Irregular or random waves representing various stochastic sea states are characterized as the sum or combination of multiple wave elements defined from a suitable wave spectrum. This article includes the Joint North Sea Wave Project (JONSWAP) spectrum for the irregular wave in order to specify the wave spectrum. The one-sided JONSWAP spectrum is described in the IEC 61400-3 design standard as follows (Jonkman, 2007): S ζ 1 - Sided ω = 1 2 π 5 16 H s 2 T p ω T p 2 π − 5 × exp − 5 4 ω T p 2 π − 4 1 − 0.287 ⁡ ln γ × γ exp − 0.5 ω T p 2 π − 1 σ ω 2 where H s and T p are the significant wave height in meters and the peak spectral period in seconds, respectively. γ , σ and ω represent the peak shape parameter of an imposed irregular sea state, a scaling factor and the frequency of incident waves, respectively. According to the IEC 61400–3 design standard recommendation, the scaling factor can be defined in Equation 1 as follows: σ ω = 0.07  for    ω ≤ 2 π T p 0.09  for    ω > 2 π T p (1) and the peak shape parameter can be expressed as Equation 2: γ = 5  for    T p H s ≤ 3.6 exp 5.75 − 1.15 T p H s  for    3.6 < T p H s ≤ 5 1  for    T p H s > 5 (2)

After defining the equation of the waves, the general equation of motion of the FOWT in frequency-domain can be described as: I FOWT ω x ̈ ⃗ + B FOWT ω x ̇ ⃗ + C FOWT x ⃗ = f ⃗ FOWT ω + f ⃗ PTO ω (3) where I FOWT , B FOWT and C FOWT represent the inertia elements, the damping elements and the stiffness matrix, respectively. f ⃗ FOWT ( ω ) defines the hydrodynamic force and waves viscous drag on the platform and f ⃗ PTO ( ω ) expresses the load craeted by the power take-off (PTO) equipment from the OWCs. The system states in Equation 3 are defined in Equation 4 as follows: x ⃗ = s u r g e s w a y h e a v e r o l l p i t c h y a w f o r e - a f t s i d e - t o - s i d e (4)

Note that the states correspond to the rigid-body motions of the structure. The inertia elements of the FOWT are specified in Equation 5 as following: I FOWT ω = A Hydro ω + M Platform + M Tower (5) the term M Platform is the platform mass and M Tower is the tower mass with the nacelle-rotor-blades assembly. A Hydro stands for the platform’s added mass in frequency domain, which may be calculated using WAMIT from the panel radiation program.

The stiffness element C FOWT is given by Equation 6 as following: C FOWT = C Hydro + C Mooring + C Tower (6) where C Hydro is the platform hydrostatic restoring matrix, C Mooring is spring stiffness of the mooring lines and C Tower is the tower stiffness matrix.

The damping elements can be obtained by Equation 7 as follows: B FOWT ω = B Hydro ω + B Tower + B viscous + B chamber (7) here B Hydro , B Tower and B viscous describe the platform damping matrix, the tower damping matrix and the nonlinear viscous drag, respectively. B chamber is the PTO’s effect as e xternal force. Note that the internal free surface has a behavior like a piston so that the pressure is uniform within the chamber (Aubault et al., 2011). Hence, the external force can be given by Equation 8: f PTO ω = − p ω S (8) where p is the pressure drop inside the turbine and S is the internal free surface area of water. Assuming that the air is an ideal gas and the air compression/decompression is an isentropic process, the time-dependent air density may be described as: ρ = ρ a p p 0 1 γ (9) the terms ρ 0 and p 0 describe the density and pressure that signify the state of the chamber at rest, respectively. γ expresses the heat capacity ratio of air. After linearizing the time derivative of Equation 9, Equation 10 can be obtained as follows: ρ ̇ = ρ 0 γ p 0 p ̇ (10)

The linearized mass flow inside the turbine is described in Equation 11 as (M’zoughi et al., 2018): m ̇ = d ρ V d t = ρ a γ p a p ̇ V a + ρ a V ̇ (11) where V and V 0 define the air volume within the chamber and the air volume of the chamber in the rest state, respectively.

Considering non-dimensional turbo-machinery nomenclature, a Wells turbine with diameter D and rotational velocity N is examined through a linear connection between the pressure and flow coefficients as described in Equation 12: Ψ = K Φ (12) here the pressure and flow coefficients can be defined in Equations 13, 14 as: Ψ = p ρ a N 2 D 2 (13) Φ = m ̇ ρ a N D 3 (14)

Assuming the pressure drop is proportional to the flow rate, non-dimensionalization is applied. Thus, the linear relation can be obtained as: Ψ c = K c Φ c (15) where the pressure and flow coefficients are described as follows: Ψ c = p ρ a g H (16) Φ c = 2 π m ̇ ρ a ω S H (17) the term g is the gravity acceleration and H is the wave height. Incorporating Equations 15–17 into Equation 11, the mass flow inside the turbine is given by: m ̇ ω = S ω p 2 π g K c (18)

Combining Equations 11, 18, the pressure complex amplitude can be described as: p ̂ ω = i ω Γ S ω 1 + ε Γ 2 V ˆ − ω 2 ε Γ 2 S ω 1 + ε Γ 2 V ̂ (19) where V ˆ represents the complex amplitude of the air volume oscillation and the constants Γ and ε are defined in Equations 20, 21 as following: Γ = 2 π ρ a g K c (20) ε = V 0 γ p 0 S (21)

Considering Equations 8, 19, the PTO force can be described in Equation 22 as: f ˆ PTO ω = − i ω B PTO q ˆ r + ω 2 K PTO q ˆ r (22) where x ˆ r describes the complex amplitude of the relative displacement. Based on Equation 19, the PTO damping and stiffness elements can be expressed in Equations 23, 24 as following: B PTO ω = Γ S ω 1 + ε Γ 2 (23) K PTO ω = ε Γ 2 S ω 2 1 + ε Γ 2 (24)

Hence, the frequency domain equation of motion of the system defined in Equation 3 is given by Equation 25 as follows: I FOWT ω x ̂ ⋅ ⋅ + B FOWT ω + B PTO ω x ̂ ⋅ + C FOWT + K PTO ω x ̂ = f ⃗ FOWT ω (25) where f ⃗ FOWT includes f ⃗ Hydro and f ⃗ viscous . f ⃗ viscous defines the viscous force and f ⃗ Hydro stands for the hydrodynamic force of the waves on the platform.

In order to evaluate the performance of the proposed platform, two approaches were considered using numerical and experimental methods.

The numerical approach involves use of the mean values of RAO indicators. For this, MultiSurf, WAMIT, FAST and MATLAB software were employed.

The MultiSurf tool was used to represent various offshore body geometries. As can be seen in Figure 2, two platforms were compared, a closed and an open OWC-based platform. The platforms are equipped with four OWCs. The platform design is detailed in Table 1. Note that the water displacement for both platforms is 6,000 m 3 . The dimensions of 40 × 40 × 10 m³ were selected based on the ITI barge platform, a well-established FOWT platform design developed by the National Renewable Energy Laboratory (NREL). These dimensions are representative of a stable and buoyant structure capable of supporting the turbine while accommodating the additional integration of the OWC chambers. Furthermore, this size ensures sufficient buoyancy and stability under offshore conditions, meeting the necessary design constraints for hydrodynamic performance and structural integrity.

The platform panel geometry defined in MultiSurf can be integrated in WAMIT (Lee and Newman, 2013) in order to calculate added mass, damping, hydrostatic and hydrodynamic force matrices. Then, these matrices can be included in FAST to model the system. The characteristics of the FOWT are detailed in Table 2.

In order to analyze the behavior of the systems, RAOs were utilized to have an input-output for system responses against various wave inputs. The frequency-response function (FRF) or RAOs can be calculated through Equation 26 as (Pintelon and Schoukens, 2012): R A O = H ω = S x y ω S x x ω (26) where H ( ω ) is the FRF, S x y ( ω ) is the cross-spectral density and S x x ( ω ) is the auto-spectral density of the wave input x ( t ) and the system states y ( t ) , in the frequency domain. S x y ( ω ) and S x x ( ω ) are defined by Equations 27, 28 as following: S x y ω = 1 M ∑ s = 1 M Y s r X ̄ s r (27) S x x ω = 1 M ∑ s = 1 M X s r X ̄ s r (28) where X [ s ] is the fast Fourier transform (FFT) spectrum of segment s, M is the simulations’ quantity of the process and r is the random noise sequence. For a more detailed description of RAO calculation, please refer to (Aboutalebi et al., 2021a).

In order to conduct the experimental test, a small 3D printed prototype of the floating wind turbine was obtained. The characteristics of the FOWT prototype are detailed in Table 3.

The experimental tests were carried out on two platforms: a closed and an open moonpool-based platform. Both platforms were subjected to an irregular wave to assess the extent of oscillations in their rotational modes. The moonpools integrated into the platforms served as the air chambers for the OWCs, allowing the regulation of airflow by opening or closing the moonpools, as previously demonstrated in our earlier studies (Aboutalebi et al., 2021a; Aboutalebi et al., 2021b, Aboutalebi et al., 2022).

In this study, both numerical simulations and experimental testing have been utilized to analyze the dynamic behavior of FOWTs with and without OWCs. The numerical simulations provided a comprehensive framework to predict the RAOs under varying wave conditions, allowing us to understand the system’s performance across different configurations. To validate the numerical results, experimental tests have been conducted using a scaled model of the FOWT in a wave basin, applying Froude similarity to ensure dynamic similarity with full-scale conditions. The experimental data confirmed the numerical findings, demonstrating the effectiveness of OWCs in reducing platform vibrations, particularly in pitch and roll motions. This dual approach strengthens the reliability of the results and enhances the overall understanding of the dynamic interactions between wave forces and floating structures.

All the experimental tests presented in this article were conducted in the Automatic Control Group (ACG) laboratory in the University of the Basque Country (UPV/EHU). The wave tank, available in the laboratory, is made of stainless steel due to its continuous contact with water. The sides and base are made of glass, except for two plates at both ends of the base, which are made of methacrylate to facilitate the fixing of the structures. The wave tank is 4.750 m*0.550 m*0.650 m (L*W*H). Figure 3 shows the configuration of the floating platform, mooring lines, and water level sensors, along with the dimensions of the wave tank.

To scale the floating platform, Froude scaling factors have been employed as outlined in Table 4. The properties and components of the scale model were designed using SolidWorks CAD software and subsequently assembled. ABS plastic was chosen for its durability, low density, resilience, and water resistance to 3D print the platform components. This material choice enabled us to verify the model’s buoyancy.

When validating the design of offshore wind turbines, these scaling factors ensure that the physical tests conducted on scaled models or in simulation environments accurately reflect the real conditions experienced by full-scale turbines. This includes considerations for structural loads, aerodynamic forces, hydrodynamic forces, and operational performance under different environmental conditions.

If a wind turbine model is built at a 1:166.6 scale, the geometric scaling factor ( λ ) is 0.006, while the scaled time is the original times sqrt ( λ ) . According to the Froude scaling factors, the scaling of the FOWT is detailed in Table 5.

As can be seen in Figure 4, the wave tank consists of a wave absorber, a barge-based FOWT and a wave generator. A human-machine interface (HMI) was used to control the movements of the paddle used for forming waves inside the wave tank (see Figure 5). For the interface, Sysmac Studio software compatible with OMRON equipment has a simple and intuitive interface that is divided into two general sections, one associated with the configuration of the controllers (PLC) and the other associated with the HMI (screens, control commands, etc.), as shown in Figure 6.

A small scale prototype of the NREL 5 MW FOWT with modified platform was modeled as shown in Figure 7. The FOWT includes eight catenary mooring lines, nacelle, blade, rotor, tower and platform.

The mooring lines are made of stainless steel chain, attached to the corners of the platform (two mooring lines at each corner) to keep the platform from drifting. The platform is 24 cm × 24 cm × 6 cm. Four moonpools were integrated at the corners with a distance of 0.5 cm from the sides. Each equipped moonpool has a size of 3 cm × 3 cm × 6 cm.

To measure the rotational movements of the platform, an electronic board was placed on the top of the nacelle. The board consists of an ESP32 module, an MPU6050 module and a 5 v power source. In order to obtain rotational moments, the ESP32 was interfaced with the MPU6050. The MPU6050 is an accelerometer and gyroscope module that measures rotational angles including pitch, roll and yaw. The ESP32 was used to transfer the obtained rotational movements data via WiFi to the CPU in the computer. Finally, Matlab was the interface used to monitor the rotational movements of the platform. A schematic representation of the measurement instrumentation setup used in the experimental tests is provided in Figure 8. This figure illustrates the sensor for thre measurement of pitch, roll and yaw displacements, data acquisition systems, and other relevant devices on the platform.

OWCs primarily dampen rotational modes (pitch, roll and yaw) rather than translational modes (surge, sway, and heave) in FOWTs. Rotational motions are more sensitive to wave forces, making OWCs more effective in reducing these oscillations. In contrast, translational modes are influenced by linear hydrodynamic forces, where the effect of OWCs is smaller. This paper focuses on the rotational modes, where OWCs have the most significant impact on vibration reduction. After obtaining the rotational modes RAOs using the equations detailed in Section 2.2, three informative Figures 9A–C can be plotted, depicting the platform’s roll, yaw, and pitch RAOs, respectively. Note that the similations have been conducted in absence of wind so that the blades angle kept and locked at 0°. The RAOs are illustrated with blue and red curves corresponding to the closed-OWC platform and open-OWC platform, respectively. As may be seen in Figures 9A, B, the RAO values for the roll and yaw modes appear to be relatively small, suggesting that the oscillations induced by the imposed wave along the surge mode are minor.

In Figure 9C, the platform pitch RAO takes center stage as the most significant among the RAOs due to its considerable value. As depicted, the platform pitch RAO exhibits a progressive increase for both closed and open OWC-based platforms until they reach their respective resonance periods. Subsequently, the platform pitch RAO descends, eventually approaching around zero for long-term wave periods. In simpler terms, the platforms’ oscillations intensify as they approach their natural frequencies, followed by a gradual decrease to slightly above zero for longer wave periods. An intriguing observation is that the platforms’ pitch curves intersect precisely at the wave period of 12.32 s, representing a critical point of interaction. Analyzing the behavior for wave periods lower than 12.32 s, it becomes evident that the open OWC-based platform exhibits lower oscillations compared to the closed OWC-based platform. However, a different pattern emerges for wave periods higher than 12.32 s, where the oscillations in the closed OWC-based platform become lower in comparison.

Considering the above results, an irregular wave with a significant period of 10s and wave amplitude of 5m, based on the JONSWAP spectrum, was imposed on the 5 MW barge-based floating offshore wind turbine (FOWT) system, as illustrated in Figure 10A.

Due to the direction of the irregular wave, the platform’s roll and yaw oscillations remain relatively small, as depicted in Figures 10B, C. These observations align with the expectations drawn from the interpretation of Figures 9A, B, which illustrate the platform’s roll and yaw Response Amplitude Operators (RAOs).

However, it becomes apparent from the platform pitch RAO spectrum in Figure 9C that the platform’s pitch response significantly impacts the system’s stability. Figure 10D demonstrates the platform pitch angle for both the closed and open OWC-based platforms, represented by blue and red curves, respectively. Notably, when subjected to the irregular wave input, the open OWC-based platform exhibits lower pitch oscillations compared to the closed OWC-based platform. Specifically, the distance between the highest wave crest and the lowest wave trough reduces from P C = 17.506 degrees for the closed OWC-based platform to P O = 14.381 degrees for the open OWC-based platform, reflecting a considerable oscillation reduction.

These results highlight the significant influence of OWCs on reducing platform pitch oscillations, thus enhancing the stability and performance of the hybrid FOWT system under irregular wave conditions. The reduced pitch oscillations further underscore the potential of OWCs in mitigating the adverse effects of wave-induced motions and improving the overall reliability of floating offshore wind turbines.

The model tests were conducted at the ACG laboratory, utilizing the infrastructure to test a prototype of the barge-based floating offshore wind turbine (FOWT). The tests were performed in a wave tank with an irregular wave input, having a scaled significant wave period of 10s and wave amplitude of 0.75m, as shown in Figure 11. The benchmark tests were carried out on both a closed moonpool-based platform (represented in blue) and an open moonpool-based platform (represented in red) as depicted in Figure 12. Notably, the tests focused on the platform pitch, roll, and yaw angles.

Figures 12A, E represent the wave elevation (in centimeters) and the Power Spectral Density (PSD) of the wave, respectively. In terms of platform pitch, Figure 12B illustrates the distance between the highest wave crest and the lowest wave trough for the closed moonpool-based platform ( P C = 18.41 d e g r e e s ) and the open moonpool-based platform ( P O = 12.23 d e g r e e s ) . This comparison reveals a significant reduction in pitch oscillations for the open moonpool-based platform when subjected to the irregular wave input, highlighting the stabilizing effect of the OWCs.

Figures 12C, D showcase the platform roll angle and platform yaw angle, respectively. Both figures demonstrate relatively small oscillations for platform roll (between around −2° and 1°) and platform yaw (between around −1° and 2°), indicating that the impact of the irregular wave on these rotational modes is minor.

For better readability, Figure 12B shows the time range from 0s to 200s, while Figure 12F provides a zoomed-in view of the same data from 130s to 150s. Similarly, Figures 12C, D correspond to the zoomed-in view in Figures 12G, H respectively.

These model test results further validate the positive influence of OWCs in reducing platform pitch oscillations, enhancing the overall stability and performance of the barge-based FOWT system under wave-induced conditions. The promising outcomes pave the way for further optimization and implementation of OWCs in floating offshore wind turbine designs, potentially enhancing the reliability and efficiency of these renewable energy systems.

The RAO results for roll, yaw, and pitch responses of the floating platform are depicted in Figure 13 for two configurations: open-moonpools (dashed red line) and closed-moonpools (solid blue line). In the roll RAO (a), both configurations exhibit multiple resonance peaks, particularly around 1.5 s and 2.5 s periods, with minimal differences between the two setups. The yaw RAO (b) also shows peaks, but with lower magnitudes than roll, indicating less sensitivity to yaw motion, with the open-moonpools platform. In the pitch RAO (c), the pitch response is more significant, peaking sharply near 2 s and 3 s periods, where the open-moonpools configuration shows a noticeable reduction in response, especially between 1s and 2.5s, indicating superior pitch damping capabilities compared to the closed-moonpools platform.