a method with seven DOF, namely rotor speed, surge, sway, roll, pitch and 1st fore-aft and side-side bending modes.However, the axial symmetry of a semi-submersible type FOWT requires additional DOF to capture the resulting coupling effects under misaligned wind and waves observed in hindcast.
Finally, the National Renewable Energy Laboratory (NREL) has developed Response Amplitudes of Floating Turbines (RAFT) [13].This open-source method considers the six floater rigid body modes as DOF.Therefore, it can be applied to estimate the coupled response under non-zero wave heading.However, it is limited to only one wave component and only estimation of the fore-aft bending moment.Several extensions to the code have been applied to allow for modeling non-zero wind heading, including multiple wave systems and allow for spectral analysis of the equivalent moment using a rainflow-counting equivalent approach.
This paper presents a comparison of a frequency-domain method and a time-domain method.It shows the need for extending frequency-domain methods with improved linearization of the mooring system, including the first-tower mode as a DOF and using a full numerical approach for the hydrodynamic added mass and damping.
The following sections detail an overview of the method and a comparison against timedomain simulations performed for an industry project by Siemens Gamesa Renewable Energy (SGRE).Similarities and differences between the FD-method and TD-method are described and improvements to the method are presented.

Method set-up
This sections briefly describes the FD-method based on RAFT.A detailed description of the method is presented by Hall et al. [13].The general equation of motion for a system is given by Equation 1.In these equations ⃗ ξ(t) represents the response for the 6 floater DOF (surge, sway, heave, roll, pitch and yaw).
If the system is linearized around an operating point, this can be presented in the frequency domain using Equation 2.
The various matrices for mass (M), added mass (A), damping (B) and stiffness (C), presented in Equation 2, are the linearized system properties of the terms presented in Equation 1.Each matrix (indicated in bold) consists of 6-by-6 coefficients, considering the 6 floater DOFs.
The structural mass matrix M struc consists of diagonal terms for the mass and inertia of the structure.In RAFT a hybrid numerical/analytical approach is used for the hydrodynamic added mass.For submerged members the response can be estimated using a potential flow solver A rad or estimate the added mass/inertia using the Morison equation A Hydro,morison .For the members which are evaluated using potential flow, the radiation damping is captured in B rad .If it is decided to evaluate a column/pontoon/brace using the potential flow software (numerical) or Morison equation (analytical), the excitation ⃗ F BEM / ⃗ F Hydro,Inertia is evaluated using that approach too.In the analysis the the vertical columns are estimated using potential-flow and the connecting pontoons modelled as Morison members.
The hydrodynamic viscous damping B Hydro,drag and viscous drag excitation force ⃗ F Hydro,drag are estimated using an iterative approach.In this approach, the hydrodynamic drag is estimated from the relative velocity between the wave particle and the structure [9,14].
The aerodynamic excitation and damping are estimated from predefined blade profiles and operational settings (rpm and pitch setting at the operational wind speeds).The aerodynamic loads and derivatives at the nacelle are obtained from CCBlade [15].The aerodynamic damping B aero is assumed constant and equal to δT δv , which is derived from CCBlade at the mean wind speed.
The structural stiffness C struc is the destabilizing term (p-delta) of the system.C hydro is the stabilizing term that includes contribution of the waterplane area and buoyancy of the submerged structure.The mooring stiffness is estimated using the MoorPy package [16] and uses a quasi-static approach where the mean response is determined by the mean thrust on the rotor at that mean wind speed.
The time-domain simulations are performed using BHawC and Orcaflex.BHawC is the aero-servo-elastic software developed by Siemens Gamesa Renewable Energy (SGRE).OrcaFlex (developed by Orcina) is used to estimate the hydrodynamic, structural and mooring response.The coupling of the programs is described in more detail by Arramounet et al. [17].

Unloaded System Description
The considered floating offshore wind turbine configuration is the system developed for New England Aqua Ventus I (NEAV1).This system consists of a SG 11.0-200 DD turbine and SGRE tower mounted on the VolturnUS concrete floater.For the considered system, free-decay analysis have been performed in BHawC-Orcaflex to estimate the natural frequency of the system.In RAFT, the natural frequencies are calculated for the unloaded system.A comparison between the estimated natural frequencies is presented in Table 1.The heave, pitch and roll natural frequencies are estimated reasonably well by the FD-method.
However, the underestimation of the natural frequencies of surge, sway and yaw is very noticeable.These underestimations are due to an underestimation of the mooring stiffness as no reasonable estimation of the quasi-static stiffness of the complex mooring system could be achieved, as is explained in more detail in Section 5.The configuration used in the time-domain simulation consisted of two chain types being connected which could not be modelled by MoorPy at the time of writing.The influence of the mooring system and the total stiffness on the heave, roll and pitch motion is limited as the stiffness for these modes is dominated by hydrostatic stiffness and therefore show a comparable response.

Load case description and excitation spectra comparison
Two cases are selected from the DLC 1.2 load case set performed for the industry project (Table 2).These cases are selected to investigate the performance of the method below and above rated wind speed and for small and large misalignment angles.An illustration of the wind and wave headings with respect to the floater orientation is given in Figure 1.The time-domain signal of the simulations performed by Siemens Gamesa are limited to 600 seconds per load case.To compare the low frequencies it is assumed that at least 10 oscillations of 60 seconds should have been completed.The lower bound of the frequency for this comparison is therefore set to 1/60 ≈ 0.01667 Hz.It is acknowledged that some natural frequencies of the system are lower than this set lower bound.However, the natural frequencies of roll and pitch (main tower base load driving modes) are above this lower bound and therefore this is considered a reasonable assumption.From Figure 2 and the standard deviations Table 3 it can be observed that there is a reasonable agreement between the TD-and FD-method for the wind turbulence and wave spectra.

Mean response comparison
To compare the performance of the thrust estimation, mooring stiffness and hydrostatic stiffness estimation, the mean response of the affected modes is compared for the two load cases.Figure 1 shows that the wind approaches from 292.50 • with respect to the floater reference frame.This means mostly propagating in negative y-direction and slightly positive x-direction with respect to the floater reference frame.From Table 4 it can be observed that the mean surge and sway response is overestimated by 200 % and 74 % respectively for both cases.However, the difference between estimated mean response for roll and pitch is smaller.The roll and pitch natural frequencies were estimated within a 6 % difference, suggesting a reasonable estimation of the inertia and stiffness in those directions.Furthermore, the mean response is estimated reasonably well, this indicates that the thrust on the rotor is estimated well and that differences in the mean surge and sway response result from the underestimation of the quasi-static mooring stiffness.

Dynamic response comparison
For the two load cases the dynamic response spectra are presented in Figure 3a and Figure 3b.
The standard deviations are presented in Table 5.It can be observed that the surge shows less dynamic response for the TD-method.This difference in surge is related to the underestimation of the surge natural frequency by the FD-method as this natural frequency is far below the lower cut-off frequency of 0.01667.Furthermore, second-order wave forcing is not included in the FD-method.In the TD-method this can excite natural frequencies below the wave frequency range.The sway motion shows a smaller underestimation as the dynamic response is driven by the turbulence of the wind, which is estimated reasonably well.The heave response is dominated by the wave excitation.As there is limited influence of the aerodynamic excitation and response, the response of this degree of freedom is comparable between the two methods.The peak around the heave natural frequency corresponds to the peaks present in the wave spectra (Figure 2b) and the neglected hydrodynamic damping as the connecting pontoons are assumed as Morison members.
Roll and pitch show noticeable differences around the wind turbulence frequency range (< 0.04 Hz).The differences around the roll and pitch natural frequencies are due to an increased excitation due to difference-frequency second-order wave forces, which excite the natural frequencies below the wave-frequency range.
Furthermore, the increased response in the FD-method is due to the reduced hydrodynamic damping of the system.This is mainly the result of the Morison approach used for the connecting pontoons so radiation effects are not taken into account.
Finally, low frequency yaw response is underestimated by the FD-method.This is due to the exclusion of the second-order wave drift force in the FD-method, which can excite the yaw natural frequency in the TD-method.Furthermore, no aerodynamic yaw excitation is estimated since the aerodynamic code assumes a symmetric thrust distribution across the rotor plane.

Tower base bending moment comparison
The aim of this research is to explore the use of a computationally efficient estimate of the equivalent bending moment at the tower base (TB).This is often considered as a reference load between designs and iterations.For the two load cases, the Power Spectral Densities (PSDs) of the tower base bending moments are presented in Figure 4a and Figure 4b.For both load cases it can be observed that the misalignment between the wind and waves results in a smaller contribution of the wind-turbulence in the fore-aft bending moment (M y).In this direction, the misaligned swell shows a noticeable response around its peak periods at approximately 10 s ≈ 0.10 Hz.
From the obtained PSDs of the tower base bending moments, the equivalent stresses are estimated using the Tovo-Benasciutti method.To account for the mean stress effect due to a mean roll/pitch response, the Goodman correction is applied to the stress PSD.The equivalent moment around the tower base is estimated in the local floater reference frame, as presented in Figure 1.
For Case 1 the direction of the maximum/minimum equivalent bending moment is reasonably well estimated (Figure 5a).The equivalent bending moment in fore-aft direction is overestimated by 14 % and the side-to-side equivalent bending moment is underestimated by 22 %.The maximum equivalent bending moment is underestimated by about 21 %.The underestimation in side-to-side is due to increased response seen in the FD-method around the wave frequency range and the response around the first tower base bending frequency.In the time-domain simulations the first tower natural frequency can be excited by blade passing excitation and second-order wave forces, which is not included in the FD-method.
Case 2 (Figure 5b) shows a reasonable estimation of the equivalent tower base bending moment, within an error of 2 %.This is mainly due to the limited response around the first tower base bending frequency.The tower base bending moment response is governed by the response around the wind and wave frequencies which is estimated reasonably well by the FD-method.This shows that at and above rated wind speed the blade passing frequency is around the first tower bending frequencies.To conclude, the difference in equivalent moment between the first case is an accumulation of the differences identified in excitation spectra and dynamic response characteristics.Together with the assumption of a rigid tower this results in differences in the estimation of the equivalent moment at the tower base.However, for Case 1 (above rated wind speed) the motion response estimates are more comparable and the tower base response is governed by response around the wind and wave frequencies and the first tower bending modes are not noticeably excited.

Conclusion
For two load cases of an industry project the mean response, motion response and tower base bending moment estimates from a FD-and TD-method are compared.The results show that differences in the mooring stiffness estimation result in differences in the surge and sway natural frequencies of up to 48 %.This difference in quasi-static mooring estimation also results in differences of the mean response of up to 207 % and up to 58 % for the dynamic response.However, a better match between the response estimations of load-driving modes (pitch/roll) is observed.The mean response in the direction of the wind is estimated within 5 %.Finally. it is observed that the FD-method can estimate the equivalent bending moment within an error of 5 % for the Case 2 (above rated wind speed) as a reasonable estimation of the tower base bending response around the wind turbulence and wave frequency ranges is found.
However, for the load case below rated wind speed (Case 1) differences of up to 22 % in equivalent moment are found.This mismatch between the equivalent moment estimation for this case is mainly due to the assumption of a rigid tower, no blade passing excitation and no second-order wave forcing in the FD-method.Therefore, no response around the first tower mode are observed in the FD-method.However, as the response around the wind turbulence and wave frequency is comparable, this additional high-frequency response seen in the TD-method increases the number of cycles experienced by the tower and therefore increases the equivalent moment at the tower base.
From the findings presented in this paper the main limitations for using a FD-method in industry projects are identified: • The linearization of the mooring system results in underestimation of the mooring stiffness.
• The assumption of a rigid tower results in underestimation of the tower base bending moment response around the first tower natural frequency.
• The underestimation of hydrodynamic added mass and damping due to the hybrid analytical/numerical approach results in underestimation of radiation damping around the wave-frequency range.
Each of these individual differences add up to differences in the estimated equivalent bending moment.Due to their cumulative but coupled effects, the individual contributions of the differences are not separable without further analysis.
For future work it is recommended to increase the duration of the time-domain simulations to better compare the low-frequency response.Furthermore, it is recommended to include the first fore-aft and side-to-side tower modes as DOF in the FD-method.Moreover, it is recommended to include blade-passing as an excitation.From a modeling perspective it is recommended to model all major components of the floater (columns and pontoons) in potential flow software to properly capture the diffraction-radiation effects.For slender members a Morison equation estimate would be sufficient.

Figure 1 :
Figure 1: Illustration of relative wind and wave directions.

Figure 2 :
Figure 2: Wind and wave spectra comparison for load Case 1 and 2.

Figure 4 :
Figure 4: Comparison of the tower base bending PSDs for two load cases.

Table 2 :
Environmental data for the two selected cases.

Table 3 :
Comparison of standard deviations of wind and wave spectra for the four load cases.

Table 4 :
Comparison of the mean responses

Table 5 :
Comparison of standard deviations of motion response characteristics.