As offshore wind turbines supported on floating platforms extend to deep waters, the various effects involved in the dynamics, especially those resulting from the influence of moorings, become significant when predicting the overall integrated system response. The combined influence of waves and wind affect motions of the structure and induce tensile forces in mooring lines. The investigation of the system response under misaligned wind-wave conditions and the selection of appropriate mooring systems to minimize the turbine, tower, and mooring system loads is the subject of this study. We estimate the 50-year return response of a semisubmersible platform supporting a 13.2 MW wind turbine as well as mooring line forces when the system is exposed to four different wave headings with various environmental conditions (wind speeds and wave heights). Three different mooring system patterns are presented that include 3 or 6 mooring lines with different interline angles. Performance comparisons of the integrated systems may be used to define an optimal system for the selected large wind turbine.

## Introduction

There is a growing interest in the development of floating offshore wind turbines (FOWTs) to harness the stronger and more consistent offshore wind resource in deeper waters. A few FOWT concepts supporting an offshore baseline wind turbine model, such as the NREL 5 MW wind turbine model [1], have been proposed and studied in the past decade [2–5]. Recently, large-scale wind turbines, such as the 13.2 MW wind turbine model developed at Sandia National Laboratories (SNL) [6–8], have gained interest because of their potential to bring down overall system costs. Related studies have been carried out to evaluate the system's response including extreme and fatigue loads, global dynamic response, etc., usually under simplifying assumptions of co-aligned wind and waves. It is not uncommon, however, for wind and waves to be significantly misaligned, particularly in stable atmospheric conditions [9]. Besides, as per the requirements of IEC 61400-3 [10], multidirectional distributions of wind and wave directions can have an important influence on loads acting on the support structure. This influence, in general, is site-specific and structure-dependent.

The influence of misaligned wind-wave conditions is directly affected by the directional stiffness and damping properties of the integrated FOWT system, which is composed of the wind turbine, floating platform, and mooring system. To date, only a few studies have been conducted to investigate the influence of wind-wave misalignment on the response of floating platforms for wind turbines. Philippe et al. [11] studied the impact of wind-wave misalignment on a floating wind turbine located on a barge platform and discovered that the sway, roll, and yaw response amplitude operators increased with misalignment and were large especially for a misalignment of 90 deg. Ramachandran et al. [12] studied the importance of wind-wave misalignment on a tension-leg platform platform under a wind excitation of 10 m/s, and showed the significance of 90 deg and 180 deg misalignments on both the spectral energy content and on time series of the platform motions.

In this study, wind-wave misalignment effects are studied for a semisubmersible FOWT supporting a large-scale 13.2 MW SNL wind turbine model. The 50-year return extreme response for different response measures are obtained using the environmental contour (EC) method under selected wind-wave misalignment conditions. In addition to the structural properties of the platform, the mooring lines also provide directional stiffness and damping to the integrated system. Different mooring patterns are studied to investigate the effect of mooring system contributions under misaligned wind-wave directions and their influence on the 50-year return extreme response. Finally uncertainties in the response (conditional on wave and wind conditions) are also studied by employing a correction to the two-dimensional (2D) EC method.

## Structural Reliability and the Environmental Contour Method

### Structural Reliability Problem.

*n*-dimensional environmental random variable vector and

*Y*the response of interest. The probability of failure

*p*can be written formally as

_{f}where *Y _{T}* represents the

*T*-year return period response and $fX(x)$ denotes the joint probability density function of the environmental variables. Direct evaluation of the integral in Eq. (1) is usually expensive as it involves extensive computations to establish the short-term conditional probability, $P[Y>YT|X=x]$. Agarwal and Manuel [13] used a parametric method to estimate the extreme loads for a 2 MW offshore wind turbine based on a Gumbel short-term load distribution. Ronold et al. [14] described a quadratic Weibull distribution model to describe wind turbine load distributions that were adapted and applied in other studies, such as by Manuel et al. [15]. Recently, Lee et al. [16] proposed a Bayesian spline method to establish a nonhomogeneous generalized extreme value distribution to represent the extreme load distributions of land-based wind turbines. In this method, the short-term distribution $fY|X(\u2009y\u2009|\u2009X=x)$ was estimated, instead of fitting generalized extreme value distributions for each

**bin separately, using a spline model that connects all the bins across the random variable space. The advantage of this method is that it is not necessary to have large amounts of data to define the tails of each short-term distribution in order to reduce the uncertainty in estimation.**

*x*Note that Eq. (1) suggests a separation between the environmental random variable distribution $fX(x)$, which is site-specific from the conditional load distribution, $fY|X(\u2009y\u2009|\u2009X=x)$, which is model-specific. Accordingly, an alternate method for establishing the turbine *T*-year return period response that takes advantage of this separation is the EC method. This method belongs to the class of inverse reliability problems that can yield (approximately) the *T*-year return period response with far less computational effort than is possible with parametric methods, direct integration, or Monte Carlo simulation.

### Environmental Contour Method.

The environmental contour method was proposed by Winterstein et al. [17] and has been widely used to derive the extreme responses for fixed and floating offshore structures. Details related to the theoretical framework for the application of the method can be found elsewhere [17–19]. The EC method, which is built upon classical first-order reliability method principles, makes the assumption that the variability of the response (conditional on the environment) is relatively small compared to the variability in environmental random variables themselves. Also, in the EC method, an environmental contour associated with the target return period or target probability of failure, *p _{f}*, is constructed. The desired response

*Y*is approximately equal to the largest median response considering all environmental conditions defining the environmental contour.

_{T}**into independent standard normal space**

*X***via methods such as the Rosenblatt transformation [20]. Associated with the target probability of failure,**

*U**p*, defined before in terms of the return period,

_{f}*T*, it is easy to construct the “environmental contour” in

*U*space. It is defined by an

*n*-dimensional hypersphere, $||u||2=\u2211i=1nui2=\beta T2$, where

*β*, has the notion of a conventional reliability index and is such that $\Phi \u2009(\beta T)=1\u2212pf$, where $\Phi \u2009(\xb7)$ refers to the standard normal cumulative distribution function. Using a Rosenblatt transformation scheme, we have

_{T}In the numerical studies that follow, we employ the EC method to derive the 50-year period return response under selected wind-wave misalignment conditions for three different mooring systems. These results are compared both to assess the wind-wave misalignment influence and to evaluate alternate mooring patterns for the FOWT integrated system.

## Integrated Sandia 13.2 MW Floating Offshore Wind Turbines Model

### Integrated Floating Offshore Wind Turbines Model.

The Sandia 13.2 MW semisubmersible FOWT model, as summarized in Table 1 and depicted in Fig. 1, is the subject of ongoing study. Please refer to Liu et al. [7] for details regarding this model.

Rated generator power | 13.2 MW | Rotor configuration | 3 blades, 102.5 m |

Rotor, hub diameter | 205 m, 5 m | Cut-in, rated, cut-out wind speed | 3 m/s, 11.3 m/s, 25 m/s |

Hub height | 133.5 m | Nacelle mass | 1,030,000 kg |

Tower mass | 553,995 kg | Blade mass (per blade) | 59,047 kg |

Platform draft | 30 m | Tower base above SWL | 15 m |

Offset columns above SWL | 18 m | Upper columns length | 39 m |

Base columns length | 9 m | Main column diameter | 9.75 m |

Offset (upper) columns diameter | 18 m | Base columns diameter | 36 m |

Pontoons diameter | 2.4 m | Platform mass, including ballast | 4.547 × 10^{7} kg |

Rated generator power | 13.2 MW | Rotor configuration | 3 blades, 102.5 m |

Rotor, hub diameter | 205 m, 5 m | Cut-in, rated, cut-out wind speed | 3 m/s, 11.3 m/s, 25 m/s |

Hub height | 133.5 m | Nacelle mass | 1,030,000 kg |

Tower mass | 553,995 kg | Blade mass (per blade) | 59,047 kg |

Platform draft | 30 m | Tower base above SWL | 15 m |

Offset columns above SWL | 18 m | Upper columns length | 39 m |

Base columns length | 9 m | Main column diameter | 9.75 m |

Offset (upper) columns diameter | 18 m | Base columns diameter | 36 m |

Pontoons diameter | 2.4 m | Platform mass, including ballast | 4.547 × 10^{7} kg |

The platform model was developed for use in 200 m of water and was sized so as to support Sandia's 13.2 MW wind turbine [6] with SNL100-02 blades [21]. The wind turbine is located over the center column. There are three offset columns, with pontoons around the center column, each of which has attached catenary mooring lines. Braces are used to connect all of the columns as an integrated body. The large waterplane area moment of inertia provides good stability and stiffness, thus limiting the platform pitch angle in wind and waves. The blade pitch and generator control routines for the NREL 5 MW wind turbine [1] are modified for the larger rotor based on specific performance considerations.

Hydrodynamic loads on the platform are calculated via a combination of potential flow theory and Morison's equation. The first-order potential flow solution including added mass, radiation damping, and wave excitation is obtained using a panel method solver, WAMIT [22], in which the columns and pontoons are modeled as rigid bodies. Additional viscous forces on large-volume components are accounted for by including the nonlinear drag terms in Morison's equation.

### Alternative Mooring Systems.

Three alternative mooring system configurations are proposed based on an earlier study [7]. Figure 2 shows the three patterns, SNL300, SNL630, and SNL660, that are studied here (where the notation SNLXYY indicates a mooring system for the SNL model with X mooring lines where the angle between adjacent mooring lines is YY degrees.) SNL300 is the baseline mooring system selected in a previous study based on steady-state analysis [8]. As seen in Fig. 2, SNL630 and SNL660 have six mooring lines connected to the platform compared to the three lines used in SNL300. The integrated turbine system with these three mooring patterns is subjected to misaligned wind and wave input conditions, and the 50-year return system response is studied in each case.

## Numerical Studies

Time-domain stochastic simulations are carried out to investigate the effect of wind-wave misalignment and mooring system configuration on the integrated system with the SNL 13.2 MW turbine model. Four wind-wave misalignment conditions are considered as shown in Fig. 3; specifically, these are 0 deg wind (*β*_{wind}: 0 deg) and four wave directions (*β*_{wave}: 0 deg, 30 deg, 60 deg, and 90 deg). The rotor is assumed to be aligned with the wind (zero yaw), while the wave direction is defined relative to the wind direction.

Ten 1 h simulations (after removal of transients) are carried out for each of the selected sea states (defined later) using the analysis tools described next. Extreme load/response statistics of interest are obtained so as to identify critical sea states and to compare the different mooring system patterns.

### Dynamic Analysis Tool.

Fatigue, aerodynamics, structures, and turbulence (FAST), an open-source aero-hydro-servo-elastic time-domain nonlinear dynamic modeling tool, was developed at NREL for the analysis of two- and three-bladed horizontal-axis wind turbines, sited either onshore or offshore. Wind turbines in FAST are modeled as a combination of rigid and flexible bodies in a modal multibody formulation. Stochastic inflow 3D wind velocity fields are generated using turbsim, which employs spectral models to generate zero-mean *u*, *v*, and *w* (i.e., longitudinal, lateral, and vertical) components of the wind velocity vector over a two-dimensional vertical gridded plane that covers the rotor plane. Needed mean wind field profiles are added deterministically. The HydroDyn module in FAST uses potential flow theory, along with imported files from WAMIT, to derive hydrodynamic loads on the semisubmersible platform. It also includes viscous drag components on the platform using Morisons equation as stated before.

Loads on the mooring lines are computed using mooring analysis program [23], which is integrated into FAST. Mooring analysis program accounts for the effects of distributed cable mass, strain, and cable elasticity to provide the line profile and effective forces for a cable suspended at steady-state (static equilibrium). In other words, forces arising from inertia, bending, torsion, viscous drag, and internal damping are not included.

### Environmental Contours.

Site no. 14 in the North Sea, designated Norway 5 in Li et al. [24], with a water depth of 202 m, is selected in this study. Long-term joint distributions of mean wind speed at a 10 m height (*U _{w}*), significant wave height (

*H*), and spectral peak period (

_{s}*T*) are provided [24] by based on analytical distribution fits to hindcast data. The distributions of the environmental random variables are presented in Table 2.

_{p}Random variable | Distribution | Conditional relationships | Parameters |
---|---|---|---|

U_{w} | $Weibull\u223c(\alpha Uw,\beta Uw)$ | — | $\alpha U=2.029$ |

$\beta U=9.409$ | |||

$Hs|Uw$ | $Weibull\u223c(\alpha Hs,\beta Hs)$ | $\alpha Hs=a1+a2ua3$ | $a1=2.136,a2=0.013,a3=1.709$ |

$\beta Hs=b1+b2ub3$ | $b1=1.816,b2=0.024,b3=1.787$ | ||

$Tp|(Hs,Uw)$ | $Lognormal\u223c(\mu Tp,\sigma Tp)$ | $\mu Tp=ln[m1+\nu 2]$ | $\theta =\u22120.255$ |

$\sigma Tp=ln[\nu 2+1]$ | $\gamma =1.0$ | ||

$\nu =\sigma Tp\mu Tp$ | $e1=8.0,e2=1.938,e3=0.486$ | ||

$m=T\xafp(h)\xb7[1+\theta (u\u2212u\xaf(h)u\xaf(h))\gamma ]$ | $f1=2.5,f2=3.001,f3=0.745$ | ||

$T\xafp(h)=e1+e2\xb7he3$ | $k1=\u22120.001,k2=0.316,k3=\u22120.145$ | ||

$u\xaf(h)=f1+f2\xb7hf3$ | |||

$\nu (h)=k1+k2\xb7\u2009exp(hk3)$ |

Random variable | Distribution | Conditional relationships | Parameters |
---|---|---|---|

U_{w} | $Weibull\u223c(\alpha Uw,\beta Uw)$ | — | $\alpha U=2.029$ |

$\beta U=9.409$ | |||

$Hs|Uw$ | $Weibull\u223c(\alpha Hs,\beta Hs)$ | $\alpha Hs=a1+a2ua3$ | $a1=2.136,a2=0.013,a3=1.709$ |

$\beta Hs=b1+b2ub3$ | $b1=1.816,b2=0.024,b3=1.787$ | ||

$Tp|(Hs,Uw)$ | $Lognormal\u223c(\mu Tp,\sigma Tp)$ | $\mu Tp=ln[m1+\nu 2]$ | $\theta =\u22120.255$ |

$\sigma Tp=ln[\nu 2+1]$ | $\gamma =1.0$ | ||

$\nu =\sigma Tp\mu Tp$ | $e1=8.0,e2=1.938,e3=0.486$ | ||

$m=T\xafp(h)\xb7[1+\theta (u\u2212u\xaf(h)u\xaf(h))\gamma ]$ | $f1=2.5,f2=3.001,f3=0.745$ | ||

$T\xafp(h)=e1+e2\xb7he3$ | $k1=\u22120.001,k2=0.316,k3=\u22120.145$ | ||

$u\xaf(h)=f1+f2\xb7hf3$ | |||

$\nu (h)=k1+k2\xb7\u2009exp(hk3)$ |

The target reliability index *β*_{50}, the radius of the sphere in standard normal space, is computed as $\Phi \u22121(1/(50\xd7365.25\xd724))=4.58$ since simulations of 1 h duration are used in this study.

In this study, only the median of *T _{p}* conditional on

*U*and

_{w}*H*is used instead of using

_{s}*T*'s full conditional distribution given

_{p}*U*and

_{w}*H*. Thus, $u3=0$ in

_{s}**space, is used in this study, and the resulting 2D environmental contour for**

*U**H*and

_{s}*U*is shown in Fig. 4(b). A total of 24 sea states including a sea state corresponding to the rated wind speed is selected representing the operating range of the turbine. Since we are interested in the right (upper) tail of the distribution of extreme loads, only the higher of the two

_{w}*H*values at each

_{s}*U*is selected. These 24 sea states are presented in Table 3.

_{w}U_{hub} (m/s) | H (m)_{s} | T (s)_{p} | U_{hub} (m/s) | H (m)_{s} | T (s)_{p} | U_{hub} (m/s) | H (m)_{s} | T (s)_{p} |
---|---|---|---|---|---|---|---|---|

3 | 5.83 | 15.15 | 11 | 7.34 | 14.80 | 18 | 8.95 | 14.65 |

4 | 6.03 | 15.10 | 11.3 | 7.45 | 14.79 | 19 | 9.19 | 14.64 |

5 | 6.21 | 15.05 | 12 | 7.60 | 14.77 | 20 | 9.43 | 14.64 |

6 | 6.40 | 15.00 | 13 | 7.82 | 14.74 | 21 | 9.67 | 14.63 |

7 | 6.59 | 14.95 | 14 | 8.04 | 14.72 | 22 | 9.92 | 14.63 |

8 | 6.78 | 14.91 | 15 | 8.26 | 14.70 | 23 | 10.17 | 14.62 |

9 | 6.98 | 14.86 | 16 | 8.49 | 14.68 | 24 | 10.42 | 14.62 |

10 | 7.18 | 14.83 | 17 | 8.72 | 14.67 | 25 | 10.67 | 14.62 |

U_{hub} (m/s) | H (m)_{s} | T (s)_{p} | U_{hub} (m/s) | H (m)_{s} | T (s)_{p} | U_{hub} (m/s) | H (m)_{s} | T (s)_{p} |
---|---|---|---|---|---|---|---|---|

3 | 5.83 | 15.15 | 11 | 7.34 | 14.80 | 18 | 8.95 | 14.65 |

4 | 6.03 | 15.10 | 11.3 | 7.45 | 14.79 | 19 | 9.19 | 14.64 |

5 | 6.21 | 15.05 | 12 | 7.60 | 14.77 | 20 | 9.43 | 14.64 |

6 | 6.40 | 15.00 | 13 | 7.82 | 14.74 | 21 | 9.67 | 14.63 |

7 | 6.59 | 14.95 | 14 | 8.04 | 14.72 | 22 | 9.92 | 14.63 |

8 | 6.78 | 14.91 | 15 | 8.26 | 14.70 | 23 | 10.17 | 14.62 |

9 | 6.98 | 14.86 | 16 | 8.49 | 14.68 | 24 | 10.42 | 14.62 |

10 | 7.18 | 14.83 | 17 | 8.72 | 14.67 | 25 | 10.67 | 14.62 |

A JONSWAP wave spectrum with frequencies up to 10 rad/s is used to generated a stochastic sea surface elevation process with a time-step, $\Delta t=0.2s$. A 3D stochastic inflow wind field is generated according to the Kaimal spectrum in Turbsim, using 27 × 27 grid points on the rotor plane and a 0.1 s time-step. The normal turbulence model (NTM) with turbulence category A per IEC definitions is considered. The IEC wind profile defines shear for the mean wind field. In this study, a power law exponent 0.14 and a surface roughness of 0.03 m is used. In the absence of directional information, it should be noted that the same joint distribution for *U _{w}* and

*H*is assumed, regardless of the wind and wave directions.

_{s}## Results

For each selected sea state on the 2D environmental contour, ten 1 h simulations are carried out, and the maximum response over that duration for various response measures of interest is computed. As stated before, with the EC method, the largest median 1 h maximum response approximately defines the *T*-year response (here, *T* = 50 years).

Next, 50-year values for different response measures and for the different FOWT system models are estimated along with the associated critical sea state that caused the extreme response. First, different wave headings are investigated in order to assess the severity of the wind-wave misalignment on the 50-year extreme response. Response measures for the integrated system that are studied include the rotor thrust, blade and tower base bending moments for the wind turbine, and the fairlead anchor tension for the mooring system. In general, similar conclusions regarding extreme values and associated critical sea state are made for the different mooring systems when different wind-wave misalignments are considered. Thus, for brevity, only the FOWT system with the SNL300 mooring lines is discussed first. Later, to further understand different resistance contributions introduced by alternate mooring configurations, other response comparisons are discussed. Particular attention is paid to those response variables that show significant variation as the configuration of mooring patterns is changed.

### Wind-Wave Misalignment Effects on 50-Year Return Extreme Responses

#### Fifty-Year Return Period Rotor and Tower Loads.

Four different rotor and tower loads are selected to study the influence of wind-wave misalignment on the integrated FOWT system. These include the rotor thrust (LSShftFxa), blade root out-of-plane bending moment (RootMyc1), fore-aft tower base bending moment (TwrBsMyt), and the side-to-side tower base bending moment (TwrBsMxt). The 50-year response values are controlled either by the highest wind speed (25 m/s) and associated wave height or by a wind speed close to rated (11.3 m/s) and associated wave height. Misalignment effects are presented by using the parameter, $R\beta ,0$, which describes the ratio of the extreme response at a wave heading, *β*_{wave}, to the corresponding response when *β*_{wave} equals zero.

Since the wind direction is the same for all the selected environmental conditions, insignificant variation is expected for LSShftFxa and RootMyc1 as *β*_{wave} changes from 0 deg to 90 deg. As can be seen from Table 4, the 50-year RootMyc1 value is not affected by varying the wave heading direction and a very slight reduction in rotor thrust is seen with increasing wind-wave misalignment. For both these response measures, extremes occur for wind speeds at or around rated and with associated wave heights given by the environmental contour.

Response | β_{wave} (deg) | Y_{50–}_{yea}_{r} | $R\beta ,0$ | U_{hub} (m/s) | H (m)_{s} | Response | β_{wave} (deg) | Y_{50–year} | $R\beta ,0$ | U_{hub} (m/s) | H (m)_{s} |
---|---|---|---|---|---|---|---|---|---|---|---|

RootMyc1 ($MN\xb7m$) | 0 | 63.25 | 1.00 | 12 | 7.60 | LSShftFxa (MN) | 0 | 3.23 | 1.00 | 11.3 | 7.45 |

30 | 63.27 | 1.00 | 11.3 | 7.45 | 30 | 3.06 | 0.95 | 11 | 7.39 | ||

60 | 62.96 | 1.00 | 11.3 | 7.45 | 60 | 3.04 | 0.94 | 13 | 7.82 | ||

90 | 63.81 | 1.01 | 11.3 | 7.45 | 90 | 3.10 | 0.96 | 13 | 7.82 | ||

TwrBsMxt ($MN\xb7m$) | 0 | 160.20 | 1.00 | 25 | 10.67 | TwrBsMyt ($MN\xb7m$) | 0 | 591.90 | 1.00 | 13 | 7.82 |

30 | 308.05 | 1.92 | 25 | 10.67 | 30 | 500.10 | 0.84 | 12 | 7.60 | ||

60 | 459.30 | 2.87 | 25 | 10.67 | 60 | 481.50 | 0.81 | 13 | 7.82 | ||

90 | 357.95 | 2.23 | 25 | 10.67 | 90 | 493.80 | 0.83 | 11.3 | 7.45 |

Response | β_{wave} (deg) | Y_{50–}_{yea}_{r} | $R\beta ,0$ | U_{hub} (m/s) | H (m)_{s} | Response | β_{wave} (deg) | Y_{50–year} | $R\beta ,0$ | U_{hub} (m/s) | H (m)_{s} |
---|---|---|---|---|---|---|---|---|---|---|---|

RootMyc1 ($MN\xb7m$) | 0 | 63.25 | 1.00 | 12 | 7.60 | LSShftFxa (MN) | 0 | 3.23 | 1.00 | 11.3 | 7.45 |

30 | 63.27 | 1.00 | 11.3 | 7.45 | 30 | 3.06 | 0.95 | 11 | 7.39 | ||

60 | 62.96 | 1.00 | 11.3 | 7.45 | 60 | 3.04 | 0.94 | 13 | 7.82 | ||

90 | 63.81 | 1.01 | 11.3 | 7.45 | 90 | 3.10 | 0.96 | 13 | 7.82 | ||

TwrBsMxt ($MN\xb7m$) | 0 | 160.20 | 1.00 | 25 | 10.67 | TwrBsMyt ($MN\xb7m$) | 0 | 591.90 | 1.00 | 13 | 7.82 |

30 | 308.05 | 1.92 | 25 | 10.67 | 30 | 500.10 | 0.84 | 12 | 7.60 | ||

60 | 459.30 | 2.87 | 25 | 10.67 | 60 | 481.50 | 0.81 | 13 | 7.82 | ||

90 | 357.95 | 2.23 | 25 | 10.67 | 90 | 493.80 | 0.83 | 11.3 | 7.45 |

Tower base bending moments are of special interest since these response measures affect the overall stability of the integrated system. Critical sea states for the side-to-side tower moment (TwrBsMxt) are those with highest waves and the apparent wave force-driven effects. On the other hand, the fore-aft moment (TwrBsMyt) is largest around the rated wind speed. Large uncertainty in the tower base bending moment is seen in the wind speed range from 15 m/s to 20 m/s but not around the cut-out wind speed, as can be seen in Fig. 5 which shows the 50-year TwrBsMxt and TwrBsMyt values along with their variability as a function of wind speed and for the different wave headings and mooring systems.

As *β*_{wave} increases from 0 deg to 90 deg, 50-year TwrBsMyt values reduce by around 17% from 591.90 MN m to 493.8 MN m. In contrast, TwrBsMxt increases dramatically, by more than a factor of 2 when *β*_{wave} increases from 0 deg to 90 deg. These two tower base bending loads even become somewhat comparable when the wave heading is 90 deg. This suggests that in preliminary design steps, it is not appropriate to neglect or underestimate side-to-side tower bending moments to assess overall system performance. The triangular platform geometry is such that a misalignment (*β*_{wave}) of 60 deg leads to synchronized and in-phase broadside wave loads over one entire side of the triangle for long-crested waves, which are assumed here. In all of the other cases (0 deg, 30 deg, and 90 deg), different points on one exposed upstream side of the platform will not all experience wave loads that are in phase. This is why Table 4 shows greater response levels for $\beta wave=60$ deg versus all the other cases.

#### Fifty-Year Return Period Mooring Line Loads.

Anchor and fairlead tensions of mooring line nos. 1 and 2 are studied to investigate the effect of wind-wave misalignment. Different critical sea states were observed for two mooring lines. For mooring line no. 1, the extreme fairlead and anchor tension occur around the rated wind speed, while for mooring line no. 2, no anchor force is detected and the extreme fairlead tension occurs at the lowest wind speeds (around 3 m/s). As shown in Table 5, no significant variation is observed for TFair1, TFair2, and TAnch1 tensions as *β*_{wave} changes from 0 deg to 90 deg. Note that TFair1 and TFair2 are the tension forces at the fairleads of mooring lines 1 and 2, respectively; TAnch1 and TAnch2 are the tension forces at the anchors of mooring lines 1 and 2, respectively. The dominant effect on the integrated system is from wind forces as opposed to wave forces. The nonzero mean wind force on the system causes a net surge offset that makes a significant contribution to overall tension forces at the fairlead and anchor of mooring line 1. This is evident in Table 5.

Response | β_{wave} (deg) | Y_{50–year} | $R\beta ,0$ | U_{hub} (m/s) | H (m)_{s} | Response | β_{wave} (deg) | Y_{50–year} | $R\beta ,0$ | U_{hub} (m/s) | H (m)_{s} |
---|---|---|---|---|---|---|---|---|---|---|---|

TFair1 (kN) | 0 | 4346.0 | 1.00 | 11.3 | 7.45 | TFair2 (kN) | 0 | 2120.5 | 1.00 | 3 | 5.83 |

30 | 4361.5 | 1.00 | 12 | 7.60 | 30 | 2127.0 | 1.00 | 3 | 5.83 | ||

60 | 4363.5 | 1.00 | 12 | 7.60 | 60 | 2170.5 | 1.02 | 3 | 5.83 | ||

90 | 4349.0 | 1.00 | 13 | 7.82 | 90 | 2196.5 | 1.04 | 3 | 5.83 | ||

TAnch1 (kN) | 0 | 2487.0 | 1.00 | 11.3 | 7.45 | TAnch2 (kN) | 0 | 0.0 | — | — | — |

30 | 2509.0 | 1.01 | 12 | 7.60 | 30 | 0.0 | — | — | — | ||

60 | 2510.0 | 1.01 | 12 | 7.60 | 60 | 0.0 | — | — | — | ||

90 | 2490.0 | 1.00 | 13 | 7.82 | 90 | 0.0 | — | — | — |

Response | β_{wave} (deg) | Y_{50–year} | $R\beta ,0$ | U_{hub} (m/s) | H (m)_{s} | Response | β_{wave} (deg) | Y_{50–year} | $R\beta ,0$ | U_{hub} (m/s) | H (m)_{s} |
---|---|---|---|---|---|---|---|---|---|---|---|

TFair1 (kN) | 0 | 4346.0 | 1.00 | 11.3 | 7.45 | TFair2 (kN) | 0 | 2120.5 | 1.00 | 3 | 5.83 |

30 | 4361.5 | 1.00 | 12 | 7.60 | 30 | 2127.0 | 1.00 | 3 | 5.83 | ||

60 | 4363.5 | 1.00 | 12 | 7.60 | 60 | 2170.5 | 1.02 | 3 | 5.83 | ||

90 | 4349.0 | 1.00 | 13 | 7.82 | 90 | 2196.5 | 1.04 | 3 | 5.83 | ||

TAnch1 (kN) | 0 | 2487.0 | 1.00 | 11.3 | 7.45 | TAnch2 (kN) | 0 | 0.0 | — | — | — |

30 | 2509.0 | 1.01 | 12 | 7.60 | 30 | 0.0 | — | — | — | ||

60 | 2510.0 | 1.01 | 12 | 7.60 | 60 | 0.0 | — | — | — | ||

90 | 2490.0 | 1.00 | 13 | 7.82 | 90 | 0.0 | — | — | — |

### Alternative Mooring Systems.

In Tables 6 and 7, selected 50-year response values for all three mooring configurations and associated critical sea states for platform yaw motion are presented. The different mooring line configurations result in different directional stiffness contributions to the integrated system; specifically, SNL660 will lose stiffness in the *x* direction due to the different interline angle compared to SNL300; however, it will gain stiffness in system yaw especially when the wind and waves are misaligned. Various quantities of interest are selected while focusing on differences introduced by alternative mooring patterns. To compare extreme values for the selected response measures, a ratio of the extreme response for each mooring pattern to that for the corresponding extreme with the SNL300 system, namely, $Ri,1$ is defined, where *i* = 1, 2, 3 corresponds to SNL300, SNL630, and SNL660, respectively.

SNL300 | SNL630 | SNL660 | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Response, Y | β_{wave} (deg) | Y_{50–year} | $R\beta ,0$ | U_{hub} (m/s) | H (m)_{s} | Y_{50–year} | $R\beta ,0$ | U_{hub} (m/s) | H (m)_{s} | Y_{50–year} | $R\beta ,0$ | U_{hub} (m/s) | H (m)_{s} |

Yaw (deg) | 0 | 5.69 | 1.00 | 23 | 10.17 | 5.49 | 0.97 | 21 | 9.67 | 5.00 | 0.88 | 25.00 | 10.67 |

30 | 6.17 | 1.00 | 25 | 10.67 | 5.84 | 0.95 | 20 | 9.43 | 5.61 | 0.91 | 25.00 | 10.67 | |

60 | 5.72 | 1.00 | 23 | 10.17 | 5.50 | 0.96 | 21 | 9.67 | 5.02 | 0.88 | 25.00 | 10.67 | |

90 | 6.31 | 1.00 | 24 | 10.42 | 5.83 | 0.92 | 21 | 9.67 | 5.44 | 0.86 | 25.00 | 10.67 |

SNL300 | SNL630 | SNL660 | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Response, Y | β_{wave} (deg) | Y_{50–year} | $R\beta ,0$ | U_{hub} (m/s) | H (m)_{s} | Y_{50–year} | $R\beta ,0$ | U_{hub} (m/s) | H (m)_{s} | Y_{50–year} | $R\beta ,0$ | U_{hub} (m/s) | H (m)_{s} |

Yaw (deg) | 0 | 5.69 | 1.00 | 23 | 10.17 | 5.49 | 0.97 | 21 | 9.67 | 5.00 | 0.88 | 25.00 | 10.67 |

30 | 6.17 | 1.00 | 25 | 10.67 | 5.84 | 0.95 | 20 | 9.43 | 5.61 | 0.91 | 25.00 | 10.67 | |

60 | 5.72 | 1.00 | 23 | 10.17 | 5.50 | 0.96 | 21 | 9.67 | 5.02 | 0.88 | 25.00 | 10.67 | |

90 | 6.31 | 1.00 | 24 | 10.42 | 5.83 | 0.92 | 21 | 9.67 | 5.44 | 0.86 | 25.00 | 10.67 |

Ratio, $Ri,1$ | Ratio, $Ri,1$ | ||||||||
---|---|---|---|---|---|---|---|---|---|

Response, Y | β_{wave} (deg) | SNL300 Y_{50–year} | R_{2,1} | R_{3,1} | Response, Y | β_{wave} (deg) | SNL300 Y_{50–year} | R_{2,1} | R_{3,1} |

Surge (m) | 0 | 22.02 | 1.04 | 1.14 | Sway (m) | 0 | 3.42 | 0.96 | 0.80 |

30 | 22.19 | 1.03 | 1.15 | 30 | 4.10 | 0.95 | 0.87 | ||

60 | 22.27 | 1.03 | 1.14 | 60 | 4.36 | 0.96 | 0.93 | ||

90 | 22.15 | 1.04 | 1.14 | 90 | 4.52 | 0.97 | 0.90 | ||

Yaw (deg) | 0 | 5.69 | 0.97 | 0.88 | TwrBsMxt (kN⋅m) | 0 | 1.60 × 10^{5} | 0.99 | 0.99 |

30 | 6.17 | 0.95 | 0.91 | 30 | 3.08 × 10^{5} | 1.00 | 1.01 | ||

60 | 5.72 | 0.96 | 0.88 | 60 | 4.59 × 10^{5} | 1.00 | 1.00 | ||

90 | 6.31 | 0.92 | 0.86 | 90 | 3.58 × 10^{5} | 1.00 | 1.01 | ||

TFair1/2 (kN) | 0 | 4.35 × 10^{3} | 1.00 | 1.01 | TFair2/3 (kN) | 0 | 2.12 × 10^{3} | 1.00 | 1.07 |

30 | 4.36 × 10^{3} | 1.01 | 1.02 | 30 | 2.13 × 10^{3} | 1.02 | 1.09 | ||

60 | 4.36 × 10^{3} | 1.02 | 1.04 | 60 | 2.17 × 10^{3} | 1.01 | 1.08 | ||

90 | 4.35 × 10^{3} | 1.02 | 1.03 | 90 | 2.20 × 10^{3} | 1.01 | 1.06 | ||

TFair3/6 (kN) | 0 | 2.17 × 10^{3} | 1.05 | 1.12 | TAnch1/2 (kN) | 0 | 2.49 × 10^{3} | 1.01 | 1.01 |

30 | 2.21 × 10^{3} | 1.05 | 1.10 | 30 | 2.51 × 10^{3} | 1.03 | 1.05 | ||

60 | 2.24 × 10^{3} | 1.05 | 1.10 | 60 | 2.51 × 10^{3} | 1.04 | 1.08 | ||

90 | 2.24 × 10^{3} | 1.05 | 1.11 | 90 | 2.49 × 10^{3} | 1.04 | 1.07 |

Ratio, $Ri,1$ | Ratio, $Ri,1$ | ||||||||
---|---|---|---|---|---|---|---|---|---|

Response, Y | β_{wave} (deg) | SNL300 Y_{50–year} | R_{2,1} | R_{3,1} | Response, Y | β_{wave} (deg) | SNL300 Y_{50–year} | R_{2,1} | R_{3,1} |

Surge (m) | 0 | 22.02 | 1.04 | 1.14 | Sway (m) | 0 | 3.42 | 0.96 | 0.80 |

30 | 22.19 | 1.03 | 1.15 | 30 | 4.10 | 0.95 | 0.87 | ||

60 | 22.27 | 1.03 | 1.14 | 60 | 4.36 | 0.96 | 0.93 | ||

90 | 22.15 | 1.04 | 1.14 | 90 | 4.52 | 0.97 | 0.90 | ||

Yaw (deg) | 0 | 5.69 | 0.97 | 0.88 | TwrBsMxt (kN⋅m) | 0 | 1.60 × 10^{5} | 0.99 | 0.99 |

30 | 6.17 | 0.95 | 0.91 | 30 | 3.08 × 10^{5} | 1.00 | 1.01 | ||

60 | 5.72 | 0.96 | 0.88 | 60 | 4.59 × 10^{5} | 1.00 | 1.00 | ||

90 | 6.31 | 0.92 | 0.86 | 90 | 3.58 × 10^{5} | 1.00 | 1.01 | ||

TFair1/2 (kN) | 0 | 4.35 × 10^{3} | 1.00 | 1.01 | TFair2/3 (kN) | 0 | 2.12 × 10^{3} | 1.00 | 1.07 |

30 | 4.36 × 10^{3} | 1.01 | 1.02 | 30 | 2.13 × 10^{3} | 1.02 | 1.09 | ||

60 | 4.36 × 10^{3} | 1.02 | 1.04 | 60 | 2.17 × 10^{3} | 1.01 | 1.08 | ||

90 | 4.35 × 10^{3} | 1.02 | 1.03 | 90 | 2.20 × 10^{3} | 1.01 | 1.06 | ||

TFair3/6 (kN) | 0 | 2.17 × 10^{3} | 1.05 | 1.12 | TAnch1/2 (kN) | 0 | 2.49 × 10^{3} | 1.01 | 1.01 |

30 | 2.21 × 10^{3} | 1.05 | 1.10 | 30 | 2.51 × 10^{3} | 1.03 | 1.05 | ||

60 | 2.24 × 10^{3} | 1.05 | 1.10 | 60 | 2.51 × 10^{3} | 1.04 | 1.08 | ||

90 | 2.24 × 10^{3} | 1.05 | 1.11 | 90 | 2.49 × 10^{3} | 1.04 | 1.07 |

It is found that, in general, the different mooring configurations do not change the critical sea state for most of the response measures studied, except for the platform yaw motion. Using the 2D EC method, the critical sea state for the SNL630 model is identified to be that for winds slightly below the cut-out wind speed, i.e., around 20 m/s.

The different mooring configuration options have a relative large influence on platform surge, sway, and yaw motions. The extreme surge motion increases by about 14% when the SNL660 model is used relative to the SNL300 model, but only by about 4% increment with the SNL630 model. Reduction in both sway and yaw motions is achieved with the more spatially spread out mooring configurations; this is especially true for the yaw motion and when *β*_{wave} is large (greater wind-wave misalignment).

Different from the significant effects of misalignment on the side-to-side tower base bending moment, mooring line patterns have a limited influence on this response measure. The reason for this is that no significant platform roll motion changes are observed with the different mooring patterns.

Regardless of the reduction in platform sway and yaw motion, only slight increments are observed for the fairlead strain as the mooring lines are spread out more. The increments tend to be larger for the side mooring lines, line nos. 2–3 in SNL300 and line nos. 3–6 in SNL630 and SNL660. As the mooring lines are spread out more, to control the surge motion, greater resistance in the longitudinal surge direction is required. As a result, higher tensions and strains would occur as is shown here for the SNL630 and SNL660 models. Nonzero anchor tensions are only observed at position 1 for SNL300 and at position 1&2 for the SNL630 and SNL660 models. Only a 1% increment is found for Anchor 1/2 when $\beta wave=0$. This increment becomes larger up to about 7% as *β*_{wave} goes from 0 deg to 90 deg. However, one should note that even in the 90 deg misalignment case, no anchor tension results at positions 2–3/4–6. These findings could be potentially used to further optimize the mooring system by decreasing the mooring line lengths for nos. 2–3/4–6 and increasing them for nos. 1/1–2.

## Conclusions

The influence of misaligned wind and waves on a large floating offshore wind turbine supported on a semisubmersible platform with three alternative mooring systems and for selected environmental conditions is investigated. Of interest are 50-year return period values for various response measures. A reduced fore-aft tower base bending moment is observed for greatly misaligned wind-wave conditions (i.e., where $\beta wave=90$ deg); on the other hand, side-to-side tower base bending moments are more than two times larger at this greatest misalignment.

No significant variation is observed in mooring line tensions due to the long mooring lines used in these models. In the most severe sea states, the mooring lines are still lying mostly on the seabed. Further optimization to the mooring systems can still be achieved in principle.

Sway and yaw motion reduction could be achieved with mooring configurations that are more spatially spread; but this comes at the expense of increased extreme surge motion. Larger fairlead and anchor strains result as the interline angle increases in the mooring system.

In general, additional simulations employed with the 2D EC method and/or the consideration of response uncertainty in 3D Inverse first-order reliability method can help to further refine and justify findings claimed regarding wind-wave misalignment and alternative mooring systems.

## Acknowledgment

The authors acknowledge the technical assistance received from Edwin Thomas, Watsamon Sahasakkul, and Mohit Soni.

## Funding Data

Sandia National Laboratories (1307455).