Predictive Super-Twisting Sliding Mode Control for Maximum Power Point Tracking of Floating Offshore Wind Turbines

Predictive Super-Twisting Sliding Mode Control for Maximum Power Point Tracking of Floating Offshore Wind Turbines Mohammad Mohammadi Shahir, Moein Sarbandi, Mohammad Rasool Mojallizadeh, Mohamed Assaad Hamida, and Franck Plestan Abstract—This study focuses on designing a robust control method for floating offshore wind turbines (FOWTs), aiming to achieve maximum power generation in Region II while reducing fatigue loads on the wind turbine’s platform. Due to the complex dynamic modeling of FOWTs, a reduced-order model is chosen for controller design. However, this simplified model cannot fully capture the behavior of FOWTs under various conditions. To address this issue, a robust control method based on optimal predictive control (OPC) and super-twisting (STW) sliding mode control is introduced. OPC and STW are integrated to mitigate their respective limitations and to achieve optimal performance with robustness against perturbations. The efficacy of the proposed controller is validated through comparison with reference open-source controller (ROSCO) within the MATLAB/Simulink–OpenFAST environment. The comparison results indicate that the proposed method enhances power generation while effectively mitigating power fluctuations. Index Terms—Floating wind turbine, optimal control, supertwisting sliding mode. I. INTRODUCTION Wind turbine operation is divided into four regions based on wind speed. Among these regions, Regions II and III play a crucial role in power generation. This paper focuses on Region II, where the objective is to design a controller that adjusts the generator torque to achieve maximum power extraction, which is known as maximum power point tracking (MPPT) [1]. Wind turbines are categorized into two groups: onshore and offshore. Due to some limitations of onshore turbines, there is growing interest in floating offshore wind turbines (FOWTs), as they offer advantages such as access to stronger and more consistent wind resources [2]. These advantages enable FOWTs to extract more power in comparison to onshore wind turbines. Despite these advantages, the dynamics of FOWTs exhibit a high degree of freedom, making the design of control laws more challenging [3]. To address this problem, various control methods have been implemented for MPPT in Region II. One such approach Mohammad Mohammadi Shahir, Moein Sarbandi, Mohamed Assaad Hamida, and Franck Plestan are with Nantes Universit´ e, ´ Ecole Centrale Nantes, CNRS, LS2N, UMR 6004, F-44000 Nantes, France (emails: [email protected], [email protected], [email protected], [email protected]). Mohammad Rasool Mojallizadeh is also with Arts et M´ etiers Institute of Technology, LAMPA, F-49035 Angers, France (email: mohammad- [email protected]). is a model-free method based on artificial neural networks (ANN), which is used to enhance stability and performance of the FOWT without requiring a dynamic model of the system [4], [5]. However, these strategies suffer from high computational effort, especially for real-time applications. Another alternative method is the sliding mode controller (SMC), which is an appropriate approach for dealing with uncertainties and external disturbances and has a straightforward design [6]. However, one of the main drawbacks of SMC is chattering, which makes it difficult to achieve the desired system response and increases energy consumption [7]. To overcome the limitations of SMC, high-order sliding mode (HOSM) control has been proposed as an effective alternative [8]. Among various HOSM methods, the supertwisting (STW) sliding mode control is recognized as an effective approach to mitigate the chattering phenomenon and provide high robustness against uncertainties [9]. However, this kind of controller suffers from a long transient response time in some cases [10]. Model-based controllers, such as model predictive control (MPC), are another effective approach in this research area. To reduce the complexity of the dynamic model of FOWTs, a simplified model is employed for the design of the MPCbased controller [11], [12]. Although MPC-based methods are robust, there remains a need to further enhance their performance in the presence of perturbations [13]. To address the limitations of the control methods reviewed in this paper, STW and optimal predictive control (OPC) are combined to compensate for each other’s deficiencies. OPC ensures optimal performance, while STW effectively handles system perturbations. To develop the proposed MPPT control law, the sliding variable is defined based on the tracking error. Next, the sliding variable is predicted using the Taylor series for the next time horizon, and the optimal control law is designed by solving an optimization problem. Finally, to enhance robustness against perturbations, it is combined with the STW. To analyze the performance of the predictive super twisting sliding mode (PSTW) method, numerical simulations are conducted in Region II using the NREL 5MW FOWT model in the MATLAB/Simulink-OpenFAST environment. To validate the accuracy and performance of 2025 33rd Mediterranean Conference on Control and Automation (MED) June 10 - 13, 2025. Tangier,Morocco 979-8-3315-7719-3/25/$31.00 ©2025 IEEE 553 2025 33rd Mediterranean Conference on Control and Automation (MED) | 979-8-3315-7719-3/25/$31.00 ©2025 IEEE | DOI: 10.1109/MED64031.2025.11073465 Authorized licensed use limited to: University of Nantes. Downloaded on September 28,2025 at 09:56:04 UTC from IEEE Xplore. Restrictions apply. the PSTW, the simulation results are compared with the ROSCO controller. The ROSCO is an open-source reference control framework designed for wind turbines [14]. The comparison results show that PSTW effectively tracks the desired trajectory, increasing power generation while reducing power fluctuations under stochastic wind conditions. These results highlight that the integrated OPC and STW method is an effective approach for achieving optimal performance with sufficient robustness against perturbations. The primary contribution of this paper can be summarized as follows: •Development of a control law based on OPC and STW for a 5MW FOWT to ensure optimal performance and robustness for MPPT. •Evaluation of the effectiveness of the proposed control law using the ROSCO method within the MATLAB/Simulink–OpenFAST environment. This paper is organized into four sections. Section II reviews the dynamic model of FOWT. Section III presents the proposed controller. Section IV discusses the simulation results, and the conclusion is given in Section V. II. SYSTEM MODELING The maximum mechanical power that can be generated by the FOWT is defined by a nonlinear expression P=1 2ρπR2v3Cp(λ, β),(1) where ρrepresents the density of air, Ris the blade radius, and vdenotes the wind speed. The power coefficient, which is represented by Cp(λ, β), shows the amount of electrical power that can be extracted from wind energy. The power coefficient Cp(λ, β)can be approximated as a function of the tip-speed ratio (TSR) λand the blade pitch angle β, as follows [15]: Cp(λ, β) = cp1cp2 λ1−cp3β−cp4e −Cp5 λ1,(2) where 1 λ1=1 0.08β+λ−0.035 1+β3, and cp1to cp5are curvefitting coefficients. The TSR is given by λ=Rωr v, where ωr represents the rotor speed. Following [16], the curve-fitting coefficients are considered as cp1= 0.22,cp2= 116,cp3= 0.4,cp4= 5, and cp5= 12.5. The relationship between Cp(λ, β)and λfor various values of βis shown in Fig. 1. According to this figure, the maximum value of the power coefficient is achieved when the blade pitch angle is set to zero. Based on this analysis, β= 0◦is selected. The nonlinear reduced-order dynamic model of the FOWT is defined as [17] ˙ωr=1 Jt (Ta−NgTg−Ktωr) + Λ(·),(3) where ˙ωris the time derivative of the rotor speed. Jtand Ng are the total rotor inertia and the gearbox ratio, respectively. Ktrepresents external damping, Λ(·)represents lumped uncertainties, and Tgis the generator torque, which acts as Fig. 1: Variation of the power coefficient with respect to blade pitch angle and TSR. a control input. The aerodynamic torque, represented by Ta, is defined as: Ta=1 2 Cp(λ, β) ωr ρπR2v3.(4) This simplified model is straightforward for controller design. However, using this model for controller design cannot provide high performance under varying conditions. To address this limitation, a robust controller with optimal performance is proposed in the next section. III. CONTROL DESIGN The actual dynamic model of FOWTs incorporates a high number of degrees of freedom, making the design of an effective MPPT controller challenging [3], [18]. To solve this problem, a reduced-order model is utilized to design the optimal control law. However, the optimal control law, which is extracted from the reduced-order model, cannot provide high performance across various conditions. To address this issue, the optimal control law combines with the STW algorithm for perturbation rejection. For the design of the control law, the nonlinear dynamic model of FOWT in the presence of uncertainties can be rewritten as follows (see (3)): ˙ωr=F(·)−G(·)u, (5) where the functions F(·)and G(·)are bounded but unknown. Assumption 1: The unknown functions F(·)and G(·) are bounded in the operation domain, which are defined as F(·) = f(·) + ∆f(·), where f(·)is known and ∆f(·)is an uncertain term but bounded. Similarly, G(·) = g(·) + ∆g(·), where g(·)>0is the known part and ∆g(·)is the unknown part but bounded, where |∆g(·)| g(·)≪1[19]. The nonlinear reduced-order dynamic model of a FOWT can be written as ˙ωr=f(·)+∆f(·) |{z } F(·) −g(·)1 + ∆g(·) g(·) |{z } G(·) u, (6) 554 Authorized licensed use limited to: University of Nantes. Downloaded on September 28,2025 at 09:56:04 UTC from IEEE Xplore. Restrictions apply. where f(·) = 1 Jt(Ta−Ktwr),g(·) = Ng Jt, and u=Tg. The control input of the system is defined as u=−1 g(·)(uopt +ustw),(7) here, the term uopt represents the optimal control law that ensures the desired performance, while ustw denotes a supertwisting control law designed to compensate for all uncertainties in the system. Taking (7) into account, (6) is rewritten as ˙ωr=uopt +ustw +δ(·),(8) where δ(·) = F(·) + ∆G g[ustw +uopt]. Recalling Assumption 1, it yields that the last term of the previous equation ∆G g[ustw +uopt]is small and can be viewed as an uncertainty. A. Speed reference The 5 MW wind turbine operates in Region II when the wind speed is between 3m/s and 11.4m/s [20]. In this region, the primary objective is to maximize power output. To achieve this goal, the generator speed is regulated by generator torque to reach the optimal TSR. As a result, maximum power is extracted from the turbine [1]. Based on this definition, the desired rotor speed is defined as ω∗ r=λ∗ Rv, (9) where ω∗ ris the desired rotor speed tracked by the controller to optimize power extraction, λ∗is the optimal value for TSR, which is considered as λ∗= 7.55 [20]. B. Optimal predictive control Inspired by [21], [22], the sliding variable is predicted for the next step and optimized over a finite horizon. To do so, the sliding variable σis considered as follows: σ=ωr−ω∗ r.(10) By using Taylor series, the prediction of the sliding variable for the next time step can be calculated as σ(t+h) = σ(t) + h˙σ(t) + h2 2! ¨σ(t) + ···+hk k!σ(k)(t), (11) where his a positive constant and is considered as the prediction time. The order of the Taylor series is defined as k=ϵ+γ, where ϵis the relative degree of the system and γis the control order. According to (10), the relative degree of the system is found to be one. Furthermore, the control order is restricted to zero [23]. Therefore, the first-order Taylor series expansion for the sliding variable is sufficient. By considering this definition, (11) reads as σ(t+h) = σ(t) + h˙σ(t).(12) By replacing (10) in (12), one has σ(t+h) = ωr−ω∗ r+h[ ˙ωr−˙ω∗ r].(13) Considering the nominal model of the system without uncertainties, and ustw = 0, (13) is written as σ(t+h) = ωr−ω∗ r+h[f(·) + uopt −˙ω∗ r].(14) The performance index as a function of the optimal control law is defined as follows: j[uopt(t)] = 1 2w1[σ(t+h)]2+1 2w2u2 opt(t),(15) where w1>0,w2≥0are the design parameters. By substituting (14) into (15) and taking the partial derivative of the performance index with respect to uopt, gives ∂j ∂uopt =hw1ωr−ω∗ r+hf(.) + uopt −˙ ω∗ r +w2[uopt(t)].(16) By considering ∂j ∂uopt = 0, the optimal control law can be defined as uopt(t) = −h h2+R(σ+h(f(.)−˙ω∗ r)) ,(17) where R=w2 w1≥0is defined as the control weight ratio. By defining ψ=1 1+Rh−2, the optimal control law is defined as uopt(t) = −ψ h(σ+h(f(.)−˙ω∗ r)) ,(18) where 0≤ψ≤1. The control law is derived in a closedform expression without the need for numerical optimization, which simplifies its implementation. Theorem 1: In the presence of the optimal control law (18), the sliding variable (10) remains within a small boundary layer. Proof: By considering ustw = 0 and the actual dynamic model of the system (8), the time derivative of the sliding variable is defined as ˙σ= ˙ωr−˙ω∗ r=uopt −˙ω∗ r+δ(·).(19) Applying the optimal control law, which is extracted from the nominal model in (19), gives ˙σ=−ψ hσ+ (δ(·)−f(·)) + (1 −ψ)(f(·)−˙ω∗ r).(20) The Lyapunov function for the evaluation of system stability in the presence of an optimal controller is considered as V=1 2σ2. In accordance with (20), the time derivative of Vis defined as ˙ V=−ψ hσ2+ [(δ(·)−f(·)) + (1 −ψ)(f(·)−˙ω∗ r)]σ. (21) Considering positive constants Q1and Q2such that |δ(·)− f(·)|< Q1and |f(·)−˙ω∗ r|< Q2. Therefore, (21) can be written as ˙ V≤ −ψ hσ2+ [Q1+ (1 −ψ)Q2]|σ|.(22) 555 Authorized licensed use limited to: University of Nantes. Downloaded on September 28,2025 at 09:56:04 UTC from IEEE Xplore. Restrictions apply. Based on the inequality ab ≤ca2+b2 4c, where a, b, c are positive real constants, the terms [Q1+ (1 −ψ)Q2]|σ|with c=ψ 4hare defined as ˙ V≤ −ψ hσ2+ψ 4hσ2+ [Q1+ (1 −ψ)Q2]2ψ h ≤ −3ψ 4hσ2+ [Q1+ (1 −ψ)Q2]2ψ h. (23) With reference to the Lyapunov function, (23) is rewritten as ˙ V≤ −3ψ 2hV+ [Q1+ (1 −ψ)Q2]2ψ h.(24) Using the comparison lemma (for more details, see [21]) and solving (24) gives V=1 2σ2≤V(0) −2h2 3ψ2[Q1+ (1 −ψ)Q2]2σ−3ψ 2ht +2h2 3ψ2[Q1+ (1 −ψ)Q2]2.(25) Following (25), the first term exponentially converges to zero with respect to time. As a result |σ| ≤ √2 √3[Q1+ (1 −ψ)Q2]h ψ.(26) According to (26), it can be concluded that the sliding variable remains within a small boundary layer, which is dominated by hand ψ.■ The control law (18) is derived from the reduced-order model. Due to discrepancies between the actual and nominal models, the system includes several sources of uncertainty. As a result, the control law does not have high performance in the presence of perturbations. To address this limitation, the STW algorithm is integrated with OPC to compensate for all system uncertainties. Implementing this approach allows the control law to achieve optimal performance and a satisfactory response. More details are provided in the next section. C. Super-twisting sliding mode control According to the (8), the time derivative of the sliding variable in the presence of ustw can be rewritten as follows: ˙σ= ˙ωr−˙ω∗ r=uopt +ustw −˙ω∗ r+δ(·).(27) By applying the optimal control law (18) in (27) gives ˙σ=−ψ hσ+ (δ(·)−f(·)) + (1 −ψ)(f(·)−˙ω∗ r) + ustw. (28) By defining new variables such that η(·) = |δ(·)−f(·)| and ζ(·) = |f(·)−˙ω∗ r|, gives ˙σ=−ψ hσ+η(·) + (1 −ψ)ζ(·) |{z } κ(·) +ustw,(29) where κ(·)is proved to be bounded in Section III-B. Assumption 2: The function κ(·)is Lipschitz continuous, and its derivative satisfies |˙κ(·)|<∆, where ∆is the upper bound of the derivative of uncertainties. The STW algorithm can be defined as [9] ustw =−k1|σ|1 2sign(σ) + z ˙z=−k2sign(σ),(30) where k1and k2are chosen as follows [24]: k2>∆, k1>1.41pk2+ ∆.(31) By integrating two control laws, which are defined in (18) and (30), the proposed control law has high robustness with optimal performance. To analyze the performance of the proposed control law, the next section assesses the system results. IV. RESULT This section discusses the numerical simulation of the PSTW using MATLAB/Simulink-OpenFAST. The NREL 5MW FOWT model from OpenFAST is selected for the system simulation, which includes 24 degrees of freedom. However, the control law is derived from (3), which has one degree of freedom. The structure of the closed-loop system for the FOWT is illustrated in Fig. 2. The stochastic wind speed profile is illustrated in Fig. 3. The mean wind speed is set to 5.5 m/s, and the peak wave height is 3.25 m. The closed-loop response of the system under stochastic wind and irregular wave conditions is compared to ROSCO [14]. The comparison results between PSTW and ROSCO are shown in Fig. 4. The results show that the PSTW controller, derived from (3), has performance similar to ROSCO. However, the key finding is that the proposed method generates more power, as shown in Fig. 4a. Additionally, Fig. 4b shows the performance of the PSTW in tracking ω∗, which is similar to ROSCO, except during the initial time. Fig. 4c shows the generator torque of PSTW for MPPT. According to this, it can be seen that the PSTW is lower than the generator torque in ROSCO method, thereby reducing fatigue loads on the platform of the wind turbine. The power coefficient, which indicates the amount of electrical power that can be extracted from the wind source, is illustrated in Fig. 4d. According to this figure, the power coefficients of both PSTW and ROSCO fluctuate around the optimal value. For a more accurate comparison, the root mean square (RMS) values of key parameters such as generator speed, generator torque, power, and power coefficient, along with the power variation (Var) defined as P|Pnext −Pcurrent|, are normalized based on ROSCO, as shown in Fig. 5. This figure indicates that the power variation has decreased by 11.8%, while the power generation has increased by 5.02%. Moreover, the RMS value of Cp, which is similar to ROSCO, indicates the ability of the proposed controller to maximize power generation. Fig. 6 presents the RMS values of platform motion. It can be seen that the PSTW has the ability to better manage roll, roll rate, yaw rate, and pitch rate, while its performance in yaw and pitch remains approximately the same. Fig. 7 displays the normalized RMS value of fairlead 556 Authorized licensed use limited to: University of Nantes. Downloaded on September 28,2025 at 09:56:04 UTC from IEEE Xplore. Restrictions apply. force (FA) and anchor force (AN). It is clear from the figure that the force acting on the mooring line (AN1–AN3, FA1–FA3) is approximately the same as in ROSCO strategy. Fig. 2: Illustration of the proposed control structure. 0 200 400 600 800 1000 Time (sec) 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 Wind speed [m/s] (a) Fig. 3: Turbulent wind speed. V. CONCLUSION This study proposed a robust control strategy for a FOWT in Region II, integrating OPC with STW controller. The objective was to maximize power extraction while reducing the fatigue load on the platform structure. Simulation results demonstrate that the PSTW outperforms ROSCO in terms of power generation. Additionally, the results indicate that the PSTW approach effectively reduces the fatigue load acting on the FOWT. ACKNOWLEDGEMENTS This project has received funding from the European Union’s Horizon Europe Framework Programme (HORIZON) under the GA n. 101120278 - DENSE. 0 200 400 600 800 1000 Time (sec) 0 500 1000 1500 Power [kw] Rosco PSTW 400 420 440 460 480 500 400 450 500 550 600 (a) 0 200 400 600 800 1000 Time (sec) 5 6 7 8 9 Rotor Speed [rpm] Rosco PSTW * (b) 0 200 400 600 800 1000 Time (sec) 0 2 4 6 8 10 12 14 16 18 Generator Torque [kN. m] Rosco PSTW (c) 0 200 400 600 800 Time (sec) 0.2 0.3 0.4 0.5 0.6 Cp Rosco PSTW 600 620 640 660 0.35 0.4 0.45 0.5 0.55 0.6 Rosco PSTW (d) Fig. 4: Comparison results between the proposed controller and ROSCO: (a) Power, (b) Rotor speed, (c) Generator torque, and (d) Power coefficient. 557 Authorized licensed use limited to: University of Nantes. Downloaded on September 28,2025 at 09:56:04 UTC from IEEE Xplore. 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