The gyroscopic anti-rolling system is a motion stabilizing device based on the principle of conservation of angular momentum, which has achieved remarkable results in the field of ships. In recent years, it has gradually been applied to floating offshore wind turbine platforms. This section introduces the basic structure, working principle and parameter design of the gyroscopic anti-rolling system.

The typical gyroscopic anti-shudder system mainly consists of the following components:

The gyroscope damping system can be classified into two types based on the control method: passive and active. The precession axis of the passive gyroscope damping system is only equipped with damping devices (such as hydraulic dampers), and no additional energy input is required. When the platform sways, the gyroscope spontaneously generates precession motion, and the swaying energy is consumed by the damping devices. The passive system has a simple structure and high reliability, but its control effect is limited and cannot be adjusted. The active gyroscope damping system is equipped with servo motors on the precession axis, and can actively generate control torques. Based on the motion state of the platform, the control system calculates and outputs the optimal control torque in real time, significantly improving the damping effect and adaptability.

This study adopts an active gyroscopic anti-roll system with a dual-axis linkage and gyroscopic anti-roll arrangement. It combines the reliability of passive anti-roll with the efficiency of active control. Through the PSO-optimized fuzzy control strategy, the optimal anti-roll effect is achieved, effectively controlling the longitudinal and lateral roll of the platform. At the same time, factors such as system energy consumption and lifespan are also considered.

This study adopts the common circular ring thin-disc type gyro design. The material uses the most commonly used high-strength steel alloy, with a density of 7850 k g · m 3 . The outer diameter of the gyroscope is 4m, the inner diameter is 3m, and the thickness is 1.6m. Under these conditions, the weight of a single gyroscope is approximately 69t. Based on this, for the moment of inertia of the ring-shaped disc rotating around its active axis, the applicable formula is Equation 1:

Here, m represents the mass of the circular disc, R 1 is the outer radius, and R 2 is the inner radius.

Substitute the known values and obtain I s p i n = 215625 k g · m 2 . This is a very large value, which also explains why this large-scale gyroscope device can provide a significant stabilizing effect in the offshore wind turbine stability system. A gyroscope with such a large moment of inertia, once rotating at high speed, can provide extremely strong angular momentum, effectively resisting external disturbing torques.

For the rotational inertia of the flywheel’s oscillation axis, it is generally in a 1:2 ratio to the main axis. This is to set the rotational inertia of the oscillation axis I g x = I g y = 107000 k g · m 2 .

After comprehensively considering the decoupling effect, energy consumption, structural strength and other factors, the rotational speed of the flywheel should be designed within a reasonable range. Here, the rotational speed of the flywheel is set at 1000 rpm. This can provide a relatively stable control effect while ensuring the safety and reliability of the system.

For this gyroscope system, its static radial load F r , s t a t i c = m g = 676.9 k N , Gyroscope torque M g y r o = I g x × w s p i n × Ω p l a t f o r m = 1120.5 k N · m . Here, w s p i n represents the rotational speed of the gyroscope. Ω p l a t f o r m is platform maximum angular velocity, which is 0.1 rad/s based on the 10-second wave cycle. The bearing spacing is approximately 2 meters, then dynamic axial load F a ,   d y n a m i c = M g y r o L b e a r i n g = 560.3 k N .

Regarding centrifugal force is Equation 2:

w p r e c e s s i o n is maximum precession angular velocity, assume it to be 0.05 rad/s, r represents the radius of rotation.

Radial total load formula is Equation 3:

Equivalent dynamic load is Equation 4:

Substituting the values, we can obtain that the equivalent dynamic load F e q u i v a l e n t = 878.5 k N .

In actual engineering applications, the value can be selected as 1.5 times the equivalent load.

The power consumption of the control system mainly consists of the frictional torque of the bearings, air resistance, the precession control system, and the electrical system. The calculation process is omitted. It is concluded that the total power consumption of the dual-gyro anti-shudder system is approximately 304 kW, accounting for about 6% of the wind turbine’s power, which is a relatively reasonable figure.

For semi-submersible floating wind turbine platforms, multiple gyroscopic decoupling devices are usually installed at the corner points or the center of gravity of the platform to achieve comprehensive control of the platform’s roll and pitch motions.

When the floating platform is subjected to wave action and undergoes oscillation, the gyroscopic device will undergo precession motion within its internal frame. According to the law of conservation of angular momentum, the precession torque generated by the gyroscopic device is opposite to the oscillation direction of the platform, thereby applying an opposing torque to the platform and suppressing the oscillation motion. Through the cooperation of two counter-rotating gyroscopes, comprehensive control of the platform’s oscillation motion can be achieved.

The total angular velocity of the gyroscope rotor consists of two parts: the motion of the platform and the motion of the gyroscope itself, as shown in Equations 5, 6:

Here: ω gyroself 1 = [ q ˙ gyro 0 g s ] , ω gyroself 2 = [ − q ˙ gyro 0 − g s ] Here, q ˙ gyro represents the precession angular velocity, and g s represents the gyroscope’s rotational speed.

The Kane method is adopted to model the gyroscope. The generalized inertia force of the gyroscope F g y r o * is expressed as Equation 7:

Where ω g y r o _ r is the angular velocity component of the gyroscope’s r-th degree of freedom, I g y r o is the rotational inertia matrix of the gyroscope, α is the angular acceleration vector, and ω F 1 is the angular velocity vector of the platform coordinate system.

The passive constraint moments acting on the gyroscope include the spring moment F g y r o K (Equation 8):

Where k g y r o is the spring moment coefficient, q g y r o ( t ) is the precession angular displacement of the gyroscope, and R 01 is the transformation matrix from the platform coordinate system to the inertial coordinate system.

The damping moment F g y r o D is expressed as Equation 9:

Where d g y r o is the damping moment coefficient, and q ˙ g y r o ( t ) is the precession angular velocity of the gyroscope.

The active control moment of the gyroscope F g y r o C is expressed as Equation 10:

Where T c is the moment output by the controller.

The generalized active force of the gyroscope F g y r o _ t o t a l _ r on the r-th degree of freedom expressed in the inertial system is Equation 11:

Where ω P r is the angular velocity component of the platform on the r-th degree of freedom.

Through the above modeling method, this invention established complete gyroscopic dynamic equations, including the gyroscope’s inertial characteristics, constraint properties, and control inputs, laying a theoretical foundation for subsequent control system design.

Addressing the motion stability issues of semi-submersible floating wind turbines under wind-wave coupling effects, this paper proposes a full coupling dynamic modeling method that incorporates anti-rolling gyroscopes. By establishing a 12-degree-of-freedom model that includes the platform’s six-degree-of-freedom motion, tower flexible deformation, and anti-rolling gyroscopes, the simulation accuracy of pitch motion has been significantly improved. Compared to the traditional OpenFAST model, the gyroscopic subsystem modeling process has been added, and dynamic transfer of control torque has been achieved through generalized coordinate coupling.

During the modeling process, reference was made to the 5MW floating wind turbine model with OC4-DeepCwind semi-submersible platform developed by the National Renewable Energy Laboratory (NREL) of the United States. This model has been widely used to study the dynamic behavior of floating wind turbines, but it mainly focuses on the coupling between platform motion and wind turbine dynamics. This paper makes innovative improvements on this basis: by introducing an anti-rolling gyroscopic system, the model’s stability control capability for platform roll and pitch has been enhanced; meanwhile, the model structure has been optimized to accommodate efficient application of control algorithms. Table 1 shows the main parameters of the NREL 5MW floating wind turbine.

The semi-submersible wind turbine system consists of a platform, tower, nacelle, rotor, and anti-rolling gyroscope ( Figure 1 ). The platform employs a six-degree-of-freedom rigid body model, with the gyroscopic system located at the platform’s center of gravity, generating anti-rolling moments through a dual-axis servo mechanism. To characterize the multi-body coupling effects, the following core coordinate systems are established: the inertial coordinate system F₀(O₀-X₀Y₀Z₀), platform coordinate system F₁(O₁-X₁Y₁Z₁), tower-top coordinate system F₂(O₂-X₂Y₂Z₂), and rotor coordinate system F₃(O₃-X₃Y₃Z₃). This coordinate system fully describes the relative motion relationships between various components of the wind turbine, laying the foundation for subsequent dynamic modeling. Vector rotational transformations between different coordinate systems are implemented through rotation matrices, using Euler angle descriptions.

This study builds a fully coupled dynamic model of the floating wind turbine system based on Kane’s method, incorporating 12 degrees of freedom (as shown in Figure 1 ): three translational degrees of freedom for the platform (surge, sway, and heave) and three rotational degrees of freedom (roll, pitch, and yaw), four degrees of freedom for the first and second bending modes of the tower in the fore-aft and side-to-side directions, one rotational degree of freedom for the wind turbine rotor around the hub axis, and one precessional degree of freedom for the anti-rolling gyroscope relative to the platform.

This model achieves a comprehensive description of the fully coupled effects between platform motion, tower deformation, rotor rotation, and gyroscopic movement through the selection of generalized coordinates and the establishment of constraint equations.

The equations of motion for this model are derived using Kane dynamics. As a direct result of Newton’s laws of motion, the Kane equations for a simple complete system can be expressed as a Equation 12.

For this model, the generalized active force F i and the generalized inertia force F i * can be expressed as Equation 13:

Where F X r represents the active force, M N r represents the active moment at point X r , H ˙ E N r represents the first-order derivative of the rigid body’s angular momentum at the center of mass,   E ω i N r ,   E a i N r , and   E V i N r represent angular velocity, acceleration, and linear velocity, respectively.

Based on the established equations of motion, the complete nonlinear equations of motion for the coupled wind turbine system can be introduced as shown in Equation 14.

M i j is the mass matrix, which depends on the system’s degrees of freedom q , control input u , and time t . q ¨ j is the second-order derivative of the corresponding degree of freedom, and f i is the function of the corresponding force.

This study considered three main types of external loads acting on the floating wind turbine system: wind loads, hydrostatic forces, and hydrodynamic forces. The wind loads mainly include the axial thrust and aerodynamic moments acting on the rotor, which are expressed as Equation 15:

Where the thrust coefficient and moment coefficient are obtained through fitting experimental data. The hydrostatic forces mainly consider the linear restoring forces and moments caused by platform motion, expressed as Equation 16:

The calculation of hydrodynamic forces is based on Airy wave theory, using the Morison method to estimate the viscous effects and added mass effects of waves on the platform, expressed as Equation 17:

Where C d represents the drag coefficient, ρ f represents seawater, D represents the diameter of the platform column, C a represents the added mass coefficient, ( v → r e l ) t = ( v → ˙ f ) t − ( v → ˙ ) t represents the relative velocity of waves to the column, and ‖ ( v → r e l ) t ‖ represents the norm of ( v → r e l ) t .

By superimposing the generalized active forces and generalized inertia forces of the platform, tower, nacelle, stabilizing gyroscope, and rotor, we obtain the generalized active forces and generalized inertia forces of the entire system, and solve the equations to determine the motion state of the wind turbine, as shown in Equation 18.

To validate the consistency between the 11-degree-of-freedom catenary wind turbine model (without tuned mass dampers) and the simulation results from OpenFAST software, this study designed two typical conditions for comparison with OpenFAST software: free decay experiments under still water with no wind conditions (Condition 1) and dynamic response analysis under coupled wind-wave effects (Condition 2). The specific condition parameters are shown in Table 2 .

During the model validation process, AeroDyn v15 was used to calculate aerodynamic loads, and the map++ program was used to calculate mooring line tensions. To comprehensively evaluate model performance, initial disturbances were applied separately to surge, pitch, and tower-top fore-aft displacement in Condition 1 to observe the system’s free decay characteristics; in Condition 2, the wind direction was set consistent with the wave direction to examine the system’s dynamic response under coupled environmental loads.

Firstly, a free decay experiment was conducted under conditions of still water and no wind. The longitudinal sway, longitudinal heave and tower top front-back displacement of the wind turbine were assigned initial values, while the initial values of the other degrees of freedom were all zero. The free decay responses of the system were observed, and the results are shown in Figures 2 – 4 respectively.

As shown in the figure above, the model in this paper demonstrates good consistency with OpenFAST calculation results in surge and pitch directions. For tower-top fore-aft displacement, although there are slight differences in amplitude, the overall dynamic characteristics remain consistent. Specific data comparisons are shown in Table 3 .

Second, comparative validation was conducted under combined wind and wave conditions. According to observations from Figures 5 – 7 , under Condition 2 with a wave frequency of 0.1 Hz, the established wind turbine model shows high consistency with the OpenFAST model in natural frequency calculations, indicating high accuracy in frequency analysis. For time-domain analysis, the platform exhibits significant fluctuations in motion response for surge, heave, pitch, yaw, and tower-top fore-aft displacement degrees of freedom. These fluctuations mainly result from external excitation of wind and waves. However, as time progresses, the external loads on the wind turbine gradually approach dynamic equilibrium, causing the platform’s motion response and tower-top fore-aft displacement response to eventually stabilize in a periodic oscillation state. Specific data comparisons are shown in Table 4 .

In summary, the simulation results of the established 12-degree-of-freedom wind turbine model under still water with no wind conditions and combined wind-wave conditions show high consistency with the calculation results of OpenFAST software, with all error indicators controlled within 10%. This demonstrates that the model established in this paper has good accuracy and can be used for subsequent control strategy research.

Through comparative validation of these two typical conditions, it is fully demonstrated that the wind turbine dynamics model established in this paper can accurately simulate the dynamic response characteristics of wind turbines under different environmental conditions, laying a reliable theoretical foundation for subsequent research.

To achieve mapping from system states to control quantities, input and output variables must first undergo fuzzification. This paper uses seven linguistic values to describe pitch angle, pitch angular velocity, and control moment M: {NB, NM, NS, ZO, PS, PM, PB}, representing negative big, negative medium, negative small, zero, positive small, positive medium, and positive big, respectively.

A complete fuzzy control rule base was designed based on expert experience and system dynamic characteristics.

To improve the performance and adaptability of the control system, this paper introduces a hierarchical control architecture, dividing the entire control system into two levels: the decision layer and the execution layer:

Decision layer (upper-level control): Responsible for environmental condition recognition, control objective determination, and control parameter optimization. This layer operates at a lower update frequency, handling global and long-term system tasks, including: condition analysis and recognition based on sensor data, selection or adjustment of control parameters according to identified conditions

Execution layer (lower-level control): Responsible for the generation and execution of real-time control signals, operating at a higher frequency to ensure system real-time responsiveness, including: Real-time reception and filtering of sensor data, execution of fuzzy inference calculations to generate control signals, transmission of control signals to actuators and monitoring of execution status

The advantage of the hierarchical architecture lies in effectively separating tasks of different time scales, allowing the decision layer to focus on global optimization without affecting the real-time performance of the execution layer, while providing enhanced system adaptability and robustness. Information exchange between the two layers is carried out through parameter interfaces and state feedback, ensuring coordinated system operation.

During the PSO optimization process, the optimization objective function is first designed to achieve the goal of reducing the pitch angle. The optimization variables are the parameters of the fuzzy controller, with each particle representing a complete set of fuzzy controller parameters, including 7 membership function position parameters for input variable θ , 7 membership function position parameters for input variable θ^·, and 7 membership function position parameters for output variable M .

The operating conditions are designed as: wave height H   =   { H 1 ,   H 2 ,   H 3 } , period T   =   { T 1 ,   T 2 ,   T 3 } , and wind speed V   =   { V 1 ,   V 2 ,   V 3 } The objective function is designed as Equation 19:

θ ( t ) is the pitch angle of the system at time t , t start and t end are the start and end points of time integration, θ max is the predetermined maximum allowable pitch angle, and λ 1 is the weight of the penalty factor, used to adjust the severity of the penalty for exceeding the maximum allowable pitch angle.

The condition recognition phase involves environmental parameter measurements through real-time monitoring of wave parameters (such as wave height H ( t ) and wave period T ( t ) ) and wind parameters (such as wind speed V ( t ) and wind direction). Characteristic quantities are calculated through short-term statistical feature extraction, as shown in Equation 20.

Based on the measurement results of environmental parameters, the condition distance metric uses Euclidean distance:

Where ( H i , T j , V k ) are the preset condition points. The similarity between the current condition and preset conditions is determined, and the closest condition is matched, as shown in Equation 22.

Based on the matched condition point, the optimal fuzzy controller parameters are selected. If the current condition completely matches a preset condition, the corresponding parameters are directly used; if it falls between different conditions, weighted interpolation is used to calculate the controller parameters, with the interpolation formula being Equation 23.

Where the weight is Equation 24.

Normalization is performed through distance D i j k to ensure that closer conditions have a greater influence on interpolation. The system monitors the roll reduction effect in real-time. The short-term roll reduction effect is calculated using the integral formula J 1 ( t ) = 1 T ∫ t − T t θ 2 ( τ ) d , reflecting the mean square value of the pitch angle within the time window.

When a change in operating conditions is detected, the criterion Δ D = | D ( t ) − D ( t − Δ t ) | > ϵ is used to determine whether control parameters need to be switched. Where D ( t ) represents the Euclidean distance D i j k between the condition characteristic quantities ( H c h a r , T c h a r , V c h a r ) at the current time t and the nearest preset condition point. If the switching condition is met, a parameter smooth switching strategy is adopted, as shown in Equation 25.

Where λ is the smoothing factor, used to reduce system disturbances caused by switching.

To comprehensively evaluate the performance of different control strategies, Tables 7 , 8 summarize the key performance indicators of each control scheme under three operating conditions. This study employs three evaluation indicators for quantitative analysis of control effects:

1. RMS suppression rate calculation formula is expressed as Equation 26:

Where, R M S c is the root mean square value under controlled conditions, and R M S n is the root mean square value under uncontrolled conditions.

2. Standard deviation suppression rate calculation formula is expressed as Equation 27:

Where, σ c is the standard deviation under controlled conditions, and σ n is the standard deviation under uncontrolled conditions.

3. Peak suppression rate calculation formula is expressed as Equation 28:

Where, | x | m a x , c is the absolute peak value under controlled conditions, and | x | m a x , n ​ is the absolute peak value under uncontrolled conditions.

These three indicators evaluate the control effects of the stabilization system from the perspectives of root mean square value, standard deviation, and peak value, comprehensively reflecting the performance of control strategies under different operating conditions.

This study addresses the motion stability problem of floating offshore wind turbines under complex sea conditions by proposing a particle swarm optimization fuzzy control method based on a gyroscope-wind turbine coupled dynamics model, and verifies its effectiveness through a series of simulation experiments. The main achievements of this research are as follows:

Under free decay conditions (LC1), the pitch angle RMS suppression rate reaches 37.56%, far higher than passive control’s 14.36%; the tower-top displacement RMS suppression rate reaches 44.23%, significantly better than passive control’s 31.66%.

Under normal operating conditions (LC2), active control achieves 21.45% in pitch angle peak suppression, higher than passive control’s 13.16%; tower-top displacement peak suppression rate reaches 22.61%, better than passive control’s 18.64%.

Under extreme sea conditions (LC3), active control performs most prominently in pitch angle control, with peak suppression rate reaching 31.52%, far higher than passive control’s 15.73%; tower-top displacement peak suppression rate reaches 24.58%, better than passive control’s 19.26%.

Under the random sea condition (LC4), the peak vibration suppression rate of the active control for the roll angle reached 38.16%, which was much higher than the 13.91% of the passive control. The vibration suppression rate of the tower top displacement was 17.83%, also higher than the 5.01% of the passive control.

Despite a series of achievements in floating wind turbine stabilization control, there are still the following directions worthy of in-depth study:

In conclusion, this research provides new technical approaches and solutions for stabilization control of floating offshore wind turbines. The established gyroscope-wind turbine coupled dynamics model and PSO-optimized fuzzy control system provide theoretical foundation and technical support for research in related fields. Future research will further expand the application scope and engineering implementation pathways of control strategies, making greater contributions to the development of the floating offshore wind power industry.