The aim of this study is to determine whether, on average, ocean gravity waves affect the average power output of turbines mounted on floating platforms. We do so by comparing the power output of a turbine mounted on a platform allowed to be mechanically affected by wave motions to the same arrangement with no wave motions. We also compare the average power of a simple multi-turbine farm configuration of floating-platform turbines to a fixed-platform arrangement.

The comparison is made via simulations. For the moving platform, we employ large-eddy simulations (LESs) of a stably- and unstably-stratified atmosphere, a mechanical model for the platform, a parametric representation of a 5 MW turbine, and a spectral representation of the ocean gravity waves. The waves and platform dynamics are suppressed when we compute the fixed-platform outcomes. The novelty of the results derives from the complexity of the fully coupled model and the high spatio-temporal resolutions of the simulations.

The simulation of wind turbines has proven to be an effective way to inform the design, siting, and prediction of wind turbine power production (e.g., ). Significant fidelity has been achieved in simulations of individual and clusters of turbines as well as the interaction of turbine wakes and its environment. Offshore sites for wind farms, particularly deep-water ones, have obvious attractive features, among these are lower interference with maritime operations, potentially smaller ecological impact, the possibility of employing larger turbines, and more reliable wind conditions than shallower sites. One obvious challenge to wind turbine deep-water deployment is that they may require a floating platform.

The modeling of floating-platform offshore wind turbines is a multi-scale, multi-physics problem which can be addressed using varying levels of simulation fidelity. For instance, the work of used the low-fidelity OpenFAST model with empirical modifications to the drag forces to simulate the wave loads of semi-submersible platforms. While this method improves on the under-predictions of the low-frequency, nonlinear responses from typical engineering models, it does rely on calibration to experimental data. However, the highest-fidelity, blade-resolved simulations including the turbine and floating platform require less empirical calibration but must include boundary layer physics at very small scales. Meanwhile, the regional characterization of the wind resource requires the encapsulation of hour- and kilometer-scale phenomena in mesoscale simulations (i.e., simulations that bridge the scales between highly resolved models and earth system simulations).

The complexity and multitude of design considerations associated with the various platform configurations has shown the need for comprehensive studies on the modeling offshore wind platforms and their interactions with the environment. The Offshore Code Comparison Collaboration, Continued with Correlation and Uncertainty Project (OC6) addresses the need for model validation and comparison. Of particular interest to the current work are Phase III and Phase IV of the OC6 project, which studied the aerodynamic rotor behavior and full-system, aerodynamic and hydrodynamic interactions of floating platforms. Their results have shown that two-phase flow simulations of semi-submersible and TetraSpar floating platforms are able to be successfully compared against experiments, but there are many remaining challenges in these high-fidelity simulations and simplified models of their behavior are still necessary in many instances. Our long-term goal is to create more accurate representations of wind farms in the mesoscale simulations. For this purpose, we argue that LESs are appropriate to resolve atmospheric interactions with turbines, and the momentum and energetic exchanges critical to power generation by the turbines and their effect on the atmospheric boundary layer. LES has been used to compare the effect of moving platform turbine wakes to those of a fixed platform. For example, found that the wakes of these two configurations were similar.

Large-scale simulations using LESs of wind farms are now possible with modern computing tools . These studies allow for computation at different atmospheric conditions and are often able to connect to experimental data . Unlike blade-resolved models where individual turbines are largely represented in their geometric entirety (see ), LES models often represent the turbine via either an actuator line or an actuator disk model . The choice of the wind turbine rotor model in high-fidelity wind farm simulations is dependent on both the computational resources available and the expected wake dynamics to be encountered. For instance, in floating turbines with a surge motion frequency commiserate with the rotor speed, a vortex ring state can be encountered if the blade tips interact with the vortex emerging from the previous blade and consequently change the near-field wake. In other cases, an asymmetric loaded rotor or individual pitch control can lead to different loading on individual blades and change the downstream wake behavior. For these situations, an actuator line method or a blade-resolved method may be appropriate to resolve the tip vortex behavior or blade loading differences. However, differences in wake behavior can also be caused by changes to the turbine orientations or to rotor behavior which are not depending on resolving the individual blade loading profiles or tip vortices.

The inclusion of platform motion in the dynamics of turbines has become more prevalent in the literature recently (e.g., ). For instance, wake deflections due to turbine yaw steering or platform yaw or pitch are a steady phenomenon which can directly impact wind farm performance. Previous studies by and have also shown that lower-frequency floating-platform motions can introduce large-scale structures in the far wake which can enhance wake recovery. These effects can be captured using simpler actuator disk models without the need for resolving blade motions or blade pitch controllers, and are the focus of the current study . The wind tunnel experiments in report decreased mean power at high tip-speed ratios and high surge due to the stall onset on blades. They were also able to obtain an increase in power under certain controlled wave conditions. Their work points to the importance of having active and passive wave motion control on these floating offshore wind turbines (FOWTs). A significant effort is being made to model and simulate turbine dynamics on moving platforms. These model and simulation efforts are aimed at discerning first-order dynamic/structural issues that will inform the design and construction of these FOWTs. For example, compare wake characteristics of turbines mounted on different floating platforms and their wakes to those of fixed turbines.

Here we assess the effect of fully developed ocean gravity waves on the average power output from turbines mounted on a floating platform over a wide range of wind speeds. We focus on the power characteristics of a 5 MW NREL turbine mounted on a specific and familiar moored platform, namely the tension-leg platform. These platforms will have active motion control. We omit active motion control, since active platform stability mechanisms primarily address transient motions due to storms, swells, and developing ocean conditions, and the residual flow due to waves (for example, see ).

Our approach allows us to understand how floating-platform motions from ocean dynamics and turbulent thrust intermittency alters key elements of the flow used in mesoscale parameterizations including hub-height wind speeds and directions, turbulence, and turbine power output over the timescales present in a mesoscale simulation. First, we extend LESs to include floating turbine dynamics such that power production and subsequently turbulent kinetic energy (i.e., the quantities of interest in a mesoscale parameterization) are directly modified by the turbine motions. To do this, we have modified an existing LES tool capable of modeling a variety of atmospheric conditions using an actuator disk approach. The resolution of atmospheric dynamics of these simulations is 10 m. Second, we include explicitly how platform motions affect the wind flow encountering the turbine as well as the effects of platform motion on the mechanics to turbine. Third, we account for how gravity wave motions affect the dynamics of the platform itself.

An ocean platform would be exposed to a variety of different waves. Our study will focus on motions induced by “fully developed” ocean wave conditions. Well-established and verified empirical models for these ocean waves have been around for more than half a century. The most widely cited and tested models for these waves are the Pierson–Moskowitz model (see ) and its variant applicable to developing conditions, the more prominently peaked JONSWAP spectrum model (see ). The Pierson–Moskowitz model has as input the wind speed at a set reference height above the mean sea elevation, and it produces a multi-spectral description of the gravity wave field, presuming the sea state has reached statistically stationary conditions (fully developed) over sufficiently large expanses (fetches). The gravity waves are, by some measures, the overwhelmingly most common waves found in deep waters, where most of the turbines on floating platforms operate or plan to operate. In the ensemble, these gravity waves are stationary and can be described as having random phases. As a consequence of this choice of the types of waves under consideration, we then exclude swells generated by distant storms and disturbances which manifest themselves as episodic/transient changes in the sea surface, dissipative white-capping waves (typified by random breaking events), or steep episodic breaking waves of large amplitude that result from interactions of the wave field with the underlying ocean currents. Notwithstanding the prevalence of wind waves in places where these floating-platform ocean turbines are placed or are planned for, in the interpretation of the results that follow, it is cautioned that fully developed sea states are not common in nature and thus the Pierson–Moskowitz spectral model for these waves apply with caveats to the practical case. However, the effect on turbine power output due to waves modeled by Pierson–Moskowitz is not at all obvious and a motivation for this paper. Different geographic and weather conditions determine what types of transient oceanic disturbance are prevalent, and thus answering whether waves of this type have an impact on power output of these floating installations will require specifics of geography, time expanse, and large-scale atmospheric conditions to be meaningful.

The remainder of the document is organized as follows. In Sect. , we detail the model of the platform/turbine dynamics, responding to wind wave motions. In Sect. , we give the problem statement and make plain how the wind exposed to the turbines is affected by motions of the platform. In Sect. , we provide details on how the turbine is handled numerically and how the atmospheric initial conditions and boundary conditions are chosen for the LES formulation. The LES equations and numerical details appear in Appendix . In Sect. , we describe the results of our numerical experiments. Wherever possible we show relative quantities, comparing power output, for example, of the turbine mounted on a moving platform and the same where the platform is not subjected to ocean waves. Concluding remarks and suggestions for immediate research follow-up appear in Sect. .

Our modeling approach for the simulation of the FOWTs dynamics is adapted from . A schematic of the platform and turbine appear in Fig. . In what follows, we will be specifically using the parameters associated with an NREL 5 MW turbine, mounted on a floating platform. All of the parameters required to re-create the platform motions we use in this study are found in .

Generally, we would have 12 degrees of freedom (DOFs): surge, heave, and pitch position, which are the symmetric motions; and the antisymmetric motions are sway, roll, yaw position (the other 6 DOFs are the velocities of each aforementioned position). In what follows, however, we will always line up the platforms/turbines such that only symmetric degrees need to be considered (see for studies that focus on the effect of sway surge and yaw effects on turbine wakes). We therefore only consider 6 DOFs: surge ζ, heave η, and pitch α (and their respective velocities), as shown in Fig. b. See also Fig.  for more details on the coordinates and platform geometry.

In this model, the balance of forces and torques for the surge, heave, and pitch are 1Eq˙=F, with the overdot indicating a time, t, derivative. Here, q=[ζ,ζ˙,η,η˙,α,α˙]⊤:=[ζ,u,η,v,α,w]⊤; the inertia matrix E is defined as 2E=1000000MX000Mdcos⁡α001000000MY0Mdsin⁡α0000100Mdcos⁡α0Mdsin⁡α0J, and the forcing vector is 3F=uQζ+Mdw2sin⁡αvQη-Mdw2cos⁡αwQα. The weight forces are 4Qζ(we)=0,5Qη(we)=(MN+MP+MS)g,6Qα(we)=(MNdNv+MPdPv)sin⁡α+(MNdNh+MPdPh)cos⁡α, where MN, MP, and MS are the masses of the systems N, P, and S shown in Fig. . The dPv, dNv, dPh, and dNh are the distances between BS and BP, and between BS and BN in the direction parallel to the tower's axis v and horizontal axis h, dN=dNh2+dNv2. Buoyancy forces are defined as 7Qζb=0,8Qηb=-ρwVgg,9Qαb=ρwVgdGsin⁡α, and wave forces as 10Qζ(wa)=(ρVg+mx)cos⁡αz¨,11Qη(wa)=(ρVg+mx)sin⁡αz¨,12Qα(wa)=0. Wind forces Qwi are 13Qζ(wi)=-FA+FAN+FAT14Qη(wi)=015Qα(wi)=-FAdPvcos⁡α-dPhsin⁡α16-FANdNvcos⁡α-dNhsin⁡α17-FATdTcos⁡α 18FA=12ρACTuupstream+vζ+dPh2+dPv2wcos⁡α219FAN=12ρCdNANcos⁡αuupstream+vζ+dNwcos⁡α220FAT=12ρCdThTDTcos⁡αuupstream+vζ+dTwcos⁡α2, where uupstream is the freestream inflow velocity to the turbine actuator disk described in Eq. () and provided by the LES model, A is the turbine-swept plan area, CT is the coefficient of thrust, CdN=1 is the coefficient of drag for the nacelle, AN=9.62m2 is the nacelle area, CdT=1 is the coefficient of drag for the tower, hT=87.6m is the exposed tower height, DT=4.935m is the mean tower diameter, and dT=75.7843m is the vertical distance between BS and BT in Fig. .

We refer the reader to for the definition of tie rod forces Qtr and hydraulic drag forces Qhy. The composite forces are then 21Qζ=Qζ(we)+Qζ(b)+Qζ(wi)+Qζ(tr)+Qζ(wa)+Qζ(hy)Qη=Qη(we)+Qη(b)+Qη(wi)+Qη(tr)+Qη(wa)+Qη(hy)Qα=Qα(we)+Qα(b)+Qα(wi)+Qα(tr)+Qα(wa)+Qα(hy). The modifications we made to the platform model in are that the wind is supplied by an LES of the atmosphere, and the upstream velocity at the hub takes into account the motion of the turbine itself due to wave motions. Furthermore, we use an actuator disk approximation for the turbine; hence, some of the details on how the velocity from the LES calculation enter the platform dynamics will be fully clarified in the subsequent sections. The wave forcing Q(wa) requires the calculation of z¨, the vertical sea elevation acceleration. The ocean waves considered here are specific to a fully developed gravity wave deep-water sea model. See Appendix  for the calculation of the sea surface acceleration.

We aim to determine whether floating motions of a tension-leg platform (TLP) affect the mean power generated by wind turbines mounted on floating platforms and the wake deficits. In this study, we focus on wave motions due to fully developed conditions, which means that we exclude remotely generated swell and transient, storm-driven conditions. To accomplish this, we will be comparing the steady power output of NREL 5 MW turbines mounted on moving TLPs), to the same turbines mounted on non-moving platforms. As these turbines are usually arranged into large wind farms composed of many turbines, we also want to determine whether wave motions affect the power generation of a farm of floating-platform turbines due to changes in the waking effect of upstream turbines on the other turbines.

We will simplify the dynamics of the turbine (compared to actuator line and blade element theory techniques) at 10 m grid spacing by making use of an actuator disk approximation. Actuator line representations can better resolve tip vortices and individual blade influence by projecting blade element theory on local flow velocities using the chord length, and lift and drag coefficients at different blade segments. However, actuator lines require fine grid spacing typically on the order of 1–2 m, and the process of projecting input wind velocities onto thrust and torque contributions for each blade section is more complex. Actuator disk simulation does not require this level of resolution and takes a simpler approach in projecting thrust and torque by applying them using 1D momentum theory based on thrust and torque coefficient reference tables created from actuator line simulations. The choice of 10 m grid spacing is to allow more than 10 model grid cells across the rotor diameter to ensure it is resolved by the numerics . We confirm that the actuator disk is giving similar wake deficits as the actuator line representation in Appendix B4.

The wind will be generated by an LES approximation of compressible, non-hydrostatic atmospheric flows using a model called portUrb . The subgrid model uses a one-equation closure to account for unresolved turbulent kinetic energy. The thrust deficit applied by the turbine to the surrounding flow is derived from the general thrust expression T=12ρC|u|u, where ρ is the atmospheric density, C is the coefficient of thrust, and u is the inflow velocity. Further details are presented in Sect. . Because we are constraining the incoming velocity to be directed normal to the turbine disk, the component of the velocity affecting the thrust deficit, in the absence of platform motions, is 22u(t)⋅x^=uupstream(t), which will be taken at an upstream and on-axis location to the turbine. The computational details of finding uupstream appear in Appendix . When the platform motion is taken into account, 23u(t)⋅x^:=U(t)=uupstream(t)+uplatform(t), where 24uplatform(t)=u(t)+dPw(t)cos⁡[α(t)], where dP is the distance between BS and BP, as shown in Fig. . The right-hand terms of Eq. () are found by solving Eq. ().

The goal of the subsequent numerical simulations is to assess the relative impact of induced inflow velocity perturbations on the power output and wake deficits of turbines mounted on floating platforms. The motion of the floating platform is modeled mechanically and is subjected to fully developed ocean gravity waves. The impact is surmised by comparing the power output and wake deficits of turbines affected by floating-platform motions to that of stationary turbines – all other aspects being identical, including the turbulent inflow. The random nature of the waves and turbulent intermittent thrust dictate that we report the time average power output rather than snapshots in time. In what follows, we describe the simulation modeling and numerical conditions used in deriving numerical results.

The LES model uses compressible, non-hydrostatic, conservation-form equations conserving mass, momenta, and mass-weighted virtual potential temperature. The equations, along with numerical implementation details, are given in Appendix . See and for further details.

The sub-grid-scale (SGS) is modeled with an eddy viscosity model that assumes resolved averages of correlations of unresolved perturbations are proportional to the resolved gradient, the coefficient of proportionality being the eddy viscosity. The eddy viscosity is modeled with a one-equation closure model that actively prognoses SGS unresolved turbulent kinetic energy (TKE) (see ). TKE is evolved according to shear production, resolved transport, turbulent transport (dissipation), TKE dissipation, and buoyancy modification. At the ocean surface, surface friction is applied using Monin–Obukhov similarity theory according to a roughness length determined by the Charnock relation , which estimates roughness length of the ocean surface based on friction velocity u⋆ in an iterative process involving the following two equations: 25u⋆=κ|u|hubln⁡zhubz0,26z0=αgu⋆2, where κ=0.4 is the von Karmann constant, α=0.018 is the Charnock constant, and |u| is specified for each experiment and enforced as a horizontal average in the turbulent precursor via pressure gradient forcing at hub height.

Each simulation is set up by creating a neutral atmospheric boundary layer (ABL) precursor simulation based on a neutral stratification. Initially, a log profile of u velocity is imposed, based on the target hub-height velocity with v=w=0, using an air density of 1 kg m⁻³ and a constant potential temperature of 300 K. Turbulence is seeded by applying random potential temperature perturbations from a uniform distribution of -0.25,0.25 K in the bottom 100 m of the domain. The simulation is spun up for 4 model hours, and then statistics are gathered over another 4 model hours. The concurrent precursor simulation uses periodic boundaries in the horizontal direction and solid wall boundaries in the vertical, and it forces the turbine simulations through the inflow boundaries with open boundaries on the outflow. Pressure gradient forcing is used to force the precursor simulation toward a target hub-height velocity, and the pressure gradient computed from the precursor is also used in the forced turbine simulations. The default neutral ABL simulations form the TI1.0 simulations. The precursor turbulent velocity perturbations about the mean at a given vertical level are scaled by a factor of 2× and 0.25× to force the TI2.0 and TI0.25 simulations, respectively, to simulate the turbines over different turbulence intensities. The precursor hub-height turbulence intensities (TIs) of the TI1.0 simulations for each target hub-height velocities are reported in the last column of Table  for reference, and they agree well with the ranges of TI observed in offshore contexts from Fig. 2 of . Verification of the ABL and turbine wake response is carried out in Appendix .

We used simulations to assess how the power output of a single wind turbine, 5 MW NREL, mounted on a floating tension-leg platform as well as well as that of a cluster of the same, are affected by ocean wave motions. This assessment was done by comparing the time-averaged power of the floating-platform ocean turbines to the same wind conditions while not allowing the platform to move by oceanic disturbances.

The simulations consisted of a wind, captured by large-eddy simulations of an atmosphere with three inflow turbulence intensities; a mechanical model of a floating chain-stabilized tension-leg platform due to ; and a time-dependent empirical model for fully developed wind waves described by the Pierson–Moskowitz spectrum (see ). The dynamics of the platform, the turbines, the ocean waves and the atmosphere are coupled. The specific turbine was the NREL 5 MW design. Platforms are often motion stabilized; however, we did not take into account any active motion stabilization and control dynamics. The three turbulence intensities act as a means to understand whether the results in this study are dependent upon the pre-existing turbulence intensity, typically modulated by surface roughness length and buoyant stability of the mean atmospheric profile.

For an isolated turbine mounted on a floating platform subjected to waves, we found that the platform motions due to a fully developed ocean minimally affected the wake deficits downstream. Further, the time-averaged downstream power production was not affected in a significant way by ocean-induced and intermittent thrust-induced platform motions. The floating-platform motion-induced perturbations to inflow velocity are quite small compared to already existing turbulent intensity, even at 25 % of the naturally occurring neutral boundary layer turbulence, which is the primary reason why observed changes to wake deficits and downstream power generation due solely to floating-platform motions are small.

Figure  depicts the power spectrum of uplatform and uupstream for neutral atmospheric conditions and the Moving/Ocean platform configuration, with a hub speed of 11 m s⁻¹ and fully developed ocean wave conditions. From this figure, we surmise that the spectrum of U(t) is dominated by platform motions in the high-frequency regime and by the wind turbulence in the low-frequency regime. Furthermore, the most energetic part of the spectrum is associated with wind dynamics and turbulence. The sharp spectral peak and trough originate in the vibrational character of the particular floating platform under consideration, and thus one cannot conclude that this structure will appear in the spectra of other floating-platform arrangements. Regarding power generation assessments and whether differences in power output are significant for a fixed or moving platform, the spectra indicate that if power difference assessments are made at sufficiently long times, well over hundreds of seconds, the average power output is dominated by the wind contribution, not the combined wind/platform contribution. The random-phase nature of the waves and the dominance of the wind dynamics on U(t) over platform motions suggest that under fully developed ocean conditions, neither the single turbine mounted on a floating platform nor organized clusters of turbines of the type we investigated are affected in a significant way regarding power generation by ocean motions. Long time averages of the platform acceleration was zero, and therefore it was not expected to see, in the average, a net increase or decrease in the power output of a single turbine. However, it was not obvious at the onset that in cluster arrangements the platform dynamics would affect the wake of windward turbines and thus affect the power of the turbines downstream. However, our calculations reveal that downwind turbines are not appreciably affected by average power output. Wake disturbances were clearly apparent on turbines subjected to wave motions; however, this was at near distances that are outside of the realm of practicality.

As a check, we compared random ocean perturbations to (zero-mean) sinusoidal fluctuations of the hub speed in order to determine whether random effects in the platform dynamics had a crucial role to play in enhancing/suppressing the time-averaged power output of a moving turbine, as compared to pure sinusoidal fluctuations of the hub speed. We found that, for reasonable sinusoidal amplitude and frequency perturbations, there was little change on the power output of the turbine in the far wake, although downstream power production is lowered in the near wake where downstream turbines are unlikely to be placed.

Wind waves modeled by the Pierson–Moskowitz (PM) spectrum – and for developing seas, the JONSWAP model – represent the vast majority of wave types in deep waters; however, the model assumptions are seldom perfectly realized in nature: the PM spectrum defines a “fully developed sea”, requiring wind to blow steadily at a constant speed over considerable distances and time. Because these perfect, uninterrupted conditions seldom occur naturally, the PM spectrum serves more as a foundational theoretical limit. Despite this, the PM spectrum is extensively utilized in ocean engineering and naval architecture as it represents the absolute worst-case equilibrium for a specific wind speed, providing engineers with a conservative, standardized benchmark for designing the structural integrity of ships, oil rigs, and floating platforms.

A full assessment of the effects of wave motions on the power output of turbines mounted on moving platforms necessitates consideration of transient dynamics. As a first step, we argue, it is critical to consider how multi-spectral wind waves affect the power output of FOWTs. Some of the transient conditions to consider next are obvious: the effect of swells, changing wind conditions and storms, and developing or decaying ocean waves. Some are less obvious. For example, finite-time averages of power output could still be affected by a fully developed ocean: when the hub wind dips up and down around the minimum or maximum wind operating conditions, for example, or when wind directions are not well accounted for by an alignment of the wind and the turbine. Nevertheless, for moored platforms like the TLP design studied here, under the most generic of ocean conditions, namely sustained steady winds and a fully developed ocean, the power of floating ocean turbines does not appear, in simulations, to be affected significantly when compared to fixed turbines. The consideration of transient wave effects on power output necessitates the consideration of the types of waves and conditions found in actual geographical locations of interest to be meaningful. Our simulation framework takes as a prescribed forcing the sea wave models; hence, in principle, the formulation is easily extensible to consider transient wave cases.