Predicting Loads and Dynamic Responses of An Offshore Jacket Wind Turbine in A Nonlinear Sea

In order to improve the simulation efficiency, a novel transformed linear Gaussian model has been first proposed in this paper for generating equivalent “nonlinear” irregular waves. It is demonstrated by calculation examples that for obtaining equivalent “nonlinear” waves with the same accuracy, the transformed linear Gaussian model is about 2.7 times faster than the traditional nonlinear simulation method and is about 2.14 times faster than the method proposed by Agarwal and Manuel (2011). The loads and dynamic responses calculation results regarding an offshore jacket wind turbine in this paper demonstrate that nonlinearly simulated irregular waves with bottom effects should be considered in order to design an un-conservative support structure for the offshore wind turbine. Furthermore, by studying the calculation results in this article we have found that the loads and dynamic responses of the offshore wind turbine when inputting transformed linearly simulated waves with bottom effects are almost identical to the corresponding values when inputting nonlinearly simulated waves with bottom effects. All these calculation results clearly demonstrate the superiority and effectiveness of using our novel transformed linear Gaussian model for predicting the wave loads and dynamic responses of an offshore wind turbine operating in a realistic nonlinear sea with bottom effects.


Introduction
Renewable energy is being actively pursued in the contemporary world because it is sustainable and does not emit greenhouse gasses that are detrimental to the environment and human health. A classic example of renewable energy is offshore wind energy. An engineering device utilized for exploiting the offshore wind power is called an offshore wind turbine (OWT). In order to successfully design an offshore wind turbine, accurately simulating the irregular ocean waves in the OWT dynamic analysis process is of uttermost importance. However, until very recently, the majority of the people in the worldwide offshore wind energy research community have applied linearly simulated irregular waves in their OWT dynamic simulation processes (Jahangiri and Sun and Jahangiri, 2019;Wei et al., 2017;Saha et al., 2014;Wang, 2015Wang, , 2016Wang, , 2017Wang and Wang, 2017). The linear irregular waves simulation model has the limitations that it can only generate unrealistic waves with horizontal symmetries, i.e. the generated waves have statistically symmetric wave crests and troughs. This linear waves simulation model is only suitable for approximately simulating irregular waves from a very mild sea state in a very deep sea. However, real world ocean waves are statistically asymmetric (i.e. having sharper and higher crests but smoother and shallower troughs) in a harsh deep sea or at a shallow water coastal site. Since most of the modern offshore wind turbines are installed and operated in shallow water coastal sea areas, inputting linearly simulated irregular waves in their loads and dynamic responses analysis can therefore never be regarded as a reliable engineering practice. Therefore, the nonlinear irregular wave theory should be more appropriate for the analysis and design of these shallow water offshore wind turbines. However, in the existing literature there are only limited studies (Agarwal and Manuel, 2009, 2010, 2011Marino et al., 2013Marino et al., , 2014Marino et al., , 2015Mockutė et al., 2019;Schløer et al., 2016Schløer et al., , 2012 in which the nonlinear wave model has been applied to generate water waves as the inputs in the loads and responses analysis of offshore wind turbines. A major disadvantage of the nonlinear wave simu-lation model in Agarwal and Manuel (2009, 2010, 2011 is that it is very time-consuming. Furthermore, the sea bottom effects and their influences on the loads and dynamic responses of the offshore wind turbines had not been taken into account in the aforementioned studies (Agarwal and Manuel, 2009, 2010, 2011Marino et al., 2013Marino et al., , 2014Marino et al., , 2015Mockutė et al., 2019;Schløer et al., 2016Schløer et al., , 2012. In the field of offshore engineering it is well-known that in a shallow water coastal site the wave energy will be dissipated because of the existence of a sea bottom boundary layer. Therefore, the research results in the aforementioned studies obtained using nonlinearly simulated waves without bottom effects as simulation inputs should be treated with caution. Besides the nonlinear nature of the shallow water ocean waves, things are sometimes becoming more complex because not all sea states have narrow (or finite) spectral bandwidth and unimodal wave spectra. Oftentimes, ocean waves are mixed waves consisting of locally wind-generated high frequency waves and low frequency swell coming from distant storms. The resulting mixed sea states will have bimodal wave spectra Xia, 2012, 2013). Unfortunately, until very recently no research work has been conducted to predict the loads and dynamic responses of offshore wind turbines operating in a nonlinear sea characterized by a bimodal wave spectrum.
With the motivation to fill the knowledge gap as mentioned above, in the present study, the loads and dynamic responses of an offshore wind turbine operating in a nonlinear sea characterized by a bimodal wave spectrum will be rigorously calculated. The sea bottom effects on the loads and responses of the chosen offshore wind turbine will simultaneously be considered during the simulation process. With the aim to further improve the simulation efficiency, a novel transformed linear Gaussian model will be proposed for the first time in this article for generating equivalent "nonlinear" waves. The superiority and effectiveness of using the proposed novel transformed linear Gaussian model will be demonstrated through the subsequent calculation examples in this paper.

Nonlinear irregular waves and the transformed linear Gaussian model
In an idealized linear Gaussian sea the ocean waves have statistically symmetric crests and troughs. However, real world ocean waves are statistically asymmetric, i.e. the wave crests are becoming sharper and higher and the wave troughs are becoming smoother and shallower than expected under the linear Gaussian assumption. If x denotes the longitudinal coordinate and t denotes the time, we can then express the nonlinear sea free surface elevation η(x, t) as (Brodtkorb, 2004;Langley, 1987): r mn q mn in which are the first order linear Gaussian components and are the second order nonlinear correction components. In Eq. (1) is a sufficiently large positive integer and denotes a random complex valued wave amplitude that can be calculated based on a specific wave spectrum and the angular frequency . Furthermore, in Eq. (1) is a specific wave number that is related to through the dispersion relation, and is the uniformly distributed phase angle in the interval of [0 ]. The and terms in the above equation are second order transfer functions that can be calculated (for a constant water depth d) as follows (Langley, 1987): (2) . (3) The traditional linear irregular wave simulation method is executed by numerically implementing Eq. (1) omitting the term . The traditional nonlinear wave simulation method is executed by numerically implementing Eqs.
(1)−(3) altogether. Because there are N 2 sum frequencies components and another N 2 difference frequencies components in Eqs.
(2)−(3), the traditional nonlinear wave simula-tion method will become very time consuming when N is large. With the aim to improve the simulation efficiency, a novel transformed linear Gaussian model will be proposed for the first time in this article for generating equivalent "nonlinear" waves. The theoretical background of the proposed transformed linear Gaussian model is explained as follows.
By setting x=0 in and writing it in a simplified manner as , the nonlinear non-Gaussian wave elevation process can be modeled as the function of a standard linear Gaussian process : ) . (4) In the above equation the transformation G(·) performs the appropriate nonlinear translation and scaling so that is always normalized to have mean zero and variance one. In this research, a monotonic cubic polynomial type Hermite transformation model will be constructed for G(·) so that the first four moments of the original true process match the corresponding moments of the transformed model, i. e.: m η σ ηh 3h4 ϑ where and are the mean and standard deviation values of the second order nonlinear random waves. For N=4 moments, the expressions for the coefficients , and are (Winterstein, 1988): The coefficient in Eq. (6) is the skewness of the surface elevation process , and the coefficient in Eq. (7) is the kurtosis of the surface elevation process . The above Eqs. (5)−(8) indicate that the G(·) function is related to the values of , , , and of the second order nonlinear waves which can be calculated as follows (Langley, 1987).
In Eqs. (9)−(12), the values of the coefficients and can be calculated based on a specific wave spectrum according to the theories in Langley (1987). Then the func-tional transformation G(·) can be obtained by using Eq. (5).

The chosen offshore jacket wind turbine
Our calculation examples are with regards to the National Renewable Energy Laboratory (NREL) 5MW OC4 jacket offshore wind turbine placed in shallow water nonlinear waves as shown in Fig. 1.
The overall OC4 jacket offshore wind turbine system is composed of the NREL offshore 5 MW baseline wind turbine, the tower and the jacket foundation (Vorpahl et al., 2013). Table 1 summarizes the properties of the NREL offshore 5MW baseline wind turbine (Jonkman, 2007). The NREL 5MW OC4 jacket offshore wind turbine is sited in 20 m of water. Marc Seidel had once pointed out (Seidel, 2007) that "Jacket substructures are an attractive solution in water depths of about 20 m to 50 m for offshore wind turbines". Therefore, 20 m is a fairly practical water depth of a coastal site where an offshore jacket wind turbine can be installed and operated. The four legged OC4 jacket has four levels of X-braces, accordingly mud braces and four central piles with a penetration depth of 45 m being grouted to the jacket legs. The transition piece (TP) between the OC4 jacket and tower is a block of concrete that is penetrated by the upper parts of the four jacket legs. The total height of the OC4 jacket from mudline including the TP and excluding the tower is 70.15 m. The conical tower has a total length of 68 m leading to a realistic hub height over the mean sea level (MSL) of 120.55 m.
The Young's modulus of the jacket steel structure is 2.1×10 11 N/m 2 , and the Poisson's ratio of the jacket steel   (Vorpahl et al., 2013). It is noted that the water depth is only 20 m at the site where this offshore wind turbine is placed. Consequently, shallow water nonlinear irregular waves with bottom effects should be used when calculating the wave loads and dynamic responses of this offshore wind turbine. Fig. 2 is a "zoom-in" plot of the jacket support structure that also includes some detailed illustrations (Popko et al., 2012). As an example joint K1L2 identifies the double K-Joint at the upper K-joint level being part of leg 2. Joint X4S2 identifies the intersection point at the lowest level X-Joint on the jacket side 2. Joint X4S3 identifies the intersection point at the lowest level X-Joint on the jacket side 3. The outside diameter of each of the steel pipe leg of the jacket is 1.2 m, and the outside diameter of each of the steel pipe brace of the jacket is 0.8 m. The transition piece is positioned on the top of the jacket with its center in the centerline of the tower and this transition piece is a rigid concrete block with a mass of 666 t and a size of 4 m×9.6 m×9.6 m.

Calculation examples and discussions
We assume that this offshore jacket wind turbine is placed at an International Electrotechnical Commission (IEC) Class I-B wind regime site. The turbulence random field over the rotor plane is simulated using TurbSim (https://nwtc.nrel.gov/TurbSim) with a hub height wind speed of 18 m/s based on a Kaimal power spectrum and an exponential coherence spectrum. Hydrodynamic loads on the support structure of the offshore wind turbine are computed using Morison's equation expressed as follows.
and are the drag and the inertia coefficients, respectively. is the water density, and D is the diameter of a specific member of the jacket structure. The variables and are the undisturbed water particle velocity and acceleration, respectively. The variables and in Eq. (13) denote the velocity and acceleration of the support structure, which are obtained from the dynamic analysis of the entire offshore wind turbine system, including blade aero-elasticity, structural dynamics, etc. at each time step.
We have calculated the loads and dynamic responses of the aforementioned offshore wind turbine by stochastic time domain simulations implemented in FAST (Jonkman and Buhl, 2005). FAST joins an aerodynamics module (Aero-Dyn), a hydrodynamics module (HydroDyn) for offshore systems, a control and electrical system (servo) dynamics module, and a structural (elastic) dynamics module to enable coupled aero-hydro-servo-elastic analysis in the time domain. The AeroDyn module in FAST uses the wind-inflow data simulated via TurbSim and solves for the aerodynamic loads (wind loads). In our research, the shallow water nonlinear waves are simulated based on a bimodal Torsethaugen wave spectrum which is suited for the North Sea areas where most of the current offshore wind turbines are installed (https://en.wikipedia.org/wiki/Offshore_wind_pow er). The mathematical expression of the Torsethaugen spectrum is as follows (see Wang, 2014).
In the above equation, i=1 denotes the first peak, i=2 denotes the second peak, and = 4. denotes the JONSWAP wave spectrum whose mathematical expression is as follows: ω where is the wave angular frequency and In Eq. (15) denotes the significant wave height and denotes the spectral peak period. As shown in Brodtkorb (2004) The infinite water depth wave spectrum in Eq. (14) does not consider the sea bottom friction effects due to the existence of a bottom boundary layer. To overcome this shortcoming, a "corrected" finite water depth spectrum taking also the sea bottom effects into account can be obtained by using the following equation (Bouws et al., 1985): In Eq. (16), is a function with a range [0 1]. is a dimensionless angular frequency defined by . denotes the wave number that can be obtained by resorting to the following dispersion equation: T p T p In Fig. 3, the blue curve shows an infinite water depth Torsethaugen wave spectrum with H s =7.5 m and =6 s as inputs to Eqs. (14) and (15). The red curve in Fig. 3 shows a finite water depth (d=20 m) Torsethaugen wave spectrum with H s =7.5 m and =6 s as inputs to Eqs. (14)−(17). We can obviously find that the wave energy under the finite depth red curve spectrum has been reduced because of the energy dissipation due to the sea bottom effects.
The simulation results of a calculation example in Fig. 4 is for demonstrating the superiority and effectiveness of using our novel transformed linear Gaussian model for generating equivalent "nonlinear" waves.
The 650 s red curve wave time series in Fig. 4 was nonlinearly simulated using Eqs. (1)−(3) based on the red curve finite depth Torsethaugen spectrum in Fig. 3. The 650 s blue curve wave time series in Fig. 4 was transformed linearly simulated using Eqs. (4)−(12) based on the red curve finite depth Torsethaugen spectrum in Fig. 3. The "perfect fit" between the red curve and blue curve in Fig. 4 clearly demonstrates the accuracy of using our novel transformed linear Gaussian model for generating equivalent "nonlinear" waves with bottom effects. The zoom-in plot in Fig. 5 further substantiates the "perfect fit" between the red curve and blue curve time series.
The 650 s green curve wave time series in Fig. 4 was nonlinearly simulated using Eqs. (1)−(3) based on the blue curve infinite depth Torsethaugen spectrum in Fig. 3. Clearly, this green curve substantially deviates from the red curve (as well as the blue curve) in Fig. 4, and this obviously indicates that the finite depth sea bottom effects have to be considered during the simulation of nonlinear waves.
Carefully studying Fig. 4 reveals that the three wave time series have approximately the same mean value. However, the standard deviation values of the red curve and green curve wave time series in Fig. 4 are quite different. In statistics, the standard deviation is a measure that is used to quantify the amount of variation or dispersion of a set of data values. A low standard deviation indicates that the data points tend to be close to the mean value of the set, while a high standard deviation indicates that the data points are spread out over a wider range of values. The standard deviation values in Table 2 quantitatively show that in Fig. 4 the red curve from the nonlinear simulation with bottom effects fits perfectly with the blue curve from the transformed linear Gaussian model with bottom effects. These standard deviation values in Table 2 also show quantitatively that in Fig. 4 the two curves from nonlinear simulations with and without bottom effects deviate quite a lot from each other.
For predicting the long term design load of the aforementioned offshore wind turbine we usually need to carry out a full long term analysis (FLTA) in which a large num-   ber of wave time series should first be generated as the inputs to the subsequent wind turbine dynamic simulation processes. Suppose the cut-in wind speed at the hub height is 3 m/s, the cut-out wind speed is 25 m/s and the wind speed bin size is 1 m/s as shown in Table 3, we then need to consider 22 wind speed (U w ) bins in the full long term analysis. Similarly we need to consider 25 significant wave height (H s ) bins and 17 spectral peak period (T p ) bins. Therefore, we need to consider 9 350 (22×25×17) environmental state (U w , H s , T p ) bins in order to finish the full long term analysis. Suppose for each specific environmental state (U w , H s , T p ) we need to perform 20 10.5-minute stochastic time domain simulations of the offshore wind turbine, we then have to simulate 187 000 (9 350×20) 10.5-minute wave elevation time series. In Section 2 we have theoretically illustrated why generating nonlinear wave time histories using the traditional nonlinear simulation method is very time consuming. In this section we will use a specific calculation example to demonstrate that the nonlinear simulation method is indeed inefficient.
We have numerically implemented Eqs.
(1)−(3) and Eqs. (14)−(17) in MATLAB to generate 187 000 nonlinear wave sections (each 10.5 minutes long) based on the red Torsethaugen spectrum in Fig. 3. It took us about 4182 s to generate these 187 000 wave sections on a desktop computer (Lenovo PC, Intel ® Core (TM) i7-4 790 CPU @3.60 GHz 3.60 GHz 16 GB). On the other hand, we have also numerically implemented Eqs. (4)−(12) and Eqs. (14)−(17) in MATLAB to generate 187 000 equivalent "nonlinear" wave sections (each 10.5 minutes long) by using the transformed linear Gaussian model based on the red Torsethaugen spectrum in Fig. 3. It took us about 1 552 s to generate these 187 000 wave sections on a desktop computer (Lenovo PC, Intel ® Core (TM) i7-4 790 CPU @3.60 GHz 3.60 GHz 16 GB). The aforementioned calculation results demonstrate that for obtaining the equivalent "nonlinear" waves with the same accuracy, the transformed linear Gaussian model is about 2.7 times faster than the nonlinear simulation method.
In our research we have also included the comparison in terms of computational time with the method proposed by Agarwal and Manuel (2011) in order to show quantitatively the superior efficiency of the proposed method. For implementing this comparison we have utilized the HydroDyn (v2.05.01) module of the FAST software developed by the National Renewable Energy Laboratory. The nonlinear waves simulation routine in the HydroDyn module has been developed based on the work of Agarwal and Manuel (2011). From the HydroDyn user's manual (https://wind. nrel.gov/nwtc/docs/HydroDyn_Manual.pdf) we can find that "The full difference-and sum-frequency wave kinematics QTFs (Quadratic Transfer Functions) are implemented analytically following Sharma and Dean (1981), which extends Stokes second-order theory to irregular multidirectional waves." In our research we have slightly modified the Fortran source code of HydroDyn so that a bimodal wave spectrum can also be implemented in it. Utilizing Hydro-Dyn we then generated 187 000 nonlinear wave sections (each 10.5 minutes long) based on the red Torsethaugen spectrum in Fig. 3. It took us about 3 319 s to generate the 187 000 wave sections on a desktop computer (Lenovo PC, Intel ® Core (TM) i7-4 790 CPU @3.60 GHz 3.60 GHz 16 GB). The aforementioned calculation results demonstrate that for obtaining the equivalent "nonlinear" waves with the same accuracy, the transformed linear Gaussian model is about 2.14 times faster than the method proposed by Agarwal and Manuel (2011). Table 4 summarizes our predicted fore-aft shear forces on the mudline section of a leg pile of the aforementioned offshore jacket wind turbine operated in a nonlinear mixed sea with bottom effects versus without bottom effects. Please note that in Table 4 "MNSWB" means "Mean value under sea state of nonlinearly simulated waves with bottom effects". "MTLSWB" means "Mean value under sea state of Table 2 Standard deviation values of the three wave time series in Fig. 4   transformed linearly simulated waves with bottom effects". "MNSWWOB" means "Mean value under sea state of nonlinearly simulated waves without bottom effects". "SDNSWB" means "Standard deviation value under sea state of nonlinearly simulated waves with bottom effects". "SDTLSWB" means "Standard deviation value under sea state of transformed linearly simulated waves with bottom effects". "SDNSWWOB" means "Standard deviation value under sea state of nonlinearly simulated waves without bottom effects". The aforementioned codes are also applied in Tables 5−8.
The chosen specific nonlinear mixed sea state has a Torsethaugen spectrum (H s =7.5 m, =6 s, U w =18 m/s) and a water depth of 20 m. The calculation results in Table 4 have been obtained by performing stochastic time domain simulation of the offshore wind turbine in FAST. However, FAST does not have the capability of simulating nonlinear irregular waves with bottom effects that are the prerequisite inputs for the subsequent stochastic time domain simulations. Thus, in our study, we have numerically implemen-ted Eqs. (1)−(3) and Eqs. (14)−(17) in MATLAB to generate nonlinear wave sections with and without bottom effects (each 10.5 minutes long) based on the Torsethaugen spectra in Fig. 3. We have also numerically implemented Eqs. (4)−(12) and Eqs. (14)−(17) in MATLAB to generate an equivalent "nonlinear" wave section (10.5 minutes long) with bottom effects by using the transformed linear Gaussian model based on the red Torsethaugen spectrum in Fig. 3. These generated nonlinear irregular waves time series are saved as .Elev files and imported into FAST for subsequent stochastic time domain simulations of the offshore wind turbine. Further computations in FAST have led us to finally obtain the wind turbine fore-aft shear forces time series on the mudline section of a leg pile as shown in Fig. 6 and Fig. 7.
In a similar way and with regards to the same offshore jacket wind turbine we have also calculated the axial forces time series on the mudline section of a leg pile, the fore-aft deflections at X4S2 and X4S3, and the fore-aft bending moments time series on the mudline section of a leg pile, and our calculation results are summarized in Figs. 8−15 and    Tables 5−8. Please note that in the afore-mentioned loads and dynamic response calculations, the structural nonlinearity is not included. This neglection may overestimate the structural dynamic responses. This is the limitation of the present work and will be tackled in our future research. By studying the calculation results in Figs. 14−15 and Table 8, we notice that using nonlinearly simulated waves without bottom effects as inputs in the stochastic time domain simulation of the offshore jacket wind turbine will      slightly over-predict the fore-aft bending moments on the mudline section of a leg pile. However, by carefully studying the calculation results in Figs. 6−13 and Tables 4−7, we can find that using nonlinearly simulated waves without bottom effects as inputs in the stochastic time domain simulation of the offshore jacket wind turbine will significantly over-predict the fore-aft shear forces and the axial forces on the mudline section of a leg pile and the fore-aft deflections at X4S2 and X4S3. Therefore, nonlinearly simulated realistic waves with bottom effects should instead be utilized as inputs in the stochastic time domain simulation in order to design an un-conservative support structure for the offshore wind turbine. This will help offshore energy companies to develop and design more economically competitive offshore wind turbines.
Our calculation results in Figs. 4−5 and Table 2 have substantiated the accuracy and efficiency of the novel transformed linear Gaussian model for generating equivalent "nonlinear" irregular waves. On the other hand, by carefully studying the calculation results in Figs. 6−15 and Tables 4−8 we also see that the offshore jacket wind turbine fore-aft shear forces and bending moments values and axial forces values and the fore-aft deflections at X4S2 and X4S3 when inputting transformed linearly simulated waves with bottom effects are almost identical to the corresponding values when inputting nonlinearly simulated waves with bottom effects. All these calculation results undoubtly demonstrate the superiority and effectiveness of using our novel transformed linear Gaussian model for predicting the wave loads and dynamic responses of an offshore wind turbine operated in a realistic nonlinear mixed sea with bottom effects.

Concluding remarks
In order to improve the simulation efficiency, a novel transformed linear Gaussian model has been first proposed in this paper for generating equivalent "nonlinear" irregular waves. It is demonstrated by calculation examples that for obtaining the equivalent "nonlinear" waves with the same accuracy, the transformed linear Gaussian model is about 2.7 times faster than the traditional nonlinear simulation method and is about 2.14 times faster than the method proposed by Agarwal and Manuel (2011). Then, the fore-aft shear force and bending moment values and axial force values and the fore-aft deflections at nodes X4S2 and X4S3 of an offshore jacket wind turbine operated in a nonlinear mixed sea have been rigorously calculated by using the nonlinear simulation method taking into account the bottom effects. It has been found that using nonlinearly simulated waves without bottom effects as inputs in the stochastic time domain simulation of the wind turbine will always over-predict the fore-aft shear force and axial force values and the fore-aft deflections at nodes X4S2 and X4S3. Therefore, nonlinearly simulated realistic waves with bottom effects should instead be utilized as inputs in the stochastic time domain simulation in order to design an unconservative support structure for the offshore jacket wind turbine. This will help offshore energy companies to develop and design more economically competitive offshore wind turbines. Furthermore, by studying the calculation results in this article we have found that the fore-aft shear force and bending moment values and axial force values and the fore-aft deflections at nodes X4S2 and X4S3 when inputting transformed linearly simulated waves with bottom effects are almost identical to the corresponding values when inputting nonlinearly simulated waves with bottom effects. All these calculation results clearly demonstrate the superiority and effectiveness of using our novel transformed linear Gaussian model for predicting the wave loads and dynamic responses of an offshore wind turbine operated in a realistic nonlinear mixed sea with bottom effects.

Acknowledgments
Special thanks go to the anonymous reviewers whose valuable comments have led to the improvement of this paper.