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 1Pwind=12ρπR2v∞3, where ρ is the air density, R is the rotor radius, and v∞ denotes the free-stream (upstream) wind speed . However, only a portion of this energy can be converted into mechanical power owing to fundamental aerodynamic limits, reported by Betz's law . 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 Pa and torque τa read as 2Pa=12ρπR2Cpλ,βvr3,3τa=Paωr, where ωr is the rotor speed, vr denotes the REWS, and the power coefficient Cp(λ,β) is a nonlinear function of the tip-speed ratio λ and the blade pitch angle β, as depicted in Fig. .

The tip-speed ratio is defined as 4λ=ωrRvr. For readability, the notation λ is used to represent λ(ωr,vr) unless explicitly stated otherwise.

The full-order FOWT model includes a high number of degrees of freedom (24) for blade and tower bending modes, platform pitch and surge motions, and mooring dynamics. While this comprehensive model captures detailed 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 ωr is given by 5ω˙r=1Jτa-ngτg+δ(t), where J is the equivalent rotational inertia, τa and τg denote the aerodynamic and generator torques respectively, ng is the gearbox ratio, and δ(⋅) captures unmodeled dynamics and disturbances.

The control vector u consists of the generator torque and the blade pitch angle u=[τgβ]⊤, the input used depending on the operating region . In Region II, control is primarily achieved by adjusting the generator torque τg, with blade pitch angle fixed at β=0. In contrast, control in Region III is dominated by blade pitch β actuation, with the generator torque τg held constant at its rated value; i.e., τg=τg*. The objective in this paper is to design an observation solution allowing for the estimation of the REWS vr from the measurement of the rotor speed ωr. The wind speed here is viewed as a time-varying parameter whose dynamics are unknown, which gives 6v˙r=fv(t), with fv(t) being an unknown bounded function. Using ωr=(λvr)/R from Eq. () together with Eqs. ()–(), the observer-based model reads as 7ω˙rv˙r=1JρπR3vr22λCp(λ,β)-ngτg0︸f(x,u)+δ(t)fv(t)︸Δ(t), with the objective being to estimate vr from the measurement of ωr in spite of Δ(⋅). The system can be written in observation-oriented form 8x˙=f(x,u)+Δ(t)y=h(x), where x=[ωrvr]⊤ is the state vector, h(x)=ωr is the measured output, and u=[τgβ]⊤.

Remark 1. The modeling of Cp(λ,β) has been carried out extensively . In this study, an exponential model is used that approximates the power coefficient and reads as 9Cp(λ,β)≈a(λ,β)β+b(λ,β), where the coefficients a and b are defined as 10a=-c0c2exp⁡(-c4λ1-1),b=c0(c1λ1-1-c3)exp⁡(-c4λ1-1), with λ1-1=1λ+c5β-c6β3+1 and c0=0.5, c1=73.5, c2=0.4, c3=5, c4=13.125, c5=0.08, and c6=0.0035.

This section details the numerical procedure for the analysis of the observability of the system in Eq. (). We denote the operating domain O⊂IR4 in which x=[ωrvr]⊤ 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.

Given the previous assumption, the observability analysis developed in the sequel is made for the system in Eq. () without perturbation; i.e. δ(t)=0. The generic observability analysis is defined as follows.

Definition 1 . Consider the system given by Eq. (), with x=[ωrvr]⊤ and u=[τgβ]⊤ evolving in the operating domain O, and suppose Assumption 2 is fulfilled. Consider that δ(t)=0. The system formulated in Eq. () is locally observable if12Φδ(t)=0=yy˙δ(t)=0=ωr1JρπR3vr22λ(ωr,vr)Cp(λ(ωr,vr),β)-ngτgis a state coordinate transformation; i.e. z=Φ(ωr,vr,β,τg) is invertible on O.

Checking whether Φδ(t)=0 is invertible is difficult in practice; therefore, the previous definition can be reformulated by the next equivalent one.

Definition 2. Consider the system given by Eq. (), with x=[ωrvr]⊤ and u=[τgβ]⊤ evolving in the operating domain O, and suppose Assumption 2 fulfilled. The system is generically observable on O if13det∂Φδ(t)=0∂x≠0with14Φδ(t)=0=yy˙δ(t)=0=ωr1JρπR3vr22λ(ωr,vr)Cp(λ(ωr,vr),β)-ngτg.

Observability condition. The system defined by Eq. () is locally observable if the following condition is fulfilled: 16∂y˙∂vr≠0⟺∂Cp∂vr⋅vr+3Cp(λ(ωr,vr),β)≠0.

In the simulation sections, the previous condition in Eq. () is numerically and experimentally evaluated in realistic operating conditions.

Consider the system defined by Eq. () that is locally observable. As a consequence, the transformation 17z=z1z2=yy˙=Φ(ωr,vr,β,τg) is a state coordinate transformation; i.e. the state vector x=[ωrvr]⊤ can be expressed as a function of z1, z2, τg, and β. That is, 18x=ωrvr=Φ-1(z1,z2,β,τg). Furthermore, from the state coordinate transformation given in Eq. (), one gets 19z˙1=z2z˙2=ω¨r=ddtρπR3vr22JλCp(λ,β)-ngJτ˙g+δ˙(t), which can be rewritten as 20z˙=z˙1z˙2=0100︸Az+0F(⋅), with (replacing ωr and β with the state coordinate transformation in Eq. ) 21F(⋅)=ddtρπR3vr22JλCp(λ,β)-ngJτ˙g+δ˙(t). It should be noted that the structure in Eq. () corresponds to a perturbed double-integrator system. Indeed, the coordinate transformation in Eq. () yields z1=y and z2=y˙ so that z˙1=z2. All model uncertainties and unmeasured effects appear as an additive term in the second equation, namely z˙2=F(⋅). Therefore, the linear part of the dynamics corresponds to the standard chain-of-integrators form.

Assumption 3. The time derivatives of the control inputs (i.e., β˙ and τ˙g) are bounded over the operating domain O. The function F(⋅), which involves v˙r and δ˙(t), is unknown but is assumed to be bounded over O.

Given that the function F(⋅) is not well-known, it cannot appear in the observer. A solution for the observation of the system defined by Eq. () is a robust one proposed by . Thus, consider the canonical form in Eq. () that is a perturbed uncertain double integrator. From , the supertwisting-based observer reads as 22z^˙1=z^2+Lϕ11/2a1z1-z^11/2sign(z1-z^1)︸γ1(z1,z^1)z^˙2=Lϕ1a2sign(z1-z^1)︸γ2(z1,z^1), where a1 and a2 are constant values that are fixed as suggested in , a1=1.5 and a2=1.1. Additionally, 23Lϕ1>F(z1,z2,β,β˙,τg,τ˙g,t) ensures z^=[z^1z^2]⊤→[z1z2]⊤ in a finite time in spite of the perturbations and uncertainties.

Theorem 1. Consider the system in Eq. () and Assumptions 1–3 fulfilled. Suppose that it is locally observable in the sense of Definition 1. Therefore, the system (with Φ defined by Eq. )24x^˙=ω^˙rv^˙r=f(x^,u)+∂Φ∂x^-1⋅Lϕ11/2a1|ωr-ω^r|1/2sign(ωr-ω^r)Lϕ1a2sign(ωr-ω^r)is an observer of Eqs. ()–() with a1 and a2 being constant values that are fixed as suggested in , a1=1.5 and a2=1.1, and the constant Lϕ1 such that25Lϕ1>ddtρπR3vr22JλCp(λ,β)-ngJτ˙g+δ˙(t).

A drawback of the proposed approach is that the term Lϕ1 is difficult to carefully tune because determining the bound of 31ddtρπR3vr22JλCp(λ,β)-ngJτ˙g+δ˙(t) is a very hard task that could give an overestimation and then induce chattering. A solution consists of using an adaptive version of the supertwisting-based observer , which allows online tuning of the observer thanks to the evaluation on only the estimation error of ωr. The principle is the following:

If the estimation error of ωr is large, it could be due to gains that are too small versus the uncertainty/perturbation effects. Then, gain adaptation law is defined in order to increase the gains of the observers.

The simulation environment integrates MATLAB/Simulink (2023a) for implementing the observers with OpenFAST , which simulates the high-fidelity aero-hydro-servo-elastic model of the FOWT. Each test is run for 800 s under identical wind and wave conditions, with a fixed sampling time of 0.0125 s. Although the observer design is based on the reduced-order model in Eq. (), all 24 degrees of freedom available in OpenFAST are activated to ensure a comprehensive evaluation under realistic conditions.

Realistic turbulent inflow wind fields are generated using TurbSim based on the IEC Kaimal turbulence model, with a mean wind speed of 18 m s⁻¹. A logarithmic mean wind profile is employed, resulting in vertical wind shear across the rotor disk. The inflow is prescribed as a full-field turbulent wind to OpenFAST such that the aerodynamic loads are computed using the spatially varying wind field. For analysis and validation purposes, REWS is considered, while the underlying aerodynamic response is influenced by the full-field inflow. Irregular wave conditions are modeled using the HydroDyn module . The incident wave field is prescribed as a stochastic irregular process with a significant wave height of 3.25 m. Hydrodynamic loads acting on the floating platform are computed using the built-in potential-flow formulation in HydroDyn, based on precomputed WAMIT data. This formulation accounts for linear wave-excitation forces, hydrostatic restoring forces, and radiation effects through convolution-based memory terms. The hydrodynamic model is fully coupled with the aero-servo-elastic dynamics in OpenFAST such that wave-induced platform motions interact with the aerodynamic response of the rotor. Both wind and wave conditions are illustrated in Fig. . The observer design parameters have been fine-tuned to achieve the best performance as follows: for the constant-gain SOSMO in Eq. (), the coefficients are set following to a1=1.5 and a2=1.1, and the gain is selected as Lϕ1=0.01; for the adaptive-gain observer (ASOSMO) in Eqs. ()–(), the design parameters are chosen as α=10-4 and ε=10-3, with initial values of k1(0)=0.1 and k2(0)=10-4.

Prior to evaluating the performances of observers, the system's observability is verified from Eq. (). Figure  illustrates the evolution of Eq. (). Its consistently nonzero behavior confirms that Φ is invertible, thereby ensuring observability of the nonlinear system under stochastic wind conditions.

With the observability condition satisfied in Fig. , the estimation performance of the three observers in Fig.  is subsequently assessed.

In this subsection, a Monte Carlo (MC) analysis is employed as a powerful tool to assess the sensitivity of the observers to initialization conditions and to quantify their convergence time under identical operating conditions .

In this experiment, the only quantity varied from run to run is the initial condition of the observers. All other components of the simulation, including the turbulent wind field, wave excitation, OpenFAST dynamics, and control inputs, are kept identical across all MC realizations. Consequently, any observed variation in transient behavior is solely attributable to different initial observer states. A set of N=100 simulations is generated by initializing the wind and rotor speed estimates within a uniform ±30% interval around their true values. Three initialization scenarios are examined: (i) rotor speed initialization error only, (ii) wind speed initialization error only, and (iii) simultaneous initialization errors in both wind speed and rotor speed.

To compare the observers consistently, a window convergence time is used. Convergence is declared when the worst-case estimation error across all MC runs remains within a prescribed tolerance band for a continuous duration of Thold=20 s. The thresholds are chosen as εω=0.2 rpm for rotor speed and εv=1.7 m s⁻¹ for wind speed. The convergence time for wind speed is defined as the earliest time at which every MC trajectory satisfies the inequality Δv≤εv for all t values in a window of length Thold. Rotor speed convergence is defined analogously. Requiring convergence over the entire simulation would be unnecessarily restrictive. Turbulent wind excitation, platform motion, and nonlinear aerodynamic effects naturally cause short-lived error fluctuations even after the estimator has converged. The windowed criterion avoids misclassifying such fluctuations as divergence and better reflects practical control requirements.

The resulting convergence times for SOSMO, ASOSMO, and CD–EKF across all scenarios are reported in Table  and illustrated in Fig. . This metric captures the earliest time after which all realizations remain within the prescribed bounds.

The error-band plots show the dispersion of estimation errors caused solely by changes in initial observer states. In all scenarios, all observers converge. Rotor speed errors settle quickly because ωr is directly measured, which justifies the smaller threshold εω. Wind speed estimation is more difficult because vr is unmeasured and its dynamics are unknown. As a result, wind speed error bands are wider, and the threshold εv must be larger.

From Fig. , it can be seen that the CD–EKF typically drives the estimation error toward zero more rapidly during the initial transient. However, when applying the windowed convergence criterion – which requires the estimates to remain within the prescribed bounds for a continuous duration – the SOSMO achieves the shortest convergence times in most scenarios. This difference arises because the adaptive law in the ASOSMO starts with a conservative gain that increases only after sufficient excitation, leading to a slower approach to steady-state accuracy. In contrast, the SOSMO and CD–EKF employ fixed gains or explicit covariance updates, allowing them to settle more quickly once the estimation error enters the tolerance band. Overall, while the CD–EKF is fast in the early transient, the SOSMO exhibits the most favorable worst-case convergence times under the robustness metric used here, whereas the ASOSMO consistently requires longer convergence due to its gain adaptation mechanism.

To assess the computational burden associated with each observer, the execution time of every estimator block using the MATLAB/Simulink Profiler (R2023a) is measured. Importantly, the reported run-time refers exclusively to the time required for the internal computations of each observer. All measurements are obtained under identical conditions (Sect. ), ensuring fair comparison. The CD–EKF exhibited the longest run-time (18 ms), followed by the ASOSMO (11 ms) and the SOSMO (9 ms). These results reflect the higher algorithmic complexity of the CD–EKF, as expected. The ASOSMO increases complexity slightly with its adaptive gain mechanism, in contrast to the constant gain used in the SOSMO (see Fig. ).

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 . The three datasets were selected under complementary operating regimes as the following:

Case 1 (Region III only). With vr∈[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.

Figures – illustrate the experimental results for the three test cases. In all scenarios, observers are able to estimate the wind speed despite the presence of wave-induced platform motions and unmodeled dynamics.

To provide a comprehensive assessment of estimation performance, multiple complementary metrics are considered in both the time and the frequency domains. Let x(k) denote a scalar component of the state vector, namely either the rotor speed ωr(k) or the REWS vr(k), x^(k) its estimate, and e(k)=x^(k)-x(k) the corresponding estimation error at sample k=1,…,N. The following statistical indicators are used in this section: (i) the root mean square error RMSE=1N∑k=1Ne(k)2; (ii) the mean estimation error (bias) μe=1N∑k=1Ne(k); (iii) the variance of the estimation error σe2=1N-1∑k=1N(e(k)-μe)2; and (iv) the mean square error (MSE), defined as MSE=E[e2]=σe2+μe2. Together, these metrics quantify overall accuracy, systematic bias, stochastic dispersion, and the combined effect of bias and variance, as summarized in Table .

Table  shows that all three observers achieve comparable levels of accuracy across the different operating regimes, with variations depending on the wind region and excitation level. In several cases, the sliding-mode observers exhibit reduced bias or variance, while in others the CD–EKF provides similar or slightly lower dispersion. These results indicate that the proposed observers achieve performance levels on par with the reference CD–EKF while relying on fundamentally different estimation principles.

The frequency-domain characteristics of the estimation error are examined through the power spectral density (PSD), shown in Fig. . The (one-sided) PSD of the estimation error, denoted by Se(f), describes how the error energy is distributed across frequencies and is formally defined as 35Se(f)=lim⁡T→∞1T∫0Te(t)e-j2πftdt2, where f denotes frequency. For real-valued signals, a one-sided representation is used so that all spectral energy is contained on the nonnegative frequency axis. In practice, Se(f) is estimated from the sampled error sequence e(k) using Welch’s method, as implemented in MATLAB via the function. Following , the frequency-weighted PSD f⋅Se(f) is reported to emphasize the contribution of different frequency bands to the overall estimation error.

For all three cases, the three estimators exhibit a similar low-pass behavior, with small differences primarily in the low- and mid-frequency ranges. Depending on the operating condition, the sliding-mode observers and the CD–EKF alternately show lower error energy in specific frequency bands, indicating that none of the approaches systematically dominates across the entire spectrum. Importantly, all estimators preserve a significant portion of the low-frequency content relevant for wind turbine control.

The probabilistic structure of the estimation error is further analyzed through empirical probability density functions (PDFs) (shown in Fig. ), Specifically, the empirical distribution of the estimation error e(k) is evaluated using a normalized histogram and compared with a Gaussian probability density function parameterized by the sample mean μe and variance σe2, given by 36pGauss(e)=12πσeexp⁡-(e-μe)22σe2. In all cases, the error distributions are approximately symmetric and well-approximated by Gaussian fits. Differences between observers mainly appear in the spread and centering of the distributions, consistent with the bias and variance values reported in Table .

Finally, Fig.  summarizes the normalized wind speed error metrics for the three observers across the three test cases. The radar representation provides a compact overview of performance trade-offs across RMSE, mean error, variance, and MSE. The results illustrate that the observers exhibit comparable overall performance, with each method showing relative strengths depending on the operating regime and metric considered.

Overall, the experimental results demonstrate that the proposed second-order sliding-mode observers achieve estimation performance comparable to that of the widely used CD–EKF while offering alternative robustness and tuning characteristics. The multi-metric analysis highlights that no single estimator uniformly outperforms the others across all conditions but rather that each approach provides a viable and reliable solution for REWS estimation in FOWTs under realistic experimental conditions.