Dynamic Response Analysis and Vibration Control for A Fixed-Bottom Offshore Wind Turbine Subjected to Multiple External Excitations

For the offshore wind turbines installed in earthquake areas, their operation is affected by seismic loads in addition to wind and wave loads. Therefore, it is necessary to study the dynamic responses and vibration control of the wind turbines. In previous studies, the structural responses of offshore wind turbines are usually investigated in the parked case, while the blade rotation effect is usually not considered. The evaluation on the structural responses may be inaccurate under this condition, further affecting the evaluation on the vibration control performance of a control system. In view of it, this paper established a complete multi-body model of a fixed-bottom offshore wind turbine considering pile-soil interaction, and then performed simulations when the wind turbine was subjected to multiple external excitations. Continued, a single tuned mass damper (STMD) system and a multiple tuned mass dampers (MTMDs) system were applied to control structural vibrations of the wind turbine. Then, based on the construction of a simplified main structure-TMD system, TMD parameters were optimized. Finally, twelve load cases including operating and parked conditions were selected to perform simulations. Results show that the influence of the seismic excitation on blade responses is greater under the parked condition than that under the operating condition. Moreover, STMD/MTMDS exhibit better performance under the parked condition than that under the operating condition. Compared with STMD, MTMDS can better suppress the vibrations at both the fundamental and highorder modes, and exhibits significant robustness under the condition of changing soil parameters.


Introduction
Owing to the increasing requirement for the power and electricity in the world, the cost of fossil fuels is also increasing (Ettefagh, 2015). Recently, opportunities for building high-power offshore wind turbines have been created (Wang et al., 2018). Offshore wind farms have ampler wind resource than the onshore sites, so they can provide more electricity (Wang et al., 2019). Many types of substructure such as monopile, tripod, and jacket can be used to support the fixed-bottom offshore wind turbines (Hussan et al., 2017). Particularly, the monopile offshore wind turbine studied in this work still has great application potentiality, although some larger offshore wind turbines are deployed at deeper water sites (Seidel, 2014).
For the offshore wind turbines, they must be able to safely experience the effect of marine environmental factors such as wind and wave over their lifetimes (Antonutti et al., 2014). Additionally, some offshore wind turbines are installed in earthquake-prone regions such as USA, Japan, and China (Katsanos et al., 2016). Seismic excitations can in-duce the structural vibration of wind turbines (Katsanos et al., 2016). Under this condition, the offshore wind turbines may be simultaneously affected by wind, wave, and seismic loadings, which may result in significant vibrations of the wind turbines. Therefore, it is necessary to explore the influence of combined external loadings on the offshore wind turbines to design effective approaches to suppress the structural vibration.
In order to mitigate the dynamic response of structures subjected to various vibration sources, many vibration control devices are developed, including tuned mass dampers (TMDs) (Ding et al., 2019), tuned liquid column dampers (TLCDs) (Colwell and Basu, 2009), roller dampers (Zhang et al., 2014), etc. TMD, which is the research focus of this paper, has been widely used in buildings and bridges (Rahman et al., 2015). Specifically, the passive TMD is adopted in the present work because it is easier to implement and needs no external power input compared with the semi-active (Arrigan et al., 2011;Van-Nguyen et al., 2016) and active (Fitzgerald et al., 2013) control modes. In addition, according to the number of TMDs, TMD systems can be divided into single TMD (STMD) and multiple TMDs (MT-MDs). For the application of STMD in the wind turbine vibration control, Stewart and Lackner (Stewart and Lackner, 2013) developed a set of optimal passive TMDs for four different offshore wind platforms. Based on an updated version of the FAST code (Jonkman and Buhl Jr, 2005), named FAST-SC (Lackner and Rotea, 2011), simulations were performed under combined wind and wave conditions to evaluate the performance of TMD. For the application of MTM-Ds, Zuo et al. (2017) applied an MTMDs system to reduce the vibration of a tower subjected to wind, wave, and seismic excitations, but only modeled the tower under the parked condition. In the above studies, MTMDs has been proved to have better vibration control and robust performances than STMD. Therefore, MTMDs is worthy of further study.
Note that the dynamic characteristics of the offshore wind turbine are greatly affected by its operating states (Ghassempour et al., 2019). Therefore, an accurate structural response analysis of the offshore wind turbine suffered external excitations should be carried out considering its operating states. Moreover, the blade effects should not be ignored in the dynamic response analysis, even if this analysis is aiming at a tower (Gonzalez, 2016). Unfortunately, there are relatively few studies on the dynamic response analysis of the normal operating fixed-bottom wind turbines subjected to combined effects of wind, wave, and earthquake loadings. Motivated by this research potential and based on the remarkable pioneering research on application of TMD in offshore wind turbines (Lackner and Rotea, 2011;Zuo et al., 2017;Ghassempour et al., 2019;Stewart and Lackner, 2014), the focus of this work is to investigate (1) the influence of seismic loadings on the structural re-sponse of the monopile offshore wind turbine under normal operating and parked conditions, and (2) the control effects of STMD and MTMDs systems on vibrations of the wind turbine subjected to combined wind, wave, and seismic excitations. Based on this, the structure of this paper is organized as follows.
Section 2 introduces the research object. In Section 3, a multi-body model of the monopile offshore wind turbine with pile-soil interaction included is established. In Section 4, several load cases are chosen. In Section 5, the structural responses of the original wind turbine with and without considering the seismic excitation are carried out. In Section 6, the system dynamics of the wind turbine coupled with TM-Ds is explored. The robustness of MTMDs is also investigated. Finally, the conclusions are provided.

Research object
In the present work, the National Renewable Energy Laboratory (NREL) 5-MW wind turbine with a monopile supporting structure (Jonkman and Musial, 2010) is modeled. This wind turbine is a three-bladed upwind variable-speed variable-pitch turbine (Jonkman et al., 2009). Tower is connected to a monopile supporting structure with a diameter of 6 m and a thickness of 60 mm. The Young's modulus and shear modulus of the tower are 210 GPa and 80.8 GPa, respectively. The density of the tower is set as 8500 kg/m 3 for considering the paint, bolts, welds and flanges that are not considered in the thickness data (Jonkman and Musial, 2010). The tower base is located at 10 m above the mean-sea level. The monopile extends from the tower base down to the sea floor, which is at 20 m below the mean-sea level. The density, Young's modulus and Poisson's ratio of the monopoile are 7500 kg/m 3 , 210 GPa and 0.3, respectively. The detailed geometrical, material, structural and mass properties are available in References (Jonkman and Musial, 2010;Jonkman et al., 2009). Fig. 1 shows the topology of the multi-body model of the 5-MW monopile wind turbine. The major components are modeled as rigid or flexible bodies. These bodies are connected by joints with various degrees of freedom (DOFs) and spring-damping force elements. In order to satisfy particular demands of the installation distribution and dynamic simulations, various components such as the nacelle, tower, blades and gearbox will be provided to some type of the wind turbine. Therefore, it is necessary to model these components as subsystems in establishing the multibody model.

Multi-body modeling of the wind turbine
The tower is modeled as a flexible body and is rigidly connected to the foundation. The hydrodynamic loads acting on the supporting structure are generated by the Hydro-Dyn module, which is developed by NREL (Simpack, 2016). The spring-damping force element between foundation and ground is used to model pile-soil interaction. The seismic excitation is applied between ground and reference frame. The nacelle is modeled as a rigid body and is mounted at the tower top. The main shaft used to support the rotor is installed in the nacelle and elastically coupled with the gearbox. The gearbox housing and the generator are installed in the nacelle. The output shaft of the gearbox and the generator shaft are respectively connected to the brake disc by joints and spring-damping force elements. Three blades are modeled using the Euler-Bernoulli beam and are connected to the hub. The blade pitch angles are controlled by a pitch controller. The blade aerodynamic loads are provided by the AeroDyn module developed by NREL (Simpack, 2016). Fig. 2 shows the simplified pile-soil interaction models in the fore-aft and side-to-side directions of the wind turbine. According to the research of Sun and Jahangiri (2018), the pile-soil interaction is represented with the translational springs with coefficients of k x and k y as well as the rotational springs with coefficients of k α and k β . The damping properties of the soil are represented by the translational and ro-tational dash-pot dampers with coefficients of d x , d y , d α , and d β . According to Sun and Jahangiri (2018), k x and k y are set to 3.89×10 9 N/m and k α and k β are set to 1.14×10 11 Nm/rad to represent the clay soil condition. Soil damping properties d x , d y , d α and d β are chosen such that the corresponding damping ratios are 0.6% (Sun and Jahangiri, 2018). It needs to be noted that the vertical pile-soil interaction is not considered because it is not the focus of the present work.

Load case setting
In order to comprehensively explore the dynamic characteristics of the monopile offshore wind turbine under normal operating conditions, eleven wind−wave combinations representing normal sea conditions are selected (Carswell, 2015), as shown in Table 1 (Load cases 1−11). Wind speeds of 4 m/s and 24 m/s are near the cut-in wind speed and the cut-out wind speed, respectively. The significant wave height is considered as the function of a Weibull probability density function (DNV, 2013), which is fit to the wave data associated with each wind bin data.
Additionally, 50-year extreme wind and wave are also considered for the parked conditions of the wind turbine. The mean value of hub-height wind speed is set to 37 m/s with an intensity of 11%, and the significant wave height is set to 13.8 m with a period of 19 s (Si et al., 2014), which are also listed in Table 1 (load case 12). Under the parked condition, the drivetrain is fully locked and the blade pitch angle is set to the maximum value. Fig. 3 shows the acceleration time history and the power spectral density (PSD) of the seismic excitation, which is derived from the 1940 El Centro Earthquake. The peak   ground acceleration is 0.22g and the duration time is 53.735 s.

Structural responses of the original wind turbine
It is of interest to explore the effect of earthquake loadings on dynamic responses of the wind turbine system without TMD, as shown below.
The seismic excitation starts at 400 s and ends at 453.735 s in simulations. The structural responses of the original wind turbine system (without TMD) are evaluated with and without considering the earthquake load. The standard deviation (STD) and the 95th percentile of the blade-root out-of-plane shear force (RootFc), blade-root out-of-plane bending moment (RootMt), tower-top fore-aft shear force (YawBrFc), tower-top fore-aft bending moment (YawBrMt), tower-base fore-aft shear force (TwrBsFc) and tower-base fore-aft bending moment (TwrBsMt) are calculated as the evaluation indices. Figs. 4 and 5 show the statistics of these evaluation indices with and without consider-   XIE Shuang-yi et al. China Ocean Eng., 2022, Vol. 36, No. 1, P. 50-64 53 ing the seismic loading. The load increase percentages from the case without considering the seismic loading are shown in Fig. 6. Several remarks on these data are obtained as follows.
(1) Generally, the earthquake load significantly influences the dynamic responses of the wind turbine. Specifically, the STD and the 95th percentile of structural loads generally become larger when the earthquake load is applied. This is expected because when the seismic excitation is considered, huge additional power comes in, which in turn results in larger dynamic responses of the turbine system.
(2) In general, the seismic excitation has the most significant impact on the tower-base loads under the normal operating conditions. Concretely, the increase percentages of the STD and the 95th TwrBsFc under Load case 1 reach up to approximately 987% and 624%, respectively (Fig. 6).
(3) For the normal operating cases, the increase percentages of all the evaluation indices are almost the largest under Load case 1 when the seismic excitation is considered (Fig. 6). This is attributed to the relative small values of structural loads and small aerodynamic damping for this case.
(4) Under the parked condition, the pitch angles of blades are feathered to 90° and the turbine rotor is fixed. Therefore, the STDs and the 95th percentiles of the structural loads are relatively smaller than those under the normal operating cases. In addition, it is observed that the influence of the earthquake loadings on the blade loads is greater under the parked condition than that under the normal operating conditions. This indicates that the aerodynamic damping plays a significant role in the influence of the seismic excitation on the structural responses of the wind turbine (Prowell et al., 2012;Asareh et al., 2016).
For brevity of this paper, only the time histories and the corresponding frequency spectra of structural loads under Load case 1 and the parked condition within the duration of the seismic loadings are represented (Figs. 7 and 8). It is obviously observed that the fluctuation of wind turbine dynamic responses becomes more obvious due to the application of the seismic load. This change is represented by the increase of STDs and the 95th percentiles of structural loads. Moreover, it is found from the frequency spectra that the seismic excitation not only increases the vibration energy at low frequencies, but also enables the high-order structural modes of the wind turbine system.

System dynamics of the wind turbine coupled with TMDs
6.1 Design of the STMD/MTMDs system When designing TMD, the parameters need to be determined including the mass, spring stiffness and damping constant. In this paper, by considering that the tower bending modes are crucial for the monopile offshore wind turbines (Lackner and Rotea, 2011), the first and second structural modes of the supporting structure, which would be simultaneously enabled under the seismic excitation, are therefore controlled. The TMDs are installed at the first two maximum amplitudes of the modal shapes. In the present paper, a TMD mass ratio of 5%, i.e. 44000 kg, is selected according to the study by Ghassempour et al. (2019). Note that the study on the selection of TMD mass ratio is not the focus in our paper, which will be further investigated in the future. Moreover, in order to suppress the tower displacement, STMD is mounted at the tower top. Compared with the STMD system, MTMDs is relatively complex to design. Fig. 9 shows the simplified schematic diagram of the main structure−MTMDs system. In the figure, , and represent the mass, stiffness and damping coefficients, respectively. Subscript p represents the main structure and subscript TMDn represents the n-th TMD in the MTMDs system.
For the simplified main structure−TMD system, its dynamics equation is as follows: where x is the vector with N+1 dimensions, which repres- ents the displacement response relative to the foundation. The first component corresponds to the displacement of the main structure, and the rest corresponds to the displacement of TMD. The matrices M, D and K represent the mass matrix, damping matrix and stiffness matrix of the system respectively.
As the setting in references (Zuo et al., 2017;Li and Liu, 2003), two assumptions are made in this paper to facilitate the design of the MTMDs system: (1) the natural frequencies of the MTMDs system are uniformly distributed around their mean natural frequency; (2) each TMD has the identical mass and damping coefficient. For a main structure-MTMDs system, the natural frequency of the main structure can be determined as . The natural frequency of the j-th TMD in MTMDs can be calculated using the first assumption: represents the mean natural frequency of the MTMDs system, and is the non-dimensional frequency interval.
The mass ratio of the main system is defined as , then the mass of each TMD is determined by: The stiffness and damping coefficients of each TMD in MTMDs are: is the mean damping ratio of the MTM-Ds system, and is the damping ratio of the j-th TMD. In order to suppress the tower displacement response under external excitations, the mean square displacement of the tower is taken as the optimization objective, which is expressed as (Xu and Igusa, 1992): where is the power spectral density of the excitation and is the transfer function. In this work, the combined wind, wave, and seismic excitations are taken into account. Wind and wave loadings excite the fundamental mode of the system, while the seismic loading mainly excites the second mode of the system. For the base excitation and the dynamic pressure acting on the supporting structure, the transfer functions are different (Zuo et al., 2017).
For the base acceleration excitation, the transfer function is expressed as (Xu and Igusa, 1992): where For the dynamic pressure on the tower, the transfer function is expressed as (Asareh et al., 2016): μ Based on equations above, the optimal TMD parameters can be calculated via a numerical searching approach by minimizing the mean square displacement of the tower when the mass ratio is determined. In the present paper, the simplex coding genetic algorithm (SCGA) is used for TMD parameter optimization (Hedar and Fukushima, 2003). SCGA combines the genetic algorithm and simplexbased local optimization algorithm named Nelder-Mead method (Kelley, 1999), which converges fast with low computational cost (Si et al., 2014). Table 2 shows the parameters of the STMD and MTM-Ds systems. In the table, STMD indicates that a single TMD is installed at the tower top to mainly suppress the first tower fore-aft structural mode (Fig. 10a). 2-MTMDs means two TMDs are simultaneously adopted to suppress both the first and second tower structural modes (Fig. 10b). 10-MT-MDs denotes that ten TMDs are simultaneously applied to control the first two tower fore-aft structural modes, with five TMDs used to mitigate the first tower mode and another five adopted to mitigate the second tower mode, as shown in Fig. 10c.

Dynamic responses of the wind turbine coupled with TMDs
In simulations, the external loadings are assumed acting in the fore-aft direction of the wind turbine. When STMD, 2-MTMDs and 10-MTMDs systems are adopted respect- ively, the displacement time histories at both the tower top and Position 2 for twelve load cases are obtained. The mitigation percentages of STDs and the 95th percentiles of displacements versus the case without TMD control are summarized in Figs. 11 and 12. In the figures, the negative percentages indicate that the results obtained with TMD are larger than those obtained without TMD. Note that for the sake of simplicity, only the time histories of displacements under load case 1 and the parked condition within the duration of the seismic excitation are represented. Based on these results, some remarks are obtained.
(1) Firstly, the displacement reduction percentages of the three different TMD arrangements under the parked condition are much larger than those under the normal operating conditions. This is because when the wind turbine is parked, its dynamic behaviors are not influenced by the operating state; therefore, the vibration energy is mainly concentrated near the first and second structural modes. The tuned STMD/MTMDs can play a significant role in restraining the displacements of the tower.
(2) Secondly, it is found that among the three TMD arrangements, 10-MTMDs has the best mitigation effect on the displacements of the two positions, followed by 2-MT-MDs. This can be explained by the following reasons. When the seismic excitation is applied, not only the fundamental vibration mode but also the higher structural modes of the wind turbine system are enabled. However, the STMD system can only control the vibration at the fundamental frequency.
(3) Thirdly, as shown in Figs. 13 and 14, in the first few seconds after applying earthquake loads, the STMDM/MT-MDs systems do not seem to function. This is because the STMDM/MTMDs does not begin to move relative to their original positions, which needs to take time. After this, the STMDM/MTMDs systems can effectively reduce the displacements of the tower. In order to more comprehensively examine the vibration control effect of STMD/MTMDs, the mitigation percentages of STDs and the 95th percentiles of structural loads from the case without TMD for different TMD arrangements are shown in Figs    XIE Shuang-yi et al. China Ocean Eng., 2022, Vol. 36, No. 1, P. 50-64 59 1 and the parked condition, and Figs. 18 and 20 show the corresponding frequency spectra. Based on these figures, some important conclusions are obtained. When the STMD system is applied, the vibration at the fundamental frequency can be mitigated. Figs. 18 and 20 illustrate that, as expected, the STMD system can suppress the vibration at the fundamental frequency, but the suppression effect of STMD on high-order vibration is not satisfact-ory. This is reflected in Figs. 15 and 16 as well as the timedomain diagrams. It is observed that the amplitudes at highorder frequencies under STMD are almost coincident with the case without TMD. When the 2-MTMDs system is adopted, the structural loads are further mitigated compared with the case with STMD. This can be explained by the fact that the high-order vibration modes are suppressed by another TMD located at Position 2. Therefore, multiple peaks

60
XIE Shuang-yi et al. China Ocean Eng., 2022, Vol. 36, No. 1, P. 50-64 Fig. 18. Frequency spectra of structural loads under load case 1. XIE Shuang-yi et al. China Ocean Eng., 2022, Vol. 36, No. 1, P. 50-64 61 in the time histories of structural loads are reduced and the response curves become smoother. When the 10-MTMDs system is installed, the structural vibrations of the wind turbine system are further suppressed to varying degrees under different load cases. In addition, the frequency spectra of blade-root loads obviously exhibit that the 10-MTMDs system can significantly suppress the peaks at main frequency contents, except for the energy of the tower-top fore-aft bending moment at approximate 1 Hz.
To summarize, the high-order structural modes are excited when the seismic load is applied. The utilization of the MTMDs system can significantly reduce these high-order vibration modes and further suppress the dynamic responses of the wind turbine system. In addition, splitting a large TMD into multiple small TMDs will be conducive to the installation of TMDs because the mass of each TMD in MTMDs becomes smaller.

Robust performance of the MTMDs system
In this section, the robust performance of the MTMDs system proposed in this paper will be explored under the cases with various soil stiffness and soil damping constants. For brevity, this robustness study is performed only for Load case 1 and the parked condition. The coupled simulation model is run for a case with a soil spring coefficient that is 10 times and 10% the initially determined value, and a case with soil damping coefficient that is 10 times and 10% the original setting parameter. Figs. 21 and 22 list the mitigation percentages of STDs of structural loads from the cases without TMD control (the soil parameters are adjus-ted accordingly). It can be found from figures that the 10-MTMDs system can still suppress the structural loads when the soil parameters are changed under the two conditions, which verifies the robustness of the 10-MTMDs system. Particularly, the vibration mitigation effect of 10-MTMDs is better under a change of soil damping constant than under a change of soli stiffness constant. This is because modifying the soil spring coefficient changes the natural frequency of the wind turbine system, and the initial tuning frequencies of 10-MTMDs are no longer fully suitable. Specifically, when the soil stiffness constant becomes larger or smaller, the first tower fore-aft natural frequency increases or decreases accordingly, as shown in Figs. 23a−23b and Figs. 24a−24b. On the contrary, the influence of changing the soil damping constant on the first tower fore-aft mode can be negligible. Moreover, similar with the analysis in Section 5, the load reduction effect of 10-MTMDs under the parked condition is better than that under the normal operating condition even though the soil parameters are changed. Another important finding of this robustness study is that when TMD control is not applied, the change of soil parameters causes the change of the dynamic responses of the wind turbine system. Note that the influences of changing the soil stiffness or damping constants on the wind turbine structural loads are different. Restricted by the length of this paper, the content will be further studied in the future.

Conclusions
This work adopts the single TMD (STMD) and the multiple TMDs (MTMDs) systems to control the structural vibration of a fixed-bottom offshore wind turbine subjected to combined effects of wind, wave, and seismic loadings. In view of the possible effects of rotation of the wind turbine rotor, both the normal operating and parked conditions are simulated in this paper. Firstly, through the SIMPACK framework, a complete multi-body dynamics model of the offshore wind turbine was built, where the pile-soil interaction was considered. Secondly, the full-system eigenfrequency analysis was performed to explore the effect of pilesoil interaction on the structural modes. Thirdly, the external loadings including aerodynamics, hydrodynamics, and seismic loadings are modeled to construct the coupled aerohydro-elastic-servo simulation model. Fourthly, the influence of the earthquake excitation on structural responses of the original wind turbine system was analyzed. Fifthly, the dynamic responses of the wind turbine coupled STMD/MT-MDs were analyzed under the combined wind, wave, and earthquake excitations to evaluate their vibration suppression performances. Finally, the robustness of MTMDs was investigated. Main conclusions of this paper are given as follows.
(1) Generally, the seismic excitation has the most significant impact on the tower-base loads under the normal operating conditions. Specifically, the increase percentages of  the STD TwrBsFc and the 95th TwrBsFc under Load case 1 reach up to approximately 987% and 624% compared with the case without applying the seismic excitation, respectively.
(2) Owing to the significant role of aerodynamic damping, the influence of the seismic excitation on the structural responses of blades under the parked condition is greater than that under the normal operating conditions. In addition, the vibration suppression effect of the STMD/MTMDs systems under the parked condition is better than that under the normal operating conditions.
(3) Compared with the single TMD, multiple TMDs can better suppress both the fundamental and high-order vibration modes when the wind turbine is subjected to the combined wind, wave, and earthquake loads. Moreover, the MTMDs system has significant robust performance under the condition of changing soil parameters, and is more conducive to install due to the smaller mass of each TMD than that of the STMD system. Noted that the main contribution of this work is to investigate the vibration control effect of STMD and MTM-Ds on the wind turbine subjected to wind, wave, and earthquake loadings under the normal operating and parked conditions. Therefore, more comprehensive research such as the  XIE Shuang-yi et al. China Ocean Eng., 2022, Vol. 36, No. 1, P. 50-64 experimental testing is necessary to adopt STMD or MTM-Ds in the practical engineering applications. Moreover, the influence of various soil parameters on dynamic characteristics of the wind turbine will be investigated in detail in the future.