Dynamic Responses of an Offshore Floating Photovoltaic Array Under Wind and Wave Conditions

Abstract

To clarify the relationship between offshore floating photovoltaic (FPV) platform motions and mooring loads under combined wind and wave conditions, a numerical model based on three-dimensional potential flow theory was established to study the dynamic responses of a 4 × 4 offshore FPV array. Time-domain statistics and spectral analysis show that higher wind speed increases the mean surge offset. As wind speed rises from 10 to 15 m/s, the maximum and standard deviation (STD) of mooring line tension increase by 59.4% and 117.8%, respectively, while the low-frequency surge response weakens. Higher turbulence intensity increases the low-frequency peaks in the surge and mooring line tension spectra. As the spectral peak period increases, the heave and pitch spectral peaks shift toward lower frequencies. The wave-frequency surge response also becomes more pronounced. Higher significant wave heights amplify heave and pitch motions and mooring line tension, while weakening the low-frequency surge response. Across all cases, the end platforms have larger heave ranges; as surge displacement increases in the wave propagation direction, the wave-facing platform descends slightly and the wave-sheltered platform rises.

1. Introduction

As photovoltaic power generation expands, offshore floating photovoltaic (FPV) systems offer a new approach to alleviating land-use constraints [1]. Mountainous terrain can amplify wind speeds and modify wind fluctuations [2], whereas open offshore environments are less affected by such topographic effects. Compared with single platforms, modular arrays are better suited to deployment over large sea areas and facilitate fabrication, transportation, and installation [3]. A 5 MW nearshore FPV farm in Singapore demonstrates the practical application of modular systems in coastal waters [4]. Model tests of soft-connected arrays showed similar motion responses among modules under the investigated regular and irregular wave conditions [5]. Numerical analyses of multibody systems showed that the effects of multibody interactions on module motions varied with position [6]. PV module reliability may be affected by moisture ingress, dust, temperature variations, wind loads, vibrations, and corrosive gases [7]. In the offshore case examined by Ikhennicheu et al., wave loads accounted for approximately half of the estimated total environmental load [8]. The safety of platform connections and mooring systems under wind and wave loading is also an important consideration in offshore FPV design [9]. Accordingly, understanding array motion responses, differences in platform motions across the array, and mooring loads under combined wind and wave loading provides an important foundation for offshore FPV array design.

Experimental and numerical studies have shown that increasing wave height or wave steepness may amplify platform motions, wave loads, or mooring loads [10,11]. He et al. [12] further found that, for a hinged modular raft-type FPV system under regular waves, heave displacement and roll angle increased approximately linearly with wave height, whereas connector and mooring forces increased superlinearly. Wave period also affects platform motion characteristics. Gomis et al. [13] found that platforms follow wave motions more closely under long-period waves, whereas wave-facing platforms exhibit larger motion amplitudes under short-period waves. Wave overtopping and mooring dynamics can also jointly influence platform motion trajectories [11]. The dominant peaks in mooring load spectra generally coincide with the wave frequency [13]. At the array scale, Delacroix et al. [14] conducted regular- and irregular-wave tests on an FPV array and observed reduced motions of the rear floaters under short waves. For a six-module FPV system, Ji C. et al. [15] found that ball joints arranged along the wave-propagation direction helped reduce rotational motions and mooring tensions.

Building on studies of wave-induced responses, some researchers have introduced wind loading to investigate the motions and mooring loads of floating structures under combined wind and wave loading. Studies of floating offshore wind turbines have shown that aerodynamic forces affect the surge and pitch responses of semi-submersible platforms, while steady wind modifies the heave responses of tension-leg platforms [16,17,18]. Xu et al. [19] conducted fully coupled numerical simulations of three multi-floater array configurations under combined wind, wave, and current conditions. Their results showed that the rectangular arrangement exhibited better overall stability, lower mooring forces, and lower sensitivity to changes in environmental loading direction. Li et al. [6] investigated an FPV array consisting of four hexagonal platforms by varying the regular-wave period at a constant wind speed of 30 m/s. They found that the mean tension in the primary load-bearing mooring lines varied little over the investigated period range, while higher-order frequency components were more pronounced in mooring tensions than in platform motions. For a two-platform FPV system, Fu et al. [20] found that wind loading increased the surge RAO under long-period waves by up to 83% compared with wave-only conditions. Wind loading had limited effects on the heave and pitch RAOs. More recently, Wang et al. [21] investigated a four-module, flexibly connected FPV array under operational, design, and extreme wind–wave conditions using a two-way CFD/CSD coupling method. They found that the wave-facing platform exhibited larger motion responses than platforms farther inside the array, while mooring-line tensions showed a multi-peak distribution. Shi et al. [22] developed a frequency-domain hydroelastic model for a flexible offshore FPV structure under combined wind and wave loads, accounting for non-uniform wind-load distribution. Their results demonstrated that structural flexibility affects the predicted motion responses and internal force distribution. These studies show that wind loading affects the motion amplitudes and mooring tensions of floating structures.

A study of large continuous flexible FPV platforms found that the relationship between the characteristic wind-field scale and platform length affects surge responses. When wind loads distributed along the platform undergo phase cancellation, the overall motion response decreases [22]. Calculations for an integrated floating wind–aquaculture platform showed that, under the investigated sea conditions, low-frequency surge responses arise mainly from the combined contributions of second-order wave forces and turbulent wind loads [23]. Meanwhile, mooring constraints modify the relationship between motions and loads. Increasing the mooring stiffness of a soft-connected modular FPV array can reduce peak surge motions while increasing mooring tensions and their fluctuations [24]. Statistical results for a hinged multibody FPV array also showed that introducing random wind loads increased the maximum mooring tension in the system under the investigated normal sea state [25]. Li et al. [26] found that mooring tensions in a flexible pipe-type FPV array exhibited multi-peak nonlinear distributions, while inter-row interference led to non-uniform vertical displacements of photovoltaic panels. Although existing studies have examined FPV motions and mooring responses under different loading conditions, the relationship between array-wide motions and mooring loads under separately varied wind and irregular-wave parameters remains less clearly resolved for a fixed array configuration. Table 1 compares representative offshore FPV studies with the present study in terms of array configuration, environmental loading, and the primary comparison.

Table 1. Comparison of representative offshore FPV studies and the present study.

Reference	FPV Configuration	Environmental Loading	Primary Comparison
Jiang et al. (2023) [5]	Soft-connected lattice array	Regular and irregular waves	Module motions under regular and irregular waves
Li Y. et al. (2024) [6]	Four hexagonal platforms	Combined waves and wind	Multibody versus single-body models: differences in platform motions and mooring loads
Delacroix et al. (2023) [14]	4 × 4 articulated array	Regular and irregular waves	Motion differences between front and rear floaters at different wavelengths
Ji C. et al. (2024) [15]	Six-module semi-submersible system	Regular and irregular waves	Three connector designs for a six-module system
Xu et al. (2024) [19]	Six floats in three layouts	Wind, waves, and current	Rectangular, linear, and honeycomb-like array layouts
Fu et al. (2025) [20]	Two-platform system	Waves with and without wind	Wave-only versus wind–wave responses of a two-platform system
Wang et al. (2025) [21]	Four-module flexible floating-tube array	Combined wind and waves	Responses across operational, design, and extreme wind–wave conditions
Jin et al. (2026) [25]	Mooring-hinged multi-floater system	Irregular waves with and without wind	Module responses under irregular waves with and without wind
Li Q. et al. (2025) [26]	Flexible floating-pipe array	Fluctuating wind and random waves	Dynamic responses and mooring tensions at five wind–wave angles
Present study	4 × 4 ball-jointed array with four-point mooring	Irregular waves with and without turbulent wind	Time-domain statistics and spectra assess how mean wind speed, turbulence intensity, significant wave height, and peak period affect motion–mooring relationships and along-wave response differences.

Accordingly, this study uses a 4 × 4 offshore FPV array to examine the effects of mean wind speed, turbulence intensity, significant wave height, and spectral peak period. The relationship between platform motion responses and mooring-load variations under different environmental conditions is delved into through time-domain statistical and spectral analyses. Motion differences among platforms at different positions along the wave propagation direction are also revealed. The research workflow is shown in Figure 1. Section 2 introduces the coupled hydrodynamic and mooring numerical method based on three-dimensional potential flow theory, along with model validation and additional damping correction. Section 3 presents the dynamic response and mooring-load results. Section 4 summarizes the main conclusions.

Figure 1. Flowchart of the dynamic response analysis of an offshore FPV array under combined wind and wave loads.

2. Numerical Methods and Model Validation

2.1. Numerical Formulation

2.1.1. Three-Dimensional Potential-Flow Theory

The numerical model was developed based on three-dimensional potential flow theory, in which the fluid surrounding the platform is assumed to be ideal, irrotational, inviscid, and incompressible. Under these assumptions, the velocity potential at any point in the fluid domain satisfies the Laplace equation:

∇²φ=0

(1)

The velocity potential must satisfy the boundary conditions on the wetted body surfaces, free surface, and seabed, as well as the radiation condition at infinity. The total velocity potential is decomposed into the incident-wave, diffraction, and radiation potentials. The incident-wave potential is known, whereas the diffraction and radiation potentials are determined using the Green’s function method subject to the governing equation and corresponding boundary conditions. For a multibody FPV system, inter-platform hydrodynamic interactions, including radiation and shielding effects, must also be considered. The unsteady velocity potential of the system can then be obtained by extending the velocity potential formulation for a single FPV platform [20].

$$\phi (X,Y,Z;t)=\left[({\phi }_{\mathrm{I}}+{\phi }_{\mathrm{D}})+{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{j=1}^6{\phi }_{\mathrm{R}j}^{(m)}{X}_j^{(m)}}}\right]{\mathrm{e}}^{-\mathrm{i}\mathsf{\omega }\mathrm{t}}$$

(2)

where (X, Y, Z) is the position vector defined in the earth-fixed coordinate system; t denotes time; M is the total number of floating platforms; ${\phi }_{\mathrm{R}j}^{(\mathrm{m})}$ is the radiation potential induced by the motion of platform m in the jth degree of freedom, with all the remaining platforms held fixed; i is the imaginary unit; and ω is the wave angular frequency. The potential-flow model does not directly resolve viscous flow around the platforms. Additional heave and pitch damping is introduced to approximately represent local viscous dissipation, as described in Section 2.3.

2.1.2. Equations of Motion

The frequency-domain equations of motion for the platform can be expressed as:

$$\left\{-{\omega }^2\left[M+A(\omega )\right]-\mathit{i\omega B}(\omega )+C\right\}\tilde{X}={\tilde{F}}_j^m$$

(3)

where M denotes the platform mass matrix; A(ω) and B(ω) denote the added-mass and damping matrices, respectively; C is the hydrostatic stiffness matrix; ${\tilde{F}}_j^m$ is the complex wave-excitation force vector; and $\tilde{X}$ is the complex-valued motion-response vector in the frequency domain.

Based on the frequency-domain results, the time-domain equation of motion for the platform can be formulated in convolution-integral form.

$$\left[M+A_{\infty }\right]\ddot{X}(t)+{\displaystyle {\int }_0^{t}h}(t-\tau )\ddot{X}(\tau )\mathrm{d}\tau +\mathit{CX}(t)=F(t)$$

(4)

where A_∞ denotes the platform added-mass matrix; X(t) and $\ddot{X}(\tau )$ represent the displacement matrix and acceleration matrix of the platform, respectively; F(t) refers to the external excitation force matrix, which covers wave forces, mooring forces, wind forces and other environmental loads; and h(t) denotes the retardation function, which can be expressed as:

$$h(t)={\displaystyle \frac{2}{\pi }}{\displaystyle {\int }_0^{\infty }B}(\omega ){\displaystyle \frac{\sin (\mathit{\omega t})}{\omega }}\mathrm{d}\omega ={\displaystyle \frac{2}{\pi }}{\displaystyle {\int }_0^{\infty }\left[A(\omega )-A_{\infty }\right]}\cos (\mathit{\omega t})\mathrm{d}\omega$$

(5)

2.1.3. Wind Load

The wind-induced load is evaluated using a quadratic drag formulation based on the relative velocity between the ambient wind and the floating structure:

$$F_w=C_{d,w}\left|U_w-\dot{x}\right|\left(U_w-\dot{x}\right)$$

(6)

where F_w is the wind-induced force, C_d,w is the dimensional wind-force coefficient with units of N/(m/s)², U_w is the wind velocity, and $\dot{\mathrm{x}}$ is the velocity of the floating structure in the wind direction. The quadratic drag formulation represents the resultant wind load without resolving detailed airflow around the platforms.

2.1.4. Mooring System Model

Using the lumped-mass method, the mooring system is discretized into a series of nodal masses connected by line elements, with the mass of each mooring-line element concentrated at its corresponding nodes. By considering an arbitrary mooring-line element, the governing equation of motion can be established based on force equilibrium as follows:

$${\displaystyle \frac{\partial \mathbf{T}}{\partial {\mathit{S}}_e}}+{\displaystyle \frac{\partial \mathbf{V}}{\partial S_e}}+\mathit{w}+{\mathbf{F}}_{\mathit{mh}}=m{\displaystyle \frac{{\partial }^2\mathbf{R}}{\partial t^2}}$$

(7)

$${\displaystyle \frac{\partial \mathbf{M}}{\partial {\mathrm{S}}_e}}+{\displaystyle \frac{\partial \mathbf{R}}{\partial S_e}}\times \mathit{w}=-\mathit{q}$$

(8)

where T is the tension vector; M is the bending-moment vector; V is the shear-force vector; S_e denotes the total length of the mooring-line element; w and F_mh are the gravitational and hydrodynamic load vectors acting on the mooring-line element, respectively; m is the mass of the element; R denotes its position vector; and q is the distributed moment vector acting along the element.

2.2. Numerical Model Validation

To validate the established multi-body hydrodynamic model, the three-module model analyzed in the studies of Chen et al. [27] and Zheng et al. [28] was reproduced. The model consists of three identical rectangular floating modules arranged in series along the wave-propagation direction. In Figure 2, w_gap denotes the gap width between adjacent modules, which is 5 m. O marks the coordinate origin, and the arrows indicate the coordinate axes and incident-wave direction. The horizontal line represents the still-water level, and the inset shows the top view of a module.The geometric dimensions, mass properties, and water depth used in the simulations are summarized in Table 2. The calculated heave and pitch response amplitude operators (RAOs) of each module were compared with the numerical results reported in these two studies.

Table 2. Floating-module properties.

Module Characteristic	Value
Length, L (m)	100
Breadth, B (m)	50
Draught, D (m)	2
Water Depth (m)	50
Center of gravity above base, KG (m)	2.5
Radius of roll gyration, Rxx (m)	14.5
Radius of pitch gyration, Ryy (m)	28.9

Figure 2. Schematic of the three-module validation model (side and top views) [27].

Figure 2. Schematic of the three-module validation model (side and top views) [27].

The heave and pitch RAOs of the three modules were calculated over an angular frequency range of 0.01–2.51 rad/s and compared with the numerical results reported by Chen et al. [27] and Zheng et al. [28], as shown in Figure 3. The present results agree well with the reference results in terms of the variation in each module’s motion responses with wave frequency. Over the common comparison frequency range, the normalized mean absolute differences relative to Chen et al. [27] are 0.47–0.70% for heave and 1.09–4.47% for pitch. Relative to Zheng et al. [28], the corresponding differences are 0.61–0.95% and 1.59–2.08%, respectively. This agreement supports the model’s ability to calculate the heave and pitch responses of the three-module reference case under waves.

Figure 3. Comparison of (a) pitch and (b) heave RAOs of the three modules between the present study and the numerical results reported by Chen et al. [27] and Zheng et al. [28].

The reference case assesses the calculation of multi-body hydrodynamic motion responses, including their frequency dependence and differences among modules. Although the reference configuration differs in dimensions and geometry from the target FPV platform, both are analyzed within the same three-dimensional potential-flow framework using their respective geometries and properties. The target-platform responses are calculated independently rather than obtained by scaling the reference results. This comparison supports the model’s predictions of wave-induced multi-body motion responses, while the combined wind–wave mooring responses are presented as numerical predictions for the specified conditions.

2.3. Additional Damping Correction

Potential-flow models primarily account for radiation damping and have limited ability to capture energy dissipation caused by fluid viscosity, flow separation, and vortex shedding during platform motions. Heave and pitch have well-defined hydrostatic restoring stiffness, so their free-decay responses can be used to calibrate additional damping. Without moorings, displacement release does not produce an oscillatory surge decay response for calibration. Therefore, additional damping is calibrated only for heave and pitch. The corrected potential-flow free-decay responses are compared with CFD results. The CFD free-decay model was built in STAR-CCM+ for an isolated 7 m × 7 m platform (draft 0.339 m), without moorings or inter-platform connections. An overset mesh combined background and moving regions in a 60 m × 40 m × 30 m computational domain. The 14 m × 10 m × 9 m locally refined background region and 8 m × 8 m × 4.5 m overset region used mesh sizes of 0.20 m and 0.08 m, respectively, with approximately 2.82 million cells in total. Separate heave and pitch decays were initiated in still water at −0.30 m and 2°, respectively. The root mean square error (RMSE) between the corresponding motion time histories is used to quantify the overall agreement.

The additional damping calibrated for an isolated platform without moorings or connections is assigned to each module in the array to represent local viscous dissipation. Multibody hydrodynamic interactions and connection constraints are accounted for separately in the array model. As shown in Figure 4 and Table 3, increasing the additional damping ratio initially improves the agreement between the heave and pitch free-decay responses and the CFD results. Among the tested damping ratios, the RMSE is lowest at 12% for heave and 18% for pitch. Further increases in the damping ratio slightly raise the RMSE for both motions. With the selected corrections held constant, the RMSEs during the stages with larger and smaller amplitudes in the free decay histories are 0.0308 and 0.0382 m for heave and 0.1457° and 0.2009° for pitch, respectively. These corrections are treated as equivalent damping values calibrated over the complete decay histories. Therefore, additional damping ratios of 12% for heave and 18% for pitch, calibrated for the isolated 7 m × 7 m platform with a draft of 0.339 m, are adopted in the subsequent calculations of the 4 × 4 array.

Figure 4. Comparison of heave and pitch free-decay responses at different additional damping ratios with CFD results.

Table 3. RMSE comparison for heave and pitch free-decay responses at different additional damping ratios.

Heave				Pitch
Damping Ratio	RMSE (m)	Damping Ratio	RMSE (m)	Damping Ratio	RMSE (deg)	Damping Ratio	RMSE (deg)
0%	0.0897	4%	0.0567	0%	0.8162	4%	0.5811
6%	0.0471	8%	0.0406	6%	0.4739	8%	0.3857
10%	0.0367	12%	0.0352	10%	0.3142	12%	0.2585
14%	0.0355	16%	0.0368	14%	0.2173	16%	0.1911
-	-	-	-	18%	0.1793	20%	0.1795

2.4. Assessment of Stochastic Variability and Numerical Sensitivity

2.4.1. Effect of Random Seeds

To assess uncertainty in maxima and spectral trends, simulations last 1850 s, with the first 50 s discarded. Three realizations use seeds 12345, 54321, and 34521, varying wave seeds for wave-only cases and wind seeds for combined cases while fixing the wave seed at 12345. PSDs use Welch’s method with a 4096-point Hamming window, 50% overlap, and a 16384-point FFT. Table 4 and Figure 5 show that changing random seeds alters motion extremes and spectral peak amplitudes, while the dominant peak frequencies remain broadly unchanged. Thus, the principal frequency-domain response characteristics are retained under the prescribed environmental conditions, although peak magnitudes vary with the random realization.

Table 4. Motion statistics for three random realizations.

Case Type	Random Seed	Surge (m)			Heave (m)			Pitch (deg)
		Mean	STD	Max	Mean	STD	Max	Mean	STD	Max
OW	12345	0.822	0.657	2.135	−0.012	0.147	0.577	−0.013	0.978	3.507
OW	34521	0.811	0.712	2.386	−0.011	0.149	0.816	−0.015	0.989	5.325
OW	54321	0.807	0.676	2.306	−0.012	0.147	0.688	−0.013	0.976	3.874
WW	12345	2.975	0.333	1.485	−0.013	0.145	0.568	−0.042	0.965	3.519
WW	34521	2.983	0.308	1.741	−0.013	0.145	0.573	−0.042	0.965	3.190
WW	54321	2.989	0.299	1.401	−0.013	0.145	0.570	−0.042	0.964	3.222

Figure 5. Comparison of surge, heave, and pitch response PSDs for three random realizations under wave-only and combined wind–wave loading.

2.4.2. Sensitivity to Numerical Discretization

To assess the sensitivity of the numerical results to discretization settings, the panel-mesh size, number of frequency points, time step, and mooring-line discretization were varied individually from their baseline settings of 0.15 m, 35 frequency points, 0.01 s, and 100 elements per mooring line, respectively. The resulting motion-response and mooring-tension statistics were evaluated over 50–1850 s, as summarized in Table 5. Compared with the baseline case, the changes in mean module-motion STDs were all below 0.8%, while the changes in peak mooring tension did not exceed 1.25%. These results indicate that the calculated motion responses and mooring tensions are relatively insensitive to the tested numerical discretization settings.

Table 5. Sensitivity of motion responses and mooring tension to numerical discretization.

Parameter	Setting	Mean Module STD			Tension STD (kN)	Peak Tension (kN)	Peak Tension Change (%)
		Surge (m)	Heave (m)	Pitch (deg)
Original	0.15/35/ 0.01/100	0.33503	0.14463	0.96590	5.9531	56.8295	-
Panel-mesh	0.2	0.33570	0.14448	0.96379	5.9516	56.6167	−0.3744
Frequency	50	0.33246	0.14470	0.96498	5.9447	56.6593	−0.2995
Time-step	0.012	0.33549	0.14348	0.95864	5.9302	56.1233	−1.24
Mooring	80	0.33502	0.14463	0.96590	5.9530	56.8405	+0.0194

3. Effects of Combined Wind and Wave Loading on the Dynamic Responses of the FPV Array

3.1. Array Model Parameters and Simulation Conditions

3.1.1. Array Model and Mooring Layout

To investigate the dynamic responses of offshore FPV systems under combined wind and wave loading, this study establishes a numerical model of a 4 × 4 FPV array. Each FPV platform has plan dimensions of 7 m × 7 m, with floaters beneath the platform measuring 0.678 m in diameter. The platform draft is 0.339 m, and the displacement volume is 4.5664 m³. Each platform has a mass of 4680.6 kg, with moments of inertia about its center of gravity of 34,512, 34,512, and 56,330 kg·m² about the X, Y, and Z axes, respectively. The center of gravity is located at the horizontal center of each platform, approximately 0.903 m above the still-water level. The freeboard is 2.670 m, the clear gap between adjacent platforms is 0.8 m, and the water depth is 15 m.

Adjacent platforms are connected by ball-and-socket joints located midway between them at the still-water level. The joints maintain array connectivity by constraining relative translations while permitting relative rotations, with identical rotational stiffness and damping of 100 kN·m/° and 20 kN·m/(°/s), respectively, about each global axis. Joint loads are not analyzed in this study.

A four-point mooring system is adopted, with each mooring line measuring 63 m in length. The array is secured by catenary mooring lines with equivalent properties comparable to steel chains. Each line has an equivalent diameter of 0.107 m, a submerged weight of 95.2 N/m, and an axial stiffness of 90,000 kN. The initial platform-end tension of each mooring line is 4.844 kN. One anchor is located at (−56.7, −56.7, −15) m, with the others arranged symmetrically about the global X and Y axes. The horizontal seabed is at z = −15 m, with the mud-layer depth of 2 m. Both inline and lateral seabed friction coefficients are set to zero. The array geometry and mooring layout are shown in Figure 6b.

Figure 6. Model and mooring layout of the 4 × 4 offshore FPV array: (a) plan view of the array; (b) side view and mooring system.

The panel inclination is 0°. Wind-force coefficients are calculated by integrating the projected contributions of exposed surface elements, using drag coefficients of 1.28 for the main PV surfaces and 1.20 for the remaining surfaces. For wind along the X direction, the total projected area of the exposed surface elements is 16.334 m² per platform, giving a horizontal wind-force coefficient of 12 N/(m/s)². The equivalent wind forces are applied at each platform’s center of gravity, and the corresponding aerodynamic moments are calculated about the same point; the pitch-moment coefficient is 4.318 N·m/(m/s)². All platforms use the same coefficient set, without aerodynamic shielding corrections between platforms.

3.1.2. Natural Modes and Simulation Conditions

Modal analysis is first performed on the 4 × 4 FPV array to determine its natural frequencies and mode shapes.

Figure 7 shows the eight mode shapes of the 4 × 4 FPV array and their corresponding natural frequencies. Modes 1 and 2 have natural frequencies of 0.1530 and 0.1532 rad/s, respectively, and both exhibit coupled surge, sway, and yaw motions. Mode 3 has a natural frequency of 0.2087 rad/s and primarily exhibits coupled sway and yaw motions. Modes 4–6 have natural frequencies ranging from 0.2129 to 0.2138 rad/s and are dominated by surge motion. These six modes lie in a relatively low-frequency range. In contrast, Modes 7 and 8 have higher natural frequencies of 1.6312 and 2.0108 rad/s, respectively. Mode 7 primarily exhibits coupled roll and pitch motions, whereas Mode 8 exhibits coupled heave, roll, and pitch motions.

Figure 7. Natural frequencies and dominant motion modes of the 4 × 4 FPV array.

To investigate the effects of different environmental parameters on the motion responses of the FPV array, six groups of irregular-wave cases are defined: OW- H_s, OW- T, WW-H_s, WW-T, WW-U and WW-TI. Here, OW denotes wave-only conditions, and WW denotes combined wind and wave conditions. The first four groups are used to examine the effects of significant wave height and spectral peak period, while the remaining two groups are used to examine the effects of mean wind speed and turbulence intensity. The parameters for each case are listed in Table 6. Irregular waves are generated using the JONSWAP spectrum, with a peak enhancement factor of γ = 3.3. The full difference-frequency QTF matrix was not calculated; therefore, second-order difference-frequency wave-drift loads were not included in the present simulations. Turbulent wind histories follow an NPD wind spectrum referenced to 10 m above the still-water level, with prescribed turbulence intensity. For the 10 m/s wind case with 17% turbulence intensity, the longitudinal integral scale calculated from the wind-velocity autocorrelation is 99.7 m. The longitudinal integral scale was calculated by integrating the normalized wind-velocity autocorrelation function up to its first zero crossing and multiplying the integral by the mean wind speed. All 16 modules receive the same wind history, assuming fully coherent fluctuations without spatial variation or time delays. For the combined wind and wave cases, the prescribed wind and irregular-wave inputs are applied simultaneously in the time-domain simulations.

Table 6. Wind and wave conditions considered in the simulations.

Case Type	Hs (m)	Tp (s)	U (m/s)	TI (%)
OW-Hs	0.4/0.7/0.9/1.3/1.8	5.5	-	-
OW-T	0.9	4.5/5.5/6.5/7.5/8.5/9.5	-	-
WW-Hs	0.4/0.7/0.9/1.3/1.8	5.5	10	17
WW-T	0.9	4.5/5.5/6.5/7.5/8.5/9.5	10	17
WW-U	0.9	5.5	6/8/10/12/15	17
WW-TI	0.9	5.5	10	8/11/14/17

In this study, motion responses are evaluated for each platform individually. For irregular-wave and combined wind–wave conditions, the standard deviation (STD) of each platform’s motion time history is calculated to quantify its fluctuation intensity about the mean position. The STDs of the 16 platforms are then arithmetically averaged to obtain the mean module STD, which represents the average fluctuation intensity across the array rather than collective array motion. Power spectral density (PSD) analysis is also performed to examine the frequency content of the motion responses and identify dominant spectral peaks [29]. Differences in responses among platforms along the wave-propagation direction are examined separately in Section 3.4.

3.2. Effects of Wind Field Characteristics and Irregular Wave Parameters on the Motion Responses of the FPV Array

The effect of mean wind speed on array motions is examined with H_s = 0.9 m and T_p = 5.5 s. Mean wind speed U ranges from 0 to 15 m/s. The U = 0 case represents wave-only conditions. A constant turbulence intensity of 17% is adopted for the turbulent wind cases. Figure 8 presents the surge, heave, and pitch time histories of the array and the corresponding PSD results, focusing on changes in the mean offset and responses in different frequency ranges with mean wind speed.

Figure 8. Time histories and PSD results for the surge, heave, and pitch responses of the FPV array at different mean wind speeds.

As shown in Figure 8, under wave-only conditions, the surge time history exhibits pronounced slow variations with superimposed local fluctuations, whereas heave and pitch oscillate around zero. As the mean wind speed increases, the mean surge position shifts in the positive direction, and the slow fluctuations about the mean position gradually diminish. The heave and pitch time histories remain similar across the cases, with only small differences in the timing and amplitudes of local peaks and troughs. Neither exhibits the pronounced shift in mean position observed in surge.

As shown in Figure 8b, the surge PSD is mainly concentrated below 0.5 rad/s. Across the wind-speed cases, the dominant surge spectral peaks lie between 0.104 and 0.215 rad/s. The upper end of this range is close to the reported surge-dominated natural frequencies. As the mean wind speed increases, the magnitudes of the dominant low-frequency surge peaks decrease, while the response near the wave spectral peak remains small. In Figure 8d,f, the dominant heave and pitch spectral peaks occur at approximately 1.120 rad/s, close to the wave spectral peak angular frequency of 1.143 rad/s. Their responses are mainly concentrated in the wave-frequency range, with only small changes in peak locations and overall spectral shapes across the wind-speed cases. To further distinguish changes in mean offset from changes in motion range, Figure 9 compares the mean surge positions and the maximum and minimum displacements at different mean wind speeds, spectral peak periods, and significant wave heights. This indicates that the low-frequency surge response is associated with the low-frequency horizontal modes of the interconnected array.

Figure 9. Mean, maximum, and minimum surge displacements of the FPV array under different wind and wave conditions.

In Figure 9a, an increase in mean surge offset is accompanied by a narrowing of the fluctuation range. The additional mean surge offset relative to the wave-only case rises from 1.27 m at U = 6 m/s to 2.54 m at U = 15 m/s. Its rate of increase slows at higher wind speeds. The lower displacement bound rises substantially, whereas the upper bound changes relatively little. As mean wind speed increases from 0 to 15 m/s, the surge displacement range decreases from approximately 3.6 to 1.6 m, a reduction of about 56%. The next section examines mooring tension time histories, PSD results, and statistics to assess how mooring loads change as the surge displacement range narrows.

As shown in Figure 9b, the mean surge offset under wave-only conditions reaches its maximum at T_p = 5.5 s and then decreases slightly as the period increases. Under combined wind and wave loading, the mean surge position remains around 3.0 m and changes little with increasing spectral peak period. Across spectral peak periods of 4.5–9.5 s, the surge displacement range under combined wind and wave loading remains around 2 m and is smaller than that under the corresponding wave-only condition at each period. Both the mean surge position and the fluctuation range are relatively insensitive to changes in spectral peak period.

In Figure 9c, the mean surge offset increases with significant wave height under both loading conditions, with a greater increase under wave-only conditions. As the significant wave height increases from 0.4 to 1.8 m, the difference between the maximum and minimum displacements increases from approximately 1.4 to 4.2 m under wave-only conditions, whereas it remains within 2.0–2.2 m under combined wind and wave loading. At smaller wave heights, the fluctuation range under combined wind and wave loading is larger than that under wave-only conditions. At larger wave heights, the range under wave-only conditions expands and exceeds that under combined wind and wave loading. The surge fluctuation range is therefore markedly less sensitive to wave height when wind loading is included.

Figure 10 compares the surge, heave, and pitch PSDs at different spectral peak periods alongside the corresponding incident wave spectra.

Figure 10. PSD results for the motion responses of the offshore FPV array at different wave spectral peak periods.

In Figure 10a, the surge response is mainly concentrated below 0.5 rad/s, and the wave-frequency response is weak at shorter spectral peak periods. As the spectral peak period increases, the wave spectral peak shifts toward lower frequencies, and a distinct local peak appears in the surge response near the corresponding wave spectral peak, indicating an increased contribution from wave-frequency components. At longer periods, the additional response peak is clearly separated from the low-frequency spectral peak, indicating the coexistence of low-frequency and wave-frequency surge responses.

In Figure 10b,c, the main spectral peaks of the heave and pitch responses shift toward lower frequencies together with the wave spectral peak. As the spectral peak period increases from 4.5 to 9.5 s, the wave spectral peak angular frequency decreases from approximately 1.396 to 0.661 rad/s, and the heave and pitch spectral peaks generally become more pronounced under longer-period conditions. Unlike surge, which consistently retains a distinct low-frequency component, the main heave and pitch responses are concentrated near the wave spectral peak, and their response frequency ranges vary primarily with the wave period.

Figure 11 compares motion PSDs at different significant wave heights to assess the response of each degree of freedom.

Figure 11. Motion response PSDs of the offshore FPV array at different significant wave heights.

In Figure 11a, as the significant wave height increases, the peak magnitude of the wave spectrum increases, and the local peak in the surge PSD near the wave spectral peak frequency also gradually increases. In contrast, although multiple local peaks appear in the surge PSD within the low-frequency range of 0.104–0.215 rad/s, their magnitudes decrease with increasing significant wave height.

In Figure 11b,c, the main spectral peaks of the heave and pitch responses are both concentrated near the wave spectral peak. Their magnitudes increase substantially with significant wave height, while their peak frequencies remain nearly unchanged. At a significant wave height of 1.8 m, the main heave and pitch spectral peaks reach approximately 0.22 m²/(rad/s) and 12 deg²/(rad/s), respectively.

Unlike the shifts in spectral peak frequencies caused by changes in spectral peak period, increasing significant wave height primarily enhances the wave-frequency response without substantially changing the dominant response frequency range. At the largest tested wave height of 1.8 m, the maximum wave elevation at the available probe was 1.355 m, below the nominal freeboard of 2.670 m, and the maximum relative pitch between adjacent platforms was 4.53°; thus, overtopping was not expected at the monitored location.

The preceding results show that changes in wave parameters primarily affect the wave-frequency heave and pitch responses, while significant wave height also influences the low-frequency surge response. To further examine the effects of wind-field fluctuations, Figure 12 compares the motion response PSDs at different turbulence intensities, with the mean wind speed U = 10 m/s, significant wave height H_s = 0.9 m, and spectral peak period T_p = 5.5 s held constant.

Figure 12. Motion response PSDs of the offshore FPV array at different turbulence intensities.

In Figure 12a, the surge PSD increases with turbulence intensity primarily at frequencies below 0.5 rad/s. At higher turbulence intensities, several local peaks in the low-frequency surge PSD increase in magnitude. The surge PSD curves near the wave spectral peak frequency remain close to one another and show no substantial increase with increasing turbulence intensity, indicating that the changes in surge response occur primarily in the low-frequency range.

In Figure 12b,c, the heave and pitch PSD curves at different turbulence intensities nearly overlap. Their main spectral peaks are located near the wave spectral peak frequency, with magnitudes of approximately 0.06 m²/(rad/s) and 3.3 deg²/(rad/s), respectively. The spectral peak frequencies and overall spectral shapes change little. Whereas increasing significant wave height substantially increases the wave-frequency heave and pitch spectral peaks, increasing turbulence intensity mainly enhances the low-frequency surge response and has little effect on the wave-frequency heave and pitch responses.

3.3. Effects of Wind Field Characteristics and Irregular Wave Parameters on the Mooring Loads of the FPV Array

All mooring-tension plots show Line A, the windward-side mooring line selected for analysis under the fixed incident-wave direction and symmetric mooring layout. Wind loading shifts the mean surge position of the array and changes its motion range. To examine the corresponding mooring response, Figure 13 presents the mooring tension time histories and PSD results at different wind speeds.

Figure 13. Mooring tension time histories and PSDs at different mean wind speeds.

As shown in Figure 13a, under wave-only conditions, the overall mooring tension is relatively low, and its fluctuation range is small. As the mean wind speed increases, both the overall tension level and the peak-to-trough range increase substantially, with more pronounced instantaneous peaks at higher wind speeds. Large tension peaks occur at similar times across the wind-speed cases, while their magnitudes increase with mean wind speed. Therefore, the significant wave height and spectral peak period of irregular waves should be considered when analyzing the effects of wind speed on mooring tension fluctuations.

As shown in Figure 13b, the mooring tension PSD generally decreases with increasing frequency. The increase in PSD values with wind speed is mainly concentrated in the low-frequency range below 0.4 rad/s. A weaker wave-frequency response is also present at approximately 0.8–1.2 rad/s and becomes more pronounced at higher wind speeds. Together with Figure 8, these results show that the dominant low-frequency surge PSD peak decreases with increasing mean wind speed, whereas the low-frequency mooring tension PSD values increase. To quantify changes in tension fluctuations, Figure 14 further compares the STD of mooring tension at different mean wind speeds, spectral peak periods, and significant wave heights.

Figure 14. Effects of wind speed and wave parameters on the STD of mooring tension: (a) wind speed; (b) spectral peak period; (c) significant wave height.

As shown in Figure 14a, the STD of mooring tension increases with mean wind speed. When mean wind speed increases from 10 to 15 m/s, mooring tension STD rises to 2.18 times its value at 10 m/s. This agrees with the wider tension range in Figure 13a and higher low-frequency PSD values in Figure 13b. Together, Figure 8, Figure 9 and Figure 14a show that mooring tension STD continues to increase as the growth in mean surge offset slows and the low-frequency surge response weakens. Thus, as mean wind speed increases, weaker low-frequency surge motion does not necessarily imply lower mooring-tension fluctuations.

To further interpret this trend, the surge-direction tangent stiffness of the mooring system increased from 25.46 to 60.61 kN/m as the mean wind speed increased by 20% (from 10 to 12 m/s). The equivalent relative-wind damping per platform increased from 240.1 to 288.1 N·s/m. The increased damping suppresses surge fluctuations, while the higher mooring stiffness makes tension more sensitive to the remaining motion; the greater mean wind load also contributes to the increase in maximum mooring tension.

In Figure 14b, the mooring tension STD under wave-only conditions increases slightly from T_p = 4.5 to 5.5 s, then gradually decreases as the period increases. Under combined wind and wave loading, mooring tension STD generally increases with spectral peak period. As T_p increases from 4.5 to 9.5 s, the difference from the wave-only condition rises overall by approximately 46.4%, from 4.83 to 7.07 kN. Under combined wind and wave loading, the mean surge position changes little with spectral peak period, but mooring tension STD increases. Period changes therefore affect tension fluctuations even when the mean surge position remains nearly unchanged.

As shown in Figure 14c, the tension STD increases with significant wave height under both loading conditions, with a more pronounced increase at larger wave heights. The values under combined wind and wave loading remain higher than those under wave-only conditions, and the difference between them increases overall from 4.50 to 5.64 kN. Taken together, Figure 9c and Figure 14c show that, under combined wind and wave loading, the surge displacement range changes little as significant wave height increases, whereas mooring tension fluctuations increase substantially.

Mooring tension STD quantifies load fluctuations, while maximum tension identifies the peak load. Figure 15 shows how maximum mooring tension varies with mean wind speed, spectral peak period, and significant wave height.

Figure 15. Effects of wind speed and wave parameters on maximum mooring tension: (a) wind speed; (b) spectral peak period; (c) significant wave height.

In Figure 15a, as mean wind speed increases from 10 to 15 m/s, maximum mooring tension under combined wind and wave loading rises to 1.59 times its value at 10 m/s. This agrees with the more pronounced tension spikes at higher wind speeds in Figure 13a. Together, Figure 9a, Figure 14a and Figure 15a show that mean surge offset grows more slowly as mean wind speed increases, while mooring tension STD and maximum tension continue to rise.

In Figure 15b, maximum tension under combined wind and wave loading varies non-monotonically with spectral peak period. It reaches relatively high values at T_p = 6.5 and 8.5 s, with a decrease at 7.5 s. It remains substantially higher than that under wave-only conditions at all periods considered. This trend differs from the generally increasing tension STD in Figure 14, indicating that maximum tension also depends on the formation of instantaneous peaks and cannot be determined solely from the overall level of fluctuations.

In Figure 15c, the maximum tension increases substantially with significant wave height under both loading conditions, with a more pronounced increase at larger wave heights. The difference between the two loading conditions reaches 43.4 kN at H_s = 0.9 m and is 30.4 and 30.2 kN at H_s = 1.3 and 1.8 m, respectively. Although this difference decreases, the maximum tension continues to rise under both loading conditions. Beyond H_s = 0.9 m, maximum mooring tension increases more rapidly under wave-only conditions, although it remains higher under combined wind and wave loading.

Wind loading shifts the array in the positive surge direction. As mean wind speed increases, the mean surge offset grows more slowly and the dominant low-frequency surge PSD peak decreases, while mooring tension STD and maximum tension continue to rise.

To isolate the effect of wind speed fluctuations, turbulence intensity is varied from 8% to 17% while mean wind speed, significant wave height, and spectral peak period are held constant. Figure 16 compares the mooring tension time histories, PSDs, and statistics across these cases.

Figure 16. Mooring tension responses at different turbulence intensities: (a) time histories; (b) power spectral densities (PSDs); (c) statistical characteristics.

As shown in Figure 16a, the mooring tension time histories follow similar overall patterns across turbulence intensities, with large peaks occurring at broadly corresponding times. As the turbulence intensity increases, some peaks rise and some troughs fall, widening the tension fluctuation range, while mean tension changes little.

As shown in Figure 16b, the mooring tension PSD generally decreases with increasing frequency, and differences between the cases are mainly concentrated below 0.4 rad/s. As the turbulence intensity increases, the PSD values within this frequency range generally rise, whereas the curves at higher frequencies remain close to one another. This agrees with the increase in low-frequency surge PSD values in Figure 12a. Increasing turbulence intensity therefore mainly enhances the low-frequency surge and mooring tension responses, with little effect on wave-frequency components. Figure 16c shows that mean tension remains around 16 kN, while tension STD increases with turbulence intensity. The maximum tension is approximately 49 kN at both 8% and 11% turbulence intensity and rises to approximately 57 kN at 17%, an increase of about 16% from the 8% case. Thus, increasing turbulence intensity amplifies mooring tension fluctuations and peak loads while leaving mean tension nearly unchanged. Together with the surge results in Figure 12a, this indicates that increased turbulence primarily enhances low-frequency fluctuations, with little change in the mean surge position or mean mooring tension.

3.4. Differences in Platform Motion Responses Within an FPV Array Along the Wave Propagation Direction Under Combined Wind and Wave Loading

Platforms F(1,2)–F(4,2) are selected to represent different positions along the wave propagation direction. Their vertical-plane motion trajectories are compared across mean wind speeds, turbulence intensities, significant wave heights, and spectral peak periods. The wave-only case is included in the mean wind speed comparison. Each trajectory represents platform motion about its mean surge position.

Figure 17a shows that the trajectories of F(1,2) and F(4,2) tilt slightly in opposite directions. As the mean wind speed increases from 0 to 15 m/s, the surge motion ranges of all four platforms gradually decrease. The trajectories of F(1,2) and F(4,2) become more vertically oriented, while those of F(2,2) and F(3,2) become more concentrated. In contrast, Figure 17b shows that increasing turbulence intensity from 8% to 17% generally expands the surge motion ranges of all four platforms. Changes in mean wind speed and turbulence intensity do not alter the overall heave pattern: F(1,2) and F(4,2) generally have larger heave motion ranges than F(2,2) and F(3,2).

Figure 17. Vertical-plane motion trajectories of platforms in the 4 × 4 array under different conditions: (a) mean wind speeds; (b) turbulence intensities.

Figure 18a shows that increasing significant wave height markedly expands the heave motion ranges of all four platforms. The trajectories of F(1,2) and F(4,2) tilt toward opposite sides. Figure 18b shows that increasing the spectral peak period also expands the motion trajectories, and the loop-shaped trajectory of F(1,2) becomes more pronounced. Across the wave height and spectral peak period cases considered, F(1,2) and F(4,2) generally have larger heave motion ranges than F(2,2) and F(3,2).

Figure 18. Vertical-plane motion trajectories of platforms in the 4 × 4 array at different significant wave heights and spectral peak periods.

Taken together, Figure 17 and Figure 18 show that increasing mean wind speed narrows the surge motion range of each platform, whereas increasing turbulence intensity widens it. Increasing significant wave height mainly expands the heave motion ranges, while a longer spectral peak period makes the motion trajectory more pronounced. The trajectories show that F(1,2) generally descends slightly as its surge displacement increases in the wave propagation direction and rises as it moves back. F(4,2) shows the opposite pattern: it rises as its surge displacement increases and descends as it moves back. Despite changes in motion ranges and trajectory shapes, F(1,2) and F(4,2) generally retain larger heave motion ranges than F(2,2) and F(3,2).

Following the trajectory analysis of four representative platforms, statistical measures are used to examine the motion responses of all 16 platforms.

Figure 19 reports the 16-platform averages of the mean, STD, and maximum–minimum range of surge, heave, and pitch. As mean wind speed rises from 0 to 15 m/s, mean surge offset increases from 0.82 to 3.36 m, while surge STD falls by 75.3% and its range narrows from 3.61 to 1.55 m. Raising turbulence intensity from 8% to 17% changes mean surge offset by only 0.052 m, but increases surge STD and range to 1.67 and 1.70 times their initial values. Heave and pitch STDs change by no more than 2.2% across either set of wind cases.

Figure 19. Statistical characteristics of platform motion responses under different wind and wave parameters.

As significant wave height rises from 0.4 to 1.8 m, heave and pitch STDs reach 4.57 and 4.37 times their initial values. Their ranges expand from 0.39 to 1.80 m and from 2.61° to 11.83°, while the surge range stays between 1.99 and 2.13 m. As spectral peak period rises from 4.5 to 9.5 s, heave STD reaches 1.60 times its initial value and its range widens from 0.86 to 1.16 m. The pitch range falls by 21.3%, while mean surge offset stays between 2.88 and 2.99 m.

Mean wind speed shifts and narrows the surge response; turbulence intensity widens it. Significant wave height amplifies heave and pitch, while spectral peak period increases heave variability and reduces the pitch range.

4. Conclusions

Based on three-dimensional potential flow theory, a numerical hydrodynamic model was established for a 4 × 4 FPV array. Time-domain statistics and spectral analysis were used to systematically examine the relationship between array motions and mooring loads under different wind and wave parameters. The conclusions are summarized as follows.

(1) Compared with wave-only conditions, wind loading induces a pronounced mean surge offset of the array. As mean wind speed increases, mean surge offset grows more slowly, while the surge displacement range and dominant low-frequency PSD peak decrease. This indicates that weaker low-frequency surge motion does not necessarily correspond to lower mooring-tension fluctuations. Mooring tension STD and recorded maximum tension continue to increase. As mean wind speed rises from 10 to 15 m/s, tension STD increases more in relative terms than maximum tension, reaching 2.18 times its value at 10 m/s compared with 1.59 times for maximum tension. Higher turbulence intensity has little effect on mean surge position or mean mooring tension, but increases low-frequency fluctuations in both responses and raises peak tension. Thus, under the investigated conditions, turbulence intensity mainly affects the fluctuating rather than the mean responses.

(2) Under combined wind and wave conditions, the mean surge position changes little with wave conditions, but the surge displacement range is substantially reduced relative to wave-only conditions, with reductions of 45.2% at T_p = 6.5 s and approximately 52.2% at H_s = 1.8 m. As the spectral peak period increases, the main heave and pitch response peaks shift toward lower frequencies, while a distinct surge response peak appears near the wave spectral peak. Increasing significant wave height raises the main wave-frequency heave and pitch spectral peaks, lowers the low-frequency surge spectral peak, and generally increases mooring tension STD and maximum tension.

(3) Across all wind and wave conditions examined, the end platforms F(1,2) and F(4,2) have larger heave motion ranges than the interior platforms F(2,2) and F(3,2). As surge displacement increases in the wave propagation direction, F(1,2) descends slightly, whereas F(4,2) rises. Higher mean wind speed narrows the surge motion range of each platform, while higher turbulence intensity widens it. Increasing significant wave height mainly expands the heave motion range, with a greater increase at the end platforms. As the spectral peak period increases, the motion trajectories show a noticeable reduction in small-scale fluctuations and generally span a wider range.

This study does not calibrate additional surge damping. Unlike heave and pitch, surge has negligible hydrostatic restoring stiffness when unmoored and therefore lacks a suitable oscillatory free-decay response. A moored decay test would involve coupled platform–mooring dynamics, making the platform’s hydrodynamic damping difficult to isolate. Therefore, only the frequency-dependent radiation damping from three-dimensional potential-flow theory was retained. The influence of uncalibrated viscous surge damping on the predicted motion responses and mooring tensions has not been quantitatively assessed and warrants further investigation. In addition, the effects of spatially varying and partially coherent turbulent winds on surge responses and mooring-tension spectra were not assessed. Future work will examine the effects of spatially correlated wind fields and conduct further investigations through wave basin experiments.