Frequency-Domain Multi-Dynamic Analysis Using Response-Based Approach for FOWT
Article information
Abstract
This paper proposes a novel frequency-domain methodology for analyzing floating offshore wind turbines (FOWTs) through stress transfer functions that consider wave and wind loads. The frequency-domain analysis model included linearizing methods of critical parameters (i.e., rotor thrusts by wind, restoring forces by the mooring system, and hydrodynamic pressures by waves). A reference wind turbine, the OC3-Hywind spar platform, was selected as the target model, and industry-standard software packages, OpenFAST and ABAQUS, were used for validation. The wave-induced motion response amplitude operators (RAOs) showed agreement with 2%, 4%, and 1% differences for the surge, heave, and pitch modes under the head–sea condition, respectively. Similarly, wind-induced motions exhibited 2% differences in the surge and pitch modes in the colinear condition. The obtained results were used to calculate the radiation pressures as an integrated form, and stress transfer functions were yielded by considering them with incident and diffraction pressures. The proposed approach showed good agreement with conventional time-domain methods in the stress spectrum, showing a 92% decrease in computational time with only a 4% difference in results. This computationally efficient methodology eliminates the need for time-domain coupled-load simulations and offers potential applications in fatigue analysis and the initial design stage of FOWTs.
1. Introduction
The global energy sector is transitioning towards renewable sources, driven primarily by the urgent need to address environmental concerns and reduce carbon emissions. This shift is particularly evident when adopting renewable energy technologies, with wind power emerging as a leading alternative to conventional fossil fuels because of its minimal environmental impact and sustainability (Greiner et al., 2014; Mohamad et al., 2023). Offshore wind technology, especially for the floating offshore wind turbine (FOWT), represents a substantial advance over traditional fixed-type wind turbine installations, offering unique advantages in harnessing wind energy in deeper waters where wind resources are typically more abundant (Firouzi et al., 2022). FOWTs are designed to operate in deeper waters, which allows them to access more consistent wind resources than fixed-type wind turbines (Viselli et al., 2015). Deploying these turbines at greater distances from shore enhances the energy production potential and alleviates the concerns related to noise pollution and visual impact, making them more acceptable to coastal communities (Ma et al., 2014). On the other hand, the offshore environment presents distinct engineering challenges, particularly the complex interactions of wave and wind loads acting on the FOWT rather than fixed-type platforms (Lackner and Rotea, 2011).
Conventional approaches to assessing the structural performance and safety of FOWTs rely primarily on time-domain simulations. These methods are computationally demanding because they must account for intricate and time-dependent interactions between the structural dynamics and environmental loads (Robertson et al., 2014). According to IEC 61400-3, one-hour Integrated Load Analyses (ILAs) should be performed for at least six random seeds (IEC, 2009). On the other hand, at least three hours of simulations are needed to achieve statistically reliable results for moored structures under a short-term sea state. A long-term assessment of the floating structure is necessary to ensure the fatigue performance of mooring lines. The American Petroleum Institute (API) suggests running simulations spanning five to ten three-hour durations with varying seeds for floating systems (API, 2005). Specifically, in the time-domain approach, a single one-hour simulation typically requires 24 hours of computation time (Karimirad and Moan, 2012; Lerch et al., 2018).
Researchers have explored alternative analysis methods to address the computational challenges of time-domain simulations. Frequency response analysis using numerical tools has become a promising approach. Hydrodynamic analysis tools, such as ANSYS-AQWA and WAMIT, are used widely to analyze wave-induced motions and forces on floating platforms (ANSYS Inc., 2020; WAMIT Inc., 2023). Building upon these tools, frequency response hydrodynamic analysis was conducted using the commercial potential flow software ANSYS-AQWA to determine the acceleration response amplitude operator (RAO) (Park and Choung, 2023). This analysis aimed to predict the dynamic responses of the structure under various wave conditions, evaluating the effects of wave frequencies and incident angles to provide critical data on the motion and acceleration characteristics of the FOWT. This study excluded the hydrodynamic pressure acting on the structure and did not fully consider the coupling effects between the aerodynamic and hydrodynamic loads.
In other studies, the hydrodynamic pressure effects were examined in greater detail. For example, a fully coupled time-domain structural analysis was applied to evaluate the combined effects of aeroelastic, hydrodynamic, and inertial loadings on FOWTs. This paper highlights the computational intensity of this approach and proposes a lodal response-based method to enhance the computational efficiency significantly (Lee et al., 2023). Similarly, a response-based time-domain analysis method was proposed to synthesize the stress responses for fatigue damage evaluation, successfully addressing the loadings from waves, wind, and mooring systems (Kyoung et al., 2019, 2020). Furthermore, an advanced methodology that combined pseudo-spectral response synthesis with pre-processed lodal responses was introduced to enhance the efficiency of time-domain structural assessments. This approach revealed its utility in buckling and strength assessments for FOWTs, significantly reducing computational time (Moon et al., 2022). On the other hand, these studies calculated the radiation pressure using the six-degree of freedom (DOF) modes of motion results from WAMIT rather than adopting a single transfer function approach. In addition, they considered concentrated loads, such as aerodynamic forces and mooring line tensions, in a time-series format, which increased the computational time and resources.
This paper presents a novel approach to evaluate the stress responses of the FOWT in the frequency domain, giving a solution to overcome the above limitations. The hydrodynamic solver examined in this study included pressure calculation modules for the frequency and time domains, where frequency-domain pressure was computed as a transfer function and time-domain pressure was determined through the impulse response function (IRF) Cummins (1962) suggested. The proposed methodology enabled calculations of the motion responses, hydrodynamic pressures, fairlead tensions, and rotor thrust in the frequency domain, significantly reducing the computational requirements while maintaining accuracy. The accuracy of the calculated motion responses, fairlead tensions, and rotor thrust was validated by a comparison with the OpenFAST simulation results. Furthermore, the proposed methodology was verified using the commercial finite element analysis software ABAQUS through comparative validation against conventional structural analysis approaches in the time and frequency domains. The incident and diffraction pressures were calculated based on wave trains, while the radiation pressures were derived from structural motion responses. Structural analysis was conducted using transfer functions representing incident, diffraction, and radiation pressures. In addition, the transfer function for radiation pressure was calculated by distinguishing the effects of waves and wind.
The approach can be simplified for conventional offshore structures where wave forces dominate environmental loading to focus exclusively on wave-induced responses. For FOWTs, however, where wind forces are predominant, the methodology incorporates the effects of radiation pressure to derive comprehensive structural responses. The significance of this methodology lies in its ability to enhance computational efficiency while maintaining engineering accuracy in FOWT structural analysis. This approach can be extended to fatigue analysis and initial design stage analysis of FOWTs, providing a versatile framework applicable to various offshore installations.
2. Proposed Methodology
2.1 Transfer Function of Pressure in Frequency Domain
Many studies use velocity potential theory to analyze the hydrodynamics in the frequency domain (Maruo, 1960; Newman, 1974; Pinkster, 1980). The velocity potential at each panel can be separated into all six-DOF modes of motion and the incident and diffracted wave fields, as shown in Eq. (1).
where Φ, φi, φd, φrj and ζj are the velocity potential, incident wave potential, diffraction wave potential, six-DOF radiation potentials, and six-DOF motion responses, respectively. Fig. 1 presents the decomposition of wave fields around a floating body into incident, diffraction, and radiation components. The pressure transfer function can be calculated using the linearized Bernoulli’s equation, as shown in Eq. (2).
Wavefield decomposition
where ρ, ζ0, and ω are the fluid density, incident wave amplitude, and frequency, respectively. The first term is the transfer function of incident and diffraction pressure, Pi + d(ω), and second term is the transfer function of radiation pressure, Pr(ω). Both transfer functions can be represented as the pressure per unit wave amplitude.
2.2 Pressure Time History
The pressure in the time domain is used for comparative validation. Cummins (1962) discussed using the IRF to simulate the time domain analysis of the floating body. The IRF can be obtained from Eq. (3).
where hp(t) is the impulse response function. In this approach, each wave amplitude was assumed to be an impulse reaction, while the wave exciting force can be calculated using a convolution integral. From this approach, the time history of the incident and the diffraction pressure at each panel can be calculated using Eq. (4).
where η(t) is the wave elevation and Pi + d(t) is the incident and diffraction pressure in the time domain. The incident and diffraction effects can be considered using IRF because they are forces generated by waves. On the other hand, radiation effects cannot be calculated using IRF because they originate from the structural motion itself. The radiation pressure is induced by inertia and damping, which are forces from the added mass at infinite frequency and the retardation function, respectively, as shown in Eq. (5).
where Fr(t), Ma(∞), R(t), and X are the radiation force in the time domain, added mass at infinite frequency, retardation function, and displacement, respectively, while the dot notation means a derivation with respect to time. Subsequently, the time series of radiation pressure can be yielded by dividing Fr(t) with the panel area and the normal vector, as shown in Eq. (6).
where Pr(t), S0, and n are the radiation pressure in the time domain, panel area, and normal vector, respectively.
2.3 Linearization Process
The mooring system and rotor thrust restoring coefficients are linearized to calculate the structural motion. The calculated motion is then used to derive the radiation pressure transfer function.
2.3.1 Mooring dynamic analysis
The methodology considers the geometric linearity and elasticity of mooring lines (Pevrot and Goulois, 1979). The restoring coefficients of the mooring system were calculated based on the variations in the mooring line tensions resulting from small displacements applied to the offshore structure in each DOF. The equilibrium configuration of the mooring lines is determined using catenary equations, and the horizontal and vertical components of the tension are calculated iteratively, ensuring equilibrium at the fairlead point. Table 1 presents the procedure for calculating the restoring coefficients, and Fig. 2 shows the static offset test configuration with the mooring lines.
Calculation procedure for restoring coefficients
Configuration of the mooring line system for static offset test analysis
2.3.2 Aerodynamic analysis
The aerodynamic loads were analyzed using OpenFAST under simplified conditions that allowed only surge motion for the rotor, controlled by a simple torque controller. This setup focused on generator torque adjustment to maintain the optimal rotor speed, providing a straightforward approach to analyze the rotor thrust characteristics. Fig. 3 shows the configuration of the simulation conditions, illustrating the three-dimensional wind field generated by TurbSim and the corresponding rotor thrust under the one-DOF condition.
Configuration of the rotor thrust calculation
The numerical setup followed the linearization procedure for the rotor thrust transfer function (Table 2). TurbSim was used to generate three-dimensional full-field wind conditions, which were then implemented into OpenFAST.
Calculation procedure of rotor thrust transfer function
The rotor thrust was derived from 55 simulation cases, using wind speeds ranging from 5 m/s to 15 m/s in 1 m/s increments with five different random seeds (Lee et al., 2024). The transfer function of the rotor thrust is linearized, as shown in Eq. (7).
where Hwind(ω), R(ω), and S(ω) are the transfer function of the rotor thrust, output spectrum, and input spectrum, respectively. Consequently, the transfer function of the rotor thrust can be expressed by Hwind(ω), as shown in Eq. (8).
where Vhub is the wind velocity at the hub height, and ω* is the critical frequency where the transfer function changes its behavior. For simple torque control, the coefficients were determined as follows: a1 = 0.218, a0 = −1.052 for the case of ω<ω*, and b1 = 11.234 and b0 = 0.203 for ω≥ω*. The critical frequency ω* was defined with coefficients c1 = 0.2 and c0 = −0.4.
2.4 Analysis Procedure
Fig. 4 shows the overall workflow of this study. The process begins with extracting the coefficients from hydrodynamic solvers, mooring dynamics, and aerodynamic analyses. In the frequency domain analysis, each load component (rotor thrusts by winds, restoring forces by the mooring system, and hydrodynamic pressures by waves) acting on the FOWT was linearized independently, excluding the coupling effects between these loads. The mooring dynamics were linearized to obtain the mooring stiffness coefficient, while aerodynamic analysis provided the rotor thrust. These coefficients were used to calculate the motion RAOs, which then served as inputs for calculating the radiation pressure transfer functions. The pressure transfer functions were derived using the pressure module, and structural analysis of these pressures produced the corresponding stress transfer functions. Finally, the results are expressed as a stress spectrum by multiplying the stress transfer functions with the environmental loading conditions.
Flowchart of the methodology for structural analysis considering the hydrodynamic pressure
3. Numerical Analysis Results
3.1 Specification of the Model & Simulation Conditions
The model used for validation in this study was the NREL 5MW OC3-Hywind spar (Jonkman, 2010), which is used widely as a reference model in the offshore wind energy research community. This FOWT consisted of a 5MW wind turbine mounted on a spar-type floating platform. Fig. 5 and Table 3 show the specifications and configuration of the wind turbine, respectively.
OC3-Hywind spar
Properties of OC3-Hywind spar platform system
In this study, the Joint North Sea Wave Project (JONSWAP) spectrum (Hasselmann et al., 1973) was selected for an irregular wave condition, which can be expressed as Eq. (9).
where Swave(ω), HS, ωp, and γ are the JONSWAP spectrum, significant wave height, peak frequency, and overshooting parameter, respectively.
The IEC-61400-1 standard recommends using the Kaimal turbulence model (Kaimal et al., 1972) to evaluate the dynamic characteristics of wind turbines (IEC, 2005). Following this recommendation, the wind spectrum in this study was implemented based on the Kaimal turbulence model, as described in Eq. (10).
where Swave(ω), σ′, Vhub, L, and Iref are the Kaimal turbulence model, velocity component standard deviation, mean velocity at hub height, velocity component integral scale parameter, and turbulence intensity, respectively. Table 4 lists the environmental conditions and parameters for the design load cases (DLCs) under normal operating conditions. The overshooting parameter (γ) of 3.3 and the turbulence intensity (Iref) of 0.16 are used as fixed values.
Environmental conditions for the operating DLCs
3.2 Wave-Induced Motion RAO and Mooring Dynamics
Based on the methodology presented in Table 1, the motion RAO was validated by calculating the restoring coefficients of the mooring system. The validation of the wave-induced motion RAO is necessary for calculating the second term in Eq. (2), specifically the transfer function of radiation pressure. Figs. 6 and 7 show the wave-induced motion RAO and mooring line tension at the fairlead locations, respectively. The results were obtained through the proposed numerical method, which uses Eq. (11) to solve the system and validated against FFT analysis from OpenFAST simulations.
Wave-induced motion RAO comparisons
Comparisons of the mooring line tension at fairlead locations
where M, Ma(ω), and B(ω) are the mass matrix, added mass from the real part of the radiation potential integral, and radiation damping coefficient determined from the imaginary part, respectively. K, Km, and Fwave(ω) are the hydrostatic restoring stiffness matrix, restoring coefficient at the mooring system, and wave exciting force. The terms ̈(ω), ̈(ω), and X(ω) represent the acceleration, velocity, and displacement response vectors in the frequency domain, respectively.
Table 5 compares the natural frequency results between the proposed methodology and OpenFAST simulations.
Comparison of the natural frequency in wave-induced motion RAO
The comparison of natural frequencies showed 2%, 4%, and 1% differences for the surge, heave, and pitch modes, respectively. Under the following sea conditions, the first mooring line exhibited maximum tension, while the other mooring lines showed uniform tension distributions due to symmetric loading.
3.3 Wind-Induced Motion RAO and Aerodynamics
Following the analysis procedure of Table 2, the motion RAO was validated by calculating the transfer function of rotor thrust. This validation of wind-induced motion is essential for determining the transfer function of radiation pressure. Figs. 8 and 9 show the rotor thrust transfer function and wind-induced motion RAO, respectively. The results obtained using Eq. (12) in the proposed method were validated against FFT analysis from the OpenFAST simulations.
Rotor thrust transfer function
Wind-induced motion RAO comparisons
where Fwind(ω) is the rotor thrust. Specifically, Fwind(ω) is the product of the rotor thrust transfer function and a wind speed of 1 m/s. The structural motion was calculated by incorporating the rotor thrust transfer function as the external force term in Eq. (12). Only the surge-directional rotor thrust transfer function was calculated because of the one-DOF condition. The pitch response was determined by applying the moment arm to this surge component. Table 6 compares the natural frequencies between the proposed method and OpenFAST.
Comparison of the natural frequency in wind-induced motion RAO
The natural frequency differences were 2% for the surge and pitch modes. The differences in peak values in Figs. 6 and 9 results from the different analysis methodologies. Although the OpenFAST results are obtained by transforming nonlinear time-domain data to the frequency domain, the proposed method uses a linearized approach in the frequency domain. This linearization process cannot account for specific nonlinear dynamic characteristics, resulting in lower peak values.
4. Structure Analysis Results
The subsequent analyses focus on panels near the waterline, where hydrodynamic effects are most dominant. This analytical approach is based on two fundamental hydrodynamic principles. First, the incident and diffraction pressures induced by waves are most prominent near the surface, as the orbital motion of water particles attenuates with depth. Second, the radiation pressure, resulting from structural motion, reaches its peak magnitude near the waterline because of the maximum moment arm from the center of gravity. Therefore, the structural analysis results are presented for these waterline regions. Fig. 10 shows the stress distribution across all structural and the locations of selected elements for structural analysis. The stress is distributed unevenly across the elements (Fig. 10(a)), with significant stress concentrations observed particularly at the waterline. The thickness of the structure was 0.0372 m, and the ballast was 49.416 m. The steel material properties were used with Young’s modulus E = 206 GPa, density ρ = 7,850 kg/m3, and Poisson’s ratio ν = 0.3. The spar was modeled using 4,128 finite shell elements. A coarse mesh modeling approach was adopted instead of fine mesh analysis to validate the proposed methodology.
Structural elements of OC3-Hywind spar
The proposed method analyzed the model across 180 discrete frequencies, ranging from 0.01 to 1.8 rad/s, with 0.01 rad/s increments. The conventional method used a time series of 10,800 seconds (three-hour duration) for the simulation. Fig. 11 shows the validation process comparing the proposed and conventional methods. The incident and diffraction pressure transfer functions were derived from the first term of Eq. (2), while their corresponding pressure time series were calculated using Eq. (4). In the proposed methodology, stress transfer functions are obtained through structural analysis of incident and diffraction pressure transfer functions. The stress spectrum was then calculated by multiplying these transfer functions with the wave spectrum of Case #11 from Table 4. In contrast, the conventional methodology conducts structural analysis on the pressure time series to derive the stress time series. These stress time series are subsequently converted to the stress spectrum. Fig. 11(a) presents the verification process, comparing the incident and diffraction stress spectrum obtained from the two methodologies. The validation process for the radiation stress spectrum in Fig. 11(b) followed a similar procedure, with the additional step of motion calculations. Figs. 6 and 9 present the motion RAO induced by wave and wind, respectively. The motion results in the time domain were obtained using OpenFAST. The radiation pressure transfer functions were obtained from the second term of Eq. (2), and their corresponding pressure time series were calculated using Eq. (6). The radiation stress spectrum was obtained by multiplying the transfer functions with their corresponding environmental spectrum. For the environmental effects, the wave spectrum was used for wave-induced radiation stress, while the wind spectrum was applied for wind-induced radiation stress.
Flowcharts comparing the proposed and conventional methods for the calculations
Figs. 12 and 13 show the pressure transfer functions and corresponding time histories for incident and diffraction, wave-induced radiation, and wind-induced radiation, respectively.
Pressure transfer function
Pressure time history
Figs. 14 and 15 present the stress responses derived from structural analysis as transfer functions and time histories, respectively.
Stress transfer function
Stress time history
The stress spectrum was obtained by multiplying the stress transfer functions with the spectrum from Case #11 in Table 4. The total stress spectrum (Fig. 16) combines the incident and diffraction, wave-induced radiation, and wind-induced radiation components calculated using the methodologies presented in Fig. 11.
Comparison of the total stress spectrum between the proposed and conventional methods
The total stress spectrum analysis between the proposed and conventional methodologies showed agreement, with area-based spectral differences of 2%. The previous results were derived from the values at the waterline. On the other hand, additional validation was performed at other positions because wave excitation forces, motions, damping, and added mass vary depending on the location. Three panels were selected along the vertical length of the spar platform (panels #1, #2, and #3), as shown in Fig. 17; Fig. 18 presents the corresponding stress spectra.
Analysis location for stress spectrum on the FOWTs
Comparison of the stress spectrum
The stress spectrum at different locations obtained using the proposed methodology quantitatively showed agreement with those from the conventional approach. On the other hand, the differences in peak values between the conventional and proposed methods arise from the limited consideration of nonlinear effects because the proposed method analyzes the linearized wave exciting forces, rotor thrust, and structural motions.
5. Conclusions
This study developed and validated frequency-domain structural analysis methodology by linearizing the transfer functions of the rotor thrust, mooring line tension, and structural motion. The effectiveness of the proposed approach was validated through numerical analyses of the OC3-Hywind spar platform. The rotor thrust was linearized using a probabilistic approach by decoupling the wind spectrum influence from thrust data obtained from OpenFAST. The transfer function of the rotor thrust was derived specifically in the surge direction. For mooring line tension, linearization was implemented by applying the static offset test to obtain the appropriate coefficients. The structural motion transfer function, expressed as a motion RAO, was constructed using these coefficients and validated against OpenFAST simulation results. The natural frequency differences for the wave-induced motion RAO were 2%, 4%, and 1% in the surge, heave, and pitch modes, respectively. The wind-induced motion RAO exhibited 2% differences in the surge and pitch modes.
The developed frequency-domain pressure module was used to calculate the transfer functions of the incident, diffraction, and radiation pressures. These transfer functions were then applied as inputs to a commercial structural analysis program, ABAQUS, to calculate the corresponding stress transfer functions. The stress spectra were then derived by post-processing the stress transfer functions with environmental loading conditions. For validation purposes, the conventional methodology calculates the pressures using the developed time-domain pressure module, with motion time series derived from OpenFAST. Similarly, the stress time series were obtained through ABAQUS and transformed into spectral form for validation and comparison.
Comparative analysis between the conventional and proposed methodologies showed agreement for total stresses, showing differences of 2%. A similar agreement was observed at other locations, with less than 4% differences. The proposed methodology showed significant improvements in computational efficiency. The analysis time was reduced from 12 hours to one hour, representing a 92% reduction in computational costs. In addition, this approach provides substantial cost advantages by enabling the derivation of rotor thrust, mooring line tension, and structural motion transfer functions without reliance on commercial software, enhancing its scalability and applicability for large-scale analyses.
The frequency-domain structural analysis methodology is particularly effective for fatigue and initial design stage analysis of ships and offshore structures in wave-dominated environments. For FOWTs, where the wave and wind effects are significant, this methodology is highly effective for initial design stage analysis and proves efficient for fatigue analyses involving multiple case studies. Nevertheless, as with other frequency-domain approaches, this methodology inherently cannot account for nonlinear dynamic effects by linearizing wave exciting forces, rotor thrust, and structural motions. Accordingly, this methodology provides a framework for advancing the structural analysis and design of FOWTs by addressing the challenges associated with complex environmental loads while ensuring high accuracy and efficiency.
Notes
The authors declare that they have no conflict of interest.
This study was supported by the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korean government (MOTIE) (Project Number:202400450063, Project Name: Development of the Mooring Load Reduction and Monitoring System for 15MW+ class Korea Floating Offshore Wind Farm).