Study on the Structural Vibration Control of a 10 MW Offshore Wind Turbine with a Jacket Foundation Under Combined Wind, Wave, and Seismic Loads

Abstract

As offshore wind power continues to develop, with increased capacity and ability to function in deeper waters, jacket-type offshore wind turbines (OWTs) are becoming increasingly challenged by complex environmental loads and significant structural vibration issues. This study focuses on a 10 MW jacket foundation OWT and proposes an optimization approach for tuned mass damper (TMD) parameters based on the artificial bee colony (ABC) algorithm. A fully coupled model of the OWT and TMD system is developed, and the TMD parameters are optimized through frequency-domain analysis and time-domain simulations. The vibration control performance of the optimized TMD is then evaluated under combined wind, wave, and seismic excitations. The results show that the passive TMD achieves substantially greater vibration suppression under seismic loading compared to combined wind and wave conditions. In addition, the optimized TMD reduces the standard deviations of tower-top displacement and tower-base bending moment by more than 50%, significantly enhancing the dynamic response of the structure and contributing to an extended fatigue life.

1. Introduction

With the global energy transition advancing toward decarbonization, offshore wind power has emerged as a key area of research due to its favorable characteristics, including remote siting, away from urban centers, and substantial development potential [1]. According to the GWEC [2], the wind energy sector is projected to achieve a compound annual growth rate of 8.8%, with global installed wind power capacity expected to increase by 981 GW by 2030. In recent years, the rated capacity of individual wind turbines has steadily grown, with units of 10 MW class and larger becoming the industry mainstream. Jacket-type foundations, known for their high adaptability to deep-water conditions, have been widely employed in OWT installations [3]. However, as OWT capacities scale up, structural vibration challenges in complex marine environments have become increasingly prominent. In particular, low-frequency modal vibrations can lead to fatigue damage in both the tower and the foundations, thereby posing serious risks to the long-term structural integrity and economic viability of OWT systems [4].

Research on vibration control strategies for OWTs can generally be classified into two primary approaches. The first approach involves actively adjusting the turbine’s operational parameters, such as blade pitch angles and generator torque, in order to regulate the aerodynamic loads acting on the OWT blades. The second approach encompasses the use of vibration control devices such as tuned mass dampers (TMD), tuned liquid column dampers (TLCD), and others, which act through passive, semi-active, or active mechanisms to dissipate energy and reduce structural vibrations [5].

The first approach to controlling OWTs involves dividing operational strategies into three regions based on incoming wind speeds [6]: Region I, Region II, and Region III. In Region I, where the wind speed is below the cut-in threshold, the turbine blades remain stationary, and no active control is applied. Once the wind speed exceeds the cut-in value in Region II, the turbine begins power generation, typically employing a maximum power point tracking method to maximize energy capture. In Region III, when the wind speed reaches or surpasses the rated level, pitch angle regulation is used to maintain a constant power output. Extensive research has focused on control methodologies within Regions II and III. Jonkman et al. [7] proposed a baseline controller widely implemented in the open-source FAST V7/8 simulation software for OWTs, with subsequent studies validating its effectiveness in managing turbine dynamics [8,9]. Building upon this foundation, the controller evolved into the ROSCO controller, incorporating features such as discrete Kalman filtering for wind speed estimation, set-point smoothing for torque and pitch transitions, and tip speed ratio tracking for generator torque control [10]. The ROSCO controller has since been extensively adopted in OWT control simulations [11,12]. Moreover, the Wind Energy Controller [13], developed by DTU, has been widely applied in OWT operational control. It is noteworthy that most of these control schemes employ relatively straightforward algorithms.

Some researchers have explored advanced control theories to design strategies targeting specific control objectives, aiming to mitigate structural loads while improving power generation efficiency. Mozayan et al. [14] proposed an improved sliding mode control method for a 4 KW wind turbine, which effectively mitigated flutter issues during operation and was validated through experimental testing on a 2 KW prototype. Bagherieh and Nagamune [15] developed linear quadratic regulator (LQR) and linear parameter-varying (LPV) control algorithms and compared their performance with the baseline controller using a 5 MW barge-type floating wind turbine model. The results demonstrated superior control performance and a more effective trade-off between power output and platform motion suppression. Shah et al. [16] employed a model predictive control (MPC) algorithm for a semi-submersible wind turbine, showing through simulation that the approach significantly reduced platform motions. Civlek et al. [17] designed a fuzzy logic controller using rotor speed deviation and its time derivative as input variables and demonstrated its superiority over traditional proportional integral control in managing turbine operating states. Moreover, advanced techniques such as incorporating neural networks into control algorithms [18,19] and applying individual pitch control strategies [20,21] have also been introduced into control frameworks. While these advanced control algorithms exhibit remarkable performance in tracking power outputs and regulating rotor speed with high precision, they exhibit fundamental limitations in suppressing critical structural dynamic loads. The primary goal of control is to ensure a rapid and accurate response to power or speed commands, with control law design focused mainly on optimizing tracking performance. As a result, the control system primarily adjusts the aerodynamic loads acting on the blades, striving to balance energy production with system efficiency. However, it lacks effective mechanisms for addressing structural dynamic responses induced by turbulent wind excitations, aeroelastic interactions, and the control actions themselves. Therefore, implementing structural control strategies beyond conventional operational control systems is imperative for achieving comprehensive load mitigation and enhancing the structural performance of OWTs.

In contrast to the first approach, which focuses on controlling operational parameters to optimize power output, the second approach to OWT vibration control involves structural control strategies. Among these, passive methods are the most widely implemented [22]. Drawing inspiration from the successful deployment of tuned mass dampers (TMDs) in civil engineering for suppressing vibrations in high-rise buildings [23,24,25], the fundamental concepts of TMD-based vibration mitigation have been extended to the OWT domain. By tuning critical TMD parameters such as mass ratio, frequency ratio, and damping ratio to align with the dynamic characteristics of the wind turbine, it is possible to achieve targeted suppression of resonant responses, especially those occurring near structural frequencies such as the tower’s first natural mode.

Lackner and Rotea [26] applied a TMD to the nacelle of a barge-type wind turbine and proposed a parameter optimization method. Their results demonstrated that the TMD reduced fatigue loads in the fore–aft direction by approximately 10% compared to the uncontrolled case. Similarly, Yan et al. [27] conducted vibration control studies on a 5 MW barge-type turbine, incorporating time-delay effects into the state-space formulation, and evaluated the damping performance using a simplified degrees-of-freedom model, which demonstrated that the TMD achieved excellent vibration suppression by significantly reducing structural responses under dynamic loads. Luo et al. [28] developed a 14-degree-of-freedom multibody dynamic model for a 5 MW semi-submersible OWT and validated the model by comparing its results with OpenFAST simulations. The comparison between controlled and uncontrolled configurations showed a significant reduction in tower-top displacement with the application of TMD. Li et al. [29] used the OC4 semi-submersible wind turbine model to investigate the influence of wind-wave misalignment on vibration control performance, and reported that TMDs installed within the nacelle and tuned to frequencies near the fore–aft and side–side tower modes effectively suppressed tower responses while targeting the pitch mode.

In addition to floating OWTs, numerous studies have investigated TMD applications in bottom-fixed OWTs. Stewart and Lackner [30] examined a 5 MW monopile turbine and evaluated the TMD’s effectiveness under various wind-wave incident angles. Their results indicated a 4–6% reduction in structural loads in the TMD-actuated direction. Zhang et al. [31] conducted a detailed investigation into the damping performance of TMDs installed in both the X- and Y-directions, which revealed that the tower-top displacement was reduced by nearly 25% under shutdown conditions of OWTs. Building upon traditional TMDs, recent research has expanded into tuned mass dampers with inerter (TMDI), which have attracted increasing attention, as demonstrated by Hua et al. [32], who modeled three simplified inerter configurations for a 5 MW barge-type wind turbine in the FAST-SC platform and showed that TMDI not only provided superior vibration mitigation performance but also effectively reduced the TMD stroke. By modifying the OpenFAST codes, Zhang et al. [33] investigated the performance of TMDI under combined wind, wave, and seismic excitations and demonstrated that placing TMDI at locations corresponding to higher modal amplitudes, particularly near the second mode, significantly enhanced damping performance. Beyond TMDI, researchers have proposed novel TMD configurations inspired by pendulum dynamics. Sun and Jahangiri [34] explored the optimal parameter selection of three-dimensional nonlinear tuned mass dampers (3D-PTMDs) under combined wind, wave, and seismic loading using a 5 MW monopile turbine model. Additionally, Jahangiri and Sun [35] analyzed the vibration control performance of 3D-NTMDs for spar-type floating wind turbines and found that setting the mass ratio to 1.5% resulted in approximately 15% and 25% reductions in roll and pitch responses, respectively. Luo et al. [36] developed an OC4 semi-submersible wind turbine model within the SIMPACK platform to analyze and compare the performance of 3D-PTMDs and dual-TMD configurations (2D-TMDs), demonstrating that 2D-TMDs effectively suppressed roll and pitch mode amplitudes of the floating platform while 3D-PTMDs more efficiently reduced the tower’s natural frequencies, with vibration mitigation effects being more pronounced in the frequency domain compared to the time domain under wave-dominated excitation conditions.

Traditional passive vibration control methods, which are characterized by energy independence and low maintenance requirements, have been well established in onshore wind power but remain insufficiently explored for application in jacket foundation OWTs. Existing studies have primarily focused on monopile foundations or floating wind turbines, while research into the dynamic response characteristics, optimization, and vibration mitigation performance of jacket foundation OWTs with TMD based on a fully coupled model is still limited. To address this gap, this study focuses on a 10 MW jacket foundation OWT and investigates an optimization design methodology for passive TMD based on an ABC algorithm. Numerical simulations are performed to assess their effectiveness under realistic operational conditions. The results aim to offer novel insights into improving the structural safety and dynamic performance of large-scale OWTs.

2. Fully Coupled Analysis Theories of OWT with Jacket Foundation

2.1. Kane’s Equations of Motion

Kane’s equations [37] provide an efficient and systematic framework for analyzing the dynamic behavior of OWTs, which are inherently complex and coupled multibody systems. Grounded in D’Alembert’s and virtual power principles, this method avoids the cumbersome computation of scalar energy functions and their derivatives required by the traditional Lagrange approach, while also eliminating the need to explicitly solve constraint forces inherent in the Newton–Euler formulation. The advantage of Kane’s equations lies in their ability to automatically eliminate ideal constraint forces and formulate the equations of motion directly in terms of generalized coordinates and partial velocities. This makes the method particularly suitable for systems with a large number of degrees of freedom, complex constraints, and strong nonlinearities. Moreover, Kane’s equations accommodate rigid–flexible, multibody, and environmental coupling effects, offering a comprehensive dynamic framework for analyzing the dynamic responses of OWTs, as shown in Equation (1).

$$F_{\mathrm{i}}+F_{\mathrm{i}}^{\ast }=0\left(i=1,2,\dots ,P\right)$$

(1)

Here, $F_i$ and $F_i^{\ast }$ are the generated active and inertial forces, respectively, which are expressed as follows:

$$F_i={\sum }_{r=1}^W{\displaystyle {\vartheta }_{E,i}^{X_r}}·F^{X_r}+{\omega }_{E,i}^{N_r}·M^{N_r} \left(i=1, 2,\dots ,P\right)$$

(2)

$$F_{\mathrm{i}}^{\mathrm{*}}={\sum }_{r=1}^W{\vartheta }_{\mathrm{E},\mathrm{i}}^{X_{\mathrm{r}}}·\left(-m_{\mathrm{r}}{\alpha }_{\mathrm{E}}^{X_{\mathrm{r}}}\right)+{\omega }_{\mathrm{E},\mathrm{i}}^{N_{\mathrm{r}}}·\left(-{\dot{H}}_{\mathrm{E}}^{N_{\mathrm{r}}}\right)\left(i=1,2,\dots ,P\right)$$

(3)

where N_r denotes reference frame, E denotes inertial frame, and W denotes a set of rigid bodies characterized; $\left\{F^{X_r}\right\}$ and $\left\{M^{N_r}\right\}$ denote the three-component active force and moment vectors, respectively; X_r is the CM (the center of mass) point location; $\left\{{\alpha }_E^{X_r}\right\}$ denotes the three-component acceleration vector of the CM point X_r; and $\left\{{\dot{H}}_E^{X_i}\right\}$ denotes the first-time derivative of the angular momentum of rigid body N_r about X_r in the inertial frame (E) three-component vector. The three-component vector quantities, $\left\{{\alpha }_E^{X_r}\right\}$ and $\left\{{\omega }_E^{N_r}\right\}$, represent the partial linear velocities of CM point X_r and the partial angular velocities of rigid body N_r in the inertial frame, respectively.

2.2. Wind Loads

OWTs operate in complex marine environments where wind speed and direction vary randomly over time and space, and the actual wind speed at any moment is the sum of the mean wind speed and its fluctuations, as shown in Equation (4).

$$V\left(x,y,z,t\right)=\overline{V}\left(z\right)+v\left(x,y,z,t\right)$$

(4)

where $\overline{V}\left(z\right)$ represents the mean wind speed and $v\left(x,y,z,t\right)$ denotes the turbulent wind speed.

Due to the effects of air viscosity and ground frictional damping, the mean wind-speed profile varies with height following a power-law distribution. The mean wind speed at a specified height can be calculated using Equation (5).

$$U\left(z\right)=U_{10}{\left({\displaystyle \frac{z}{10}}\right)}^{\alpha }$$

(5)

where $U\left(z\right)$ represents the mean wind speed at a specified vertical height z above sea level; $U_{10}$ denotes the mean wind speed at a vertical height of 10 m above sea level; and α is the ground roughness exponent and should be taken as 0.12.

2.3. Wave Loads

For the jacket foundation OWTs, the ratio of the structural characteristic dimension to the wavelength is less than 0.2, indicating that the influence of the structure on the wave can be neglected. Therefore, the Morison [38] equation is applicable. In the Morison equation, the wave force acting on the structure is divided into two components: one is the inertia force, caused by the wave acceleration field; the other is the drag force, resulting from the viscous effects between the fluid and the structure. The detailed expression is given in Equation (6).

$$F_{Hydro}={\rho }_w\pi {\displaystyle \frac{D^2}{4}}\ddot{u}+{\rho }_w{C}_m\pi {\displaystyle \frac{D^2}{4}}\left(\ddot{u}-\ddot{v}\right)+0.5{\rho }_w{C}_{D}D\left\{\left(\dot{u}-\dot{v}\right)\left|\dot{u}-\dot{v}\right|\right\}$$

(6)

Here, ρ_w denotes the seawater density, C_m is the inertia coefficient, C_D is the drag coefficient, D is the diameter of the structural member, $\ddot{u}$ represents the acceleration of the water particle, $\dot{u}$ is the water particle velocity, $\ddot{v}$ is the structural member acceleration, and $\dot{v}$ is the structural member velocity. The first term on the right-hand side of Equation (6) represents the Froude–Krylov force; the second term accounts for the effect of added mass on the hydrodynamic load; and the third term reflects the influence of drag force. Based on linear wave theory in deep water, the water particle acceleration can be expressed as follows:

$$\ddot{u}={\omega }_{wave}^2\zeta e^{kz}\cos \left({\omega }_{wave}t-kx\right)$$

(7)

where (x, z) denotes the coordinates of the water particle, and the vertical coordinate z is positive upwards. ${\omega }_{wave}$ is the wave angular frequency, $\zeta$ is the wave amplitude, k is the wave number, and t is the time.

2.4. TMD Motion Equation

To suppress the vibration responses of OWTs, TMDs are commonly employed, as illustrated in Figure 1. A TMD [39] consists of two independent single-degree-of-freedom linear mass-spring-damper elements. The position vectors of the TMD in two reference frames are expressed as shown in Equation (8).

$${\overrightarrow{r}}_{TMD/O_G}={\overrightarrow{r}}_{P/O_G}+{\overrightarrow{r}}_{TMD/P_G}$$

(8)

where O denotes the origin of the global coordinate system, and G represents the direction of the global coordinate axes. P denotes the origin of the non-inertial reference frame fixed inside the nacelle where the TMD is at rest. ${\overrightarrow{r}}_{TMD/O_G}$ represents the position vector of the TMD relative to point O in the global coordinate system. ${\overrightarrow{r}}_{TMD/P_G}$ represents the position vector of the TMD relative to point P in the global coordinate system. ${\overrightarrow{r}}_{P/O_G}$ represents the position vector of the nacelle relative to point O in the global coordinate system. The schematic of the coordinate systems is illustrated in Figure 2.

When the wind turbine nacelle is taken as the reference frame, the position vector of the TMD is expressed as shown in Equation (9).

$${\overrightarrow{r}}_{TMD/O_N}={\overrightarrow{r}}_{P/O_N}+{\overrightarrow{r}}_{TMD/P_N}$$

(9)

where ${\overrightarrow{r}}_{TMD/O_N}$ represents the position of the TMD relative to point O in the nacelle coordinate system, ${\overrightarrow{r}}_{TMD/P_N}$ denotes the position of the TMD relative to point P in the nacelle coordinate system, and ${\overrightarrow{r}}_{P/O_N}$ indicates the nacelle position vector relative to point O in the nacelle frame. By rearranging terms in Equation (9), Equation (10) can be derived.

$${\overrightarrow{r}}_{TMD/P_N}={\overrightarrow{r}}_{TMD/O_N}-{\overrightarrow{r}}_{P/O_N}$$

(10)

The expression upon differentiating Equation (10) with respect to time is given in Equation (11).

$${\dot{\overrightarrow{r}}}_{TMD/P_N}={\dot{\overrightarrow{r}}}_{TMD/O_N}-{\dot{\overrightarrow{r}}}_{P/O_N}-{\overrightarrow{\omega }}_{N/O_N}\times {\overrightarrow{r}}_{{TMD/\mathrm{P}}_N}$$

(11)

where ${\overrightarrow{\omega }}_{N/O_N}$ denotes the angular velocity of the nacelle. ${\dot{\overrightarrow{r}}}_{TMD/P_N}$ represents the velocity of the TMD relative to point P in the nacelle coordinate system. ${\dot{\overrightarrow{r}}}_{TMD/O_N}$ denotes the velocity of the TMD relative to point O in the nacelle coordinate system. ${\dot{\overrightarrow{r}}}_{P/O_N}$ indicates the velocity of the nacelle relative to point O in the nacelle coordinate system.

By further differentiating Equation (11), the acceleration of the TMD relative to point P can be obtained.

$$\begin{array}{c}{\ddot{\overrightarrow{r}}}_{TMD/P_N}={\ddot{\overrightarrow{r}}}_{TMD/O_N}-{\ddot{\overrightarrow{r}}}_{P/O_N}-{\overrightarrow{\omega }}_{N/O_N}\times \left({\overrightarrow{\omega }}_{N/O_N}\times {\overrightarrow{r}}_{{TMD/P}_N}\right)-{\overrightarrow{\alpha }}_{N/O_N}\times {\overrightarrow{r}}_{{TMD/P}_N}-2{\overrightarrow{\omega }}_{N/O_N}\times {\overrightarrow{\dot{r}}}_{{TMD/P}_N}\end{array}$$

(12)

where ${\overrightarrow{\alpha }}_{N/O_N}$ denotes the angular acceleration of the nacelle, $\overrightarrow{\dot{r}}$ represents the velocity vector, and $\ddot{\overrightarrow{r}}$ represents the acceleration vector.

The acceleration in the inertial reference frame can be determined through the application of force equilibrium.

$${\ddot{\overrightarrow{r}}}_{TMD/O_N}={\left[\begin{array}{c}\ddot{x}\\ \ddot{y}\\ \ddot{z}\end{array}\right]}_{TMD/O_N}={\displaystyle \frac{1}{m}}{\left[\begin{array}{c}{F}_x\\ F_y\\ F_z\end{array}\right]}_{TMD/O_N}={\displaystyle \frac{1}{m}}{\overrightarrow{F}}_{TMD/O_N}$$

(13)

Here, m denotes the mass of the TMD and ${\overrightarrow{F}}_{TMD/O_N}$ represents the force acting on the TMD in the nacelle coordinate system. The expression for ${\overrightarrow{F}}_{TMD/O_N}$ is given in Equation (14).

$$F_{TMD/O_N}=-c_x\dot{x}-k_{x}x+m{a}_x+F_{ext}+F_{stop}$$

(14)

where $c_x$ and $k_x$ represent the damping and stiffness of the TMD spring, respectively; $a_x$ denotes the acceleration of the TMD; $F_{ext}$ is the external force additionally applied to the TMD; and $F_{stop}$ refers to the stopping force exerted on the TMD when its stroke reaches the limit.

By substituting Equation (13) into Equation (12), the general equation of motion for the TMD is obtained, as expressed in Equation (15).

$$\begin{array}{c}{\ddot{\overrightarrow{r}}}_{TMD/P_N}={\displaystyle \frac{1}{m}}{\overrightarrow{F}}_{TMD/O_N}-{\ddot{\overrightarrow{r}}}_{P/O_N}-{\overrightarrow{\omega }}_{N/O_N}\times \left({\overrightarrow{\omega }}_{N/O_N}\times {\overrightarrow{r}}_{{TMD/P}_N}\right)-{\overrightarrow{\alpha }}_{N/O_N}\times {\overrightarrow{r}}_{{TMD/P}_N}-2{\overrightarrow{\omega }}_{N/O_N}\times {\overrightarrow{\dot{r}}}_{{TMD/P}_N}\end{array}$$

(15)

2.5. Seismic Load

Under seismic loading, the equilibrium equation of a single-degree-of-freedom system can be expressed as follows:

$$m{\ddot{v}}^{\prime }\left(t\right)+c\dot{v}\left(t\right)+kv\left(t\right)=0$$

(16)

where m, k, and c represent the mass, stiffness, and damping of the single-degree-of-freedom system, respectively; v′_(t) denotes the displacement relative to the global coordinate system, which is given by Equation (17).

$$v^{\prime }\left(t\right)=v\left(t\right)+v_g\left(\mathrm{t}\right)$$

(17)

Substituting Equation (17) into Equation (16), the expression is as shown in Equation (18).

$$m\ddot{v}\left(t\right)+m{\ddot{v}}^g\left(t\right)+c\dot{v}\left(t\right)+kv\left(t\right)=0$$

(18)

where v_(t) denotes the displacement of the single-degree-of-freedom system relative to the ground, and v^g_(t) represents the seismic ground displacement.

The equation of motion for a single-degree-of-freedom system under seismic excitation expressed in terms of relative velocity is given by Equation (19):

$$m\ddot{v}\left(t\right)+c\dot{v}\left(t\right)+kv\left(t\right)=-m{\ddot{v}}^g\left(t\right)$$

(19)

2.6. Equation of Motion for Offshore Wind Turbine Structure Considering the Combined Effects of Wind, Waves, Earthquake, and TMD

The general form of the equation of motion for an OWT under the combination of aerodynamic, hydrodynamic, and seismic loads is expressed as follows:

$$\left\{\begin{array}{c}\left[M+m\right]\left\{\ddot{v}\left(t\right)\right\}+\left[C\right]\left\{\dot{v}\left(t\right)\right\}+\left[K\right]\left\{v\left(t\right)\right\}=\left\{f_{wind}\right\}+\left\{f_{hydro}\right\}+\left\{f_{seismic}\right\}+\left\{f_{TMD}\right\}\\ m_{TMD}{\ddot{x}}_{TMD}+c_{TMD}{\dot{x}}_{TMD}+k_{TMD}{x}_{TMD}=0\end{array}\right.$$

(20)

Assuming that seismic effects on the seawater flow field are negligible, the Morison equation subjected to seismic excitation can be expressed as

$$\left\{f_{hydro}\right\}={\rho }_w\pi {\displaystyle \frac{D^2}{4}}\ddot{u}+{\rho }_w{C}_m\pi {\displaystyle \frac{D^2}{4}}\left(\ddot{u}-\ddot{v}-{\ddot{v}}_g\right)+{\displaystyle \frac{1}{2}}{\rho }_w{C}_{D}D\left\{\left(\dot{u}-\dot{v}-{\dot{v}}_g\right)\left|\dot{u}-\dot{v}-v_g\right|\right\}$$

(21)

$$\left\{f_{seismic}\right\}=-\left[M+m\right]\left\{{\ddot{v}}_g\right\}$$

(22)

2.7. TMD Parameter Optimization Method Based on the Artificial Bee Colony Algorithm

The Artificial Bee Colony (ABC) algorithm [40] is an optimization technique inspired by the intelligent foraging behavior of honeybee swarms. In solving multivariable function optimization problems, ABC demonstrates superior robustness and faster convergence compared to Genetic Algorithms, Particle Swarm Optimization, and Evolutionary Particle Swarm Optimization [41,42]. Its key advantage lies in its ability to perform both global and local searches simultaneously during each iteration, effectively avoiding entrapment in local minima. The algorithm involves three types of bees—employed, onlooker, and scout bees—that collaboratively explore and exploit potential solutions through a four-stage search process.

This study aims to minimize the standard deviation of structural displacement obtained from a simplified two-degree-of-freedom OWT TMD model under white noise random excitation. The ABC algorithm is employed to optimize the stiffness and damping parameters of the TMD for controlling the first-order bending mode of the OWT. The proposed optimization design procedure is illustrated in Figure 3.

3. OWT Parameters and Environmental Conditions

3.1. DTU 10MW Wind Turbine Parameters

In this study, a DTU 10 MW jacket foundation OWT is taken as the research object. It was jointly developed by the Technical University of Denmark and Vestas Wind Technology Company and has been widely adopted both domestically and internationally for research purposes due to its representative design, including a rotor diameter of 178.3 m, a rated rotational speed of 9.6 rpm, and a rated wind speed of 11.4 m/s, as summarized in

Table 1. DTU 10 MW OWT parameters.

Parameter	Value
Rated power	10 MW
Rotor configuration and orientation	3 blades, upwind
Control strategy	Variable speed, variable pitch
Mass of single blade, hub, nacelle	41,732, 105,520, 446,036 kg
Rated thrust	1500 kN
Rotor diameter, hub diameter	178.3, 5.6 m
Hub height	119 m
Cut-in, rated, cut-out wind speeds	4.0, 11.4, 25.0 m/s
Cut-in, rated rotor speeds	6.0, 9.6 rpm

.

3.2. Jacket Foundation Parameters

To ensure the stable operation of the wind turbine system at a water depth of 40 m, a jacket-type foundation structure is designed, as illustrated in Figure 4. Accordingly, the bottom elevation of the jacket platform is set at 30.15 m, resulting in an adjusted tower height of 85.48 m. The tower top has an outer diameter of 5.50 m with a wall thickness of 0.020 m, while the base has an outer diameter of 7.57 m and a wall thickness of 0.034 m. The total height of the jacket foundation structure is 70.15 m, with a base dimension of 15 × 15 m. The tower is made of Q355 steel, and the foundation structure is made of DH36 steel. Both steel types have an elastic modulus of 2.1 × 10¹¹ N/m² and a Poisson’s ratio of 0.3. Considering coatings and attached components, the steel density is corrected to 8500 kg/m³. To account for the effects of pile–soil interaction without modeling the soil continuum explicitly, this study adopts a boundary condition based on the equivalent pile length method [43], in which the pile is extended to a length equivalent to seven times its diameter.

3.3. Marine Environment Conditions

The offshore wind farm site is planned in the southeastern sea area of China. The joint distribution of measured wind speed and wave parameters in this area is illustrated in Figure 5.

Referring to the offshore wind turbine design standard DNV GL-ST-0437, combined with measured sea condition data from a southeastern sea area in China and the operational wind speed range of the DTU 10 MW reference wind turbine, typical combined wind, wave, and seismic load cases were selected, as shown in

Table 2. Selected typical combined wind, wave, and earthquake load cases.

Load Case No.	Wind-Wave Direction	Wind Speed (m/s)	Wave Height (m)	Peak Period (s)	Seismic Wave	Operational Condition	TMD Location
DLC1	Aligned at 0°	6.0	0.45	6.00	Humbolt Bay	Operation	nacelle/tower base
DLC2		11.4	2.00	8.50			nacelle/tower base
DLC3		16.0	3.75	12.50			nacelle/tower base
DLC4		6.0	0.45	6.00	Borrego		nacelle/tower base
DLC5		11.4	2.00	8.50			nacelle/tower base
DLC6		16.0	3.75	12.50			nacelle/tower base
DLC7		6.0	0.45	6.00	El Centro		nacelle/tower base
DLC8		11.4	2.00	8.50			nacelle/tower base
DLC9		16.0	3.75	12.50			nacelle/tower base

. The design water depth is 40 m. Wind speed time histories were generated based on the IEC Kaimal spectrum [44], with three mean wind speeds of 6 m/s, 11.4 m/s, and 16 m/s, representing wind conditions below rated speed, at rated speed, and above rated speed, respectively. Irregular wave time histories were generated using the JONSWAP spectrum [45], covering both short- and long-period waves. Seismic waves were obtained from the PEER, including the Humbolt Bay, Borrego, and El Centro records. The peak ground acceleration is 1.5 m/s², with frequencies ranging from 0 to 10 Hz, spanning low to high frequencies to effectively excite the OWT modes. The total simulation duration is 630 s, with seismic excitation applied starting at 150 s. The selected wind speed, wave, and seismic time histories and frequency domain representations are shown in Figure 6, Figure 7 and Figure 8.

4. TMD Parameter Analysis and Model Validation

4.1. TMD Parameter Analysis

Based on previous studies by relevant scholars, the mass ratio of the TMD to OWT typically ranges from 0.5% to 2%. In this study, the total mass of the OWT is 3,005,237 kg. A TMD mass ratio of 1.5% is adopted, resulting in a TMD mass of 45,078 kg. The stiffness optimization range for the TMD, using the ABC algorithm, is set between 50,000 and 300,000 N/m, and the damping optimization range is from 1000 to 80,000 N·m/s. According to the optimization algorithm described in Section 2.7, after 30 iterations, the optimized stiffness and damping values for the TMD with a mass of 45,078 kg are 159,646 N/m and 26,027 N·m/s, respectively. The iterative process of searching for the optimal damping and stiffness is illustrated in Figure 9. In this study, TMD is applied only in the X-direction.

4.2. Model Validation

To validate the modeling accuracy, an identical jacket structure was modeled in ANSYS 19.2, and a comparative analysis of natural frequencies and mode shapes was conducted against the model developed in FAST. The comparative results of natural frequencies are summarized in

Table 3. Frequency comparison between the FAST and ANSYS models.

Modal	ANSYS (Hz)	FAST (Hz)	Relative Error (%)
First in F-A	0.290	0.298	2.68%
First in S-S	0.290	0.298	2.68%
Second in F-A	1.05	1.08	2.78%
Second in S-S	1.05	1.08	2.78%

, while the mode shape comparisons are illustrated in Figure 10, which depicts the first- and second-order fore–aft (F-A) and side-to-side (S-S) vibrational modes. The results indicate that the relative errors in the first two natural frequencies between the FAST and ANSYS models are 2.68% and 2.78%, respectively. These minor discrepancies can be attributed to differences in the modeling simplifications adopted by the two software tools; nevertheless, the overall consistency remains satisfactory. Furthermore, as observed in Figure 10, the first two mode shapes exhibit strong agreement between the two models, thereby providing additional confirmation of the validity of the FAST model.

5. Vibration Mitigation Analysis of OWT Considering the Combined Wind, Wave, and Seismic Loads

5.1. Tower-Top Displacement

The time-history and frequency-domain responses of the tower-top displacements in the X- and Y-directions of the jacket foundation OWT under the combined action of wind, wave, and seismic excitations are illustrated in Figure 11 and Figure 12, respectively.

Table 4. Statistical values of tower-top displacement in the X-direction.

Load Case No.	TMD Location	Mean	Std.	95%th Max	Mean	Std.	95%th Max
		150–200 s			100–600 s
DLC1	No TMD	0.0850	0.0725	0.2346	0.0865	0.0469	0.1903
	Nacelle	0.0850	0.0310	0.1556	0.0869	0.0376	0.1655
	Tower base	0.0850	0.0720	0.2338	0.0866	0.0468	0.1902
DLC2	No TMD	0.2515	0.0480	0.3323	0.2743	0.0632	0.3885
	Nacelle	0.2524	0.0396	0.2455	0.2754	0.0612	0.3875
	Tower base	0.2516	0.0403	0.3349	0.2743	0.0631	0.3885
DLC3	No TMD	0.1964	0.0339	0.2620	0.1868	0.0375	0.2790
	Nacelle	0.1972	0.0326	0.2577	0.1876	0.0364	0.2769
	Tower base	0.1965	0.0325	0.2574	0.1868	0.0372	0.2786
DLC4	No TMD	0.0850	0.0711	0.2318	0.0866	0.0466	0.1900
	Nacelle	0.0850	0.0292	0.1533	0.0869	0.0374	0.1654
	Tower base	0.0850	0.0707	0.2286	0.0866	0.0465	0.1896
DLC5	No TMD	0.2515	0.0405	0.3356	0.2743	0.0632	0.3886
	Nacelle	0.2524	0.0392	0.3264	0.2754	0.0612	0.3876
	Tower base	0.2516	0.0396	0.3350	0.2743	0.0630	0.3885
DLC6	No TMD	0.1965	0.0315	0.2515	0.1868	0.0373	0.2784
	Nacelle	0.1973	0.0301	0.1963	0.1876	0.0361	0.2765
	Tower base	0.1965	0.0311	0.2558	0.1868	0.0371	0.2784
DLC7	No TMD	0.0850	0.0809	0.2797	0.0865	0.0482	0.1942
	Nacelle	0.0850	0.0556	0.2261	0.0869	0.0403	0.1746
	Tower base	0.0851	0.0697	0.2415	0.0866	0.0464	0.1890
DLC8	No TMD	0.2514	0.0596	0.3638	0.2743	0.0647	0.3891
	Nacelle	0.2523	0.0579	0.2498	0.2754	0.0626	0.3882
	Tower base	0.2515	0.0483	0.3407	0.2743	0.0636	0.3885
DLC9	No TMD	0.1964	0.0520	0.2991	0.1868	0.0395	0.2840
	Nacelle	0.1972	0.0503	0.2969	0.1875	0.0383	0.2819
	Tower base	0.1964	0.0411	0.2764	0.1868	0.0380	0.2805

summarizes the mean, standard deviation, and 95th max values of the X-direction tower-top displacement. These statistics were obtained from two distinct periods: the steady-state phase between 100 s and 600 s (excluding the initial 100 s transient response) and the seismic excitation phase between 150 s and 200 s.

As shown in Figure 11a, the nacelle-mounted TMD substantially reduces the X-direction tower-top displacement when the seismic excitation begins at 150 s, compared to the uncontrolled case. During the post-earthquake stage (after 200 s), the nacelle TMD continues to suppress tower vibrations; however, a slight amplification of the response is occasionally observed. This behavior is consistent with the findings of McNamara et al. [46], who reported that, for a 5 MW monopile wind turbine, the uncontrolled configuration can sometimes exhibit better performance due to the dynamic characteristics of the tower and the specific loading conditions. Regarding the Y-direction response, although the nacelle-mounted TMD effectively mitigates vibrations in the X-direction, it leads to a slight amplification in the Y-direction since no active control is applied along that axis.

The statistical data in Table 4 further highlight the distinctive performance characteristics of the two TMD configurations. Both dampers exhibit a more pronounced effect on the standard deviation and 95th max values than on the mean displacement, with the nacelle-mounted TMD providing substantially greater mitigation than the tower-base configuration. This trend is particularly evident during the seismic excitation interval from 150 to 200 s. For instance, under DLC1 conditions, while the mean X-displacement remains approximately 0.0850 m across all configurations, the standard deviation decreases dramatically from 0.0725 m without a TMD to 0.0310 m with the nacelle-mounted TMD, whereas the tower-base configuration achieves only a minor reduction to 0.0720 m. Similar trends are observed for the 95th max values, reinforcing the performance differential.

Figure 13 and Figure 14 provide further support for this behavior. Figure 13 indicates that the reduction in standard deviation under TMD control is more pronounced than the reduction in the 95th max, confirming the TMD’s particular effectiveness in suppressing fluctuations of the dynamic response, in agreement with previous studies (Xie et al., 2020 [47]). Figure 14 highlights the strong dependence of the tower-base TMD on excitation type. During the 100–600 s wind-wave-dominated interval, the tower-base TMD achieves only a 3.73% reduction in standard deviation, whereas during the 150–200 s seismic window, its mitigation efficiency rises sharply to 20.96%. This contrast demonstrates that the tower-base TMD is more effective under broadband, high-energy seismic excitations than under relatively narrow-band wind and wave loads.

Frequency-domain responses (Figure 12) reveal that the dynamic behavior of the jacket-supported OWT under combined wind, wave, and seismic loading is governed primarily by the first- and second-order structural frequencies, along with the 3P. In the X-direction, the nacelle-mounted TMD effectively suppresses the first-order frequency amplitude while minimally affecting the second-order mode. In contrast, the tower-base TMD predominantly attenuates the second-order frequency component. Due to the relatively low energy content of the second-order mode compared with the fundamental frequency, the limited effectiveness of the tower-base TMD in reducing time-domain vibrations is clarified.

For the Y-direction, both TMD configurations have little effect on the second-order frequency. However, the nacelle-mounted TMD slightly amplifies the first-order frequency amplitude, consistent with the observed increase in time-domain vibrations, indicating modal energy redistribution. Under El-Centro excitation (Figure 12e), the nacelle TMD shows limited influence on the first-order frequency but significantly suppresses the second-order component, whereas the tower-base TMD exhibits the opposite behavior. This divergence is attributed to the distinct spectral characteristics of the El-Centro record, which contains high-frequency content and elevated amplitude, activating alternative dynamic mechanisms. These observations underscore the importance of considering ground motion spectral characteristics in TMD design, as performance is highly sensitive to seismic input. The frequency-domain trends align with previous studies, including those by Liu et al. (2024) [48] and Wang et al. (2020) [49].

Overall, the results demonstrate that both TMD configurations, and especially the nacelle-mounted variant, achieve superior vibration mitigation under seismic conditions compared with routine wind-wave loading. This discrepancy arises from the operational principle of TMDs, which attain maximum efficiency when the structure undergoes large-amplitude vibrations near the tuned frequency, a condition commonly met during strong seismic events. These findings quantify TMD performance under combined loading and advance understanding of the relationship between load type, TMD location, and control effectiveness. While conventional designs favor nacelle placement for dominant vibration modes, hybrid strategies such as deploying complementary TMDs at both the nacelle and tower base to address multiple structural modes may offer comprehensive vibration suppression under multi-hazard scenarios.

5.2. Tower-Base Bending Moment

The time histories of tower-base bending moments in the X- and Y-directions under combined wind, wave, and seismic loads are presented in Figure 15a–f, and the correlated responses in the frequency domain are shown in Figure 16. The variation trends are similar to those observed for tower-top displacements. For the tower-base bending moment in the X-direction, the application of the TMD at the tower base results in limited reduction in the bending moment. In contrast, when the TMD is installed on the nacelle, a significant amplification of the tower-base response is observed. However, under the El Centro seismic excitation, the vibration mitigation effect of the TMD installed at the tower base is markedly superior to that of the nacelle-mounted configuration. Regarding the Y-direction tower-base bending moment, the TMD demonstrates more pronounced vibration reduction during the seismic loading period (150–200 s). Under combined wind and wave loading, it is observed that at certain time instants, the TMD may exacerbate the structural time history response, as illustrated in Figure 15b.

These observations are further corroborated by the statistical data summarized in

Table 5. Statistical values of tower-base bending moment in the Y-direction.

Load Case No.	TMD Location	Mean	Std.	95%th Max	Mean	Std.	95%th Max
		150–200 s			100–600 s
DLC1	No TMD	37.4145	28.4283	95.7504	37.0894	18.4138	78.2758
	Nacelle	37.3874	12.0692	66.0747	37.1883	14.8976	68.9055
	Tower base	37.4272	28.3716	95.9072	37.0902	18.4145	78.2899
DLC2	No TMD	101.6430	15.8349	135.2358	110.2957	25.2618	156.2874
	Nacelle	101.9068	15.3333	132.4681	110.6470	24.4185	155.4948
	Tower base	101.6570	15.7495	134.6816	110.2945	25.2522	156.2965
DLC3	No TMD	75.2627	13.1137	100.7264	73.6194	14.9428	111.7320
	Nacelle	75.5349	12.5113	99.0051	73.8730	14.4567	110.9582
	Tower base	75.2738	12.8634	99.7868	73.6188	14.8638	111.6897
DLC4	No TMD	37.4184	27.9799	94.8522	37.0904	18.3298	78.1225
	Nacelle	37.3916	11.6188	65.0457	37.1894	14.8512	68.8532
	Tower base	37.4258	27.9256	94.1754	37.0898	18.3348	78.0695
DLC5	No TMD	101.6446	15.7249	135.0917	110.2950	25.2603	156.2891
	Nacelle	101.9119	15.1552	131.3715	110.6467	24.4084	155.5014
	Tower base	101.6576	15.5704	135.0228	110.2954	25.2375	156.2844
DLC6	No TMD	75.2702	12.6169	98.9661	73.6195	14.8996	111.6597
	Nacelle	75.5443	12.0065	97.1414	73.8732	14.4114	110.9057
	Tower base	75.2798	12.5678	99.2185	73.6189	14.8697	111.6763
DLC7	No TMD	37.4298	28.9081	104.3831	37.0897	18.4802	78.2265
	Nacelle	37.3851	17.4318	80.7891	37.1880	15.4191	36.8900
	Tower base	37.4410	26.6670	94.7320	37.0904	18.1548	77.1172
DLC8	No TMD	101.6117	20.3121	138.5718	110.2916	25.5905	156.3217
	Nacelle	101.8791	19.4210	137.5667	110.6440	24.7104	155.5664
	Tower base	101.6426	18.0034	135.2906	110.2931	25.3999	156.2861
DLC9	No TMD	75.2479	17.4619	108.5187	73.6175	15.3834	112.3833
	Nacelle	75.5248	16.5537	107.2995	73.8712	14.8566	111.5355
	Tower base	75.2616	15.2771	104.4341	73.6179	15.1204	111.9657

. Taking DLC4 as an example, during the seismic interval of 150–200 s, the mean, standard deviation, and 95th max of the Y-direction tower-base bending moment without TMD are 37.4184 MN·m, 27.9799 MN·m, and 94.8522 MN·m, respectively. When the TMD is installed on the nacelle, the corresponding values are 37.3916 MN·m, 11.6188 MN·m, and 65.0457 MN·m, whereas for the tower-base-mounted TMD, the values are 37.4258 MN·m, 27.9256 MN·m, and 94.1754 MN·m, respectively. The TMD has a more significant effect on reducing the standard deviation. Similar patterns can be observed across other load cases. As shown by the variation rates of statistical values in Figure 17 and Figure 18, changes in standard deviation can reach up to 58% in some cases, indicating a substantial load-reduction effect of the TMD on the structure. This systematic investigation demonstrates that the control effectiveness of tuned mass dampers exhibits remarkable location dependence and frequency-spectrum sensitivity. The tower-base TMD exhibits limited effectiveness due to insufficient kinematic excitation, while the nacelle-mounted configuration, despite its global vibration suppression capability, amplifies the tower-base response through modal coupling effects. Notably, under broadband seismic excitation, the tower-base TMD demonstrates superior vibration suppression performance, revealing the crucial influence of load spectral characteristics on the control mechanism. These findings establish the principle of “location-frequency spectral adaptability,” providing a significant theoretical foundation for vibration control design in offshore wind turbine structures.

Frequency-domain analysis of the tower-base bending moment in the X-direction, as illustrated in Figure 16, reveals that the tuned mass damper (TMD) predominantly influences the structural first-order frequency component. Specifically, the nacelle-mounted TMD induces a marked amplification of the first-order frequency amplitude, consequently intensifying the corresponding time-domain response. In contrast, the tower-base TMD demonstrates negligible modification of the first-order frequency characteristics. This phenomenon elucidates the location-dependent nature of TMD effectiveness: while the nacelle position effectively controls global vibrations, it may amplify specific frequency components through modal coupling, whereas the tower-base location, limited by insufficient kinematic excitation in the first-order mode, fails to achieve meaningful control at this frequency. These findings further validate the significance of the “position–frequency spectrum adaptability” principle in vibration control of offshore wind structures, providing a theoretical basis for optimizing TMD configuration strategies. For practical engineering applications, it is recommended to select appropriate TMD locations according to target control frequency bands or consider implementing hybrid control systems to achieve more comprehensive vibration suppression.

6. Conclusions and Future Studies

A fully coupled dynamic model, developed in this study for a 10 MW jacket foundation OWT under combined environmental loading, incorporates a TMD to mitigate structural vibrations induced by wind and wave conditions, taking into account the dynamic response characteristics of the OWT. The stiffness and damping parameters of the TMD are optimized using the ABC algorithm. A comprehensive dynamic analysis is then conducted under combined wind, wave, and seismic excitations to evaluate the coupled response behavior of the structure and to validate the effectiveness of the proposed TMD configuration in vibration mitigation. The following conclusions are inferred from this study:

(1)

Under combined wind, wave, and seismic loading, the application of a TMD significantly improves the vibration mitigation performance of the structure. Compared to placing the TMD at the tower base, positioning it within the nacelle yields better damping effectiveness.

(2)

Compared to the combined wind and wave loading condition, the TMD demonstrates enhanced vibration mitigation performance under combined wind, wave, and seismic loading. When seismic excitation is applied, significant variations are observed in both tower-top displacement and tower-base bending moment. The presence of the TMD primarily influences the standard deviation and 95th max, with reductions in standard deviation reaching 50% in certain cases, while its effect on the mean value remains minimal, generally below 1%.

(3)

Based on the frequency-domain analysis, it is observed that when the TMD is installed at the nacelle, it primarily affects the first-order natural frequency of the structure. Specifically, the first-order frequency amplitude in the X-direction is significantly reduced, whereas the corresponding amplitude in the Y-direction increases. In contrast, when the TMD is located at the tower base, its influence is mainly exerted on the second-order frequency. Since the structural response is predominantly governed by the first-order frequency, the impact of base-mounted TMDs on the time-domain response and statistical parameters is generally limited; however, under seismic excitations with broad frequency content and high spectral amplitudes such as the El Centro earthquake, the tower-base TMD demonstrates significant vibration mitigation effects.

It is important to acknowledge two primary limitations of this study that also define avenues for future work. First, the analysis of the tower-base TMD was performed using parameters optimized for the nacelle position. A dedicated optimization targeting higher-order modes, for which the tower base is more effective, was beyond the scope of this research, which focused on fundamental mode control. Second, the pile–soil interaction was represented using a simplified boundary condition method rather than a more advanced model. Future studies should incorporate sophisticated soil–structure interaction models to more accurately quantify their influence on the structure dynamics response and TMD performance.