The time domain simulations for the investigations in this work are conducted using the aero-hydro-servo-elastic code FASTv8 from the National Renewable Energy Laboratory . A soil model by and that enhances the FASTv8 code is considered. The NREL 5 MW reference turbine with the OC3 monopile and soil is used as the wind turbine model. For the soil model, the required soil matrices are based on the lateral soil model of and on the axial soil model of FUGRO . These choices follow the work of . In accordance with , the soil matrices are calculated using initial conditions; i.e. no loads were applied.

The turbulent wind field is calculated using TurbSim . Here, the Kaimal turbulence model is used, which is one of the turbulence models recommended in the IEC 61400-1 guideline and which is frequently used, for example by , , and . The irregular waves are calculated using the JONSWAP spectrum. The JONSWAP spectrum is commonly used for offshore wind turbine simulations, e.g. by , , and .

Five scattering parameters (mean wind speed vs, turbulence intensity TI, significant wave height Hs, wave peak period Tp, and wind–wave misalignment θmis) are considered in this work. For these five parameters, the statistical distributions by are used, which were fitted to the measured environmental conditions at the FINO 3 research platform in the German North Sea.

The simulation length of each simulation is 10 min in addition to the run-in times to be determined in the next section.

Before creating the database for the training and testing of the meta-models, the first step is the determination of the run-in times for the NREL 5 MW reference turbine on the OC3 monopile in idling conditions using the FASTv8 software. Since previous investigations have shown that run-in times depend on wind speed , the run-in times are determined as a function of the wind speed.

The run-in times are determined by investigating the internal forces at two representative locations at the wind turbine, which are visualised in Fig. . The internal forces of the monopile are investigated at the mud line, and the internal forces of the rotor blade are investigated at its root. An overview of the considered internal forces can be found in Table . During the course of this paper, the meta-models are created and analysed for these internal forces.

To obtain the run-in times as a function of the wind speed, wind speeds of 1 to 33 m s⁻¹ with a step width of 2 m s⁻¹ are considered. A total of 10 000 simulations are carried out for each wind speed. In each of the 10 000 simulations, the random wind and wave seeds and the initial azimuth angle ψ (initial position of the rotor blades) are varied. All other scattering parameters, Hs, Tp, TI, etc., are kept constant, whereby the mean values of the corresponding probability distribution are used. This means that the 10 000 simulations conducted for every wind speed are identical except for the values of the random wind and wave seeds and the initial azimuth angle ψ.

A convergence study is conducted to determine the run-in times. Each of the 10 000 simulations per wind speed has a simulation length of 60 min. This leads to a maximum of 50 min for the initial transients as the usable simulation length is 10 min. Based on the study of on initial transients of an offshore wind turbine, it is assumed that this duration is sufficient for an idling wind turbine.

For the convergence study, the run-in time is increased step by step in 10 s intervals, starting with a minimum run-in time of 10 s. Then, in each step, for each of the 10 000 simulations, the run-in time is cut off, and a short-term DEL is calculated for a simulation length of 10 min according to Eq. (). 1Seq=∑niSimNref1m Here, ni is the corresponding number of cycles for each load amplitude Si determined by rainflow counting according to the ASTM E1049-85 standard . Nref=600 is the number of equivalent cycles for a frequency of 1 Hz for a 10 min time series, and m is the Wöhler exponent. The Wöhler exponent m=3 is chosen for steel and m=10 for the composite material of the rotor blade. The load range Si is corrected according to .

The resulting 10 000 short-term DELs Seq are then averaged for each step of the convergence study. The run-in time is considered sufficiently long if the deviation of the current mean short-term DEL from the mean short-term DEL of the maximum analysed run-in time of 50 min is less than 5 %.

As described above, the 10 000 simulations conducted for every wind speed are identical except for the values of the random wind and wave seeds and ψ. ψ is varied in each simulation in addition to the random wind and wave seeds, as during the investigations, it turned out that the run-in times, especially for the flapwise bending moment at the blade root My,Root, could only be determined if ψ is varied in each simulation. This can be explained as follows. In the case of an idling wind turbine, the blades are pitched out. Due to this pitch angle and the very low rotational speed of the rotor, the flapwise (idling: in-plane) bending moment is significantly affected by the weight of the rotor blade itself and therefore depends on the position of the rotor blade. This dependency is shown in Fig. , where the time series of the azimuth angle and My,Root are shown for one example simulation with a mean wind speed of vs = 17 m s⁻¹. It becomes clear that the value of My,Root is significantly influenced by the value of the azimuth angle. Furthermore, it is noticeable that the values of My,Root only change very slowly. This is due to the fact that the azimuth angle, i.e. the rotor position, only changes very slowly, caused by the low rotor speed (approximately one rotation per hour). If only the random wind and wave seeds are varied, while ψ has the same value in all 10 000 simulations, the resulting time series of My,Root differ only due to different rotor speeds caused by different stochastic wind fields. These differences are – especially for My,Root – not large enough to determine a convergence of the run-in times, as the time series are too similar, and the mean values of My,Root,DEL are therefore too strongly affected by the rotation of the rotor. This issue can be avoided by varying ψ in each simulation. Due to the different initial rotor positions, the time series of My,Root are more different. Consequently, the influence of the rotation of the rotor in My,Root,DEL is averaged out when calculating the mean value of My,Root,DEL. This enables the run-in times to be determined. For this reason, the azimuth angle is varied randomly to determine the run-in times.

Figure  shows an example of the convergence behaviour for wind speeds from 1 to 9 m s⁻¹ for My,DEL and My,Root,DEL. For the remaining internal forces and wind speeds considered, the convergence behaviour is shown in Appendix  in Figs.  to . It is clear that My,DEL has a better convergence behaviour than My,Root,DEL and that longer run-in times are required when looking at My,Root,DEL. For My,DEL, it can be seen that the run-in time depends on the wind speed. The higher the wind speed, the shorter the run-in time required. This dependence of the run-in time on the wind speed can be observed for all analysed internal forces, although the run-in times of the bending moments at the rotor blade roots deviate from this in some cases. For My,Root,DEL, for example, it can be seen that the results for 3 and 5 m s⁻¹ deviate from this trend, as shown in Fig. . The reason for these deviations could not be determined in this study. As the simulation of My,Root,DEL additionally requires the highest run-in times for most wind speeds, 800 s run-in time is assumed for wind speeds < 16 m s⁻¹ to be on the conservative side. For very high wind speeds, the investigations show required run-in times of less than 10 s. Here, to be on the safe side, a minimum run-in time of 60 s is specified. The run-in times defined in this work are shown in Table .

Comparing the results in Table  with the results of for the same wind turbine in operation, it is noticeable that the trend that the required run-in time becomes shorter with increasing wind speed was observed previously. The results also agree well for wind speeds of less than 3 m s⁻¹, where the turbine idles. However, the results differ significantly in one aspect. Compared to 60 s in this study, a very long run-in time of 360 s was determined for the idling wind turbine for wind speeds greater than 25 m s⁻¹. One possible reason for this could be that in this work only one combination of the environmental parameters per wind speed is calculated 10 000 times with different random seeds and varying initial azimuth angles. Here, the mean values of the probability distributions of the remaining scattering parameters are used. The determined run-in times therefore apply to average combinations of environmental parameters. However, rarely occurring combinations of input parameters, which may lead to higher run-in times, are not taken into account. In , in contrast, 10 000 different combinations of environmental parameters per wind speed bin were calculated to determine the run-in times. This means that the wind speed and other scattering parameters, such as turbulence intensity, significant wave height, and wave peak period, were varied in each of the 10 000 simulated 10 min realisations per wind speed bin. As a result, less frequently occurring combinations of input parameters were also taken into account, which may lead to higher run-in times. To summarise, it can therefore be concluded that the run-in times in this work apply to average input parameter combinations, while the run-in times from can be used to be on the conservative side.

In this work, Kriging is used as a meta-model for the simulation model of the offshore wind turbine in idling conditions. Kriging uses a combination of a regression equation to model the mean or general trend in the data and a Gaussian process with a zero mean to model the deviations from the general trend . The equation of Kriging is given as follows by : 2g(x)=f(x)+h(x)Tβ. h(x)Tβ represents the general trend in the data with known regression or basis functions h(x) and unknown regression coefficients β. f(x) is a Gaussian process with a zero mean and the covariance function or kernel function k(x,x′): 3f(x)∼GP(0,k(x,x′)). As mentioned before, using the Gaussian process (GP), the deviations from the general trend, also called residuals, are modelled. Here, the covariance function k(x,x′) describes the similarity between different data points (x and x′). It is assumed that data points whose input values are close to each other, i.e. are similar, also have similar output values. Covariance functions can be isotropic or anisotropic. They differ in the correlation length used. The correlation length indicates how distant data points may be from each other in order for them to still be correlated. For isotropic covariance functions, there is a joint correlation length for all input parameters. In contrast, for anisotropic covariance functions, there is a separate correlation length for each input parameter. This can be beneficial when the different input parameters have different influences on the output parameters . In this work, the anisotropic covariance functions are named with the prefix ARD (automatic relevance determination).

For more information on Kriging, the reader is referred to and .

The Kriging meta-model is used in this work to predict fatigue loads, i.e. short-term DELs Seq, based on the input parameters vs, TI, Hs, Tp, and θmis. These environmental parameters were identified as significant in sensitivity analyses for operating wind turbines by , , and . Due to their significant influence on the fatigue loads of operating wind turbines, it is assumed in this work that these parameters also have a significant impact on the fatigue loads in the case of an idling wind turbine.

It should be mentioned that a separate meta-model is created for each of the six internal forces analysed (see Table ). To train the meta-models, the short-term DELs Seq, determined with the aeroelastic simulation model f depending on the input parameters, are used. 4Seq=f(vs,TI,Hs,Tp,θmis) A Kriging meta-model is a mathematical model which, in principle, can also make unphysical predictions, such as negative short-term DELs. To avoid the prediction of negative DELs, the logarithm of the short-term DELs is used to train the meta-models g. 5ln⁡(Seq)=ln⁡(f(vs,TI,Hs,Tp,θmis)) As the meta-models g were trained with ln⁡(Seq), the meta-models g also predict values for ln⁡(S^eq), where S^eq is the approximated short-term DEL. 6ln⁡(S^eq)=g(vs,TI,Hs,Tp,θmis) The values of Eq. () can then be used to calculate S^eq as follows. 7S^eq=exp⁡(ln⁡(S^eq)) It should be noted that a Kriging meta-model generally predicts a mean and a corresponding standard deviation of the short-term DEL. For the prediction of the short-term DELs in this work, only the predicted mean values are used, as the mean values are the central values of the predictive distributions and provide a clear and unambiguous estimate of the expected short-term DELs.

Using the previously defined initial transients, the training and test data sets for meta-modelling can be created. Here, the same simulation model and environmental conditions are used as described in Sect. . The five environmental input parameters for the meta-models, vs, TI, Hs, Tp, and θmis, are considered scattering parameters. In addition to these variables, ψ and the random wind and wave seeds are also varied randomly in each simulation. For the other environmental variables, which are also scattered in reality (e.g. wind shear exponent), the mean values of their probability distributions are used.

In the sampling process, correlations between the input parameters are taken into account, as shown in Fig. . This means that combinations of environmental variables that do not occur in reality, such as a high wind speed in combination with a very low wave height, are not considered in this work. The correlations were considered using the statistical distributions of , which contain the correlations between the different environmental variables.

A total of 10 000 samples are created using a Halton sequence. The Halton sequence is also known as a quasi-random sampling method. Here, samples are generated in the interval of [0, 1] with the help of an equation. The created samples are then transformed using the inverse cumulative probability distribution. The Halton sequence was selected because it led to the best results in terms of the required number of samples compared to other methods in an earlier study for an offshore wind turbine in operation. In addition, the Halton sequence was also used by and .

The resulting time series for the different internal forces (see Table ) are subsequently transformed into short-term DELs using Eq. (). The data set is then split into a training data set and a test data set. Here, 85 % of the samples are used to train the Kriging meta-model, and 15 % of the samples are then used to test the meta-model.

As already described, idling conditions can certainly have a significant impact on the lifetime, as wave loads have a more significant impact on structural behaviour due to the almost-nonexistent aerodynamic damping. Thus, wave loads excite the structure more strongly. To understand the magnitude of the internal forces of the idling conditions compared to the internal forces of normal operation, the internal forces of the idling wind turbine and the internal forces of the operating wind turbine are compared first. This comparison also helps to check whether it is appropriate to create a meta-model for all the internal forces considered in this work or whether some of the internal forces in idling conditions are so small that no meta-model is required, as they have almost no influence on the lifetime.

For this investigation, the generated training and test data sets from the previous section are compared with the training and test data sets for meta-modelling for normal operation of the same offshore wind turbine (i.e. the same simulation model) from . For both data sets, the wind speed range from 3 m s⁻¹ (cut-in wind speed) to 25 m s⁻¹ (cut-off wind speed) is divided into bins with a width of 2 m s⁻¹. Subsequently, the mean of the short-term DELs is calculated for each bin and for both data sets. Example results are shown in Fig.  for Fx,DEL, Fy,DEL (see Table ), and the out-of-plane (OoP) bending moment at the blade root MOoP,Root,DEL. The comparison for the remaining internal forces is shown Appendix  in Fig. . It becomes clear that the internal forces at the monopile under idling conditions are in the same range compared to the internal forces during normal operating conditions and can be even higher than in normal operating conditions. The bending moments at the rotor blade roots are smaller but at least of a similar order of magnitude for low wind speeds. This study therefore leads to the conclusion that meta-models should be created for all six internal forces considered.

Based on the results in Sect.  showing significant impacts of ψ and the rotor speed on My,Root, this section briefly investigates whether ψ and the mean rotor speed ω need to be regarded as additional input parameters for meta-modelling. For this purpose, the meta-models are first trained with the five environmental input parameters described in Sect.  (vs, TI, Hs, Tp, and θmis) using all 8500 training samples. Subsequently, meta-models are created with ψ and ω as additional input parameters. To decide whether the additional input parameters need to be taken into account, the two meta-models created with different numbers of input parameters are compared for each considered internal force using the 1500 test samples. Here, the predicted short-term DELs of the two meta-models are compared with the short-term DELs determined with the aeroelastic simulation code FASTv8.

Example results are shown in Fig.  for My,DEL and My,Root,DEL depending on vs. From Fig. a and c, it is clear that the use of the five input parameters for the approximation of My,DEL leads to good results as both the overall trend and the scatter are represented. The meta-model of My,Root,DEL, in contrast, cannot represent the scatter of the simulated values of My,Root,DEL. Figure b and d show the approximations of the meta-models for My,DEL and My,Root,DEL with ψ and ω as additional input parameters. It is clear that there is barely any change in the approximation of My,DEL compared to the meta-model with five input parameters. For My,Root,DEL, in contrast, it becomes apparent that the approximation of the meta-model is significantly better when the two additional input parameters are employed. From these results it can be concluded, on the one hand, that it is sufficient for the internal forces at the monopile to use the five environmental input parameters that are also used for a wind turbine in normal operation. For the bending moments at the rotor blade roots, on the other hand, the two additional input parameters ψ and ω are required for an acceptable approximation quality of the meta-models.

To train and test the meta-models, the parameters ψ and ω are taken from the results of the aeroelastic simulations of the training and test data sets. Here, ψ is the initial azimuth angle at the start of the 10 min period used to calculate the DEL. ω is the mean rotor speed during the entire 10 min period. For a later prediction with the meta-models, a value between 0 and 360° can be chosen arbitrarily, as the rotor position does not depend on any other parameter. However, ω depends on, among other factors, the wind field of the aeroelastic simulation and is therefore actually an output parameter of the simulation. In order to be able to use ω as an input parameter, the relationship between the input parameters, vs, TI, Hs, Tp, θmis, and ψ, and the output parameter, ω, is therefore modelled using an additional Kriging meta-model. To train this meta-model for ω, the same simulation data are used as for training the meta-models for the prediction of the short-term DELs. Since the rotor speed strongly depends on the random seeds (see also Sect. ), it is not sufficient for the prediction of ω to only use the mean value of the Kriging meta-model for ω (as done for the meta-models for the prediction of the short-term DELs). Using only the mean value for the prediction, the scattering of ω cannot be properly modelled (see Fig. a). For this reason, the standard deviation is used in addition to the mean value to predict the short-term DELs. For this purpose, each predicted short-term DEL is randomly sampled from the normal distribution resulting from the predicted mean value of the corresponding short-term DEL and the standard deviation given by the Kriging meta-model of ω. Including the standard deviation of the Kriging meta-model for ω allows for a sufficiently accurate mapping of the relationship between ω and the input parameters (vs, TI, Hs, Tp, θmis, and ψ). This is shown in Fig. b.

Having defined the input parameters for meta-modelling, the next step is to analyse which meta-model settings lead to the best approximation of the simulation model. Here, four different basis functions and five covariance functions are analysed for the investigation of the meta-model settings for the Kriging meta-model. The analysed basis and covariance functions have already been used for Kriging meta-models in wind energy. For the five covariance functions, both the option with equal correlation lengths (isotropic covariance functions) and the option with different correlation lengths (anisotropic covariance functions) for each input parameter are evaluated. The investigated basis and covariance functions are listed in Table . More detailed information on the covariance functions can be found in .

For every possible combination of the investigated basis and covariance functions, a Kriging meta-model is created to select a meta-model setting for all further investigations in this paper. As a measure of error, the normalised root mean square error (NRMSE) is used. 8NRMSE=1E(Seq(x))∑i=1Ntest(S^eq(xi)-Seq(xi))2Ntest. Here, Seq(x) is the simulated short-term DEL, and S^eq(xi) is the ith short-term DEL predicted by the meta-model. Ntest is the number of test samples, and E(Seq(x)) is the expected value of the short-term DELs Seq(x). One advantage of this error metric is the possibility to compare the meta-models for the different fatigue loads directly with each other. Therefore, it is quite simple to determine whether there are internal forces that can be better represented by a meta-model than others.

It turns out that the approximation quality mainly depends on whether an isotropic or anisotropic covariance function is chosen. The anisotropic covariance function, i.e. different correlation lengths for the different input parameters, leads to a better approximation of the simulation model than the isotropic covariance function. An example can be seen in Fig.  for My,DEL. This result was to be expected, as the five input parameters (fatigue loads at the monopile) or seven input parameters (fatigue loads at the rotor blade root) each have a different impact on the investigated internal forces.

To be able to decide which settings lead to the best meta-model – also with regard to the required number of samples – a convergence study is carried out. Here, the number of samples is increased step by step up to the maximum number of training samples of 8500. Due to the result that the approximation quality is better when using anisotropic covariance functions, only the anisotropic covariance functions are considered further for the convergence study. Figure  shows the convergence of the NRMSE for Mx,Root,DEL and My,Root,DEL for the five different anisotropic covariance functions in combination with all analysed basis functions as a function of the number of samples used. The results of the remaining internal forces are shown in Appendix A in Fig. . It becomes clear that the choice of the covariance function has an influence on the number of samples required for meta-modelling, while the influence of the choice of the basis function is less significant for the most internal forces.

Since the ARD matern 3/2 covariance function leads to the best results overall for all fatigue loads considered, this covariance function is used from this point onwards in this paper. For the basis function, the use of no basis function, a constant basis function, or a linear basis function is recommended, as in combination with the ARD matern 3/2 covariance function, these lead to a good approximation for all internal forces analysed. Therefore, in the following, a linear basis function is used.

Furthermore, it is clear from Figs.  and that the value of the NRMSE has not yet converged for all internal forces (e.g. Mx,Root,DEL and Mx,DEL), even when all 8500 training samples are used. In order to achieve convergence, further samples would be necessary. However, as this is only useful if a convergence of the NRMSE is required for a sufficiently accurate meta-model, the next section first analyses how suitable the meta-models created in this work, using all 8500 samples, are in comparison to the simulation model.

In this section, the previously raised question of how good the meta-models created are in terms of their approximation quality and whether they should or could be more accurate is addressed. Here, it should be noted that the aeroelastic simulations themselves also contain uncertainties, e.g. due to the random seeds used for the generation of the wind and wave fields. This seed-to-seed uncertainty and the uncertainty in the meta-models compared to the aeroelastic simulation model are analysed in the following. The aim is to be able to rate the approximation quality of the meta-model in comparison to the seed-to-seed uncertainty of the simulation model. For this purpose, the test data set (see Sect. ) is simulated nine additional times with varied wind and wave seeds using the aeroelastic simulation code FASTv8. For the resulting 10 test data sets, which are identical except for the wind and wave seeds, the NRMSE is calculated for all possible combinations of the 10 test data sets to determine the seed-to-seed uncertainty of the simulation model (NRMSE_sim, 45 combinations in total). A schematic overview of the combinations considered can be found in Table . The mean value NRMSE_sim,mean is then calculated from the 45 values of NRMSE_sim. At the same time, the predictions of the meta-models are compared with the simulated test data sets. For the meta-models of the rotor blades, the predictions are compared with the test data sets in the same way that the test data sets were previously compared with each other (see Table ). Here, only the “S” on the righthand side is replaced by an “M” for meta-model prediction. For M2 to M10, separate meta-model predictions are generated. The number of the meta-model prediction, e.g. M2, indicates that the values of ψ from test data set 2 (S2) are used for this prediction. For ω, the rotor speeds are predicted using the meta-model for ω with the values of ψ from S2. Unlike the rotor blades, there is no dependence on ψ and ω of the meta-models for the monopile. For this reason, there is only one prediction from the meta-model for the test data for each internal force. This reduces the number of combinations with the test data sets to a total of 10 possible combinations (S1 vs. M1 to S10 vs. M10). NRMSE_meta is calculated for each combination of meta-model prediction and test data set. Then, in the next step, the mean values NRMSE_meta,mean are determined from the 45 (rotor blades) and 10 (monopile) values of NRMSE_meta. The resulting mean values of the NRMSE are then compared for all considered internal forces (see Table ). Additionally, a visual comparison, in which the first test data set (S1) is plotted against the second test data set (S2) and against the predictions of the meta-model (M2), is conducted. An example is shown in Fig. for Fx,DEL and My,Root,DEL.

The results in Table  show that, with the exception of My,Root,DEL, NRMSE_meta,mean is comparable to NRMSE_sim,mean for all considered internal forces. However, the deviation for My,Root,DEL is very small at less than 3 %, meaning that the NRMSE values are similar. For this reason, it can be concluded that the meta-models are sufficiently accurate, as NRMSE_meta,mean is comparable to or smaller than NRMSE_sim,mean. This is also clear from Fig. .

Based on these results and an additional convergence study, where the determination of NRMSE_meta,mean was repeated for training sample sizes from 500 to 8500 (see Fig. ), the findings of the previous section regarding the required sample size and the choice of the meta-model settings must be re-evaluated. Assuming that the values of NRMSE_meta do not have to be smaller than the values for NRMSE_sim, the results show that significantly fewer simulations than the 8500 simulations carried out in this work are enough to create sufficiently good meta-models. When using a linear basis function and an ARD matern 3/2 covariance function as recommended in the previous section, as few as 2500 samples are sufficient to create the meta-models for every considered internal force, although the NRMSE (see Figs.  and ) has not yet converged. Here, the NRMSE_meta,mean has almost converged for all internal forces (deviation of NRMSE_meta,mean using 8500 samples is less than 5 %). Of course, the choice of the sample size depends on the requested accuracy of the meta-models. Even with 500 samples, a very good approximation can be achieved. It can therefore be summarised that significantly fewer samples are required to create a sufficiently accurate meta-model compared to the results of the convergence study in Sect. .

Figure  also clearly shows that the seed-to-seed uncertainty has a different impact on the considered internal forces. For My,Root,DEL in particular, the influence of the seed-to-seed uncertainty is significant and it becomes clear that a change of the random seeds alone (while keeping all other input parameters unchanged) results in a large scattering of the short-term DELs. This large scattering can be explained by the fact that the different random seeds result in different wind fields, which cause the rotor to rotate at different speeds during the simulation (see also Sect. ).