Nomenclature
V10: Wind speeds at 10 m above sea surface (m/s)
Vhub: Wind speeds at the hub height (m/s)
Iref: Wind turbulence intensity
Hs: Significant wave height (m)
Tp: Peak wave period (s)
T: Sea state duration (h)
R1mpm: Most probable maximum von-Mises stress corresponding to 3-h sea state duration from 1-h stress data
P: Probability level
δ: Location parameter
β: Scale parameter
γ: Shape parameter
R1max: Maximum von-Mises stress during 1-h simulation (MPa)
R2max: Maximum von-Mises stress during 3-h simulation (MPa)
N: Recurrence period (yr)
R2mpm: Most probable maximum von-Mises stress corresponding to 3-h sea state duration from 3-h stress data
1. Introduction
Floating offshore wind turbines (FOWTs) have attracted the most attention as one of the energy sources for decarbonization. At the present time, which is the beginning of the fourth industrial revolution, it is obvious that energy consumption will increase exponentially in developing artificial intelligence (AI) by collecting, storing, and processing big data as AI-based technological changes occur in all areas. Thus, the supply and demand of energy will have a significant impact on the international competitiveness of major domestic industries, including semiconductors and automobiles. When international barriers to carbon-based energy, such as RE100, are rising, a rapid and successful transition to eco-friendly energy is increasingly important, and floating wind farms can also be considered as a key part. Given such industrial requirements, various studies have been conducted on FOWTs. Among them, studies on structural response calculation and structural safety assessment techniques for the FOWT platform have been dealt with relatively recently.
Faraggiana et al. (2022) comprehensively summarized various methods and cases to numerically model and optimize the FOWT platform. They analyzed and compared the characteristics of each numerical analysis tool applied in each stage of FOWT analysis, showed the importance of using proper techniques and tools, and presented the importance of verifying numerical models. They also described various characteristics and constraints to be considered for platform optimization and analyzed the convenience of the frequency domain technique and the reliability of the time domain technique via cases. Wang et al. (2022) confirmed the importance of reliability-based design for offshore wind turbine substructures and presented structural integrity monitoring and condition monitoring based on the reliability index as the future direction of maintenance. Wang et al. (2023a) referred to the semi-submersible concrete platform OO-Star wind floater of 10-MW class FOWTs and examined the intact stability, motion, structural stress, and buckling of FOWTs under the assumption of a steel platform. Specifically, they conducted integrated load analysis in the time domain of 4,000 s for 12 operational conditions and one stationary condition, and secured structural response by mapping the load results to the structural model and conducting structural analysis in the time domain. Furthermore, Wang et al. (2023b) analyzed the internal stress and load of the substructure considering various environmental conditions and wind-wave misalignment and determined that the substructure was significantly affected by turbulent wind loads and irregular wave loads. Wind loads caused the largest internal force under the rated conditions. The inertial force of the floating body by irregular waves caused the significant dynamic load effect of the platform, and its impact further increased under the stationary condition with a high wave height. They proved that the magnitude of structural response can be properly obtained under the application of all load factors at the same time, and they determined that the behavior and viscous drag by second-order wave loads had insignificant impacts on the internal force and moment of the semi-submersible platform. Although the direction mismatch between the wind and wave had no significant impact under the rated conditions, it caused variability to some extent under stationary conditions even though there was no average impact. Meanwhile, Park and Choung (2023) adopted a more practical approach. They derived the structural response of the semi-submersible FOWT platform and confirmed its effectiveness through the frequency-domain analysis based on dominant load parameters (DLPs), which have been widely used in the structural evaluation of ships and marine plants. Specifically, they defined DLPs using the thrust by wind and the acceleration of the nacelle, and they found that they are useful in identifying structural vulnerabilities. Dong et al. (2024) proposed a simplified structural strength evaluation technique for the initial structural design of FOWT platform. They conducted integrated load analysis over the short term by applying seven design wave conditions, which considered three-axis acceleration and four platform structure characteristics responses and wind loads that correspond to long-term extreme wind speed conditions. Furthermore, they derived the structural response for the ultimate load estimated by applying the Gumbel function to the obtained load component. Wang and Moan (2024a) derived internal forces acting between the local elements of the floating body (e.g., columns and pontoons) from integrated load analysis by modeling the FOWT platform as multiple objects divided into several parts and screening important load cases. Based on this, they presented an efficient platform local design process to derive the thickness of the column structure and the ring stiffener spacing. Additionally, Wang and Moan (2024b) presented a simplified method that applied only the axial load and lateral bending moment for effective use in the initial design stage considering gravity, wind loads, and pitch motion as opposed to six-degree-of-freedom load components, and confirmed its efficiency.
Meanwhile, Li et al. (2024) used the ultimate stress values estimated based on the 90% quantile by applying the Gumbel distribution to the time-series stress data obtained from 1-h load analysis and structural analysis to overcome the uncertainty of the short-term analysis results. Estimation of extreme values by applying the Gumbel distribution is one of the commonly used methods. However, it was applied without explaining the stochastic consistency among 3-h wave analysis, the corresponding marine condition and design load conditions, and 1-h load and structural analysis results, which is required to secure the statistical characteristics of irregular waves. Therefore, in this study, a statistical stress estimation technique is proposed to more efficiently and reasonably evaluate the structural strength that corresponds to the environmental probability level considered during DLC calculation. Furthermore, the applicability of the estimated stress to ultimate strength evaluation is verified based on the integrated load analysis and structural analysis.
2. Research Background
2.1 Industrial Background
Floating wind farms currently in operation include Hywind Scotland, Hywind Tampen, Windfloat Atlantic, and Kincardine. The Hywind-type FOWT developed by Equinor, a state-run firm of Norway, has the spar-type substructure with a monopile while the Windfloat-type FOWT developed by Principle Power has a semi-submersible substructure. Hywind Scotland is the world’s first floating wind farm where five 6-MW monopile spar-type FOWTs are installed, and Hywind Tampen is the largest floating wind farm with a power generation capacity of approximately 95 MW in which 11 concrete spar-type 8.6MW FOWTs are connected using a shared anchor system. Windfloat Atlantic has three 8.4MW FOWTs (total power generation capacity: 25 MW) that have the world’s first semi-submersible substructure, and Kincardine is the largest semi-submersible floating wind farm in which five Windfloat-type 9.5MW FOWTs are installed. Additionally, three floating wind farms are under construction in France, and floating wind farms are under development in Italy (four farms), England (two farms), and Scotland and Norway (multiple farms). In Asia, China aims to operate a floating wind farm with a power generation capacity of 100 MW in 2025 for the first time. Taiwan is developing a floating wind farm with a power generation capacity of 1 GW. Japan has recently completed the revision of the law to expand the FOWT industry from the fixed type to the floating type. In South Korea, five large-scale floating wind farm projects are underway mainly in the East Sea, including Munmubaram (1.26 GW), East blue power (1.13 GW), Haeuli (1.5 GW), Banditburi (800 MW), and Gray whale (1.5 GW).
Fig. 1 (Stehly and Duffy, 2021) shows the Levelized Cost of Energy (LCOE) contribution by FOWT component evaluated under the premise of 25-year operation for the reference FOWT developed by the National Renewable Energy Laboratory (NREL). The “foundation and substructure” represented approximately 27.1%, which is similar to the “maintenance during the entire life cycle (25 years)” (27.5%). Given that its level is much higher than the “turbine that includes the Rotor and Nacelle Assembly (RNA) and tower” (17.6%), it should be considered important for the rapid soft landing of the floating wind farm industry.
2.2 Theoretical Background
2.2.1 Probability level and simulation conditions
The International Electrotechnical Commission (IEC) suggests that various types of design load cases (DLCs) should be considered. For such DLCs, dozens to thousands of sub-load cases should be evaluated by combining the wind direction, wind speed, wave height, and wave direction. Given that irregular phases must be considered even for one sub-load case, more analyses are required. Additionally, a nonlinear integrated load analysis in the time domain and subsequent structural strength evaluation should be performed to consider the nonlinear load generated by the wind turbine. Thus, considerable time is also required for analysis. These are the representative issues at the present time for safe FOWT design and evaluation.
Meanwhile, IEC presents the design recurrence period of environmental conditions for each DLC. In the joint probability density (Fig. 3) of environmental conditions defined with met-ocean data, as shown in Fig. 2, design environmental conditions are selected according to the probability level for the sea state duration. Therefore, structural strength evaluation should also correspond to this probability level. The darker color indicates a higher frequency in the figure.
For example, when DLC is presented in a recurrence period of N years as shown in Fig. 4, the probability level p is determined as the ratio of T (sea state duration) to the recurrence period.
where N denotes the design recurrence period and T denotes the sea state duration. The boundary on the joint probability distribution can be expressed as an environmental contour, such as a red line. This implies that performing simulation for time T by extracting a point on the line corresponds to reproducing one of the most severe cases for time T in the recurrence period of N years. Based on this, the sub-load cases of DLC are defined as shown in Fig. 4. Therefore, the largest case among all events or stress that occurred by conducting T time analysis under the defined sub-load cases can be understood as the largest stress that may occur during the entire recurrence period of N years. In summary, if the sea state duration that implements the targeted recurrence period of DLC is determined and environmental conditions suitable for the probability level are simulated by the sea state duration, the largest value in the simulation results can be used as the maximum value during the recurrence period. In conclusion, based on this background, the structure can be evaluated via the maximum value among the stress results of analyzing sub-load cases.
Given that the period for implementing the statistical characteristics of irregular sea waves is usually three hours, it is intuitive to conduct 3-h nonlinear load analysis and structural analysis. Furthermore, Annex F.2 (IEC, 2019) of IEC 61400-3-1 presented the environmental contour of the 50-year recurrence period for 3-h sea state duration as an example. In chapter 7 of the document, however, the analysis of 10-min waves six times is basically proposed, and 1-h wave analysis or six 1-h wave analyses are recommended with respect to DLCs for which wave loads are important. Thus, there is discrepancy between the definition of the marine environment and analysis time. Therefore, in this study, stochastic extrapolation was utilized to solve this discrepancy via extreme value estimation.
2.2.2 Extreme value estimation
The extreme value estimation method can be applied to various processes, such as the calculation of extreme wave heights, design loads, and design stress. This chapter briefly describes the process of deriving (estimating) extreme values at a certain probability level through various statistical postprocessing methods based on the time-series data of certain variables such as the wave height, force, and stress.
The data measured through experiments or calculated through simulations should be post-processed for design values. In this study, peak values within the cycle were extracted from the von-Mises stress time-series data applied to design evaluation by applying the peak counting method as shown in Fig. 5 and Fig. 6. In Eq. (1), the sampled peak values (x) must be sorted according to the magnitude. Here, N denotes the number of the peak values extracted from the measured data.
To derive extreme values at the desired probability level from the data measured through experiments or calculated through simulations, it is first necessary to assume a specific extreme value distribution function model for the sampled peaks. The 3-Parameter Weibull distribution function is one of the most commonly used extreme value distribution functions, and the probability density function (PDF) of Eq. (2) and cumulative distribution function (CDF) of Eq. (3) are as follows:
The range of variable x is δ≤x <∞. There are various methods to obtain each parameter (δ, β, γ), and the method used by many researchers is the method of moment. The method of estimating the parameters of the Weibull distribution function under the application of the method of moment is shown below. The method of moment estimates parameters by matching three moments, i.e., mean (μ̂), variance (σ̂2 ), and skewness (γ̂1) of the peak values obtained from the analysis, to the mean (μ), variance (σ2), and skewness (γ1 ) of the model. The mean, variance, and skewness of the model are obtained using Eqs. (4) to (6) below.
where pi denotes the i-th largest observed value among n samples. The mean, variance, and skewness of the Weibull CDF model are obtained using Eqs. (7) to (9) below.
where Γ(x) denotes the gamma function. The shape parameter γ can be obtained by matching the skewness of the samples, γ̂1, to that of the model, γ1. The scale parameter β can be obtained by matching the variance of samples, σ̂2, to that of the model, σ2. Finally, the location parameter δ can be obtained by matching the mean of the samples, μ̂, to that of the model, μ. After finding an extreme value function that can best express the accumulated data, the most extreme value for the particular return period can be calculated using Eq. (10).
where ȳNR denotes the most probable extreme value in the NR -th measurement, and NR denotes the number of measurements during the given return period (R). Based on the measured data, NR is determined by Eq. (12).
N denotes the number of data points measured during the data accumulation time, T denotes the data accumulation time, and TR denotes the time corresponding to the return period.
3. Development of Structural Assessment Procedure for FOWT platforms
Fig. 7 shows the structural strength evaluation process performed in this study for the FOWT platform. First, wind speed and wave data are secured, and the two environmental conditions are analyzed using the joint probability distribution. The environmental contour corresponding to the 3-h probability level is obtained using the inverse first order reliability method (IFORM) for the recurrence period presented by DLC, and sub-load cases are established. Integrated load analysis for FOWT is then conducted to calculate the 1-h time-series load that acts on the platform. The 1-h time-series structural response is obtained from the load through the combination method for unit response by load type proposed by Lee et al. (2023). The most probable maximum (MPM) value corresponding to the 3-h probability is estimated by extracting the peak values of the obtained 1-h structural response.
4. Analysis and Results
To verify this technique, DLC 6.1 with a recurrence period of 50 years was analyzed. Wave conditions were defined such that they could correspond to a 3-h sea state duration. In the case of the wind speed, the reference wind speed of turbine grade I (50 m/s) was used as the wind speed at the hub height because IEC suggests that the reference wind speed should be used in DLC 6.1. In the case of turbulence, the turbulence intensity corresponding to class B was applied. Additionally, misalignment among the wave direction, wind direction, and yaw angle of the turbine must be considered in DLC 6.1, 42 direction cases were defined considering the misalignment of the yaw angle based on the wind and wave directions at 30-degree intervals as shown in Fig. 8. Furthermore, IEC 61400-3-1 suggests that 1-h wave simulation with different phases should be performed at least six times for DLC 6.1 (IEC, 2019). Accordingly, a total of 252 sub-load cases were established by applying wind speeds and waves with six irregular phases for conditions in each direction. The finite element model (FE model) of the platform has approximately 45,000 target elements. These analysis conditions are summarized in Table 1.
To compare and verify the effects of this study, two comparative analysis conditions (Condition 1 and Condition 2) are defined as listed in Table 2. Analysis was conducted under the conditions of Table 1 for both Conditions 1 and 2, but a 1-h analysis was conducted, as suggested by IED, for Condition 1, while a 3-h analysis was conducted to correspond to sea state duration for Condition 2. The effect of the analysis time on the maximum stress was examined in the same sub-load cases, and the characteristics and effectiveness of the MPM von-Mises stress estimated from 1-h and 3-h time-series stress were analyzed. The maximum von-Mises stresses obtained from Conditions 1 and 2 were expressed as R1max and R2max while the MPM von-Mises stresses obtained were expressed as R1mpm and R2mpm, respectively.
Fig. 9 shows the maximum stress of each element. Furthermore, R1max and R2max show the largest stress among the time-series stresses of all sub-load cases while R1mpm and R2mpm exhibit the maximum stress among the MPM stresses for each sub-load case. A comparison between R1max and R2max shows that R2max generated larger stress than R1max for most of the elements. This is due to the fact that larger stress can likely occur as the analysis time increases. Excessively high stress, however, was observed from some elements located in the lower part of the wind tower, confirming that the distribution of the maximum stress according to the element, i.e., the maximum stress distribution according to the platform location, was significantly different from R1max.
A comparison between R1max and R1mpm reveals that the stress of R1mpm is slightly higher. Given that 1-h time-series stress was extrapolated with a probability corresponding to three hours, stress is highly likely to increase in most cases. Moreover, abnormal stress increases were not observed unlike R2max, and it can be observed that the stress distribution tendency according to the element is very similar to R1max. In the case of R2mpm, the rapid stress increase of some elements observed from R2max was not observed. This implies that the proposed technique estimates consistent maximum stress despite the presence of outliers in the actual data. This is because the technique is less sensitive to outliers as it estimates extreme values via the statistically fitted curves of the time-series data. A comparison between R2mpm and R1mpm shows that the magnitude of stress by element and stress distribution according to the element are consistent. Hence, the cases of converting 1-h and 3-h stresses using the MPM technique exhibit high consistency, and this will be the basis for efficient strength evaluation through reduced analysis time.
Fig. 10 shows the tendencies of the maximum von-Mises stress and MPM stress according to the 42 direction cases shown in Fig. 8. Hence, the largest stresses among the six time-series stresses with irregular phases for direction cases across all elements were expressed as R1max and R2max. In the same manner, the largest stresses among the six MPM stresses by direction for all elements were indicated by R1mpm and R2mpm. In all of the four cases, the trend of the maximum stress by direction was very similar. Therefore, the observation that the stress tendency by direction was similar despite the increased analysis time or conversion into MPM stress is also judged to be a significant result. High stress, however, was observed from the 25th case of R2max unlike other cases. An outlier occurred in one case among the six simulations in the direction, and it was also confirmed that the tower-based force component was momentarily very high at that time point. This also resulted in the large stress increase of the elements in the lower part of the tower observed from R2max of Fig. 10. The tower-base force probably results from the wind load, but this outlier is unlikely to occur considering the applied turbulence intensity. Therefore, it is estimated to be a numerical abnormality in the integrated load analysis process, and more detailed analysis will be required.
Table 3 shows the average tendency of the stress change rate when compared to the 1-h maximum von-Mises stress by element. As the analysis time increased from one to three hours, the maximum von-Mises stress increased by 4.76% on average (R1max→R2max). When the MPM von-Mises stress at the 3-h probability level was calculated using the 1-h analysis results, however, it increased by 5.36% on average when compared to the maximum von-Mises stress (R1max→R1mpm). This implies that MPM stress is more conservative than the maximum stress that occurs from the long analysis time on average.
Fig. 11(a) shows the relationship between R2max and R1mpm of each element in all sub-load cases. It can be observed that the two stresses have a linear relationship based on the overall tendency despite dispersion between them. Fig. 11(b) shows the difference ratio between the two stresses. As the magnitude of stress increased, the difference ratio between the two results decreased. For the elements with a stress of approximately 230 MPa or higher, a standard deviation of approximately 4.6% occurred.
Fig. 12 shows the ratio of the maximum R2max to the minimum R1mpm for the six irregular phases of direction cases. As stress increased, the difference between the two results also decreased. For a stress of 230 MPa or higher, a standard deviation of approximately 1.6% occurred, confirming that the maximum deviation was 9.3%. Consequently, it is expected that structural strength can be evaluated more efficiently if the MPM von-Mises stress is estimated by conducting 1-h analysis once instead of conducting 3-h analysis six times for the environmental conditions of 3-h sea state duration and an appropriate margin of safety is applied.
5. Conclusion
In this study, a statistical stress estimation technique that utilizes the most probable maximum (MPM) von-Mises stress was proposed to evaluate the structural strength of the floating offshore wind turbine (FOWT) platform more efficiently. Furthermore, a structural strength evaluation methodology was established to extrapolate the results of 1-h simulation at the probability level corresponding to 3-h sea state duration based on the guidelines of IEC 61400-3-1 and numerical analysis data.
When the proposed methodology was verified based on 1-h and 3-h analysis results, it was determined that MPM von-Mises stress was not sensitive to the analysis time, outliers, and irregular environmental conditions and that consistent tendencies could be derived. Furthermore, the maximum MPM von-Mises stress showed a tendency similar to that of the maximum von-Mises stress. Specifically, the MPM von-Mises stress results of 3-h sea state duration obtained through 1-h simulation showed a high linear relationship with the 3-h analysis results. The difference ratio of the two values based on the elements with high stress had a standard deviation of approximately 4.6%, confirming that the maximum stress of each element can be effectively predicted. Additionally, larger stress than the 3-h analysis results was derived on average. The lowest case among the maximum MPM von-Mises stresses obtained from different irregular phases was also not more than 10% lower than the 3-h maximum von-Mises stress based on the elements with high stress. This laid the foundation for reducing the time and cost required to consider different irregular phases by applying an appropriate margin of safety to the MPM von-Mises stress.
In conclusion, the structural strength of the platform can be efficiently evaluated while covering irregularities in the marine environment. In future research, it is necessary to optimize the time-series stress extraction technique and probability distribution fitting technique and specify the margin of safety considering more conditions to improve the accuracy and reliability of the proposed method.
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