RAFT uses a linear, frequency-dependent formulation to solve for the steady-state dynamic response of a floating system . It also uses a nonlinear quasi-static formulation to find the system's mean response or offsets. External forces are modeled as a linear superposition of a mean force f¯ and a Fourier series of complex amplitudes f̃(ω), which represent both magnitude and phase at each frequency ω. Because RAFT assumes a linear dynamic response, the harmonic response at each frequency can be solved independently, and then the total system response can be formed by summing the mean response ξ¯ and the frequency-dependent complex amplitudes ξ̃(ω).

The complex amplitudes are obtained by solving the frequency-dependent equation of motion: 1(-ω2[Mstruc+Ahydro(ω)+Aaero(ω)]+iω[Bhydro(ω)+Baero(ω)]+Cstruc+Chydro+Cmoor)ξ̃(ω)=f̃(ω), where M is the inertia matrix of the floating structure, A is added mass, B is damping, C is stiffness, and f̃(ω) is the excitation due to wind or waves. The matrix subscripts correspond to contributions from the structure, hydrodynamics and hydrostatics, aerodynamics, and mooring system. Although not explicitly indicated, some of the coefficients in Eq. () are obtained from the linearization of inherently nonlinear phenomena, introducing a nonlinear dependence on ξ̃(ω) that is handled iteratively.

The system's dynamic response, as described by Eq. (), is solved about a mean equilibrium point that corresponds to the mean response of the system for a given load case. This mean response is obtained by solving the static equilibrium of the system: 2(Chydro+Cstruc)ξ¯=f¯aero+f¯hydro+f¯moor(ξ¯), where ξ¯ is a six-component vector with the mean displacements of the platform along its 6 rigid-body degrees of freedom (surge, sway, heave, roll, pitch, and yaw), Chydro is the hydrostatic stiffness matrix without the contribution of weight, Cstruc is the weight contribution to the hydrostatic stiffness matrix (the hydrostatic matrix is given by two separate components to correspond to RAFT variable names), f¯aero is the mean aerodynamic load, f¯hydro is the mean hydrodynamic load (due to waves and current), and f¯moor(ξ¯) is the nonlinear reaction force due to the moorings (including the effective mooring stiffness).

The way that RAFT models the floating platform hydrodynamics is important to how the wave loads are represented in the context of an array with shared mooring lines. RAFT represents a floating platform as a combination of circular or rectangular cylinders, called members . It includes a strip theory formulation that discretizes each member axially and uses Morison's equation  to model the hydrodynamic loads at each strip based on the local fluid kinematics and structure motion.

In this strip theory formulation, RAFT computes a local added mass matrix AL for each strip based on its orientation and volume: 3AL=ρvCa,p1p1p1T+Ca,p2p2p2T, where ρ is the fluid density, v is the strip's volume, p1 and p2 are the orthogonal unit vectors that are transverse to the axis of the member, and Ca,p1 and Ca,p2 are added mass coefficients along p1 and p2. For rectangular cross-sections, the coefficients Ca,p1 and Ca,p2 can be specified independently to allow for different added mass in the two orthogonal directions. RAFT sums the added mass matrices of all strips, accounting for strip position and orientation, to obtain the total added mass matrix of the floating substructure Asub.

RAFT uses the same factors to compute the inertial excitation forces on each strip based on the frequency-dependent local fluid acceleration iωũL(ω) as follows: 4f̃I,L(ω)=ρv(1+Ca,p1)p1p1T+(1+Ca,p2)p2p2TiωũL(ω). Here, ũL(ω) is the complex amplitude of wave velocity vector, which contains the wave x, y, z complex velocity components: 5ũL(ω)=ωζ(ω,x,y)cosh⁡(k(ω)(z+h))sinh⁡(k(ω)h)cos⁡βωζ(ω,x,y)cosh⁡(k(ω)(z+h))sinh⁡(k(ω)h)sin⁡βiωζ(ω,x,y)sinh⁡(k(ω)(z+h))sinh⁡(k(ω)h), where ζ is the wave elevation at the horizontal position (x,y), z is the vertical coordinate, h is the water depth, k is the wave number, and β is the wave heading.

Axial added mass and inertial excitation forces are computed using similar expressions as their transverse counterparts but adopting a reference volume based on a hollow hemisphere with the same inner and outer diameters as the axial exposed area of each strip. The reference axial volume for a strip is 6vq=π12(d+Δd)3-(d-Δd)3, where d is the diameter at the midpoint of the strip and Δd is the diameter change over the strip length. This formulation also works for member ends, in which case it represents a solid hemisphere. For rectangular members, the diameters in Eq. () are approximated from the average of the two side lengths.

Dynamic pressure loads are calculated based on the axial exposed area, 7aend=πdΔd. RAFT approximates the quadratic drag term in Morison's equation using a linearization procedure that results in a damping term and a viscous wave excitation force. Both terms involve the computation of a local damping matrix that depends on the relative fluid velocity at each strip: 8BL=8/π12ρσu,p1ap1Cd,p1p1p1T+σu,p2ap2Cd,p2p1p2T+σu,qaqCd,q+σu,qaendCd,endqqT, where q is the axial unit vector of the member, a is the effective area of the strip (in the respective direction), Cd is the drag coefficient (in the respective direction), and σu is the standard deviation of the relative water velocity (which depends on the local fluid kinematics and their phase relative to the platform motions). The term with aq refers to the surface area of the strip side, representing a skin friction drag effect, while the term with aend refers to drag that depends on axial exposed area. Similar to the added mass matrix, the damping matrix of each strip is added to the system damping matrix after accounting for the position and orientation of the strip.

The complex amplitude of the viscous wave excitation force acting on each node is computed as follows: 9fD,L(ω)=BLũL(ω). The way the strip theory formulation considers the wave kinematics at the specific location of each node on the substructure is an important aspect for array-level hydrodynamic features that will be discussed later.

To support multiple floating offshore wind turbines (FOWTs), RAFT makes multiples of the logic it uses for representing individual FOWTs through its FOWT class (Fig. ). Each FOWT object constructs its own 6×6 matrices. These matrices are then combined into an array-level 6n×6n matrix so that the full system response can be solved for in a coupled fashion.

To relate individual FOWT properties to the full array, each FOWT object is given x and y reference coordinates in the vector rref that describe its undisplaced position relative to the global origin (Fig. ). The calculation of each FOWT's response is relative to its reference position. For example, from the mean offset ξ¯, a FOWT's mean position in global coordinates rglobal can be found by 10rglobal=ξ¯+rref.

For array applications, RAFT uses a two-level mooring system structure similar to FAST.Farm. Each FOWT can have its own independent system modeled by an individual MoorPy instance. In addition, the entire array can have an array-level mooring system that can include shared mooring lines or power cables running between FOWTs. Accordingly, for an array of n FOWTs, RAFT can have n+1 MoorPy instances. Each of these instances is optional – the only requirement is that each FOWT has mooring line attachments to provide station keeping. For example, Fig.  shows a case of two turbines with a shared mooring line in which RAFT could have three MoorPy instances: one for each of the two FOWT's independent mooring lines and one array-level instance for the shared mooring line between the FOWTs. Alternatively, all the mooring lines could be represented in the array-level instance.

To account for potential couplings between FOWTs, the process for finding equilibrium in RAFT has been revised. The equilibrium solution is now determined at the array level in RAFT rather than by MoorPy. This allows for the use of multiple MoorPy instances for the mooring systems in the array and will allow other system properties such as hydrostatic forces or wake effects to be updated in future work.

The equilibrium solution process uses an iterative Newton's method solver to determine the values of the 6n system degrees of freedom that equilibrate the forces and moments on the FOWTs. During each iteration, the positions and orientations of all FOWTs in the array are set based on the array's combined degrees-of-freedom vector rarray. Next, the forces and stiffnesses on each FOWT are updated based on the new positions and orientations, and the position adjustment is computed for the next iteration. This process can take either a linear or nonlinear approach, as outlined in Fig. .

The linear approach creates a hydrostatic stiffness matrix for each FOWT (Chs) up front and then uses that same matrix to compute updated hydrostatic forces with each iteration: 11f¯hs(ξ¯)=Chsξ¯. The nonlinear approach recomputes the hydrostatic forces of each FOWT based on the instantaneous position and orientation: 12f¯hs(ξ¯)=f¯struc(ξ¯)+f¯hydro(ξ¯), where f¯struc is the structure weight and f¯hydro is the buoyancy. This new approach provides better support for unconventional platform geometries with nonlinear hydrostatics and, in the future, highly flexible substructures.

Similarly, the mean environmental loads from wind and currents (f¯env) can either be calculated once and kept constant or updated every iteration based on how changes in the platform position or orientation affect the loads: 13f¯env(ξ¯)=f¯wind(ξ¯)+f¯current(ξ¯). The calculation of these mean force components is described in previous work . This approach would account for the reduction in turbine thrust force when the platform and turbine are pitched. It could also be used when including wake effects that depend on how much turbines offset into or out of each other's wakes (a phenomenon explored with RAFT in ).

Because mooring systems are highly nonlinear, the mooring loads are always computed in a way that accounts for their nonlinearity. First, the mooring forces and moments of each FOWT's mooring system (if it exists) is computed directly from MoorPy: 14f¯moor,i(ξ¯i)=MoorPy(ξ¯i), where the introduced subscript i identifies a specific turbine in the array. Next, the mooring forces on each FOWT from the array-level MoorPy instance (if it exists) are calculated: 15f¯moor,array(ξ¯all)=MoorPy(ξ¯all), where the subscript “all” denotes the combined force or position vector of all turbines in the array.

Finally, the whole system's mean force vector is computed by combining the above terms: 16f¯net(ξ¯all)=f¯hs,1(ξ¯1)f¯hs,2(ξ¯2)…+f¯env,1(ξ¯1)f¯env,2(ξ¯2)…+f¯moor,1(ξ¯1)f¯moor,2(ξ¯2)…+f¯moor,array(ξ¯all). For efficiency, the solution process uses analytic Jacobians (or stiffness matrices) rather than finite differencing. Stiffness contributions are computed at the same time as each of the force contributions described above. The hydrostatic and mooring stiffness computations are inherently available in the MoorPy and RAFT calculation methods. Stiffness effects from environmental loadings acting at changing positions or orientations are currently neglected to avoid the need for numerical gradient calculations, since these terms are significantly smaller and unnecessary for convergence to the equilibrium. The stiffness terms from each FOWT and the array-level mooring system are combined in the equivalent format to the force combination process, Eq. (), as follows: 17Carray(ξ¯all)=Chs,1000Chs,2000…+Cmoor,1(ξ¯1)000Cmoor,2(ξ¯2)000…+Cmoor,array(ξ¯all), where the hydrostatic stiffnesses can be kept constant or recomputed as a function of displacement, corresponding to Eqs. () and (), respectively.

In the present work, the environmental forces on each FOWT are computed independently; no array-level hydrodynamic interactions or wake effects are considered. This paper primarily focuses on shared mooring systems, which is discussed more in the next section.

The main dynamic excitation affecting shared mooring lines is the phasing of wave loads between different turbines because the relative FOWT motions play a large role in the shared mooring system loads. Out-of-phase excitation from wind turbulence can also play a role, but wind turbulence is not considered in the present work. The relative phasing of wave loads on floating platforms in an array depends on the wave velocities and the platform's relative positions so the propagation of waves through the array needs to be accounted for.

We updated RAFT so that the wave loads for each platform are adjusted based on the mean platform positions and the wavelength vectors of each wave-frequency component. The wave phasing takes two forms according to whether hydrodynamics are modeled with strip theory or potential flow theory. For potential flow theory, a simple phase adjustment is applied to the wave excitation forces based on the FOWT's mean position, as follows: 18ζ̃(ω,x,y)=ζ̃0(ω)e-ik(ω)(xcos⁡β+ysin⁡β), where ζ̃ is the Fourier transform of the wave elevation at the platform's mean position (x, y), ζ̃0 is the wave elevation at the farm origin, ω is wave frequency, k is wave number, and β is wave heading.

For strip theory hydrodynamics, the proper wave phasing is achieved by using the existing formulation described in Sect.  but providing the global coordinates when computing the local wave kinematics acting on the substructure. In other words, the wave kinematics of Eq. () are computed using a phase-shifted wave elevation as in Eq. (), which inherently captures the proper phasing based on the position in the array.

Because RAFT does not resolve absolute phases, wave phase differences between turbines is only a relative phenomenon in RAFT. As a result, the implication of these phase differences is only for phenomena that are excited by relative motion, such as the loads felt by shared mooring lines, shared anchors, or fully suspended intra-array dynamic power cables.

Wave headings are handled directly by the strip theory approach, which is formulated in a 3-D way. For potential flow (linear) hydrodynamics, the frequency-dependent wave-excitation forces and moments can be computed by PyHAMS (through a built-in process in RAFT) or a priori with any preprocessor that produces excitation output files in the WAMIT format. Excitation data can be produced at a user-specified range of wave headings (such as at every 10°). Internally, RAFT interpolates the data to each wave heading that is being simulated.

Calculating the dynamic response for arrays of floating wind turbines with possible couplings, outlined in Fig. , is considerably more complex than RAFT's original method for individual FOWTs . First, RAFT computes the first-order response of each FOWT to wave excitation, including linearization of quadratic drag loads. This is done on a per-turbine basis and does not include the contribution of the array-level mooring system. In this way, the most computationally demanding step – the iterations when linearizing quadratic drag – is done on small system matrices for computational efficiency. For mooring systems that have a large influence on wave-frequency response, such as tension-leg platforms, the mooring system should be included in the FOWT-level MoorPy instance.

Once each FOWT response to wave excitation has been iteratively solved, the corresponding impedance matrix for each FOWT is saved: 19Zi(ω)=-ω2Mi(ω)+iωBi(ω)+Ci(ω). This matrix represents the combined effects from all reaction-force phenomena acting on the FOWT (mass, hydrodynamics, hydrostatics, aerodynamic damping, etc.).

Next, a system-level impedance matrix is constructed by combining the FOWT-level impedance matrices, along with the stiffness terms of the array-level MoorPy instance if it exists: 20Zarray(ω)=Z1(ω)000Z2(ω)000…+Kmoor,array. This final impedance matrix can then be used with the array-level excitation vector to solve for the coupled dynamic response of all FOWTs in the array to all sources of excitation, including different sea states as well as wind excitation. Each source of excitation is assumed to be uncorrelated, so the system response is computed independently for each source. The squares of the frequency-dependent response magnitudes from each excitation source are then summed to compute the overall response statistics. For efficiency, the effect of shared mooring lines are neglected when computing each FOWT's motion amplitudes during the iterative drag linearization. Afterward, when computing the final full system response, all coupling effects are included.

Mooring system tensions in RAFT are modeled by superimposing mean tensions, which are computed directly by MoorPy, with tension amplitudes. The tension amplitudes are complex frequency-dependent values found by multiplying the FOWT response amplitudes by mooring tension Jacobians, which are computed at the mean mooring system state. The mooring tension Jacobian is the variation in mooring line end tension relative to the platform's degrees of freedom: 21J=∂τ∂ξ, where τ is the vector of mooring line end tensions. This is calculated for each MoorPy instance, whether at the FOWT level or the array level.

Mooring line tension amplitudes are then computed for each mooring line based on the solved system response amplitudes: 22τ̃i(ω)=Jiξ̃i(ω) and 23τ̃array(ω)=Jarrayξ̃all(ω). Because each FOWT's entries in ξ̃all(ω) include the relative phase information based on each FOWT's position in the wave field, the tension responses of shared lines inherently capture the effect of relative motions between FOWTs.

Table  and Fig.  present the mean offsets and fairlead tensions calculated by RAFT and FAST.Farm. Mean tensions are included in RAFT's mean offset solution process. Figure  explains the nomenclature used for identifying the turbines and mooring lines in these results. The results show that RAFT matches the displacements and mooring line tensions computed by OpenFAST well, as would be expected considering that the model formulations are very similar for steady-state operation. This good level of agreement in both surge and pitch – and sway and roll for nonzero wind headings – indicates that the thrust force, the hydrostatic stiffness, and the mooring system stiffness are all in good agreement between models. The largest differences correspond to platform yaw, and they are still within the level of agreement expected from a low-fidelity model such as RAFT.

The dynamic response of the coupled shared mooring floating array depends on the phasing of wave excitation on each platform and both the absolute and relative wave-induced motions. Figure  compares the power spectral density (PSD) of wave elevation between RAFT and FAST.Farm to confirm that the realization of the sea state is similar between models. The level of agreement is good, as expected, because both models set up the spectrum in a way that ensures the overall energy of the spectrum will match the targeted significant wave height.

Figure  shows the PSDs of the 6 platform degrees of freedom for the first turbine from both RAFT and FAST.Farm. The figure shows generally good agreement, similar to what was seen in previous studies with a single turbine. The differences observed in roll and pitch can be explained by the strip theory hydrodynamic model in RAFT, which neglects wave radiation and wave scattering effects. These effects are included in FAST.Farm by using precomputed hydrodynamic coefficients obtained with WAMIT. RAFT can also use precomputed hydrodynamic coefficients, and the results shown from RAFT with WAMIT coefficients illustrate the closer agreement in the vertical degrees of freedom (heave, roll, and pitch) when including the wave radiation and wave scattering effects from WAMIT.

Figure  shows the PSDs of the mooring line tension at the six attachment points to the floating platforms. The results show generally good agreement in both magnitude and frequency distribution, except for well-understood differences caused by RAFT's quasi-static model not accounting for mooring dynamics. Specifically, RAFT's tension predictions have less high-frequency content than OpenFAST's because MoorPy neglects hydrodynamic drag. Also, the RAFT results do not have shared mooring line excitation for the perpendicular wave direction because the distance between the turbines stays constant, and hydrodynamic drag on the mooring line is neglected. The level of agreement in the mooring line tensions is better than was seen previously in verification with the normal VolturnUS-S mooring system, which has a catenary chain at 200 m water depth. The better agreement is likely due to the large water depth and use of synthetic ropes, which generally have less dynamic effects and can therefore be better predicted by the quasi-static mooring model used in RAFT.

The good agreement in the shared mooring line tension PSDs at lower frequencies (around 0.06 Hz) suggests that the new shared mooring capabilities in RAFT are modeled correctly. The shared mooring line tensions in the 0 and 45° cases have a distinct comb-like frequency response, which we explore more in Sect. . Comparing the wind and wave headings in Fig. , the differences can be explained as follows:

In the 0° case, the waves are in line with the turbine arrangement, which results in relative motions in all excited degrees of freedom. This also results in strong excitation of the shared mooring line because it extends in the same direction as the relative motion.