Wind speed estimation using second-order sliding-mode observers: simulation and experimental validation on a floating offshore wind turbine

Wind speed estimation using second-order sliding-mode observers: simulation and experimental validation on a floating offshore wind turbine Moein Sarbandi1, Matis Viozelange1, Mohamed Assaad Hamida1, and Franck Plestan1 1Nantes Université, École Centrale Nantes, CNRS, LS2N, UMR 6004, Nantes, F-44000, France Correspondence: Moein Sarbandi ([email protected]) Abstract. Wind speed estimation is crucial for the control and performance optimization of floating offshore wind turbines (FOWTs). This paper introduces a robust estimation framework based on second-order sliding-mode observers (SOSMOs), developed in both constant-gain and adaptive versions. The observers are developed using a reduced-order dynamic model and validated in the OpenFAST simulation environment when all degrees of freedom are activated. Their performances are compared with the continuous-discrete extended Kalman filter (CD-EKF) used in the reference open-source controller (ROSCO).5 The proposed approach is assessed under stochastic wind/wave conditions through OpenFAST simulations and further validated experimentally using a scaled software-in-the-loop (SIL) setup. Simulation results indicate that the proposed observers perform comparably to the CD-EKF in terms of estimation accuracy, while offering robustness, simpler implementation, and reduced computational complexity. 1 Introduction10 The increasing global demand for electricity has necessitated the exploration of sustainable energy solutions, with offshore wind energy emerging as a key contributor. Floating offshore wind turbines (FOWTs) offer access to vast, underutilized wind resources located in deep waters, which account for approximately 80 % of the global offshore wind potential, as reported by (Global Wind Energy Council, 2022). Compared with fixed-bottom turbines, FOWTs benefit from stronger and more consistent winds; however, the floating structure introduces additional degrees of freedom, such as platform motions, which can cause15 negative damping and exacerbate power fluctuations. In extreme cases, this instability could lead to system failure. Consequently, conventional strategies developed for onshore wind turbines are not sufficiently effective for floating ones. Therefore, advanced estimation and monitoring approaches are required to support the stability and efficiency of FOWTs (McCoy et al., 2024; Stockhouse et al., 2024). The operation of wind turbines is typically divided into four regions based on the prevailing wind speed (Stockhouse et al.,20 2024). In Region I (below the cut-in wind speed), the turbine sits idle waiting for the wind speed to increase, as the available wind energy is insufficient to operate the turbine. In Region IV (above the cut-out wind speed), the turbine also stops operating to prevent potential damage. In contrast, power generation occurs in Region II and Region III, each employing distinct control strategies. In Region II, the objective is to maximize the power coefficient to optimize energy capture whereas in Region III, 1 https://doi.org/10.5194/wes-2025-206 Preprint. Discussion started: 21 November 2025 c Author(s) 2025. CC BY 4.0 License. the objective is to keep the power at its nominal value. Indeed, maintaining power at its rated level is essential to protect the25 turbine ensure its longevity and operational stability. In the operation of FOWTs, accurate information about wind speed is a fundamental requirement for control system design, real-time monitoring, and ensuring the safe and efficient performance of the turbine (Soltani et al., 2013). Wind speed information serves multiple critical functions depending on the control strategy employed. For example, in Region II, wind speed is used to compute the optimal rotor speed reference based on the desired tip-speed ratio whereas in Region III, it plays a30 central role in blade pitch control action (Stockhouse et al., 2024). Furthermore, wind speed measurements are a key input for feed-forward control algorithms. The quality of wind speed information thus has a direct impact on the overall performance and longevity of FOWTs. Different methods exist in the literature regarding wind speed measurement or estimation on FOWTs, including sensor-based, observer-based, and neural network-based approaches.35 LiDAR use. An advanced sensor-based method commonly used for wind speed estimation is light detection and ranging (LiDAR) (Shu et al., 2016). A considerable amount of literature is using LiDAR such as (He et al., 2025; Moldenhauer and Schmid, 2025; Li and Geng, 2024; Mahdizadeh et al., 2021). Despite the widespread adoption of LiDAR systems for wind speed measurement, some disadvantages have to be mentioned. One of the most apparent limitations is the cost and the maintenance demand of these systems (Jena and Rajendran, 2015). LiDAR devices, particularly those used in offshore and floating40 structures, are expensive to acquire and install, and their operation in harsh marine environments imposes high standards on longevity, autonomous operation, and regular maintenance to guarantee data quality. In addition, a primary technical limitation lies in the vulnerability of LiDAR measurements to motion-induced errors. Floating platform motions distort the LiDAR’s line of sight, introducing systematic biases and increased uncertainty in wind speed estimation (Gräfe et al., 2023). Such disturbances can lead to errors in real-time control. These limitations highlight the need for alternative or enhanced wind speed45 estimation techniques that are accurate, sensorless, and therefore more cost-effective. Neural-networks based methods. Alternatively, some recent studies rely on neural network-based methods for wind speed estimation and forecasting (Zhang et al., 2024; Sierra-García and Santos, 2021; Pan et al., 2022). These methods typically require an offline training phase using large datasets that must accurately represent the system’s operating conditions (Chen and Han, 2022). However, deep learning models behave like black boxes, offering limited interpretability and making it difficult to50 guarantee and formally prove stability or robustness of the closed-loop system. Additionally, the generalization of these models to unseen conditions remains a significant challenge. Kalman filter solution. Another widely adopted alternative is the Kalman filter (KF) and its variants. In (Soltani et al., 2013), both linear and nonlinear KFs are used for wind speed estimation. The simulation results also showed that the performance of nonlinear KF is better than the other at the transient state for the reason that the time response of nonlinear KF is much smaller55 than that of linear KF. KFs provide model-based state estimation by integrating a system’s dynamic equations with available sensor measurements. In wind turbine applications, they have been employed to estimate wind speed by combining turbine output data with linear aerodynamic models (Boukhezzar and Siguerdidjane, 2011). However, since wind turbine systems are inherently nonlinear, standard KFs do not perform well in dynamic operating conditions. To address this, extended Kalman 2 https://doi.org/10.5194/wes-2025-206 Preprint. Discussion started: 21 November 2025 c Author(s) 2025. CC BY 4.0 License. filters (EKFs) have been developed to handle nonlinearities more effectively. A wind speed estimation method based on EKF60 was introduced in (Song et al., 2017) to improve the efficiency of wind turbine operation. By integrating this algorithm with optimal tip-speed ratio tracking, the study demonstrated enhanced control of maximum power output. This paper reported that the proposed method could raise annual energy output by around 0.8 %. In (Hernández et al., 2014), the application of an EKF for wind speed estimation was demonstrated using real experimental data. This study is particularly noteworthy, as it validates the reliability of the EKF-based estimation method with real-world operating data.65 Furthermore, some studies, such as (Chen et al., 2025; Knudsen et al., 2011), use an indirect method for wind speed estimation. In these approaches, aerodynamic torque is first estimated, allowing then the estimation of wind speed. In (Kim et al., 2024), two methods of wind speed estimation are used and compared. The first one is based on the drive-train model using measured rotor speed, pitch angle, and generator torque as inputs, and the second one involves applying the estimated wind speed using a 3D look-up table and is compared with a continuous-discrete extended Kalman filter (CD-EKF).70 Despite their widespread use, EKF-based methods for estimating wind speed have several limitations that restrict their applicability in FOWTs. One key challenge lies in the tuning of process and measurement noise covariance matrices, which is often heuristic and lacks a systematic procedure. Improper tuning can lead to divergence (Chen et al., 2025; Song et al., 2017). Additionally, EKFs require approximation of the model around operating points, making them sensitive to variations in system dynamics and reducing their accuracy in highly nonlinear or time-varying conditions. This is particularly problematic75 in FOWTs, where platform motions introduce significant nonlinearity. Furthermore, the EKF also suffers from poor robustness to model mismatch and unmodeled dynamics, which are common in offshore environments. Finally, the formal proof of stability of the closed-loop including KF/EKF solutions is not trivial. These drawbacks highlight the need for more robust, model-insensitive alternatives for wind speed estimation. Observer based on sliding mode theory. Among observer-based approaches, sliding mode observers (SMOs) have attracted80 significant attention due to their inherent robustness to uncertainties and disturbances, which are particularly prevalent in offshore environments. Unlike KFs, which rely heavily on accurate statistical models and noise characteristics, SMOs exploit the system nonlinear structure and discontinuous logic to force estimation errors to converge in finite time (Ma et al., 2024). This makes them well-suited for FOWTs, where system dynamics are often poorly known and subject to unpredictable perturbations. Furthermore, recent studies have demonstrated the potential of higher-order sliding mode observers to achieve estimation85 even in the presence of uncertainties, while reducing the negative effect (chattering (Davila et al., 2005)) induced by discontinuity appearing in the correction term. For example, in (Barambones et al., 2021), the authors estimated aerodynamic torque to be used as a reference in calculating the turbine’s optimal rotor speed for maximizing wind power capture. Although numerous studies in the field of FOWTs assume perfect knowledge of wind speed, the current paper proposes the use of a second-order sliding mode observer (SOSMO) structure for wind speed estimation, applied to FOWTs. Furthermore,90 the proposed solution includes an adaptive second-order sliding mode observer (ASOSMO) that is a novelty in the context of wind turbines. Indeed, tuning SMOs/SOSMOs remains a persistent challenge, as it typically requires prior knowledge of the bounds of perturbations, the use of adaptation laws to evaluate the gain (as shown in (Plestan et al., 2010) for adaptive sliding mode control) allows to obtain very performant solutions requiring reduced tuning effort and limited knowledge of the model. 3 https://doi.org/10.5194/wes-2025-206 Preprint. Discussion started: 21 November 2025 c Author(s) 2025. CC BY 4.0 License. It is important to notice that, in the sequel, a formal analysis of observability is made to verify that the wind estimation can be95 evaluated from the single measurement of the rotor speed; it is rarely made in the context of (FO)WTs. In the sequel, the approach is validated through simulations using the National Renewable Energy Laboratory (NREL) 5 MW FOWT within the OpenFAST simulation framework (Jonkman et al., 2009), and its performance is compared with the CD-EKF implemented in the reference open-source controller (ROSCO) (Abbas et al., 2022). It is also evaluated on a software-in-theloop located in LHEEA lab, Nantes, France and dedicated to reduced-scale model of a FOWT.100 The main contributions and original points of the present paper are summarized as follows –A numerical method for observability analysis is proposed by supposing that the estimated variable is the wind speed and the single measured variable is the rotor speed. –Then, observers based on sliding mode theory are proposed for wind speed estimation, from a single measurement that is the rotor speed, and are compared to a CD-EKF used in ROSCO. –Two SOSMOs are designed: a constant-gain structure and an adaptive-gain one (allowing dynamic tuning of the gain without any information on the system uncertainties and perturbations). –The observers are developed using a reduced-order model but validated within the OpenFAST simulator when all degrees of freedom of the FOWT are activated. –Experimental validation is conducted using a scaled software-in-the-loop (SIL) test setup replicating realistic wind and wave conditions. This paper is organized as follows: Sect. 2 presents the reduced-order dynamic model of the FOWT; Sect. 3 develops the proposed SMOs, including observability analysis, observer formulation, and adaptive gain design; Sect. 4 reports simulation105 studies and comparative evaluations with the CD-EKF under different wind conditions; Sect. 5 describes the experimental validation using a SIL setup; and Sect. 6 concludes by summarizing the main findings and outlining directions for future research. 2 Observation-oriented model The present study focuses on the NREL 5-MW FOWT OC4, which is supported by a semi-submersible floating platform and110 simulated using OpenFAST (Jonkman et al., 2009). 4 https://doi.org/10.5194/wes-2025-206 Preprint. Discussion started: 21 November 2025 c Author(s) 2025. CC BY 4.0 License. 2.1 Aerodynamic and drive-train modeling Wind turbines harness the kinetic energy of the wind to generate mechanical power through aerodynamic interaction between the wind and the rotating blades. The theoretical power available in the wind stream is given by Pwind =1 2ρπR2v3(1)115 where ρis the air density, Ris the rotor radius, and vis the wind speed upstream of the rotor. However, only a portion of this energy can be converted into mechanical power owing to fundamental aerodynamic limits, such as Betz’s law (Manwell et al., 2009). The efficiency of this conversion is described by the power coefficient Cp, which quantifies the fraction of the wind’s kinetic energy that is captured by the rotor. As a consequence, aerodynamic power Paand torque τaread as Pa=1 2ρπR2Cp(λ,β)v3(2)120 τa=Pa ω(3) where ωis the rotor speed, and the power coefficient Cp(λ,β)is a nonlinear function of the tip-speed ratio λand the blade pitch angle β, as depicted in Fig. 1. The tip-speed ratio is defined as λ=ωR v(4) Figure 1. Power coefficient Cpwith respect to tip-speed ratio λand blade pitch angle β(Sarbandi et al., 2025). 125 2.2 Reduced-order observation-oriented model of a FOWT The full-order FOWT model includes a high number of degrees of freedom (24), taking into account for blade and tower bending modes, platform pitch and surge motions, and mooring dynamics. While this comprehensive model captures detailed 5 https://doi.org/10.5194/wes-2025-206 Preprint. Discussion started: 21 November 2025 c Author(s) 2025. CC BY 4.0 License. turbine behavior, its complexity makes it unsuitable for control design and real-time estimation. Therefore, a reduced-order model is used for observer development. The equation of motion for the rotor speed ωis given by130 ˙ω=1 J(τa−ngτg) + δ(t),(5) where Jis the equivalent rotational inertia, τaand τgrespectively denote the aerodynamic and generator torques, ngis the gearbox ratio, and δ(·)captures unmodeled dynamics and disturbances. The control vector consists of the generator torque and the blade pitch angle u= [τgβ]⊤, the used input depending on the operating region (Aslmostafa et al., 2025). In Region II, control is primarily achieved by adjusting the generator torque τg,135 with blade pitch angle fixed at β= 0. In contrast, Region III control is dominated by blade pitch βactuation, with the generator torque τgheld constant at its rated value, i.e.,τg=τ∗ g. The objective in this paper is to design an observation solution allowing the estimation of the wind speed vfrom the measurement of the rotor speed ω. The wind speed is here viewed as a time-varying parameter whose dynamics is unknown that gives 140 ˙v=fv(t),(6) fv(t)being an unknown bounded function. To summarize, the observer-based model reads as   ˙ω ˙v =   1 JρπR3v2 2λCp(λ,β)−ngτg 0    | {z } F(x,u) +  δ(t) fv(t)  | {z } ∆(t) (7) the objective being to estimate vfrom the measurement of ωin spite of ∆(·). The system can be written in observation-oriented form145 ˙x=F(x,u) + ∆(t) y=H(x) (8) with x= [ω v]⊤the state vector, H(x) = ωthe measured output and u= [τgβ]⊤. Remark 1. The modeling of Cp(λ,β)has been intensively made (Castillo et al., 2023). In this study, an exponential model is used that approximates the power coefficient and reads as Cp(λ,β)≈a(λ,β)β+b(λ,β),(9)150 where the coefficients aand bare defined as a=−c0c2exp(−c4λ−1 1), b=c0(c1λ−1 1−c3)exp(−c4λ−1 1),(10) 6 https://doi.org/10.5194/wes-2025-206 Preprint. Discussion started: 21 November 2025 c Author(s) 2025. CC BY 4.0 License. with λ−1 1=1 λ+c5β−c6 β3+ 1 155 and c0= 0.5,c1= 73.5,c2= 0.4,c3= 5,c4= 13.125,c5= 0.08,c6= 0.0035. Remark 2. In the case of FOWT, platform motions and mooring dynamics are not taken into account in Eq. (8). The proposed estimation methods in the paper are developed on this simplified system and can then be applied to floating (or not) offshore (or not) wind turbines. In the sequel, the observation solutions for estimating the wind speed vare validated by supposing that only the rotor speed ωis measured, and through two separate steps: first using the full-order OpenFAST simulator, and then160 the experimental setup. This two-stage evaluation emphasizes the observer’s robustness versus simplification of the model. 3 Supertwisting-based observer Ideally, to achieve high performance in the state/parameter estimation, having an accurate model of the system is a key point. However, modeling the exact dynamics of FOWTs is highly challenging. Therefore, it is crucial to develop estimation methods that are sufficiently robust against system perturbations and modeling uncertainties. In this section, a robust observer based165 on the supertwisting algorithm (Levant, 1993) is presented for estimating wind speed by using rotor speed measurement; this observer is based on the reduced-order model presented in the previous section. Additionally, the novelty of the proposed estimation algorithm lies in the fact that the gains are dynamically adapted, allowing an easier tuning. Assumption 1. The wind speed v(t)is assumed to be unmeasured and dynamically unknown. Nevertheless, v(t)remains bounded and positive for all t≥0, such that170 0< v(t)< Vmax ∀t≥0,(11) where Vmax >0is a constant that represents an upper bound within the turbine’s operational regions. Given the uncertain nature of the system described in Eq. (8), an observer inspired from (Shtessel et al., 2014) is proposed. However, the first step is to analyze the observability of Eq. (8) in the operational domain, possibly detecting singularities. 3.1 Observability analysis175 This section is detailing the numerical procedure for the analysis of the observability of the system in Eq. (8). Denote the operating domain O ⊂ IR4in which x= [ω v]⊤and u= [τgβ]⊤are physically evolving. All the results detailed in the rest of the paper are verified only in this domain. Assumption 2. The perturbation term δ(t)and its derivative are bounded. Furthermore, δ(t)has no influence on the system observability.180 Given the previous assumption, the observability analysis developed in the sequel is made for the system in Eq. (8) without perturbation, i.e. δ(t)=0. The generic observability analysis is defined as follows. Definition 1. (Krener and Respondek, 1985) Consider the system given by Eq. (8) with x= [ω v]⊤and u= [τgβ]⊤evolving in the operating domain Oand suppose Assumption 2 is fulfilled. Consider that δ(t)=0. The system formulated in Eq. (8) is 7 https://doi.org/10.5194/wes-2025-206 Preprint. Discussion started: 21 November 2025 c Author(s) 2025. CC BY 4.0 License. locally observable if185 Φδ(t)=0 =   y ˙y   δ(t)=0 =    ω 1 JρπR3v2 2λ(ω,v)Cp(λ(ω,v),β)−ngτg   (12) is a state coordinate transformation, i.e. z= Φ(ω,v,β,τg)is invertible on O. The previous property is evaluated if Φδ(t)=0 can be inverted that is a very hard task in practice. It is why the previous definition can be reformulated by the next equivalent one. Definition 2. Consider the system given by Eq. (8) with x= [ω v]⊤and u= [τgβ]⊤evolving in the operating domain Oand190 suppose Assumption 2 fulfilled. The system is said generically observable on Oif Det∂Φδ(t)=0 ∂x = 0 (13) with Φδ(t)=0 =   y ˙y   δ(t)=0 =    ω 1 JρπR3v2 2λ(ω,v)Cp(λ(ω,v),β)−ngτg   (14) 195 Applying Eq. (13) to Eq. (14), it is obvious that the first line of the Jacobian ∂Φ/∂x equals [1 0]. Observability condition. The system defined by Eq. (8) is locally observable if the following condition is fulfilled ∂˙y ∂v = 0 →∂Cp ∂v ·v+ 3Cp(λ(ω,v),β)= 0 (15) In the simulation sections, the previous condition in Eq. (15) will be numerically and experimentally evaluated in realistic operating conditions.200 3.2 Observer design Consider the system defined by Eq. (8) that is locally observable. As a consequence, the transformation z=  z1 z2  =  y ˙y = Φ(ω,v,β,τg)(16) is a state coordinate one, i.e. the state vector [ω v]⊤can be expressed as a function of z1,z2,τgand β,i.e. x=  ω v = Φ−1(z1,z2,β,τg)(17)205 Furthermore, from the state coordinate transformation given in Eq. (16), one gets ˙z1=z2 ˙z2= ¨ω=d dtρπR3v2 2Jλ Cp(λ,β)−ng J˙τg+˙ δ(t) (18) 8 https://doi.org/10.5194/wes-2025-206 Preprint. Discussion started: 21 November 2025 c Author(s) 2025. CC BY 4.0 License. that can be rewritten as ˙z=  ˙z1 ˙z2  =  0 1 0 0   | {z } A z+  0 F(·) (19) with (replacing ωand βwith the state coordinate transformation in Eq. (17))210 F(·) = d dtρπR3v2 2Jλ Cp(λ,β)−ng J˙τg+˙ δ(t)(20) Assumption 3. The time derivatives of the control inputs (i.e., ˙ βand ˙τg) are bounded over the operating domain O. The function F(·), which involves ˙vand ˙ δ(t), is unknown but is assumed to be bounded over O. Given that the function F(·)is not well-known, it can not appear in the observer. A solution for the observation of the system defined by Eq. (19) is a robust one proposed by (Levant, 2003). Thus, consider the canonical form Eq. (19) that is a perturbed215 uncertain double integrator. From (Levant, 2003), the supertwisting-based observer reading as ˙ ˆz1= ˆz2+L1/2 ϕ1a1|z1−ˆz1|1/2sign(z1−ˆz1) | {z } γ1(z1,ˆz1) ˙ ˆz2=Lϕ1a2sign(z1−ˆz1) | {z } γ2(z1,ˆz1) (21) with a1and a2constant values fixed as suggested in (Levant, 2003), a1= 1.5and a2= 1.1and Lϕ1>F(z1,z2,β, ˙ β,τg,˙τg,t)(22) ensures ˆz= [ˆz1ˆz2]⊤→[z1z2]⊤in a finite time in spite of the perturbations and uncertainties.220 Theorem 1. Consider the system in Eq. (8) and Assumptions 1-4 fulfilled. Suppose that it is locally observable in the sense of Definition 1. So, the system (with Φdefined by Eq. (16)) ˙ ˆx=  ˙ ˆω ˙ ˆv =F(ˆx,u) + ∂Φ ∂ˆx−1 · L1/2 ϕ1a1|ω−ˆω|1/2sign(ω−ˆω) Lϕ1a2sign(ω−ˆω) (23) is an observer of Eqs. (7)-(8) with a1and a2constant values fixed as suggested in (Levant, 2003), a1= 1.5and a2= 1.1, and the constant Lϕ1such that225 Lϕ1> d dtρπR3v2 2Jλ Cp(λ,β)−ng J˙τg+˙ δ(t)(24) Proof of Theorem 1. The observer represented by Eq. (21) has been designed based on the nominal system of Eq. (19) without the uncertain/perturbed term F(·)in its definition; the gain tuning is based on the bound of F(·). By a similar way, the writing 9 https://doi.org/10.5194/wes-2025-206 Preprint. Discussion started: 21 November 2025 c Author(s) 2025. CC BY 4.0 License. 18 11 9 CD-EKF ASMO SMO 0 5 10 15 20 Execution time [ms] Figure 9. Execution time (in milliseconds) of different observers measured in MATLAB simulations under identical conditions. 5 Experimental results The proposed observers have been experimentally validated on a SIL setup at École Centrale Nantes, France. The experimental platform consists of a 1/32-scale semisubmersible FOWT, based on the OC4-DeepCwind concept, deployed in the wave tank of the LHEEA Laboratory (LHEEA Laboratory, 2025).320 As shown in Fig. 10, aerodynamic loads are emulated in real time by a six-fan thrust generator mounted on the turbine’s tower, driven by aerodynamic inputs from the OpenFAST simulation. Simultaneously, wave generators in the tank reproduce hydrodynamic conditions, ensuring realistic environmental forcing. A six degrees of freedom actuation system replaces the drive-train, enabling closed-loop testing under dynamic conditions (Ferahtia et al., 2025). Table 2 outlines selected technical characteristics of both the numerical emulator (OpenFAST) and the scaled experimental325 setup implemented in the laboratory. The table highlights key parameters of the reference full-scale FOWT alongside those used in the physical test environment. 5.1 Test conditions and scenarios Three test scenarios have been conducted to evaluate the performance and robustness of the proposed observers under various wind and wave conditions, as reported in Table 3. The three datasets were selected under complementary operating regimes as330 – Case 1 (Region III only). With v∈[11.41,25.37] m s−1, the turbine operates fully above rated only, highlighting estimator behavior under above-rated operation and strong pitch activity. – Case 2 (Transition Region II↔III). With v∈[8.20,14.42] m s−1, it covers the transition region, testing robustness to region switching. – Case 3 (Region II/III). With v∈[8.43,19.24] m s−1, serving as a general verification across variable conditions from335 low wind speed to high wind speed. 16 https://doi.org/10.5194/wes-2025-206 Preprint. Discussion started: 21 November 2025 c Author(s) 2025. CC BY 4.0 License. Figure 10. Experimental SIL test setup of the 5 MW 1/32-scale semisubmersible OC4 FOWT at École Centrale Nantes (Aslmostafa et al., 2024). Table 2. Key specifications of the experimental setup, including both full-scale and corresponding 1/32-scale parameters. Parameter Real : model scale Unit Floater type Semi-submersiblea– Nominal powerb5 MW Rotor diameterb126 m Platform height 30 : 0.9375 m Tower height 70.528 : 2.204 m Tower mass 2.5×105: 8 kg Rotor thrust 8.0×105: 24.4 N Test tank size 50 ×30 ×5m Notes. aBased on the OC4-DeepCwind platform under IEA Wind Task 30 (Robertson et al., 2014) bEmulated via software-in-the-loop (SIL). The three cases collectively address various conditions to ensure a balanced comparison of CD-EKF, SOSMO, and ASOSMO against the actual wind speed. 17 https://doi.org/10.5194/wes-2025-206 Preprint. Discussion started: 21 November 2025 c Author(s) 2025. CC BY 4.0 License. Table 3. Test conditions for experimental validation, including wind and wave ranges and region classification based on turbine operating regimes. Wind speed (m s−1) Wave elevation (m) Regiona Test case Min Max Min Max II III Case 1 11.41 25.37 −4.62 5.54 – ✓ Case 2 8.20 14.42 −2.48 2.89 ✓ ✓ Case 3 8.43 19.24 −2.35 2.92 ✓ ✓ Note. aRegion II: 3≤v < 11.4m s−1; Region III: 11.4≤v≤25 m s−1(Jonkman et al., 2009). 5.2 Results and discussion Figures 11–13 illustrate the experimental results for the three test cases. In all scenarios, observers are able to estimate the wind340 speed despite the presence of wave-induced platform motions and unmodeled dynamics. 11 11.5 12 12.5 13 Rotor speed [rpm] 0 100 200 300 400 500 600 700 800 Time [s] 15 20 25 30 Wind speed [m·s-1] Figure 11. Experimental results for case 1: rotor speed ω, wind speed v, and their estimated values under turbulent wind and wave conditions. Evaluation of root mean square of estimation error v−ˆvis reported in Table 4. In all three test cases, both sliding mode observers outperformed the CD-EKF. Specifically, SOSMO achieves RMSE reductions of 6.4 %, 18.5 %, and 13.6 % in Cases 1 through 3, respectively, compared with CD-EKF. The adaptive version (ASOSMO) also demonstrates improvements in cases 1, 2 and 3, with RMSE reductions of 1.64 %, 15.7 % and 10.6 %, respectively.345 To facilitate comparison across the three cases, Fig. 14 presents the normalized RMSE values for each observer, with normalization performed with respect to CD-EKF. 18 https://doi.org/10.5194/wes-2025-206 Preprint. Discussion started: 21 November 2025 c Author(s) 2025. CC BY 4.0 License. 10.5 11 11.5 12 12.5 Rotor speed [rpm] 0 100 200 300 400 500 600 700 800 Time [s] 8 10 12 14 Wind speed [m·s-1] Figure 12. Experimental results for case 2: rotor speed ω, wind speed v, and their estimated values under turbulent wind and wave conditions. 10.5 11 11.5 12 12.5 Rotor speed [rpm] 0 100 200 300 400 500 600 700 800 Time [s] 10 15 20 Wind speed [m·s-1] Figure 13. Experimental results for case 3: rotor speed ω, wind speed v, and their estimated values under turbulent wind and wave conditions. Table 4. Evaluation of observer performance based on root mean square of v−ˆvof wind speed estimates in dynamic experimental conditions. Method Case 1 Case 2 Case 3 CD-EKF 1.88 1.21 1.72 SOSMO 1.76 0.99 1.48 ASOSMO 1.85 1.02 1.54 Note. RMSE computed on v−ˆv; lower is better. Overall, the experimental findings align well with the simulation results. The proposed SOSMOs exhibit robustness, low computational cost, and reduced tuning complexity, combined with estimation accuracy, that make SOSMO and ASOSMO attractive for practical deployment in FOWT control frameworks.350 19 https://doi.org/10.5194/wes-2025-206 Preprint. Discussion started: 21 November 2025 c Author(s) 2025. CC BY 4.0 License. 111 0.936 0.818 0.86 0.984 0.843 0.895 Case 1 Case 2 Case 3 0 0.2 0.4 0.6 0.8 1 1.2 Normalized wind speed RMSE CD-EKF SOSMO ASOSMO Figure 14. Normalized wind speed RMSE comparison across three estimators (CD-EKF, SOSMO, and ASOSMO) for the three experimental test cases. Each bar represents the RMSE normalized w.r.t. the CD-EKF solution. 6 Conclusions This paper proposed robust wind speed estimation methods based on a second-order sliding mode observer: a constant-gain second-order sliding mode observer (SOSMO) and its adaptive version (ASOSMO). The estimation framework is built on a reduced-order nonlinear model and is validated not only on the OpenFAST simulator but also through experimental tests where all degrees of freedom are activated.355 The two observers are evaluated against the standard continuous–discrete extended Kalman filter (CD-EKF), and they demonstrate accurate tracking of wind dynamics. Unlike CD-EKF, the SOSMO-based methods not only eliminate the need for tuning noise covariance matrices but also avoid the linearization of system dynamics, thereby reducing implementation complexity and improving reliability under modeling uncertainties. Moreover, the adaptive version allows for very limited knowledge of the model.360 To summarize, the proposed observers provide a simple yet effective solution for accurate wind speed estimation and can be integrated into advanced control strategies. This integration promises improved system stability and reduced fatigue loads, contributing significantly to the performance of FOWTs. These results mark an initial step toward a comprehensive robust estimation and control framework. As future work, a fully integrated adaptive observer/controller scheme will be developed to further improve the overall365 performance and resilience of FOWTs. Appendix A: Continuous-discrete extended Kalman filter The nonlinear state-space form of a FOWT can be written as: ˙x=f(x,u) + w(t), yk=h(xk,uk) + vk(tk),(A1)370 20 https://doi.org/10.5194/wes-2025-206 Preprint. Discussion started: 21 November 2025 c Author(s) 2025. CC BY 4.0 License. where x=hωrvtvmi⊤is the state-space vector, vtand vmdenote the turbulent and mean components of the wind speed, respectively, and the control input is u=β. Additionally w(t)and vkare continuous white noise and discrete-time white noise, respectively, which are defined as w(t)∼ N(0,Q), vk∼ N(0,Rk),(A2)375 where Qis the process noise covariance matrix and Rkis the measurement noise covariance matrix. The extended Kalman filter for a continuous–discrete nonlinear system generally consists of two main steps: (i) time update (prediction) and (ii) measurement update (correction), as described in (Abbas et al., 2022; Knudsen et al., 2011) as •Step 1: Time update ˆx+ 0=E(x0),(A3)380 ˆ P+ 0=E(x0−ˆx0)(x0−ˆx0)⊤,(A4) ˙ ˆx(t) = f(ˆxk−1|k−1,uk),(A5) ˙ P(t)=F(t)Pk|k+Pk|kF⊤(t) + Qk−Kk−1RmK⊤ k−1,(A6) where F(t) = ∂f ∂xˆ xk−1|k−1,uk is the Jacobian matrix f(x,u). additionally ˆ Pis the estimation error covariance. In the above equations, the term ˆx+represents the estimate of xkusing the information from yk, while ˆx−denotes the estimate of xkusing385 the measurement from yk−1. •Step2: Measurement update Kk=Pk|k−1H⊤ kHkPk|k−1H⊤ k+Rm−1,(A7) ˆ xk|k=ˆ xk|k−1+Kk(yk−h(ˆxk|k−1)),(A8) Pk|k= (I−KkHk)Pk|k−1,(A9)390 where Hk=∂h ∂xˆ xk|k−1is the Jacobian matrix of h(xk), and Kkis the Kalman gain. By defining Q= diag1×10−5,πv3 mt2 i L,4 600, Rm= 0.02,(A10) and using the relation ˆv=vt+vm, the estimation of the wind speed is calculated. 21 https://doi.org/10.5194/wes-2025-206 Preprint. Discussion started: 21 November 2025 c Author(s) 2025. CC BY 4.0 License. Appendix B: Nomenclature Abbreviations395 ASOSMO Adaptive second-order sliding mode observer CD-EKF Continuous–discrete extended Kalman filter EKF Extended Kalman filter FOWT Floating offshore wind turbine KF Kalman filter NREL National Renewable Energy Laboratory RMSE Root mean square error ROSCO Reference open-source controller SIL Software-in-the-loop SMO Sliding mode observer SOSMO Second-order sliding mode observer STW Supertwisting Symbols and parameters α,ε Design parameters of the adaptive law [–] βBlade pitch angle [rad] λTip-speed ratio [–] ω, ˆωReal and estimated rotor speed [rads−1] ρAir density [kg m−3] τaAerodynamic torque [N m] τg,τ∗ gGenerator torque, rated value [N m] CpPower coefficient (function of λ,β) [–] JTotal rotational inertia [kg m2] k1,k2Adaptive observer gains [–] ngGearbox ratio [–] PaAerodynamic power extracted by the rotor [W] Pwind Theoretical wind power [W] RRotor radius [m] uControl input vector v, ˆvTrue and estimated wind speed [m s−1] zObserver coordinate vector 22 https://doi.org/10.5194/wes-2025-206 Preprint. Discussion started: 21 November 2025 c Author(s) 2025. CC BY 4.0 License. Author contributions. Moein Sarbandi: Writing – original draft, supervision, validation. Matis Viozelange: Methodology, writing. Mohamed Assaad Hamida: Methodology, conceptualization. Franck Plestan: Conceptualization, writing – review and editing, supervision,400 project administration. Competing interests. The contact author has declared that neither he nor any of the co-authors have any competing interests. Acknowledgements. This project has received funding from European Union’s Framework Programme for Research and Innovation Europe Horizon Europe (HORIZON) Marie Skłodowska-Curie Actions Doctoral Networks (MSCA-DN) under the Grant Agreement No. 101120278 - DENSE. The authors also thank the experimental team and researchers at LHEEA / École Centrale Nantes – CNRS, who contributed to405 the experimental campaign and corresponding database developed within the ANR project CREATIF (ANR-20-CE05-0039), France. 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