A Hybrid Fuzzy LQR-PI Blade Pitch Control Scheme for Spar-Type Floating Offshore Wind Turbines

Abstract

Floating offshore wind turbines (FOWTs) experience unbalanced loads and platform motion due to the coupling of variable wind and wave loads, which leads to output power fluctuation and increased fatigue loads. This paper introduces a new blade pitch control strategy for FOWTs that combines fuzzy logic with a linear quadratic regulator (LQR) and a proportional-integral (PI) controller. The fuzzy PI controller dynamically adjusts the PI control gains to regulate rotor speed and stabilize output power. Fuzzy LQR is employed for individual pitch control, utilizing fuzzy logic to adaptively update feedback gains to achieve stable power output, suppress platform motion, and reduce fatigue load. Co-simulations conducted with OpenFAST (Fatigue, Aerodynamics, Structures, and Turbulence) and MATLAB/Simulink under diverse conditions demonstrate the superiority of the proposed method over traditional PI control. The results show significant reductions in platform pitch, roll, and heave motion by 17%, 27%, and 48%, respectively; blade out-of-plane, pitch, and flapwise bending moments are reduced by 38%, 44%, and 36%; and the tower base roll and pitch bending moments are reduced by up to 29% and 22%, respectively. The proposed control scheme exhibits exceptional environmental adaptability, enhancing FOWT’s power regulation, platform stability, and reliability in complex marine environments.

1. Introduction

Offshore wind power is gaining significant attention in sustainable global energy development due to its advantages in avoiding the waste of land resources, reducing visual and noise pollution, and offering higher annual average wind speeds and lower turbulence intensity [1]. Floating offshore wind turbines (FOWTs) have emerged as a critical research topic in wind power generation because of their significant capacity for capturing wind energy. However, as the physical size increases and structural coupling strengthens, the dynamic response of FOWTs to aerodynamic and hydrodynamic loads also becomes more complicated. This leads to power output fluctuations, increased structural fatigue loads, and exacerbated platform motion, posing a critical technical challenge that restricts further development.

Blade pitch control technology is widely acknowledged as an effective approach for stabilizing output power, reducing platform motion, and minimizing structural loads. The operational and control objectives for FOWTs vary across different wind speed regions. Specifically, in the region above the rated wind speed (Region 3), the primary control objective is to decrease platform motion and structural fatigue loads while maintaining output power at the rated level [2]. Blade pitch control strategies are primarily categorized into collective pitch control (CPC) and individual pitch control (IPC). CPC stands for applying an identical pitch angle control command to all blades. Jonkman developed a PI controller that has been widely used as a baseline controller in numerous comparative studies. This controller has exceptional performance in terms of speed and power regulation. However, it also exhibits speed overshoot, which has a negative impact on the lifespan of FOWT [3]. Jonkman conducted additional research on the feedback tower-top acceleration approach of the CPC and found a conflict between the objectives of regulating output power and decreasing platform motion [4]. In subsequent research, the variable power CPC method proposed in [5] effectively reduced the platform pitch motion. However, it also led to increased power and speed errors. A wind speed prediction-based collective pitch feedforward control method proposed in [6] effectively reduced rotor speed and power fluctuations but did not decrease platform motion and fatigue loads. A gain-scheduled PI control strategy proposed in [7] improved power regulation but did not address the platform pitch motion issue. Additionally, optimal control methods such as linear quadratic regulator (LQR) and linear parameter varying (LPV) introduced in [8] showed good performance in power regulation and platform pitch motion reduction. However, the analysis did not include fatigue loads and still struggled with balancing power regulation and reducing platform pitch motion, indicating an impossibility to enhance both simultaneously. Research has shown that although a variety of control methods have been proposed, the CPC strategy with single-input–single-output (SISO) controllers shows its limitations when realizing multi-objective control, such as regulating power output, decreasing platform motion, and reducing fatigue loads [9]. This prompts researchers to continue exploring more effective control methods to improve the overall performance and reliability of FOWTs.

IPC provides independent pitch angle control commands for each blade of the wind turbine and has been proven to effectively reduce structural fatigue loads [10]. This approach has been extended to the application of FOWTs. The current research mainly focuses on IPC schemes utilizing multi-blade coordinate transformation techniques. This approach converts blade root bending moments in the rotating reference frame into hub pitch and yaw moments in the fixed reference frame. Subsequently, multiple SISO controllers are applied to implement IPC, thereby reducing the fatigue loads on wind turbines [11,12]. However, in the fixed reference frame, there is a phenomenon of pitch and yaw dynamics coupling that is greatly affected by the rotor speed [13,14,15].

IPC possesses the characteristics of multiple input variables, allowing for the introduction of additional control objectives, thus forming a multi-input–multi-output (MIMO) system to meet the multi-objective control requirements of FOWTs [16]. Some advanced control schemes based on IPC have been applied to achieve different control objectives. Disturbance accommodation control (DAC) is a common advanced control method suitable for MIMO systems. It has been proven to have significant advantages in power regulation and load reduction when applied to IPC for FOWTs, particularly in dealing with stochastic wind and wave conditions and system nonlinearities [17]. In [18], an IPC-based robust DAC was proposed that can simultaneously handle speed regulation and load reduction, effectively balancing the objectives of fatigue load reduction and rotor speed regulation. However, the platform motion was not mentioned in the analysis. Additionally, current methods primarily consider unstructured uncertainties and do not fully address low-frequency parameter uncertainties, potentially limiting robustness in certain operational scenarios [19]. Moreover, these methods are complex to implement and have high hardware requirements, posing challenges in execution and operation. IPC schemes based on model predictive control (MPC) can handle multi-objective control, nonlinearity, and constraints [20]. A nonlinear model predictive control (NMPC) based on lidar measurement technology was proposed in [21]. Compared to the baseline controller, it can improve the dynamic response of FOWT while reducing the pitch actuator activity. Yet, the implementation of NMPC requires an accurate model.

The use of the linearized state-space model of FOWT for MIMO control can reduce the model complexity without significantly affecting dynamic performance. This approach achieves a balance between the precision of the model and the efficiency of the computational process, thereby facilitating the design of effective control schemes [22]. State feedback control (SFC) can optimize output performance, provide solutions for stabilizing output power, and mitigate fatigue loads [23]. The LQR control, based on a linear model, has been demonstrated as an effective strategy for SFC. This approach selects feedback gains by minimizing the cost function, which significantly improves the stability of output power, decreases fatigue loads, and reduces platform motion [24,25]. As an effective optimal control method, LQR is implemented by linearizing the nonlinear model at a particular operating point, while the operating point is determined by the rotor speed and the wind speed. Yet, LQR is insufficiently robust in the presence of environmental variation and system nonlinearity, especially when the wind turbine operates away from the operating point. To ensure optimal performance of the control system in a complex environment, it is essential to update the state feedback gain to adapt to changing circumstances. The optimization control schemes based on LQR were introduced in [26,27]. These schemes primarily focus on performance in single or average environmental conditions but fail to sufficiently demonstrate their effectiveness in reducing platform motion in varying environmental situations. Thus, it is necessary to explore novel optimization methods that can adaptively adjust the feedback gain, thereby compensating for the shortcomings of the LQR strategy in terms of dynamic environment adaptability.

Fuzzy logic control has gained increasing attention due to its ability to work without relying on accurate system state-space models [28]. This method has shown significant advantages in dealing with nonlinearity and environmental variability, and has been effectively applied to the analysis and control of nonlinear systems [29]. Hence, in the design of IPC for FOWT, combining fuzzy logic with LQR is an innovative approach for blade pitch control. The fuzzy logic algorithm is used to automatically update the feedback gains of the controller, thereby enhancing the robustness of the system to parameter uncertainty and external disturbances. This will lead to a notable improvement in the stability and adaptability of the control system under diverse environmental conditions.

In short, given the high nonlinearity and uncertainty of FOWTs, developing a multi-objective blade pitch control system that efficiently meets the requirements of maintaining stable power output, suppressing platform motion, and reducing fatigue loads in diverse operating environments has become a key research challenge. Based on this, this paper proposes an innovative control scheme that combines fuzzy logic with LQR and PI. Fuzzy PI control is utilized to regulate the rotor speed of the generator, ensuring power output stability by dynamically adjusting the PI gains. Fuzzy LQR is implemented in the IPC loop, and fuzzy logic is used to adaptively update the state feedback gains of the LQR controller to maintain stable output power, while effectively reducing platform motion and fatigue loads.

The subsequent sections are as follows. Section 2 provides the structural characteristics and model of the Spar-type FOWT. Section 3 details the design of the pitch control scheme. Section 4 outlines the simulation environment and presents results in the form of diagrams to validate effectiveness of the proposed scheme, and concluding remarks and future research prediction are offered in Section 5.

2. FOWT Modeling

2.1. Spar-Type FOWT

A cylindrical ballast tank located below sea level is used as the foundation structure by the Spar-type FOWT to maintain the center of gravity of the FOWT platform below the buoyancy center. Secured with three tightened mooring lines and traction anchors, the Spar foundation can maintain stability in deep-sea environments with water depths exceeding 100 m. Figure 1 illustrates its structural and platform degrees of freedom (DOFs), while

Table 1. NREL 5 MW FOWT parameters.

Parameter	Value
Rated Power	5 MW
Number of blades	3
Rotor Orientation	Upwind
Rotor, Hub Diameter	126 m, 3 m
Hub Height	90 m
Cut-in, Rated, Cut-out wind Speed	3 m/s, 11.4 m/s, 25 m/s
Cut-in, Rated Rotor Speed	6.9 rpm, 12.1 rpm
Blade Pitch Angle setting	0°~90°
Blade Pitch rate	−8°/s~8°/s
Draft	120 m
Platform Mass	7.466 × 106 kg
Tower Mass	347,460 kg
Rotor Mass	110,000 kg
Nacelle Mass	240,000 kg
Pitch Inertia	$4.229 \times {10}^9 \mathrm{kg}\cdot$m2
Roll Inertia	$4.229 \times {10}^9 \mathrm{kg}\cdot$m2
Yaw Inertia	$1.642 \times {10}^8 \mathrm{kg}\cdot$m2
Cable Stiffness	3.842 × 108 N

provides its key parameters. For more details, it can be referred to [30].

2.2. Dynamics Modeling

The Spar-type FOWT is a complex dynamical system characterized by multi-modal, multi-physics coupling, and strong nonlinearities. The system is subjected to the coupled effects of aerodynamic and hydrodynamic loads. The blade element momentum (BEM) theory is applied to estimate aerodynamic loads, while the Morison equation is used to calculate hydrodynamic loads [31]. The power system of the wind turbine consists of three main components: the rotor blades, the drivetrain, and the generator. Aerodynamic torque is generated when the wind exerts force on the rotor blades, which is then transmitted through the drivetrain system to rotate the generator rotor and produce energy.

2.2.1. Aerodynamic Model

The wind turbine operates by using the lift or drag generated as wind flows through its rotor blades to drive its rotation. This process effectively converts the kinetic energy from the wind into mechanical energy. The Equations (1) and (2) express the aerodynamic torque and power in this process [32].

$$T_{Aero}(t)={\displaystyle \frac{1}{2{\omega }_{rt}(t)}}{\rho }_{Aero}\pi {r^2}_{rt}{v_{rp}}^3(t)C_p(\beta (t),\lambda (t))$$

(1)

$$P_{Aero}(t)=T_{Aero}(t)\cdot {\omega }_{rt}(t)={\displaystyle \frac{1}{2}}{\rho }_{Aero}\pi {r^2}_{rt}{v_{rp}}^3(t)C_p(\beta (t),\lambda (t))$$

(2)

where ${\omega }_{rt}(t)$ represents the angular velocity of the rotor ($\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}$), ${\rho }_{Aero}$ is the air density ($\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$), $r_{rt}$ is the rotor radius ($\mathrm{m}$), $v_{rp}(t)$ represents the average wind speed effective across the rotor plane ($\mathrm{m}/\mathrm{s}$), the blade pitch angle (°) is denoted as $\beta (t)$, and the blade tip speed ratio $\lambda (t)$ is defined as

$$\lambda (t)={\displaystyle \frac{r_{rt}}{v_{rp}}}{\omega }_{rt}(t)$$

(3)

The power coefficient, denoted by $C_p(\beta (t), \lambda (t))$, represents the efficiency of converting wind energy into mechanical energy. It is a nonlinear function of $\beta$ and $\lambda$, which can be expressed as [33],

$$\left\{\begin{array}{c}{C}_p(\beta (t),\lambda (t))=0.22\left({\displaystyle \frac{116}{\epsilon }}-0.4\beta (t)-5\right)e^{-12.5/\epsilon }\\ {\displaystyle \frac{1}{\epsilon }}={\displaystyle \frac{1}{\lambda (t)+0.087\beta (t)}}-{\displaystyle \frac{0.035}{{\beta }^3(t)+1}}\end{array}\right.$$

(4)

The equation for the thrust force exerted on the rotor is (5).

$$F_{Areo}(t)={\displaystyle \frac{1}{2}}{\rho }_{Aero}\pi {r^2}_{rt}{v_{rp}}^2(t)C_t(\beta (t),\lambda (t))$$

(5)

where $C_t(\beta (t),\lambda (t))$ represents the thrust coefficient, is a nonlinear function determined by $\beta$ and $\lambda$. The expression for the thrust coefficient can be derived as

$$\begin{gathered} C_t(\beta (t),\lambda (t))=0.25382-0.1369\lambda (t)+0.04345{\lambda }^2(t)-0.00263{\lambda }^3(t) \\ +(-0.008608+0.0063\lambda (t)-0.0015{\lambda }^2(t)+0.000118{\lambda }^3(t))\beta (t) \end{gathered}$$

(6)

2.2.2. Drivetrain Model

The drivetrain is essential in wind turbines, effectively transmitting the aerodynamic torque from the rotor blades to the generator. It converts the rotor low-speed, high-torque output into a high-speed, low-torque input suitable for the generator, thereby optimizing efficiency and maximizing power output. The dynamics of the drivetrain system are mathematically modeled via three first-order differential equations.

$$I_l{\dot{\omega }}_{rt}(t)=T_{Aero}(t)-S_{ds}{\theta }_{ds}(t)-(C_{dac}+C_{lvf}){\omega }_{rt}(t)+{\displaystyle \frac{C_{dac}}{N_{gr}}}{\omega }_{gen}(t)$$

(7)

$$I_h{\dot{\omega }}_{gen}(t)={\displaystyle \frac{S_{ds}}{N_{gr}}}{\theta }_{ds}(t)+{\displaystyle \frac{C_{dac}}{N_{gr}}}{\omega }_{rt}(t)-({\displaystyle \frac{C_{dac}}{{N_{gr}}^2}}+C_{hvf}){\omega }_{gen}(t)-T_{gen}(t)$$

(8)

$${\dot{\theta }}_{ds}(t)={\omega }_{rt}(t)-{\displaystyle \frac{1}{N_{gr}}}{\omega }_{gen}(t)$$

(9)

where $I_l$ represents the inertia of the low-speed shaft, $I_h$ denotes the inertia of the high-speed shaft, $C_{lvf}$ is the viscous friction coefficient of the low-speed shaft, $C_{hvf}$ is the viscous friction coefficient of the high-speed shaft, $C_{dac }$ refers to the attenuation coefficient of the drivetrain, $S_{ds}$ is the torque stiffness of the drivetrain, ${\theta }_{ds}(t)$ represents the torsional angle of the drivetrain, $N_{gr}$ denotes the gear ratio of the drivetrain, $T_{gen}(t)$ is the generator torque, and ${\omega }_{gen}(t)$ is the generator speed.

2.2.3. Generator Model

The generator transforms the mechanical energy conveyed from the drivetrain system into electrical energy. In this conversion process, the generator torque $T_{gen}(t)$ is controlled based on a preset reference value $T_{g,ref}(t)$. The dynamic behavior of the generator is commonly approximated as a first-order model shown in Equation (10).

$${\dot{T}}_{gen}(t)={\displaystyle \frac{T_{g,ref}(t)-T_{gen}(t)}{{\kappa }_{gen}}}$$

(10)

where ${\kappa }_{gen}$ represents the time constant of the generator system, which characterizes the dynamic response characteristics of the generator and indicates the speed of transition from one steady state to another. The power produced by the generator is influenced by the applied loads and the rotor speed, and is described as follows:

$$P_{Gen}(t)={\psi }_{gen}{\omega }_{gen}(t)T_{gen}(t)$$

(11)

where ${\psi }_{gen}$ is the efficiency of the generator.

2.3. Reduced-Order Model

The FOWT system exhibits strong coupling and high-order dynamics. Thus, it is necessary to establish a multi-DOF mathematical model that can fully reflect the performance of the system. Furthermore, in the controller design, a linearly simplified model that accurately describes the dynamic characteristics of the system should be developed. Equation (12) provides the mathematical representation of the FOWT.

$$M\ddot{q}+D\dot{q}+Sq=F_{Dyn}(t)$$

(12)

where $M$, $D$, and $S$ represent the system mass matrix, damping matrix, and total stiffness matrix, respectively [9]. $q$ represents the displacement of the wind turbine’s DOF, $\dot{q}$ denotes the velocity, $\ddot{q}$ is the acceleration, and $F_{Dyn}(t)$ indicates the coupling of aerodynamic and hydrodynamic load components.

The highly coupled dynamic system can accurately represent the behavioral characteristics of FOWT, but this complexity also increases the difficulty of blade pitch controller design. Presently, the PI controller is widely used for commercial blade pitch control. However, the innate nonlinear characteristics of wind turbines, coupled with the diverse wind and wave conditions, render the PI control susceptible to system response overshoot and oscillations. To resolve this issue, this research linearized the complex dynamic system of the wind turbine and designed an optimal control scheme based on LQR. Since wind turbines mainly experience aerodynamic loads in the downwind direction, this paper selects five DOFs in this direction to build a reduced-order model of the FOWT. The linear state-space equation is represented by Equation (13).

$$\begin{array}{l}\dot{x}=A_{s}x+B_c{u}_c+E_d{u}_d\\ y=Cx\end{array}$$

(13)

where $x=[q;\dot{q}]\in {\mathbb{R}}^{ 10\times 1}$ is the state vector and $q=[{\varphi }_{pfp},{\delta }_{tfa},{\delta }_{bfw1},{\delta }_{bfw2},{\delta }_{bfw3}]$, where ${\varphi }_{pfp}$ represents the pitch angle of the floating platform, ${\delta }_{tfa}$ is the fore-aft displacement of the tower, ${\delta }_{bfw1}$, ${\delta }_{bfw2}$, and ${\delta }_{bfw3}$ represent the bending displacements of blades 1, 2, and 3, respectively, along the flap-wise direction. $u_c\in {\mathbb{R}}^{ 3\times 1}$ is the input vector, and ${ u}_c=[{\beta }^1,{\beta }^2,{\beta }^3{]}^T$ represents the input for the blade pitch angle of the three blades. The vector $u_d={\tau }_{wind}+{\tau }_{wave}$ represents the environmental disturbance resulting from wind and wave loads. $y=Cx$ is the output, $A_s\in R^{10\ast 10}$ is the system state matrix, the matrix $B_c\in R^{10\times 6}$ represents the gain of the control input, which is the function of pitch angle ${\beta }^i$, the matrix ${ E}_d$ represents the disturbance input, and the matrix $C$ is the state output.

In addition, matrix $A_s=\left[\begin{array}{cc}O& E\\ -M^{-1}S& -M^{-1}D\end{array}\right]$, matrix $C=\left[\begin{array}{cc}E& O\\ O& O\end{array}\right]$, where $O=0_{5\times 5}$, matrix $E=I_{5\times 5}$. The control input gain matrix $B_c$ is obtained by linearizing the aerodynamic thrust of the rotor at the operating point. In this paper, to simplify the design of the LQR controller, assuming that matrix $B_c$ is derived around the operating point of the collective pitch angle ${\theta }^{CPC}$ and is valid for ${\beta }^i$ (i.e., $B_c({\beta }^i)\approx B_c({\theta }^{CPC})$), then matrix $B_c$ is formed by the Taylor series expansion of the aerodynamic rotor thrust Equation (5) around the operating point ($Opp$: $v_{rp}=18 \mathrm{m}/\mathrm{s}$, $\beta =14.92^{\circ }$):

$$F_{Areo}=F_{Areo}\left|{ }_{Opp}+\right.{\displaystyle \frac{\partial F_{Areo}}{\partial \beta }}\left|{ }_{Opp}\Delta \beta \right.+{\displaystyle \frac{\partial F_{Aero}}{\partial {\omega }_{rt}}}\left|{ }_{Opp}\Delta {\omega }_{rt}\right.+{\displaystyle \frac{\partial F_{Aero}}{\partial v_{rt}}}\left|\Delta {\nu }_{rp}\right.$$

(14)

where $\Delta \beta$, $\Delta {\omega }_{rt}$ and $\Delta {\nu }_{rp}$ represent the variations of pitch angle, rotor speed, and wind speed around the operating point, respectively. Assuming disregarded external disruptions and alterations in torque concerning rotor speed, the matrix $B_c$ is shown in Equation (15).

$$B_c={\displaystyle \frac{\partial F_{Aero}}{\partial \beta }}\left|{ }_{Opp}\right.$$

(15)

While the LQR controller in this paper is developed with a simplified five-DOF FOWT model, it is worth mentioning that a full nonlinear model is used for the simulations and evaluations of closed-loop performance. This model incorporates system complexity elements to guarantee the robustness and efficacy of the control strategy in practical applications.

3. Control Scheme

This section introduces a novel control scheme that combines fuzzy logic with LQR and PI controllers to address the operational challenges of FOWTs in wind speed region 3. The objective is to maintain the stable output power, suppress platform motion, and reduce fatigue loads on the blade root and tower base. Figure 2 depicts the proposed scheme configuration.

The fuzzy PI control is implemented in the CPC loop, utilizing fuzzy rules to dynamically adjust the PI gain to regulate the rotor speed of the generator, thereby ensuring stable power output. The fuzzy LQR is employed in the IPC loop, with the Linear Fusion Function (LFF) to combine the LQR control and the fuzzy logic. The fuzzy logic algorithm is used to adaptively update the state feedback gain. The IPC loop not only maintains stable power output but also effectively reduces fatigue loads and decreases platform motion. This scheme enhances system robustness to parameter uncertainties and external disturbances, further improving the stability and environmental adaptability of the control system. Equation (16) represents the control input for adjusting the pitch angle of a single blade.

$${\beta }^i(t)={\theta }^{CPC}(t)+{\beta }^{IPCi}(t)\hspace{1em}i=\mathrm{1,2},3$$

(16)

3.1. Fuzzy PI Control

For the PI control designed in this paper, the error signal is defined as the error between the generator rated speed and its actual speed, as shown in Equation (17):

$$e(t)={\omega }_{grt}(t)-{\omega }_{gen}(t)$$

(17)

where ${\omega }_{grt}(t)$ represents the rated speed and ${\omega }_{gen}(t)$ represents the actual speed. The control law for the PI controller is

$$u_{\theta }(t)=K_{P0}e(t)+K_{I0}{\int }_0^{t}e(\tau )d\tau$$

(18)

where $K_{P0}$ denotes the proportional gain and $K_{I0}$ represents the integral gain. The corresponding equations can be found in (19) and (20).

$$K_{P0}={\displaystyle \frac{2I_l{\omega }_{rt0}{\varsigma }_{\phi }{\omega }_{\phi n}}{N_{gr}\left(-\frac{\partial P_{Aero}}{\partial \theta }\right)}}$$

(19)

$$K_{I0}={\displaystyle \frac{I_l{\omega }_{rt0}{{\omega }^2}_{\phi n}}{N_{gr}\left(-\frac{\partial P_{Aero}}{\partial \theta }\right)}}$$

(20)

where ${\omega }_{rt0}$ is the rotor rated speed, ${\varsigma }_{\phi }$ and ${\omega }_{\phi n}$ are the response damping ratio and closed-loop natural frequency, respectively; ${\displaystyle \frac{\partial P_{Aero}}{\partial \theta }}$ is the sensitivity of the rotor aerodynamic power to the collective pitch angle, which is influenced by factors such as rotor speed, wind speed, and pitch angle.

In region 3, the aerodynamic power of the wind turbine exhibits a high sensitivity to changes in collective pitch angle and rotor speed. The fixed gain tuning is unable to effectively respond to different control needs in diverse wind and wave conditions, particularly in the presence of environmental factors such as turbulent wind and fluctuating waves. To address this challenge, fuzzy logic is used to adjust the gain of the PI controller dynamically. This enables the system to effectively respond to varying control demands by considering real-time fluctuations in wind speed and pitch angle, thus enhancing the system robustness. The fuzzy PI control scheme is illustrated in Figure 2.

The generator rotor speed error $e$ and the error change rate $\dot{e }$ are employed as input parameters by the fuzzy PI control. Fuzzy rules are applied to dynamically adjust the gain online and output the adjustment gains $\Delta K_P$ and $\Delta K_I$. These adjustments are then added to the initial gains $K_{P0}$ and $K_{I0}$, respectively, thereby obtaining the updated gains $K_P$ and $K_I$. The updated gains are the actual parameters output by the PI controller and are used to calculate the control signal ${\theta }^{CPC}$. This method enables adaptive adjustment of control gains based on real-time system feedback, thus enhancing the system adaptability to environmental changes and overall robustness. To maintain control system performance, selecting the appropriate domain for fuzzy sets is crucial. Based on previous research [34], the fuzzy domain for $e$ and $\dot{e }$ was set as [−6, 6], while the domain for $\Delta K_P$ and $\Delta K_I$ was set as [−1, 1] to achieve precise adjustment in speed control and maintain system stability. The input–output mapping surface plots between the input parameters $e$ and $\dot{e }$ and the output parameters $\Delta K_P$ and $\Delta K_I$ in fuzzy control are shown in Figure 3, respectively.

The CPC equation adjusted by the fuzzy logic control algorithm is shown as (21).

$${\theta }^{CPC}(t)=(K_{P0}+\Delta K_P)e(t)+(K_{I0}+\Delta K_I){\int }_0^{t}e(\tau )d\tau$$

(21)

3.2. Fuzzy LQR Control

The LQR controller is designed to decrease platform motion and reduce structural loads while maintaining stable output power. Therefore, based on the determination of the operating point in Section 2.3, a set of gains can be determined using the LQR control method. The corresponding performance index function is

$$J={\displaystyle \frac{1}{2}}{\int }_0^{\infty }(x^{T}Qx+u^{T}Ru)dt$$

(22)

where $x=[{\varphi }_{pfp},{\delta }_{tfa},{\delta }_{bfw1},{\delta }_{bfw2},{\delta }_{bfw3},{\dot{\varphi }}_{pfp},{\dot{\delta }}_{tfa},{\dot{\delta }}_{bfw1},{\dot{\delta }}_{bfw2},{\dot{\delta }}_{bfw3}]$ represents the state vector, $u={\beta }^{IPCi}(t)$ denotes the control input vector, $Q$ is the state weight matrix, and $R$ is the control input weight matrix. By adjusting $Q$ and $R$, the distribution of weight between state and control input can be balanced to meet specific performance requirements. For optimal performance, matrix $Q$ should be semi-positive definite and symmetric, and matrix $R$ should be positive definite and symmetric, i.e., $Q=Q^T\ge 0,R=R^T>0$.

According to LQR control theory, an appropriate selection of the state feedback gain matrix $\tilde{K}$ can improve the behavior of the system in closed-loop control. Consequently, the optimal control law for state feedback to minimize the performance index function $J$ is

$$u=-\tilde{K}x$$

(23)

where $\tilde{K}$ is given by

$$\tilde{K}=R^{-1}{B_c}^T{P}_c$$

(24)

where $P_c$ represents the positive definite solution of the Riccati equation shown in Formula (25).

$$Q+P_c{A}_s+{A_s}^T{P}_c-P_c{B}_c{R}^{-1}{B_c}^T{P}_c=0$$

(25)

The significant nonlinearity of wind turbines makes it difficult for a sole LQR controller to handle them accurately, and it is also susceptible to disturbances from wind speed, waves, and uncertainties in the turbine state. A control technique that combines fuzzy logic with the LQR controller is used to adaptively adjust the control gains. This approach not only ensures stable power output and platform motion, but also further reduces fatigue loads on the tower base and blade roots.

The fuzzy LQR method introduces the LFF to realize the fusion of LQR control and fuzzy logic. The state feedback gain $\tilde{K}$ of LQR is utilized to fuse the ten state vectors of the FOWT, resulting in the composite error $E$ and its rate of change $\dot{E}$. The $E$ and $\dot{E}$ serve as inputs for the fuzzy logic control, which is then used to develop the Mamdani-type fuzzy controller. The LFF is designed as

$$F_L\left(x\right)=\left[\begin{array}{ccccc}{\tilde{K}}_{pfp}& {\tilde{K}}_{tfa}& {\tilde{K}}_{bfw1}& {\tilde{K}}_{bfw2}& {\tilde{K}}_{bfw3}\\ 0& 0& 0& 0& 0\end{array}\right.\left.\begin{array}{ccccc}0& 0& 0& 0& 0\\ {\tilde{K}}_{p\dot{f}p}& {\tilde{K}}_{t\dot{f}a}& {\tilde{K}}_{b\dot{f}w1}& {\tilde{K}}_{b\dot{f}w2}& {\tilde{K}}_{b\dot{f}w3}\end{array}\right]$$

(26)

where $E$ and $\dot{E}$ are calculated as

$$\left[\begin{array}{c}E\\ \dot{E}\end{array}\right]=F_L\left(x\right)x^T=\left[\begin{array}{c}{\tilde{K}}_{pfp}{\varphi }_{pfp}+{\tilde{K}}_{tfa}{\delta }_{tfa}+{\tilde{K}}_{bfw1}{\delta }_{bfw1}+{\tilde{K}}_{bfw2}{\delta }_{bfw2}+{\tilde{K}}_{bfw3}{\delta }_{bfw3}\\ {\tilde{K}}_{p\dot{f}p}{\dot{\varphi }}_{pfp}+{\tilde{K}}_{t\dot{f}a}{\dot{\delta }}_{tfa}+{\tilde{K}}_{b\dot{f}w1}{\dot{\delta }}_{bfw1}+{\tilde{K}}_{b\dot{f}w2}{\dot{\delta }}_{bfw2}+{\tilde{K}}_{b\dot{f}w3}{\dot{\delta }}_{bfw3}\end{array}\right]$$

(27)

where $x^T=[{\varphi }_{pfp},{\delta }_{tfa},{\delta }_{bfw1},{\delta }_{bfw2},{\delta }_{bfw3},{\dot{\varphi }}_{pfp},{\dot{\delta }}_{tfa},{\dot{\delta }}_{bfw1},{\dot{\delta }}_{bfw2},{\dot{\delta }}_{bfw3}{]}^T$ and $E$ and $\dot{E }$ are expressed as Equations (28) and (29), respectively.

$$E={\tilde{K}}_{pfp}{\varphi }_{pfp}+{\tilde{K}}_{tfa}{\delta }_{tfa}+{\tilde{K}}_{bfw1}{\delta }_{bfw1}+{\tilde{K}}_{bfw2}{\delta }_{bfw2}+{\tilde{K}}_{bfw3}{\delta }_{bfw3}$$

(28)

$$\dot{E}={\tilde{K}}_{p\dot{f}p}{\dot{\varphi }}_{pfp}+{\tilde{K}}_{t\dot{f}a}{\dot{\delta }}_{tfa}+{\tilde{K}}_{b\dot{f}w1}{\dot{\delta }}_{bfw1}+{\tilde{K}}_{b\dot{f}w2}{\dot{\delta }}_{bfw2}+{\tilde{K}}_{b\dot{f}w3}{\dot{\delta }}_{bfw3}$$

(29)

The Mamdani-type fuzzy model is used to adjust the feedback gain of the IPC loop controller. The composite error $E$ and its rate of change $\dot{E}$ are used as input parameters, and the control signal $U$ is used as the output parameter. Both input and output parameters are transformed into fuzzy logic linguistic variables for expressing fuzzy rules, denoted as (N-negative, Z-zero, P-positive). The input ranges for $E$ and $\dot{E}$ are [0, 1] and [−0.05, 0.05], respectively, while the output $U$ ranges from [0, 1].

Table 2. Fuzzy rule table.

$\mathit{U}$	$\dot{\mathit{E}}$
$\mathit{E}$		N	Z	P
	N	N	N	N
	Z	N	P	P
	P	P	P	P

presents the fuzzy rules applied to the fuzzy LQR controller, and Figure 4 shows the input–output mapping surface plot between input parameters $E$ and $\dot{E }$ and output parameter $U$.

By combining the fuzzy logic with the LQR controller, the adaptive state feedback gain $\widehat{K}$ for the IPC is ultimately obtained. The pitch angle input value for the IPC loop can be determined from $\widehat{K}$ and is expressed as

$${\beta }^{IPCi}(t)=-{\widehat{K}}_{i}x(t)\hspace{1em}i=\mathrm{1,2},3$$

(30)

4. Simulation Results and Discussion

A co-simulation via OpenFAST and MATLAB/Simulink was employed to analyze the dynamic behavior of the Spar-type FOWT. The control scheme was developed in MATLAB/Simulink and implemented using the fully nonlinear dynamic model of OpenFAST, which encompasses all DOFs, structural flexibility, and complex coupling within the wind turbine components. The simulation was run for 600 s, with a time step of 0.0125 s. The constraints within the simulation included a blade pitch angle ranging from 0° to 90° and a pitch rate between −8°/s and 8°/s to emulate the realistic operating conditions of the wind turbine.

An actual wind-wave environment is used to enhance the representativeness of the simulation outcomes. Three representative wind and wave environmental cases were selected in

Table 3. Environmental cases.

Case No.	Wind Speed (m/s)	Turbulence Intensity (%)	Wave Height HS (m)	Peak Spectral Period TP (s)
1	16	13.17	3.0	10.5
2	18	12.67	3.6	10.9
3	20	12.48	4.1	11.2

based on the hindcast data collected from Site 14 in the northern North Sea, as described in reference [35]. The environmental conditions selected are intended to simulate scenarios likely to be encountered in actual operations, thereby ensuring the broad applicability of the results. Wind fields were produced by the NREL TurbSim v2.00 software [36], and wave simulations were carried out utilizing the Pierson–Moskowitz spectrum. Figure 5 displays the time series curves relating to the environmental cases.

4.1. Time-Domain Analysis

The time-domain performance of the proposed control scheme is evaluated in terms of power regulation, platform motion suppression, and structural load reduction. A comparative analysis is provided with the traditional PI-CPC scheme. The study employed Mlife v1.00 software from NREL [37] to analyze mechanical stresses at the wind turbine blade roots and tower bases. It accurately calculated the forces and moments using the damage equivalent loads (DEL). DEL values were estimated under conditions simulating a 20-year lifespan and a 1 Hz frequency. Furthermore, the normalized performance indicators for the root mean square (RMS) of platform motion, as well as the DEL at the tower base and blade root were calculated to assess the control scheme performance with greater precision. In this study, the performance indicators of the PI-CPC were set at a baseline value of one. A normalized indicator below one signifies that the performance of the proposed scheme is superior to that of the PI-CPC, whereas a value above one indicates that the performance is inferior to that of the PI-CPC.

4.1.1. Power and Speed Regulation

The dynamic responses of the output power and rotor speed are presented in Figure 6 under three diverse wind-wave environmental cases. It is observed that both the output power and rotor speed exhibit fluctuations around their rated values of 5 MW and 1173.7 rpm, respectively. The proposed scheme demonstrates notable improvements over the conventional PI-CPC, effectively mitigating power and speed fluctuations while eliminating overshoot phenomena. The proposed scheme can rapidly stabilize the system and mitigate fluctuations in power output when wind and wave conditions change.

Table 4. RMS values of the generator power and speed in Case 1, Case 2, and Case 3.

	RMS	Generator Power (kW)	Generator Speed (rpm)
Control Strategy
Case 1	PI-CPC	5512	1222.5
	Proposed scheme	5293	1173.8
Case 2	PI-CPC	5554	1231.8
	Proposed scheme	5298	1175.0
Case 3	PI-CPC	5547	1230.1
	Proposed scheme	5294	1174.2

presents the RMS values of the output power and rotor speed of the FOWT under three environmental cases for two control schemes. According to the comparison data, the output power and rotor speed are more closely matched to their rated values with the proposed control strategy. This further substantiates the significant advantages of the proposed scheme in maintaining stability in power output and rotor speed.

4.1.2. Platform Motion Response

The dynamic responses of the FOWT platform motion in three environmental cases are illustrated in Figure 7, Figure 8 and Figure 9. The subfigures (a) to (f) depict the six DOFs of the platform motion: pitch, roll, yaw, surge, sway, and heave, respectively. The results show that the proposed control scheme significantly suppresses platform motions in all cases compared to the traditional PI-CPC, particularly achieving greater reductions in pitch, roll, surge, and heave. Such suppression of platform motion is crucial for the power output and operational stability of FOWTs. Additionally, the proposed scheme exhibits rapid responsiveness to environmental changes with minimal overshoot, highlighting its significant advantages in enhancing platform stability and environmental adaptability.

As shown in Figure 10, the RMS values of the platform six DOFs motion are normalized to illustrate performance improvements under three environmental cases. Specifically, in Case 1, reductions in platform pitch, roll, and yaw were observed by 12%, 13%, and 4%, respectively, while decreases in surge, sway, and heave were 4%, 1%, and 36%. In Case 2, the reductions were 17%, 27%, and 12% in platform pitch, roll, and yaw, respectively, with surge, sway, and heave decreasing by 5%, 7%, and 48%. In Case 3, reductions included 18% in pitch, 1% in roll, and 6% in yaw, with corresponding decreases of 5% in surge and 48% in heave. Overall, the proposed control scheme significantly reduces platform motion, especially in critical DOFs related to structural safety and operational stability, such as pitch, roll, and heave.

4.1.3. Structural Loads Response

Figure 11 shows that the proposed scheme outperforms the traditional PI-CPC in terms of reducing DELs at the blade root and tower base. This is illustrated by normalized bar graphs. In Case 1, the blade out-of-plane, pitch, and flapwise bending moments decreased by 38%, 44%, and 36%, respectively. Additionally, the tower base roll and pitch bending moments exhibited reductions of 23% and 22%, respectively. In Case 2, the reductions in out-of-plane, pitch, and flapwise bending moments for the blade root were 33%, 54%, and 30%, respectively, while the roll and pitch moments of the tower base decreased by 28% and 19%. In Case 3, despite more severe wind and wave conditions, the reductions in blade out-of-plane, pitch, and flapwise bending moments were 21%, 48%, and 12%, respectively, and the roll and pitch moments at the tower base decreased by 29% and 15%. Moreover, some indicators, such as the blade edgewise bending moment and the tower base roll and yaw bending moments, even showed further decreases. These results highlight the distinct advantages of the proposed scheme in diverse environmental scenarios, particularly in reducing key parameters such as blade pitch and out-of-plane moments, as well as tower base roll and pitch moments. This performance demonstrates that the proposed scheme has strong environmental adaptability, which is essential for enhancing the structural safety of the FOWT system.

Figure 12 depicts a comparison of the change in pitch angle between the proposed scheme and the PI-CPC in three cases. Although the proposed scheme results in increased pitch activation, the blade pitch angle consistently remains within the predetermined operational range of [0°, 90°]. In addition, Figure 13 demonstrates that the maximum values of the blade pitch rate fully comply with the safety limits of [−8°/s, 8°/s]. Given that the proposed control scheme significantly improves power fluctuation, suppresses platform motion, and notably reduces DELs at the blade root and tower base, the increased pitch rate is considered entirely acceptable. The proposed control scheme is efficient and operationally safe.

4.2. Power Spectral Density Analysis

To evaluate the performance of the proposed scheme in power regulation, platform motion suppression, and fatigue load reduction at frequencies of interest, a power spectral density (PSD) analysis is performed. Figure 14 presents the frequency domain response curves for wind speed and wave height under three diverse environmental cases. The results from Case 1 reveal that the maximum energy of wind speed is primarily focused around 0.01 Hz, with wave heights predominantly within the range of 0.06 Hz to 0.365 Hz. In Case 2, the peak wind speed energy occurs at approximately 0.005 Hz, while the main frequency range observed for wave height ranges from 0.054 Hz to 0.335 Hz. In Case 3, the wind speed energy reaches its maximum at a frequency of approximately 0.015 Hz, with the main frequency range for wave height spanning from 0.047 Hz to 0.35 Hz.

4.2.1. Output Power Response

The PSD response curves of the output power in the low-frequency range in three diverse environmental cases are shown in Figure 15. Compared to the traditional PI-CPC, it is observed that in the 1P (one per revolution, 0.2 Hz) frequency range, the proposed scheme significantly reduces the PSD values of the output power at and around the peak wind speed energy. These results confirm the effectiveness of mitigating low-frequency fluctuations caused by wind and waves, as well as handling periodic fluctuations due to blade rotation. Across the frequency spectrum, the proposed scheme shows a significant advantage in reducing output power fluctuations and improving stability when compared to the PI-CPC.

4.2.2. Platform Motion Response

Figure 16, Figure 17 and Figure 18 illustrate the comparison of PSD curves for six DOF platform motions in the low-frequency range under three environmental cases. The PSD values of the proposed scheme are generally lower compared to the traditional PI-CPC, particularly in regions with higher wind energy. This highlights the significant advantages of the proposed scheme for mitigating platform motion induced by wind forces. Additionally, the proposed scheme effectively suppresses wind-induced platform motion without exacerbating it within the predominant frequency range of waves, demonstrating its capability to maintain platform stability.

4.2.3. Fatigue Load Response

Figure 19 shows the PSD curves for the blade in-plane, out-of-plane, and pitch bending moments of the FOWT. The PSD values obtained from the proposed scheme are consistently lower across the frequency range compared to those of the PI-CPC. This demonstrates that the proposed scheme effectively reduces the periodic loads resulting from blade rotation, thereby significantly enhancing its ability to mitigate fatigue loads at the blade root. Such fatigue load reduction is critical for maintaining wind turbine structural stability and extending their lifespan.

Figure 20 shows that the proposed scheme significantly reduces the PSD values of fatigue loads in the roll, pitch, and yaw at the tower base across a wide range of frequencies. Notably, in the low-frequency range of 0 Hz to 0.1 Hz, the proposed scheme effectively reduces fatigue loads induced by wind. Furthermore, in the predominant frequency range of wave activity, the PSD values under the proposed scheme are generally lower than or comparable to those of the traditional PI-CPC. This indicates that the proposed scheme is not only effective in controlling the wind-induced loads, but it also prevents exacerbating the additional loads caused by wave action.

5. Conclusions

This paper introduces a new hybrid blade pitch control method that combines fuzzy LQR and PI algorithms to suppress power output fluctuation, multi-dimensional platform motion, and fatigue loads of FOWTs. The study demonstrated that the proposed scheme outperforms the traditional PI-CPC in improving power output stability, suppressing platform motion, and reducing fatigue loads on the blade root and tower base across three diverse environmental conditions. Analysis results show that the proposed scheme achieved reductions in the RMS values of platform pitch, roll, yaw, and heave motion by 17%, 27%, 12%, and 48%, respectively. Additionally, it decreased the DELs of blade out-of-plane bending moment, blade pitch bending moment, and blade flapwise bending moment by 33%, 54%, and 30%, respectively, while the DELs of tower base roll and pitch bending moments were reduced by 28% and 19%, respectively. These reductions are essential for improving operational efficiency and ensuring the long-term stability of FOWTs. The efficacy of the proposed scheme is further substantiated via PSD analysis. Despite an increase in pitch activity, it remained within acceptable saturation limits. Given the frequent operation of the pitch actuator and its critical role in load reduction control for FOWTs, the expected research hotspots are to consider the potential faults of the pitch actuator and design corresponding to the fault-tolerant control strategies for FOWTs.