A Novel Composite Pitch Control Scheme for Floating Offshore Wind Turbines with Actuator Fault Consideration

: It is of great importance to simultaneously stabilize output power and suppress platform motion and fatigue loads in ﬂoating offshore wind turbine control systems. In this paper, a novel composite blade pitch control scheme considering actuator fault is proposed based on an augmented linear quadratic regulator (LQR), a fuzzy proportional integral (PI) and an adaptive second-order sliding-mode observer. Collective pitch control was achieved via the fuzzy PI, while individual pitch control was based on the augmented LQR. In the case of actuator fault, an adaptive second-order sliding-mode observer was constructed to effectively eliminate the need for the upper bound of unknown fault derivatives and suppress the chattering effect. This paper conducted co-simulations based on FAST (Fatigue, Aerodynamics, Structures, and Turbulence) and MATLAB/Simulink to verify the effectiveness and superiority of the proposed scheme under different environmental conditions. It is shown that platform roll was reduced by approximately 54% compared to that under PI control. For the tower fore–aft moment, load reductions of 45% or more were achievable. The proposed scheme can greatly reduce the pitch and roll of the ﬂoating platform and loads in the windward direction of the wind turbine.


Introduction
Due to the increasing physical size of floating offshore wind turbines (FOWTs) and the strong coupling of wind turbine structures, advanced pitch control methods are urgently needed to maintain a stable output power, restrain platform motion and reduce the fatigue loads [1].
Wind turbine pitch control is classified into collective pitch control (CPC) and individual pitch control (IPC).The CPC algorithms mainly include nonlinear proportional-integral derivative (PID) [2], adaptive control [3], sliding-mode control [4], etc.However, for CPC, it is assumed that all blades have similar structural characteristics and bear similar aerodynamic loads, which rarely happens in practical applications [5].As we know, maintaining a stable output power is contradictory to reducing platform motion and fatigue loads [6].IPC can effectively reduce the unbalanced loads caused by disturbance, tower shadow effect and wind shear effect while maintaining a stable output power and has drawn strong attention from wind turbine control engineers and scholars [7,8].Bossanyi performed IPC-related work [9].Roh et al. [10] proposed a PI-based IPC scheme.The bending moment at the blade root defined by the rotating coordinate reference frame is converted into the tilt bending moment and yaw bending moment at the hub under the fixed coordinate reference frame through Coleman transformation, and two parallel PI controllers are used to control the hub load.Yet, this control method is based on two independent single-input-single-output (SISO) controllers, and multiple SISO control loops may be coupled to each other and affect FOWT performance.
Considering the structural complexity and multiple control objectives of FOWTs, many advanced control methods for FOWTs, based on modern control theory, have been applied [11].Raach et al. [12] proposed an IPC method based on nonlinear model predictive control, which adjusts three blade pitch angles to achieve multiple-input-multiple-output (MIMO) control of FOWTs.Blade loads can be further reduced by this method, but platform roll, yaw and tower loads were not analyzed.Tong et al. [13] proposed a linear variableparameter MIMO control method by adjusting blade pitch angle and rotor speed.It can simultaneously adjust output power and reduce pitching motion, but the authors did not pay attention to the in-plane motion of the blade and the sideway motion of the tower.
As one of the advanced control methods of modern control theory, linear quadratic regulator (LQR) can easily achieve closed-loop optimal control for MIMO systems and has the advantages of simple construction and convenient physical implementation.Hence, it is expected to become the control method for commercial FOWTs and has been widely studied [14].Namik and Stol [15] proposed a multi-objective control algorithm based on LQR and applied it to CPC and IPC.Under different weighting matrix weights for IPC, the simulation results were compared and showed that if greater weighting matrix weight is assigned to power regulation, the platform motion will increase.On the contrary, if greater weight is assigned to reducing the platform motion, the output power fluctuation will increase.It can be seen that the choice of the LQR weighting matrix plays an important role in the stability of the FOWT.Thus, for the LQR control of FOWTs, the selection of the two penalty matrices (the state weight matrix (Q) and the control weight matrix (R)) is very important and also determines the control performance.Wang et al. [16] proposed an LQR scheme based on the Pareto multi-objective binary probability optimization algorithm, which expresses the expected control performances as multi-objective fitness functions to design a set of Pareto LQR controllers and find the optimal Q and R matrices.In addition, intelligent algorithms such as genetic algorithm [17], ant colony algorithm [18] and particle swarm optimization (PSO) [19] were also adopted to optimize the Q and R matrices.Subsequently, Namik and Stol [20] applied an LQR controller based on disturbance-regulated control to a spar-type FOWT.The fluctuation of the output power was improved.However, this method increased the pitch drive, which adversely affected the floating platform.Kim et al. [21] proposed a method to introduce the LQR as a feedforward term into PI control.Their goal was to improve the performance of traditional PI control using a method different from the optimal control method.Sarkar et al. [22] proposed an IPC method based on a wavelet LQR to reduce FOWT loads, but this method increased the fluctuation of the rotor speed and affected the power generation efficiency.The authors of [22] continued their study by combining CPC and IPC [23].CPC was obtained by controlling the generator rotor speed error via integral control to maintain a stable output power, and IPC was achieved by an LQ controller to improve the platform motion and reduce the blade 1P load.The simulation results showed that this method achieved better results than the PI control and seemed to have advantages.To sum up, there are still difficulties in the design of a pitch control system that can simultaneously meet the requirements of output power stabilization, platform motion suppression and blade and tower fatigue loads reduction for FOWT control.
The above studies were conducted under the assumption that the sensors and the pitch actuators were reliable.In actual operation, the hydraulic pitch system may suffer from faults such as hydraulic leakage, pump wear, high air content in the hydraulic oil, valve blockage and pump blockage, resulting in slow pitch action and negatively affecting the stable operation of the wind turbine.Therefore, actuator faults in blade pitch control systems have the greatest impact on wind turbines.The detection and diagnosis methods of actuator faults include signal-based, data-driven and model-based methods; Gao et al. [24] reviewed them.Effective model-based methods utilize an observer.Shi et al. [25] proposed an adaptive observer to estimate actuator fault, but in this method, the adaptive law needs to be calculated by linear matrix inequality and linear variable parameter technology, which increases the complexity of the observer design.Sun et al. [26] proposed an unknown input observer to estimate the actuator fault.This method can minimize the influence of sensor noise on the observer to estimate the fault; yet, it is necessary to use fault correction to transform the multiplicative fault into an additive fault form, which reduces the model accuracy.
The sliding-mode observer (SMO) is widely used due to its simple structure and strong robustness [27].Georg et al. [28] proposed a method based on a Takagi-Sugeno SMO to estimate and reconstruct a pitch system's actuator fault.This method can quickly detect faults, but the reconstruction accuracy needs to be improved.Cascaded SMOs were adopted to estimate the state and fault of the pitch system in [29].The estimation errors were almost zero, which allowed reconstructing the pitch system fault in finite time, but the errors increased when the fault occurred.Lan et al. [30] proposed an adaptive SMO to estimate the states of the pitch system and verified it with a CPC method.Liu et al. [31] extended the scheme in [30] using an adaptive SMO to observe faults, combined with two IPC methods based on PI and H∞.However, the simulation results showed that the IPC methods increased the standard deviation of the output power and the observation fault error compared with the CPC method.To conclude, the observer accuracy of the pitch actuator state and its fault determines the fault compensator performance.Improving fault diagnosis capability can greatly reduce the economic cost of FOWTs.
Therefore, considering the above-mentioned difficulties in the control and fault reconstruction of spar-type FOWTs, a novel pitch control scheme considering actuator fault is proposed in this paper.These are the main novelties of this method: (1) The scheme combines CPC and IPC to achieve blade pitch control.Fuzzy PI was adopted in CPC to mainly stabilize the output power, and an augmented LQR was constructed in IPC to reduce the fatigue loads of the blades and tower and simultaneously suppress platform motion and maintain output power stability.The presented augmented LQR allowed for the introduction of the blade rotor speed error into the performance index function to achieve accurate tracking of the output power; a particle swarm optimization algorithm (PSO) algorithm was used to optimize the Q and R matrices.(2) Considering the influence of actuator hydraulic leakage fault on FOWTs, an adaptive super-twisting sliding-mode (ASTSM) observer is proposed to observe the state and fault of the pitch system for fault reconstruction.The adaptive gains of this observer were used to deal with the uncertainty of actuator fault and disturbance and were adjusted dynamically according to the uncertain values.This method could effectively suppress chattering and increase the observation accuracy.
The rest of this paper is organized as follows: in Section 2, the structure and characteristics of the spar-type FOWT, as well as its complex dynamic and reduced-order model are described; Section 3 introduces the pitch control strategy and the ASTSM observer design; Section 4 describes the simulation environment and simulation settings and verifies the effectiveness of the proposed scheme on a 5 MW spar-type FOWT; and finally, conclusions are drawn in Section 5.
Table 1 shows the acronyms used in this paper for better understanding.The foundation of a spar-type FOWT is a slender column loaded at the bottom.Its ballast weight can balance the center of gravity and thus guarantee the overall stability of the wind turbine.This work considered the 5 MW OC3-Hywind spar-type FOWT as the study object and conducted co-simulations based on FAST and MATLAB/Simulink [32].The overall structure of the spar-type FOWT and its platform's degree of freedoms (DOFs) are shown in Figure 1.It includes a wind turbine, a floating platform, and a mooring line.

The Studied Spar-Type FOWT
The foundation of a spar-type FOWT is a slender column loaded at the bottom.Its ballast weight can balance the center of gravity and thus guarantee the overall stability of the wind turbine.This work considered the 5 MW OC3-Hywind spar-type FOWT as the study object and conducted co-simulations based on FAST and MATLAB/Simulink [32].The overall structure of the spar-type FOWT and its platform's degree of freedoms (DOFs) are shown in Figure 1.It includes a wind turbine, a floating platform, and a mooring line.Table 2 reports the relevant parameters of the wind turbine.The main parameters of the spar-type floating platform are shown in Table 3.

Dynamics Modeling
The spar-type FOWT is a complex dynamic system with strong nonlinearity and coupling, which suffers from fatigue loads caused by wind, waves and platform motion.In general, aerodynamic loads are estimated by the blade element momentum theory, and hydrodynamic loads are calculated via the Morrison equation [11].

Aerodynamic Model
The FOWT power depends on the interaction of its rotor with the wind.The rotor converts the kinetic energy of the wind into mechanical energy.The aerodynamic torque T a , the power P a , and the thrust F t acting on the rotor are [6] where ω r (t) is the rotor angular velocity; ρ is the air density; A is the rotor swept area; v r (t) is the effective wind speed on the rotor plane; C p (λ(t), β(t)), C t (λ(t), β(t)) are a power coefficient and a thrust coefficient, respectively; λ(t) is the tip speed ratio; and β(t) is the pitch angle, where λ(t) = (ω r (t) • R)/v r (t), and R is the rotor radius.

Pitch Actuator System and Its Fault
The pitch system consists of three identical hydraulic pitch actuators, each of which can be modeled as a second-order system ..
where β re f (t) is the pitch angle reference value, and ω n , ζ are the natural frequency and the damping ratio of the pitch actuator model.Hydraulic leakage is a potential fault that affects the system more than pump wear.It can cause changes in pitch system parameters where f hl (t) is the fault indicator, with f hl (t) ∈ [0, 1], for all t, . f hl (t) ≥ 0. The leakage is irreversible if the system is not repaired.It should be noted that the pressure is only 50% of the rated pressure at f hl (t) = 1.ζ 0 , ω n 0 are the standard parameters under normal pressure, with values of 0.6 and 11.11 rad/s; ζ hl , ω n hl are the parameters under low pressure, with values of 0.9 and 3.42 rad/s, respectively.Substitute (3) into ( 2) and rewrite it in the form of state space T is the system state, and u = β re f is the reference input of the pitch system.The fault distribution function is The unit step response of the pitch system with different values of f hl is shown in Figure 2. It can be observed that hydraulic leakage is characterized by gradual faults and affects the control performance of the FOWT. are the parameters under low pressure, with values of 0.9 and 3.42 rad s , respectively.
Substitute (3) into (2) and rewrite it in the form of state space where T T xx is the system state, and is the reference input of the pitch system.The fault distribution function is . The unit step response of the pitch system with different values of hl f is shown in Figure 2. It can be observed that hydraulic leakage is characterized by gradual faults and affects the control performance of the FOWT.

Figure 2.
Step response of the pitch actuator system with different values of hl f .

Drivetrain Model
In order to increase the rotational speed of the rotor, the aerodynamic torque was converted into a generator torque through a drivetrain to achieve the high speed required by the generator.A drivetrain can be described by the following three first-order differential equations [21]  Step response of the pitch actuator system with different values of f hl .

Drivetrain Model
In order to increase the rotational speed of the rotor, the aerodynamic torque was converted into a generator torque through a drivetrain to achieve the high speed required by the generator.A drivetrain can be described by the following three first-order differential equations [21].
where J r , J g indicate the rotational inertia of the low-speed shaft and high-speed shaft, B r , B g are the viscous friction coefficients of the low-speed shaft and high-speed shaft, B dt is the attenuation coefficient of the drivetrain, K dt is the torque hardness of the drivetrain, θ ∆ (t) is the torsion angle, N g is the gear ratio, T g (t) is the generator torque, and ω g (t) is the generator speed.

Generator Model
The generator coupled to the drivetrain performs the conversion of mechanical energy to electrical energy.The generator torque T g (t) is regulated according to its reference value T g re f (t), and its dynamics are approximated by a first-order model .
where τ g is the time parameter for the generator system.The generator power depends on the generator rotor speed and the applied loads and is expressed as where η g is the generator efficiency.

Reduced-Order Model of the FOWT
A strongly coupled dynamics system can accurately represent the FOWT behavior, but it also makes the controller design rather complicated.In this paper, the complex dynamic system of the FOWT was linearized, and a linear quadratic optimal pitch control was designed.Because the wind turbine bears most of the aerodynamic load in the downwind direction, the five DOFs in this direction were selected to deduce the reduced-order model of the FOWT.
For the purpose of optimal controller design, the linear state space equation of the FOWT is expressed as where x = [x 1 T , .
x 1 T ], and x 1 = [x p , x t f a1 , x b1 f 1 , x b2 f 1 , x b3 f 1 ]; x p is the pitch of the floating platform, x t f a1 is the fore-aft movement displacement of the tower, x bi f 1 , i = 1, 2, 3 is the displacement of the blades 1, 2 and 3 in the flapwise tip, u = [β 1 , β 2 , β 3 ] T is the three blades' pitch angle, and w = τ wind + τ wave is the environmental disturbance vector due to wind and wave loads.In order to simplify the LQR controller design, the matrix A is described by the second-order motion equation for the selected FOWT DOFs in the case of ignoring external disturbances [6,8].It is expressed as where the matrix M and matrix K are the mass coefficient and the stiffness coefficient of the FOWT linearization system, and the matrix D is the damping coefficient term which is usually relatively small [23].
In the matrix M, where H is the height of the tower; R is the length of the blade; ϕ i is the azimuth angle of the ith blade; m n , m h are the masses of nacelle and the hub; I p is the moment of inertia of the platform; I p pitch is the hydrodynamic additional mass factor associated with the selected pitch DOF of the platform; µ b (r) and µ t (h) are the mass per unit length of blade and tower; and φ b (r) and φ t (h) are the normalized, basic model shapes in the blade flapwise direction and the fore-aft displacement direction of the tower.
In the matrix K, where F Hydrostatic pitch is the hydrostatic force associated with the selected pitch DOF of the platform, k Lines mooring is the stiffness of the linearized ropes of the mooring system, k t is the elastic stiffness of the tower, k bi is the sum of the elastic stiffnesses of the blades, and EI t f a (h), EI b f (r) are the section stiffnesses in the fore-aft displacement direction of the tower and in the flapwise direction of the blade.
In the matrix D, where d p is the linear hydrodynamic damping coefficient associated with the selected pitch DOFs of the platform, the matrix B is obtained by a Taylor series expansion of the second equation in (1) around the full-load operating point (o.p.) (the operating point is considered to be the nominal wind speed of 18 m/s, corresponding to a nominal blade pitch angle equal to 15 When the external disturbances and torque change relative to the rotor speed are ignored, the matrix B is obtained by It is worth noting that although the optimal controller was designed based on the reduced-order model of the FOWT with 5 DOFs, the final proposed controller was tested based on the full-order nonlinear model of the FOWT.

Research Methodology
The focus of this study was on the region 3 (above the rated wind speed region) of the FOWT.The proposed pitch controller combines CPC and IPC.The former was used to maintain output power stability by controlling the rotor speed of the generator, and the latter was constructed by the augmented LQR method to effectively reduce the wind turbine loads and maintain the output power at the same time.Considering the pitch actuator fault, the ASTSM observer was built to observe the actuator state and its fault, and the observation result was transmitted to the fault compensator.The fault-tolerant control (FTC) system with the fault compensator restored the pitch system to a fault-free state.The structure diagram of the proposed scheme is shown in Figure 3.The control input for a single blade pitch angle is J. Mar.Sci.Eng.2023, 11, x FOR PEER REVIEW 10 of 24 turbine loads and maintain the output power at the same time.Considering the pitch actuator fault, the ASTSM observer was built to observe the actuator state and its fault, and the observation result was transmitted to the fault compensator.The fault-tolerant control (FTC) system with the fault compensator restored the pitch system to a fault-free state.
The structure diagram of the proposed scheme is shown in Figure 3.The control input for a single blade pitch angle is

CPC Design
To apply PI control to the FOWT, the error signal was chosen as

CPC Design
To apply PI control to the FOWT, the error signal was chosen as where ω g re f (t) is the rated generator speed, and ω g (t) is the actual generator speed.Then, the control law of the PI pitch controller is where k p0 , k I0 are proportional gain and integral gain.However, a practical difficulty is that the loads on a wind turbine are complex and vary when the environmental conditions, such as turbulent wind and sea wave, change.It is difficult to select the optimal gains of the traditional PI controller.A fuzzy PI controller can adjust the control parameters online and has strong adaptability and robustness.This study constructed a fuzzy PI control scheme, which is shown in the "Collective Pitch Control (CPC)" section of Figure 3.In Figure 3, k e , k ec are proportional factors, and k u1 , k u2 are quantization factors.The fuzzy PI controller uses the generator rotor speed error e and the error rate of change ec as inputs, and adopts fuzzy rules to adjust the PI parameters online; the outputs are k P1 , k I1 which are added to the initial values k p0 , k I0 to obtain the actual parameters of the PI controller outputs.The fuzzy field is usually selected in the range [ −6, 6] for wind turbine pitch control to maintain the variation of the allowable error at about 5% of the rated rotor speed, which is considered reasonable [33].The indices e and ec were set as [−6, 6], and k P1 , k I1 were set as [−1, 1] in this study.A detailed procedure for fuzzy controller design can be found in the authors' previous study [34].
Then, the pitch angle input under CPC is

Augmented LQR Controller Design
In this section, the blade rotor speed error is introduced into the performance index function, and the augmented LQR control method is presented.
Rewrite (8) as According to (22), the control's objective is to make the blade rotor speed accurately track the given signal r(t) = a × 1(t) (where a is a constant vector-here, a = 12.1 rpm-and 1(t) is a step signal with an amplitude of 1).Then, the error is e = r − y 1 (y 1 represents the actual blade rotor speed.Since the state equation of the FOWT under LQR control does not include the blade rotor speed, it is represented by y 1 ).
Then, the time derivatives of ( 22) and the tracking error e are deduced as ..
x 1 (23) Choose the augmented state vector as x = [ .

x
T , e] T = [x p , x t f a1 , x b1 f 1 , x b2 f 1 , x b3 f 1 , e] T , then the state equation of the augmented system is obtained as .
where u = , and the corresponding performance index where The above problem was transformed into a conventional LQR problem, and the optimal control law was obtained by the LQR method where P satisfies the Riccati Equation (27).
K is expressed as a block matrix K = [K .
x , K e ] according to .
x and e.Then, x K e .
x e = K .
x + K e e The integral of ( 28) is obtained as The schematic diagram of this augmented LQR control is shown in the "Individual Pitch Control (IPC)" section in Figure 3.It can be observed that the integral of the wind turbine blade rotor speed error e is fed back to the input of the controller; hence, accurate tracking of the output power can be achieved.

PSO Algorithm
The performance of an LQR controller strongly depends on the weighting matrices (Q, R); therefore, their selection is important.The PSO algorithm, which is based on swarm search to deal with optimization problems in continuous or discrete space, is applied to optimize the weighting matrices.It uses the particles constantly moving in the solution space as the optimization swarm.
Each particle adjusts its speed according to the optimal solution which has been already determined, so as to search for a better solution.Figure 4 illustrates how the PSO algorithm works.The velocity and position update formulas of each iteration particle are where v i t is the velocity of the ith particle at the tth iteration; x i t is the position of the ith particle at the tth iteration;w is the weight coefficient; c 1 , c 2 are acceleration coefficients; r 1 , r 2 are random numbers in the interval [0, 1]; and pbest, gbest are the optimal past positions of the particle itself and the particle swarm, respectively.

ASTSM Observer Design
In this section, considering the influence of the actuator hydraulic leakage fault on the FOWT, the ASTSM observer is proposed to observe the state and fault of the pitch system for fault reconstruction.The "co-design" structure is shown in Figure 3 ("Fault-tolerant control (FTC) system").
where t i v is the velocity of the ith particle at the tth iteration; t i x is the position of the ith particle at the tth iteration; w is the weight coefficient; , cc are acceleration coeffi- cients; 1 2 , rr are random numbers in the interval [0, 1]; and , pbest gbest are the optimal past positions of the particle itself and the particle swarm, respectively.

Fault Compensator
A fault compensator with a common design structure [31], which can make the pitch recover to a fault-free state in case of faults, was adopted.The fault compensator is where ωn , and x is the observation value of the ASTSM observer.Thus, the observation values of x 1 , x 2 , f hl are needed, and the observation method will be described next.

The ASTSM Observer
According to (4), the ASTSM observer is constructed as For (33), the control law v(•) can be designed by the adaptive super-twisting algorithm [35]  where s represents the sliding variable, and k 1 (t), k 2 (t) are adaptive gains where α 1 , β 1 , σ 1 , ε, k, k are arbitrary positive numbers, and k can be arbitrarily small.
e f hl = .
From Section 2.2.2, f hl is bounded; then, assume f hl ≤ L is satisfied.For (34), the sliding surface is defined as For the stability proof, refer to [35].

Simulation Results and Discussion
The effectiveness of the proposed scheme was verified.The aerodynamic, mechanical and electrical parts of the FOWT were simulated by FAST code, and the proposed controller and observer were developed in the MATLAB/Simulink environment.A fully nonlinear simulation was executed in FAST, enabling all DOFs of the FOWT.Due to physical limitations, the constraint of the pitch angle and its rate of change for the NREL 5 MW FOWT were β ∈ [0 • , 90 • ], .
β ∈ [−8, 8]( • )/s.The simulations were performed under two different wind and wave conditions.Turbulent wind fields were generated by the software TurbSim v2.00 [36], and random waves were simulated by the Pierson-Moskowitz spectrum.For each environmental condition, 600 s simulations were performed using different stochastic seeds based on time series of wind and wave loads.The environmental conditions are shown in Table 4, and the time series are shown in Figure 5.The fatigue loads for the blades and the tower were analyzed via the NREL's MLife code [37].MLife is a Matlab-based tool for post-processing results from wind turbine testing and aeroelastic dynamic simulations.The damage-equivalent loads (DELs) were obtained for a life of 20 years and a frequency of 1 Hz.
The time-series data of the first 30 s were removed in the analysis to eliminate the influence of the initial transient process.The effect of faults on the pitch actuator was simulated by modifying the natural frequency and damping ratio of the pitch system dynamics.The superiority of the presented ASTSM observer was verified by comparing it with the classical sliding-mode (CSM) observer [38] and the super-twisting sliding-mode (STSM) observer [39].
The simulations were performed under two different wind and wave conditions.Turbulent wind fields were generated by the software TurbSim v2.00 [36], and random waves were simulated by the Pierson-Moskowitz spectrum.For each environmental condition, 600 s simulations were performed using different stochastic seeds based on time series of wind and wave loads.The environmental conditions are shown in Table 4, and the time series are shown in Figure 5.

Control Performance Analysis
The values of the parameters of the CPC-PI control were chosen as k p0 = 0.01, k i0 = 0.001.The feedback gain matrix of the augmented LQR controller optimized by the PSO was  The fatigue loads for the blades and the tower were analyzed via the NREL's MLife code [37].MLife is a Matlab-based tool for post-processing results from wind turbine testing and aeroelastic dynamic simulations.The damage-equivalent loads (DELs) were obtained for a life of 20 years and a frequency of 1 Hz.
The time-series data of the first 30 s were removed in the analysis to eliminate the influence of the initial transient process.The effect of faults on the pitch actuator was simulated by modifying the natural frequency and damping ratio of the pitch system dynamics.The superiority of the presented ASTSM observer was verified by comparing it with the classical sliding-mode (CSM) observer [38] and the super-twisting sliding-mode (STSM) observer [39].

Control Performance Analysis
The values of the parameters of the CPC-PI control were chosen as 00 0.01, 0.001 pi kk == . The feedback gain matrix of the augmented LQR controller optimized by the PSO was   The proposed controller performance was investigated in terms of wind turbine output power, platform motion and fatigue DELs. Figure 7a,b shows the output power and rotor speed for case 1, and Figure 7c,d shows the responses for case 2. It can be observed that under the two environmental conditions, the fluctuations of power output and rotor speed under the proposed control method were much smaller than under CPC-PI control.The proposed controller performance was investigated in terms of wind turbine output power, platform motion and fatigue DELs. Figure 7a,b shows the output power and rotor speed for case 1, and Figure 7c,d shows the responses for case 2. It can be observed under the two environmental conditions, the fluctuations of power output and rotor speed under the proposed control method were much smaller than under CPC-PI control.CPC-PI control increased the fluctuation frequency of the output power and the rotor speed when the wind and waves changed, while the proposed method could suppress this fluctuation and maintain the stability of the output power.Figure 8a-c shows the platform pitch, roll and yaw motions for case 1, and Figure 8d-f shows the responses for case 2. It can be observed that the proposed controller significantly reduced the pitch and roll of the platform.CPC-PI control increased the fluctuation frequency of the output power and the rotor speed when the wind and waves changed, while the proposed method could suppress this fluctuation and maintain the stability of the output power.Figure 8a-c shows the platform pitch, roll and yaw motions for case 1, and Figure 8d-f shows the responses for case 2. It can be observed that the proposed controller significantly reduced the pitch and roll of the platform.CPC-PI control increased the fluctuation frequency of the output power and the rotor speed when the wind and waves changed, while the proposed method could suppress this fluctuation and maintain the stability of the output power.Figure 8a-c shows the platform pitch, roll and yaw motions for case 1, and Figure 8d-f shows the responses for case 2. It can be observed that the proposed controller significantly reduced the pitch and roll of the platform.Figure 9 reports the bar graph based on the root mean square (RMS), standard deviation (STD) and maximum absolute value (|MAX|) index, with the CPC-PI control results as the baseline.The comparisons of Figure 9a-j show the advantages of the proposed control scheme.Compared with the CPC-PI method, the RMS value of the rotor speed in case 1 with the proposed scheme could reach 1174.1 rpm, which was closer to the rated value (1173.7 rpm).With respect to platform motion, the RMS values of pitch, roll and yaw were reduced by 15%, 54% and 9%, respectively.Figure 9k-p shows the fatigue DELs of the FOWT.It is shown that the loads were suppressed or remained basically unchanged under the proposed control scheme.Compared with the CPC-PI control, the DELs of the blade out-of-plane moment and in-plane moment were reduced by 10% and 2%, and the DELs of the tower fore-aft moment and side-to-side moment were reduced by 10% and 45% in case 1.To sum up, compared with the PI control, the proposed scheme reduced platform motion and fatigue DELs, while maintaining a stable output power.
as the baseline.The comparisons of Figure 9a-j show the advantages of the proposed control scheme.Compared with the CPC-PI method, the RMS value of the rotor speed in case 1 with the proposed scheme could reach 1174.1 rpm, which was closer to the rated value (1173.7 rpm).With respect to platform motion, the RMS values of pitch, roll and yaw were reduced by 15%, 54% and 9%, respectively.Figure 9k-p shows the fatigue DELs of the FOWT.It is shown that the loads were suppressed or remained basically unchanged under the proposed control scheme.Compared with the CPC-PI control, the DELs of the blade out-of-plane moment and in-plane moment were reduced by 10% and 2%, and the DELs of the tower fore-aft moment and side-to-side moment were reduced by 10% and 45% in case 1.To sum up, compared with the PI control, the proposed scheme reduced platform motion and fatigue DELs, while maintaining a stable output power.Figure 10 shows the pitch angle change under the proposed controller and the CPC-PI controller in the two random environments.It can be observed that the proposed controller increased the frequency of the blade pitch to achieve the interacting control objectives.However, Figure 11 indicates that the maximum pitch rate was completely within the safety limit of the constraint condition of 8 deg/s.Figure 10 shows the pitch angle change under the proposed controller and the CPC-PI controller in the two random environments.It can be observed that the proposed controller increased the frequency of the blade pitch to achieve the interacting control objectives.However, Figure 11 indicates that the maximum pitch rate was completely within the safety limit of the constraint condition of 8 deg/s.

Performance in the Presence of Actuator Fault
Although FAST is an effective tool for wind turbine control verification, it does not include the dynamic response of any pitch actuator.To compensate this defect, an additional actuator dynamics module was added to the existing model.To achieve fault reconstruction the pitch system, the performance of the proposed ASTSM observer was evaluated with three pitch systems suffering from hydraulic leakage faults.Faults 1, 2 and 3 for the entire simulation time.According to (35), the parameters of the ASTSM observer were chosen as follows Pitch system 1: α 1 For convenience, only the estimation results of the ASTSM observer for actuator faults in case 1 are shown.Figure 12 shows the three pitch systems faults and the observations from the ASTSM observer.It can be seen that the ASTSM observer could accurately observe the pitch actuator faults, which verified the effectiveness of the ASTSM observer.

Performance in the Presence of Actuator Fault
Although FAST is an effective tool for wind turbine control verification, it does not include the dynamic response of any pitch actuator.To compensate for this defect, an additional pitch actuator dynamics module was added to the existing model.To achieve fault reconstruction for the pitch system, the performance of the proposed ASTSM observer was evaluated with three pitch systems suffering from hydraulic leakage faults.Faults 1, 2 and 3 persisted for the entire simulation time.According to (35), the parameters of the ASTSM observer were chosen as follows Pitch system 1:

Performance in the Presence of Actuator Fault
Although FAST is an effective tool for wind turbine control verification, it does not include the dynamic response of any pitch actuator.To compensate for this defect, an additional pitch actuator dynamics module was added to the existing model.To achieve fault reconstruction for the pitch system, the performance of the proposed ASTSM observer was evaluated with three pitch systems suffering from hydraulic leakage faults.Faults 1, 2 and 3 persisted for the entire simulation time.According to (35), the parameters of the ASTSM observer were chosen as follows Pitch system 1: Pitch system 2: In order to demonstrate the superiority of the presented ASTSM observer, our model was compared with the CSM observer and the STSM observer.The observation errors of the pitch actuator faults are shown in Figures 13-15.It was observed that with the CSM observer there was obvious chattering and a large observation error.With the STSM observer, chattering was reduced.When the ASTSM observer was adopted, the chattering effect and observation error were greatly reduced.The performance indicators in Table 5 are defined as accurately observe the pitch actuator faults, which verified the effectiveness of the ASTSM observer.In order to demonstrate the superiority of the presented ASTSM observer, our model was compared with the CSM observer and the STSM observer.The observation errors of the pitch actuator faults are shown in Figures 13-15.It was observed that with the CSM observer there was obvious chattering and a large observation error.With the STSM observer, chattering was reduced.When the ASTSM observer was adopted, the chattering effect and observation error were greatly reduced.The performance indicators in Table 5 shows a more detailed comparison of the fault observation errors.In the presence of the considered pitch system faults, these performance indexes verified the superiority of the ASTSM observer.This was a consequence of the fact that the observer gain varied according to the effect of the fault on the deviation of the sliding variable from the sliding surface (39).To clarify this, the changes in the ASTSM observer gains are shown in Figure 16.Since the adaptive parameters of the ASTSM observer varied with the tracking error, the sliding surface (39) could be infinitely close to zero, so as to effectively suppress the system chattering.Table 5 shows a more detailed comparison of the fault observation errors.In the presence of the considered pitch system faults, these performance indexes verified the superiority of the ASTSM observer.This was a consequence of the fact that the observer gain varied according to the effect of the fault on the deviation of the sliding variable from the sliding surface (39).To clarify this, the changes in the ASTSM observer gains are shown in Figure 16.Since the adaptive parameters of the ASTSM observer varied with the tracking error, the sliding surface (39) could be infinitely close to zero, so as to effectively suppress the system chattering.
The accuracy of the ASTSM observer provides a strong guarantee for the fault compensator to restore the pitch system to a fault-free state.Figure 17 shows the bar graph based on the STD and |MAX| index, with the fault-free system as the reference object; all results were normalized based on the fault-free results.In case 1, it was observed that the STD of the rotor speed increased by 43% in the presence of a fault.In terms of platform motion, the STD of pitch, roll and yaw increased by 16%, 128% and 115%, respectively.The DELs of the blade out-of-plane moment and in-plane moment increased by 53% and 4%, and the DELs of the tower fore-aft moment and side-to-side moment increased by 22% and 178% in the presence of a fault.With the help of the ASTSM observer, the fault and state of the pitch system were determined, and the fault compensator could restore the pitch system to a fault-free state and then reduce the impact of the actuator fault on the FOWT.The accuracy of the ASTSM observer provides a strong guarantee for the fault compensator to restore the pitch system to a fault-free state.Figure 17 shows the bar graph based on the STD and |MAX| index, with the fault-free system as the reference object; all results were normalized based on the fault-free results.In case 1, it was observed that the STD of the rotor speed increased by 43% in the presence of a fault.In terms of platform motion, the STD of pitch, roll and yaw increased by 16%, 128% and 115%, respectively The DELs of the blade out-of-plane moment and in-plane moment increased by 53% and 4%, and the DELs of the tower fore-aft moment and side-to-side moment increased by 22% and 178% in the presence of a fault.With the help of the ASTSM observer, the fault and state of the pitch system were determined, and the fault compensator could restore the pitch system to a fault-free state and then reduce the impact of the actuator fault on the FOWT.

Conclusions
To suppress power output fluctuation, platform motion and fatigue loads in the spartype FOWT, this paper proposes a novel composite blade pitch control scheme combining CPC and IPC.A fuzzy PI control method was designed for CPC to achieve rated rotor

Conclusions
To suppress power output fluctuation, platform motion and fatigue loads in the spartype FOWT, this paper proposes a novel composite blade pitch control scheme combining CPC and IPC.A fuzzy PI control method was designed for CPC to achieve rated rotor speed tracking and then stabilize the output power.The platform pitching motion displacement, the tower fore-aft movement displacement, the blade displacement in the along-wind direction and the tracking error of the blade rotor speed were chosen as augmented state vectors to design an LQR controller for IPC to suppress loads and platform motion and improve output power stability.The ASTSM observer was adopted in the presence of actuator fault scenarios to facilitate FTC.The simulation results indicated that the proposed scheme performed well in coping with power fluctuation, fatigue load, platform motion and actuator fault simultaneously.Under environmental conditions 1, compared to the traditional PI method, the proposed scheme reduced the root mean square of pitch, roll and yaw in the floating platform by 15%, 54% and 9%, respectively.In terms of the internal and external moments of the blade plane, the DELs were reduced by 2% and 10%, respectively.The DELs were reduced by 10% and 45%, respectively, in the longitudinal and transverse moments of the tower foundation.It should also be noted that the proposed method could significantly suppress the loads in the windward direction of the FOWT; yet, the suppression effect in the crosswind direction (blade in-plane and edgewise direction) was not obvious.Feedforward control compensation based on wind and wave forecasting and simultaneous fault observation in sensors and actuators are the breakthrough objectives of this work in the future.

Figure 2 .
Figure 2.Step response of the pitch actuator system with different values of f hl .

5
d t and d b are the structural damping coefficients of the tower and blade, ω t and ω b are the intrinsic frequencies of the tower and blade, and ζ t and ζ b are the damping ratios of the tower and blade, respectively.We have Matrix A = 0 5 * 5 I 5 * 5 −M −1 K −M −1 D , Matrix C = I

Figure 3 .
Figure 3. Diagram of the proposed scheme structure.

Figure 3 .
Figure 3. Diagram of the proposed scheme structure.

Figure 4 .
Figure 4. Working principle of the PSO algorithm.

Figure 5 .
Figure 5. (a) Wind speed and waves for Case 1, and (b) wind speed and waves for Case 2.Figure 5. (a) Wind speed and waves for Case 1, and (b) wind speed and waves for Case 2.

Figure 5 .
Figure 5. (a) Wind speed and waves for Case 1, and (b) wind speed and waves for Case 2.Figure 5. (a) Wind speed and waves for Case 1, and (b) wind speed and waves for Case 2.

Figure 6
Figure6shows the PSO convergence plot for the performance indicator function mentioned in Section 3.2.1 over 100 iterations.

Figure 6
Figure6shows the PSO convergence plot for the performance indicator function mentioned in Section 3.2.1 over 100 iterations.

Figure 6 .
Figure 6.Convergence of the PSO for the performance index function.

Figure 6 .
Figure 6.Convergence of the PSO for the performance index function.

Figure 7 . 2 Figure 8 .
Figure 7. Generator power (a) and generator speed response (b) on Case 1, Generator power (c) and generator speed response (d) on Case 2

Figure 9
Figure 9 reports the bar graph based on the root mean square (RMS), standard deviation (STD) and maximum absolute value (|MAX|) index, with the CPC-PI control results

Figure 7 .
Figure 7. Generator power (a) and generator speed response (b) on Case 1, Generator power (c) and generator speed response (d) on Case 2.

Figure 7 . 2 Figure 8 .
Figure 7. Generator power (a) and generator speed response (b) on Case 1, Generator power (c) and generator speed response (d) on Case 2

Figure 9
Figure 9 reports the bar graph based on the root mean square (RMS), standard deviation (STD) and maximum absolute value (|MAX|) index, with the CPC-PI control results

Figure 8 .
Figure 8. Platform motion response (a-c) in Case 1, and platform motion response (d-f) in Case 2.

Figure 9 .
Figure 9. Nominal performance index normalized to the CPC-PI results.

Figure 9 .
Figure 9. Nominal performance index normalized to the CPC-PI results.

J 24 Figure 10 .
Figure 10.(a) Pitch angle change for Case 1, and (b) pitch angle change for Case 2.

Figure 13 .
Figure 13.Observation error of different sliding-mode observers for pitch actuator 1 fault.Figure 13.Observation of different sliding-mode observers for pitch actuator 1 fault.

Figure 13 .
Figure 13.Observation error of different sliding-mode observers for pitch actuator 1 fault.Figure 13.Observation of different sliding-mode observers for pitch actuator 1 fault.

Figure 15 .
Figure 15.Observation error of different sliding-mode observers for pitch actuator 3 fault.

Figure 15 .
Figure 15.Observation error of different sliding-mode observers for pitch actuator 3 fault.

Figure 15 .
Observation error of different sliding-mode observers for pitch actuator 3 fault.

J 24 Figure 17 .
Figure 17.Nominal performance index normalized to the fault-free results.

Figure 17 .
Figure 17.Nominal performance index normalized to the fault-free results.

Table 2
reports the relevant parameters of the wind turbine.The main parameters of the spar-type floating platform are shown in Table3.

Table 5 .
Fault observation error comparison.

Table 5 .
Fault observation error comparison.Observation error of different sliding-mode observers for pitch actuator 1 fault.Observation error of different sliding-mode observers for pitch actuator 2 fault.
Figure 14.Observation error of different sliding-mode observers for pitch actuator 2 fault.J. Mar.Sci.Eng.2023, 11, x FOR PEER REVIEW 21 of 24