# Nonlinear control strategies for a floating wind turbine with PMSG in Region 2: A comparative study based on the OpenFAST platform

Ehsan Aslmostafa, Mohamed Assaad Hamida, Franck Plestan

► To cite this version:

Ehsan Aslmostafa, Mohamed Assaad Hamida, Franck Plestan. Nonlinear control strategies for a floating wind turbine with PMSG in Region 2: A comparative study based on the OpenFAST platform. Ocean Engineering, 2024, 300, pp.117507. 10.1016/j.oceaneng.2024.117507. hal-05114032

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Research paper

# Nonlinear control strategies for a floating wind turbine with PMSG in Region 2: A comparative study based on the OpenFAST platform

Ehsan Aslmostafa, Mohamed Assaad Hamida *, Franck Plestan

Nantes Université, École Centrale Nantes, CNRS, LS2N, UMR 6004 F-44000, Nantes, France

<figure><img src="image_3.png" /></figure>

## ARTICLE INFO

**Keywords:**

Nonlinear control

Maximum power point tracking

Floating wind turbine

Permanent magnetic synchronous generator

## ABSTRACT

This study examines new control strategies for floating offshore wind turbines (FOWTs) equipped with permanent magnet synchronous generators (PMSG), and operating in Region 2. The aim is to maximize power production while minimizing fatigue loads within wind speed limits. This is achieved by maintaining optimal tip speed ratio (TSR) using diverse control strategies. Control design based on the OpenFAST model is not trivial due to FOWTs' dynamics. Adaptive gain versions of super-twisting (STW) control methods are proposed to address this, requiring minimal system knowledge while simplifying controller gains tuning. These controllers also adapt to parameter changes, ensuring robustness. A comparative analysis of STW-based control versus baseline control demonstrates its effectiveness in maximizing power production under numerous different conditions.

## 1. Introduction

Wind energy has seen significant growth globally, with over 732 GW of installed wind turbine capacity, according to the International Renewable Energy Agency (IRENA). Offshore wind energy, particularly through FOWTs, promises increased efficiency, offering solutions to challenges posed by onshore turbines (Butterfield et al., 2007). A FOWT is a type of offshore wind energy system that utilizes a floating structure to support the turbine and tower, rather than being fixed to the seabed like traditional offshore wind turbines. FOWTs offer the advantage of tapping into wind resources in deeper waters where fixed-bottom turbines are not feasible (Zhang and Plestan, 2021b). According to the International Energy Agency (IEA), offshore wind energy has the potential to generate more than 18,000 TWh/year, about 18 times the global electricity demand Díaz and Soares (2020). The FOWTs are more complex than fixed-bottom turbines because of face larger wave and wind loads; furthermore, this class of wind turbine offers a large number of degrees of freedom. One of the challenges in designing a FOWT is selecting a stable floating platform that withstands the marine environment. Options include tension leg platforms (TLPs), spar buoys, semi-submersibles, and hybrid designs. Factors like water depth and turbine size determine the best choice. Additionally, it requires erosion-resistant turbine blades and saltwater protection for electrical components. Researchers are using realistic modeling tools, such as FAST (Jonkman et al., 2009), to optimize FOWT performances. FAST, developed by NREL, is a comprehensive simulation tool integrating fatigue, aerodynamics, structures, and turbulence modules. OpenFAST,

an open-source version offers greater flexibility for customization and is commonly used in research (Jonkman et al., 2018).

Wind turbines are used in four regions defined by wind speed. Region 1, with wind speed below the cut-in threshold, has the turbine idle due to insufficient wind. In Region 2, wind speeds (from 3 to 11.4 m/s for the studied case (Jonkman et al., 2009)) allow power production, but not at full capacity, aiming for power optimization. Region 3 is the full-load area where power is regulated through the pitch angle controller. Finally, Region 4 shuts down the turbine when speeds exceed the cut-off threshold, the objective being to avoid damage from excessive wind (Kumar and Chatterjee, 2016; Mousa et al., 2021). The aim of this paper is to investigate the power production in Region 2. In this region, various maximum power point tracking (MPPT) algorithms, such as the TSR control method have been proposed to achieve the objective of maximizing power output from wind turbines (Kumar and Chatterjee, 2016). TSR-based control continuously measures wind and blade rotational speed to maintain an optimal rotational speed using generator torque. This approach allows the turbine to operate near its maximum power coefficient in order to extract the maximum possible energy from the wind. However, achieving an efficient MPPT in FOWTs is challenging due to their complex dynamics. Therefore, advanced control strategies are being developed to optimize power generation in this context.

Wind turbines equipped with PMSG are becoming popular day by day. A PMSG is popular due to its high power density, its compactness,

<footnote>* Corresponding author.
E-mail address: mohamed.hamida@ec-nantes.fr (M.A. Hamida).</footnote>
# Nomenclature

## General terms

<table><tr><td>FOWT</td><td>Floating offshore wind turbine</td></tr><tr><td>PMSG</td><td>Permanent magnet synchronous generators</td></tr><tr><td>TSR</td><td>Tip speed ratio</td></tr><tr><td>TLP</td><td>Tension leg platforms</td></tr><tr><td>DOF</td><td>Degree of freedom</td></tr><tr><td>MPPT</td><td>Maximum power point tracking</td></tr><tr><td>SOSMC</td><td>Second-order sliding mode control algorithm</td></tr><tr><td>STW</td><td>Super-twisting control</td></tr><tr><td>ASTW</td><td>Adaptive super-twisting control</td></tr><tr><td>ASTR</td><td>Adaptive super-twisting with a reduced set of parameters</td></tr><tr><td>STAS</td><td>Self-tuning adaptive super-twisting</td></tr><tr><td>DEL</td><td>Damage equivalent load</td></tr><tr><td>RMS</td><td>Root mean square</td></tr></table>

## Machine Model

<table><tr><td>v<sub>d</sub>, v<sub>q</sub></td><td>Stator voltages in d-q coordinates</td></tr><tr><td>i<sub>d</sub>, i<sub>q</sub></td><td>Stator currents in d-q coordinates</td></tr><tr><td>i<sup>*</sup><sub>d</sub>, i<sup>*</sup><sub>q</sub></td><td>Stator reference voltages in d-q coordinates</td></tr><tr><td>L<sub>d</sub>, L<sub>q</sub></td><td>Stator inductances in d-q coordinates</td></tr><tr><td>R<sub>s</sub></td><td>Stator resistance</td></tr><tr><td>P</td><td>Pole pairs number</td></tr><tr><td>φ<sub>f</sub></td><td>Permanent magnet flux</td></tr><tr><td>λ, λ<sup>*</sup></td><td>Tip speed ratio, optimal tip speed ratio</td></tr><tr><td>C<sup>*</sup><sub>p</sub></td><td>Optimal power coefficient</td></tr><tr><td>T<sub>g</sub></td><td>Generator torque and electromagnetic torque</td></tr><tr><td>T<sub>em</sub></td><td>Electromagnetic torque</td></tr><tr><td>T<sub>a</sub></td><td>Aerodynamic torque</td></tr><tr><td>ρ</td><td>Air density</td></tr><tr><td>R</td><td>Blade radius</td></tr><tr><td>ω, ω<sup>*</sup></td><td>Rotor speed, reference rotor speed</td></tr><tr><td>J<sub>t</sub></td><td>Rotor inertia</td></tr><tr><td>K<sub>t</sub></td><td>External damping</td></tr><tr><td>φ</td><td>Platform pitch angle</td></tr><tr><td>ν</td><td>Wind speed</td></tr></table>

and its favorable torque-to-inertia ratio (Pan and Shao, 2020). Its advantages over other AC generators, like its robustness and lower weight, and specifically, the way it is engineered to operate at low rotational speeds, make this type of generator well-suited for low wind profiles (Wu et al., 2011). The design relies on a large number of poles, which enables easy adaptation of angular speed to the wind speed (Pareta and Sen, 2014).

In Region 2, various FOWT control methods are implemented, including linear solutions based on the NREL control strategy from the late 90s (Stol and Balas, 2003; Jonkman et al., 2009). However, linear methods, such as PID controllers, struggle with the system's nonlinear behavior when the conditions are changing (Boukhezzar and Siguerdidjane, 2010). The nonlinear nature of FOWTs requires extensive modeling across multiple operating points, each of them representing an approximate linear system (Hamida et al., 2022), inducing a significant effort both for modeling and control tuning. In contrast, nonlinear control strategies simplify tuning and optimize performances in Region 2 by efficiently managing wind speed fluctuations and turbulence (Corradini et al., 2015; Golnary and Moradi, 2018).

These nonlinear methods outperform linear controllers in robustness and handling uncertainties. As a nonlinear control technique, feedback linearization has been used (Wang and Chiang, 2016) in wind turbines (Region 2). A suitable feedback control law allows the application of linear control techniques. But the main drawback is due to the fact that in such a class of control, the system model should be available.

In Kahla et al. (2021), a feedback linearization approach ensures an optimal controller in different operating conditions, focusing on maintaining the TSR at its optimal value. In Mousa et al. (2019), an MPC-based control is studied with an estimated wind speed (without an anemometer), reducing costs and improving efficiency. However, the performance of MPC relies on accurate models of the wind turbine system, which can be challenging to obtain for FOWTs due to their complex dynamics and uncertain operating conditions.

The nonlinear backstepping technique (Aslmostafa et al., 2022) is also used in the control of FOWT. In Hamida et al. (2022), a novel backstepping-based speed control to achieve MPPT is presented, using the reluctant part of the electromagnetic torque via the maximum torque per ampere (MTPA) strategy. In Benakcha et al. (2019) a comparison between PI controllers and backstepping control is made, illustrating the latter's higher performance in transient modes. Additionally, adaptive backstepping control for a doubly fed induction generator (DFIG) is proposed by Rezaei (2018). In practical scenarios, such as FOWTs with uncertain dynamics, backstepping control might not be optimal, as detailed in the sequel. Hence, there is a demand for a controller that can manage uncertainties and disturbances.

Research is ongoing for robust nonlinear algorithms like sliding mode control (SMC) and higher-order sliding mode control (HOSMC). SMC is a robust control strategy that has been already applied to FOWTs (Pan et al., 2021). It offers excellent tracking performance and robustness against disturbances. In Errami et al. (2015), SMC is compared with standard PI controllers, demonstrating that SMC enhances power extraction for PMSG-equipped FOWTs with minimal impact on the power network. However, SMC may suffer from chattering, leading to increased load on turbine components.

The current paper proposes nonlinear control strategies based on the super-twisting (STW) algorithm, a subclass of higher-order sliding mode (HOSM) control, to overcome the limitations of the mentioned control methods. STW algorithms provide robustness against uncertainties, finite-time convergence, and reduction of chattering (Levant, 2003). In such a control strategy, very reduced information on the system model is required; this reduces the tuning effort while guaranteeing a high level of performance. Nonetheless, limited literature has focused on the development of such control strategies. In Matraji et al. (2018), the efficacy of a second-order sliding mode control algorithm (SOSMC) is demonstrated. This study develops a PMSG-based wind emulator to evaluate the performance of integrating SOSMC with the STW algorithm in a wind energy conversion system.

From a control design point of view, gain tuning of SMC/SOSMC is always a challenging topic because it requires evaluating the bounds of perturbation and uncertainties. That is why several approaches have been proposed to address the gain overestimation by using dynamical gains coming from adaptation laws (Shtessel et al., 2012; Taleb et al., 2021; Mirzaei et al., 2022). These algorithms are particularly well-suited for addressing FOWT control challenges, as they can online adapt the gains without any knowledge of uncertainties and disturbances. In Zholtayev et al. (2022), a single-mass model for wind turbines has been proposed, using an adaptive gain-based STW control to achieve MPPT in Region 2. The STW algorithms with various adaptive gain strategies have been discussed in Zhang and Plestan (2021a), Gutierrez et al. (2022), the performances being improved versus gain scheduling PI (GSPI) control for Region 3. It has been shown that the STW-based control successfully tracks the desired pitch angle, reducing both rotor speed and pitch actuator fluctuations, even under turbulent wind conditions. This result highlights the potential of STW-based control
methods for mitigating the adverse effects of turbulence on FOWT performance.

To the best of the authors' knowledge, no prior research has explored the implementation and comparison of various adaptive versions of STW control applied to an FOWT for low-wind conditions (Region 2). To summarize with respect to existing literature, the main contributions of the current paper are

* development of adaptive STW-based control laws for a FOWT under uncertain parameters and limited dynamic model knowledge in Region 2;

* comparative study thanks to the OpenFAST simulator to evaluate proposed STW-based control strategies under realistic conditions.

The paper is arranged as follows. Section 2 presents the system description and presents linear and nonlinear modeling approaches for FOWT. In Section 3, nonlinear controllers and their application to the FOWT are discussed in detail. Section 4 displays the results obtained with OpenFAST/SIMULINK co-simulations, along with an analysis of these outcomes. Finally, Section 5 offers a conclusion, summarizing the key findings and contributions of the paper.

## 2. System description and modeling

### 2.1. Description of FOWT

The focus of this study is made on the NREL 5MW OC3-Hywind wind turbine, which is supported by a spar-buoy platform and modeled using the OpenFAST platform (Jonkman et al., 2009). Furthermore, it is assumed that the wind turbine under consideration is equipped with a PMSG, which will be discussed later. The work is conducted in Region 2, the region associated with maximizing power over a range of wind speeds between 3 m/s and 11.4 m/s (Jonkman et al., 2009). Subsequently, in order to maintain the tip speed ratio (TSR) at its optimal value, the controller design forces the rotor speed $\omega$ to track a reference value $\omega^*$, thereby achieving MPPT. Additional information regarding the wind turbine can be found in Jonkman et al. (2009).

### 2.2. Power capture system

A wind turbine captures energy from the interaction between its blades and the wind, converting it into mechanical energy. When the wind turbine is positioned facing the wind, the mechanical power reads as Guenoune et al. (2017)

$$P = \frac{1}{2} C_p(\lambda, \beta) \rho \pi R^2 v^3 \quad (1)$$

with $\beta$ the blade pitch angle, $R$ the blade radius, $\rho$ the air density. $v$ is the wind speed whereas $\lambda$ is the TSR, defined as

$$\lambda = \frac{\omega}{v} R. \quad (2)$$

The power coefficient $C_p$ characterizes the effectiveness of wind energy conversion into mechanical energy. This coefficient can be determined through real-world experiments or accurate simulations. Various approaches can describe it, such as look-up tables (Odgaard et al., 2013) or polynomial fitting (Raach et al., 2014) that gives

$$C_p(\lambda, \beta) = c_1(c_2a - c_3\beta - c_4)e^{-c_5a} \quad (3)$$

with $a = (\frac{1}{0.08\beta+\lambda} - \frac{0.035}{1+\beta^3})$. From (3), it yields that the power coefficient $C_p$ depends on the TSR and the pitch angle $\beta$ (Guenoune et al., 2017) such that the coefficients $c_1 - c_5$ have the following fixed values (Guenoune et al., 2017)

$$c_1 = 0.22, c_2 = 116, c_3 = 0.4, c_4 = 12.5, c_5 = 21.$$

Fig. 1 displays the variation of the power coefficient $C_p$ with respect to the tip speed ratio $\lambda$ for different values of blade pitch angle. As detailed in Section 3 (control design), the aim is to generate the maximal

<figure><img src="image_4.png" /><figcaption>Fig. 1. Power coefficient $C_p$ versus the tip-speed ratio $\lambda$ for different values of pitch angle $\beta$ for NREL 5MW OC3-Hywind (Sarkar et al., 2020).</figcaption></figure>

electrical power while minimizing potential damage to the FOWT. In order to accomplish this goal, the pitch angle $\beta$ and the TSR $\lambda$ must be fine-tuned. As observed in Fig. 1, the maximum value of $C_p$ is achieved when $\beta = 0$. As a consequence, in the sequel, since the aim is to extract the maximum power, the blade pitch angle $\beta$ is set to zero. Under this condition, according to Jonkman et al. (2009), the optimal values of power coefficient and TSR for the NREL 5MW OC3-Hywind wind turbine are $C_p^* = 0.485$ and $\lambda^* = 7.55$. Additionally, the aerodynamic torque of the turbine reads as (Guenoune et al., 2017)

$$T_a = \frac{C_p(\lambda, \beta)}{\omega} \rho \pi R^3 v^2 \quad (4)$$

### 2.3. Drive train system

The drive train is used to transmit mechanical torque from the turbine to the generator (Burton and Chichester, 2001). Various drive train models exist, including single-mass, two-mass, etc. The single-mass wind turbine model is commonly used (Vidal et al., 2012) due to its simplicity and its ability to capture the drive train dynamics. Assuming a rigid rotor and neglecting shaft friction, the drive train dynamic is described as follows

$$J_t \dot{\omega} = T_a - T_g - K_t \omega \quad (5)$$

where $K_t$ is the turbine's total external damping (N m rad$^{-1}$ s), $J_t$ is the turbine total inertia (kg m$^2$), $T_a$ is the aerodynamic torque (N m), and $T_g$ is the generator torque (N m). Notice that this model is quite simple because it does not take into account some parameters or degrees of freedom (like those related to the mooring lines or the platform motions). As a consequence, it is necessary to design a controller to get higher performance for the closed-loop system. Such controllers are discussed in the next section.

### 2.4. Control-oriented modeling of the FOWT

The FOWT model used in the OpenFAST platform incorporates numerous degrees of freedom (DOFs), including platform rotation and translation, blade flap-wise/edge-wise bending mode, and tower fore-aft/side-to-side bending mode. Furthermore, it includes equations described by PDEs that are not adapted for control design. Thus, for control design, a simplified linear state model can be derived from OpenFAST. It is proposed around an operating point depending on wind and rotor speeds, and the number of DOFs can be fixed with respect to the control objectives. In the current study, the control objective will be
to force the rotor speed $\omega$ to track a reference value $\omega^*$ while limiting the tower oscillations. In this case, a two degrees of freedom (2 DOF) linear model can be obtained from OpenFAST for Region 2. The chosen degrees of freedom are the rotor rotational speed and platform pitch angle, and the taken linear model reads as (Abbas et al., 2020)

$$ \Delta\dot{x} = A\Delta x + B_{T_g}\Delta T_g + B_d\Delta M \quad (6) $$

with $\Delta x = [\Delta\varphi \ \Delta\dot{\varphi} \ \Delta\omega]^T$ showing respectively variations of pitch ($\varphi$), pitch rate ($\dot{\varphi}$), and rotor speed ($\omega$) versus operating point [$\varphi_0 \ 0 \ \omega_0]^T$. The control input $\Delta T_g$ is the variation of the generation torque $T_g$. $\Delta M$ the wind's effect, while matrices $A, B, T_g$, and $B_d$ are calculated by OpenFAST. Based on a discussion in Zhang and Plestan (2021a), Gutierrez et al. (2022), changing wind speed leads to linear models with different $A, B, T_g$, and $B_d$ matrices. Considering the operating domain of Region 2, the FOWT model reads as

$$ \dot{x} = A(t)x + B_{T_g}T_g + B_d(t)M \quad (7) $$

with $x = [\varphi \ \dot{\varphi} \ \omega]^T$ and $M$ the uncertainties/external perturbations input (including waves, wind, and system parameter variations). The matrices $A(t)$ and $B_d(t)$ change over time while matrix $B_{T_g}$ is constant and equals $B_{T_g} = [0 \ 0 \ -\frac{1}{J_t}]$ as it is discussed in Abbas et al. (2020).

## 2.5. PMSG modeling

The PMSG is considered due to its high efficiency, its reduced maintenance, and its active weight reduction by removing the gearbox. Its model is nonlinear due to the relationship between currents and speed. The model of the PMSG is written in the $(d, q)$-reference frame using the following equations (Guenoune et al., 2017)

$$ \begin{cases} i_d = -\frac{R_s}{L_d} i_d + \frac{pL_q}{L_d} \omega i_q + \frac{1}{L_d} v_d \\ i_q = -\frac{pL_d}{L_q} \omega i_d - \frac{R_s}{L_q} i_q - \frac{p\phi_f}{L_q} \omega + \frac{1}{L_q} v_q. \end{cases} \quad (8) $$

where $i_d$ and $i_q$ represent the currents along the $d$- and $q$-axes, while $v_d$ and $v_q$ denote the respective voltages and $\omega$ represents the rotor speed when the gearbox ratio is 1. Consequently, both the rotor and generator operate at the same speed. Model parameters include $R_s$, the stator resistance, $L_d$ and $L_q$, the $d$- and $q$-axis inductances, $p$ the pole pairs number, and $\phi_f$ the permanent magnet flux. It is assumed that losses are insignificant and all the input mechanical power is being efficiently converted into electromagnetic power without losses; as a consequence, the generator torque $T_g$ and electromagnetic torque $T_{em}$ are supposed to be equal, i.e., $T_g \approx T_{em}$, the electromagnetic torque reading as follows (Fitzgerald et al., 2003)

$$ T_g = T_{em} = p\left((L_d - L_q)i_d i_q + \phi_f i_q\right). \quad (9) $$

**Remark 1.** Setting the direct stator current to zero ensures the absence of stator current along the $d$-axis, consequently eliminating the reluctance torque. It leads to reducing the fatigue on the mechanical shaft (Bunjongjit and Kumsuwan, 2013).

## 2.6. An integrated model of the FOWT and PMSG

The entire system model including the wind turbine and a PMSG, is viewed as a nonlinear multiple-input multiple-output (MIMO) system. By taking the state vector $x = [\varphi \ \dot{\varphi} \ \omega \ i_d \ i_q]^T$ and recalling $x = [\varphi \ \dot{\varphi} \ \omega]^T$, the complete system is expressed as

$$ \dot{x} = \begin{bmatrix} A(t)x + B_{T_g}T_g(i_d, i_q) + B_d(t)M \\ -\frac{R_s}{L_d}i_d + \frac{pL_q}{L_d}\omega i_q \\ -\frac{pL_d}{L_q}\omega i_d - \frac{R_s}{L_q}i_q - \frac{p\phi_f}{L_q}\omega \end{bmatrix} + \begin{bmatrix} 0_{3*1} & 0_{3*1} & 0_{3*1} \\ 0 & \frac{1}{L_d} & 0 \\ 0 & 0 & \frac{1}{L_q} \end{bmatrix} \begin{bmatrix} 0 \\ v_d \\ v_q \end{bmatrix} \quad (10) $$

It is important to note that the system dynamics are affected by both the perturbation term $M$ and electrical parameters like inductance and resistance which can significantly vary with respect to temperature changes and magnetic saturation. Notice that the link between mechanical and electrical parts is made by $T_g$ that appears in $x$-dynamics and is a function of $i_d$ and $i_q$.

## 3. Control design

In Region 2, i.e., for wind speed between 3 m/s and 11.4 m/s in the current study, the goal is the maximization of power while minimizing platform fluctuations via generator torque control. To maximize power, the turbine operates near the optimal power efficiency $C_p^*(\lambda, \beta)$, typically with a 0° blade pitch angle for maximum lift (Kumar and Chatterjee, 2016) (see Fig. 1). It shows that the maximum power is achieved with a 0° blade pitch angle, and increasing the pitch angle would reduce the power extraction. The control system uses stator voltages $v_d$ and $v_q$ as control inputs. In this section, firstly, a backstepping control (Hamida et al., 2022), independent of specific operating points is developed, based on the model of (10). This controller is evaluated on the FOWT simulator provided by OpenFAST. Secondly, an adaptive STW-based controller is designed; no accurate model is required. The controller is also evaluated on OpenFAST. Notice that the controller's evaluation is made by activating the full DOFs of OpenFAST. The use of OpenFAST emphasizes the need for a robust controller versus unmodeled dynamics, uncertainties, and neglected DOFs. Furthermore, it discusses why a backstepping controller cannot struggle well to ensure MPPT and power extraction. The block diagram of the discussed approaches is shown in Fig. 2. In practice, the mechanical dynamics of wind turbines are so complex, that accurately modeling them under a control design model is challenging. To address the potential issues arising from unmodeled dynamics and perturbations, a robust control approach is proposed: an adaptive version of STW control. This control strategy not only prevents catastrophic failures in wind turbines but also ensures a satisfactory response, as demonstrated in Section 4.

### 3.1. Reference speed calculation

The reference rotor speed is determined as a function of the platform pitch velocity and the wind speed, resulting in the desired rotor speed $\omega^*$ as follows (inspired by Zhang and Plestan, 2021a)

$$ \omega^* = \frac{\lambda^*}{R} v - k\dot{\varphi}. \quad (11) $$

By this way, the tracking of $\omega^*$ will allow to optimize the power extraction while reducing the tower oscillation. Indeed, the reference rotor speed is designed to balance the rotor speed and platform pitching, where $\lambda^*$ denotes the optimal tip speed ratio and $k$ is a positive constant. When the platform pitches forward, the platform pitch velocity $\dot{\varphi}$ becomes negative, leading to an increase of the reference rotor speed $\omega^*$. As per the implemented control strategy, the aerodynamic torque also increases, causing an increase in the rotor speed. This results in an increase in the aerodynamic thrust, which prevents the platform from pitching forward. As a result, the platform pitch velocity $\dot{\varphi}$ decreases, leading the rotor speed reference to converge to its optimal value (Namik and Stol, 2013). If there is no platform pitch velocity, i.e., $\dot{\varphi} = 0$, then the reference is optimal and equal to $\frac{\lambda^*}{R} v$.

### 3.2. Backstepping control design

The backstepping approach (Krstic et al., 1995) is designed by splitting the system into subsystems, and recursively designing a controller for each of these subsystems. Designing a backstepping controller for a FOWT with a PMSG in Region 2 consists of dividing the system into the electrical (8) and mechanical subsystems (5). Define the rotor speed tracking error $\sigma_\omega$ as

$$ \sigma_\omega = \omega - \omega^*. \quad (12) $$
<figure><img src="image_5.png" /><figcaption>Fig. 2. Block diagram of control strategies in OpenFAST platform for a FOWT equipped with a PMSG.</figcaption></figure>

To achieve the desired rotor speed $\omega^*$, a virtual control signal $i_q^*$ is calculated and the control input $v_q$ will be used in an inner loop to ensure that the current $i_q$ tracks the virtual control signal $i_q^*$ that has as consequence to achieve the desired rotor speed. To achieve this objective, the time derivative of (12) reads as

$$ \dot{\sigma}_{\omega} = -\frac{p}{J_t} (L_d - L_q) i_d i_q^* - \frac{f_v}{J_t} \omega - \frac{p}{J_t} \phi_f i_q^* + \frac{1}{J_t} T_a - \dot{\omega}^* \quad (13) $$

To study the stability of the speed loop, a candidate Lyapunov function is considered

$$ V_{\omega} = \frac{1}{2} \sigma_{\omega}^{2}. \quad (14) $$

Its time derivative of $\dot{V}_{\omega}$ reads as

$$ \dot{V}_{\omega} = \sigma_{\omega} \left( -\frac{p}{J_t} (L_d - L_q) i_d i_q^* - \frac{f_v}{J_t} \omega - \frac{p}{J_t} \phi_f i_q^* + \frac{1}{J_t} T_a - \dot{\omega}^* \right) \quad (15) $$

To guarantee the stability of the speed loop, define the virtual control $i_q^*$ as

$$ i_q^* = \frac{J_t}{p\phi_f + p(L_d - L_q)i_d} \left( k_{\omega}\sigma_{\omega} - \frac{f_v}{J_t}\omega + \frac{1}{J_t}T_a - \dot{\omega}^* \right) \quad (16) $$

By replacing $i_q^*$ defined by (16) into (15), one gets $\dot{V}_{\omega} = -k_{\omega}\sigma_{\omega}^2$, with $k_{\omega} > 0$ and it value is given in Table 3. Hence, speed tracking is ensured $\sigma_{\omega} = \omega - \omega^* \to 0$. In the next step, it is necessary to define $v_q$ ensuring $i_q \to i_q^*$ that will give $\omega \to \omega^*$; in this case, MPPT achieves. In order to achieve this, the q-axis current tracking error $\sigma_{i_q}$ is defined as follows

$$ \sigma_{i_q} = i_q - i_q^*. \quad (17) $$

To analyze the stability of the current $i_q$ loop, a Lyapunov candidate function is selected as

$$ V_q = V_{\omega} + \frac{1}{2}\sigma_{i_q}^2. \quad (18) $$

The time derivative of $V_q$ reads as

$$ \begin{aligned} \dot{V}_q &= -k_{\omega}\sigma_{\omega}^2 + \sigma_{i_q} \left( i_q - i_q^* \right) \\ &= -k_{\omega}\sigma_{\omega}^2 + \sigma_{i_q} \left( -\frac{R_s}{L_q}i_q - \frac{pL_d}{L_q}\omega i_d - \frac{p\phi_f}{L_q}\omega + \frac{1}{L_q}v_q - i_q^* \right) \end{aligned} \quad (19) $$

Defining $v_q$ as

$$ v_q = L_q \left( -k_q \sigma_{i_q} + \frac{R_s}{L_q} i_q + \frac{pL_d}{L_q} \omega i_d + \frac{p\phi_f}{L_q} \omega + i_q^* \right) \quad (20) $$

with $k_q$ being a positive constant. One gets

$$ \dot{V}_q = -k_{\omega}\sigma_{\omega}^2 - k_q\sigma_{i_q}^2 \leq -\bar{K}_q V_q < 0 \quad (21) $$

where $\bar{K}_q = \min(k_{\omega}, k_q)$ and the value for $k_{\omega}$ is given in Table 3. Next, a control should be designed for d-axis side to force its current tracking

a desired value; usually, this value is taken to be zero i.e., $i_d^* = 0$ as discussed in Remark 1 (Section 2.5). Define the $i_d$ tracking error as

$$ \sigma_{i_d} = i_d - i_d^*. \quad (22) $$

One gets

$$ \dot{\sigma}_{i_d} = i_d = -\frac{R_s}{L_d} i_d + \frac{pL_q}{L_d} \omega i_q + \frac{1}{L_d} v_d \quad (23) $$

To ensure the convergence of $\sigma_{i_d}$ towards 0, the following candidate Lyapunov function is considered

$$ V_d = \frac{1}{2}\sigma_{i_d}^2. \quad (24) $$

Its time derivative reads as

$$ \dot{V}_d = \sigma_{i_d} \left( -\frac{R_s}{L_d} i_d + \frac{pL_q}{L_d} \omega i_q + \frac{1}{L_d} v_d \right). \quad (25) $$

The control law $v_d$ is chosen as

$$ v_d = L_d(-k_d\sigma_{i_d} + \frac{R_s}{L_d}i_d - p\frac{L_q}{L_d}\omega i_q). \quad (26) $$

with $k_d$ as a positive gain that gives $\dot{V}_d = -k_d\sigma_{i_d}^2 < 0$. Then, $\sigma_{i_d} = i_d - i_d^* \to 0$. The value for $k_d$ is given in Table 3. It is obvious that the backstepping design is made based on a simplified model of the wind turbine system (5): it may not encapsulate all real-world dynamics and uncertainties of a FOWT. As Fig. 3 shows, significant tracking errors can undermine MPPT performance when applied to the realistic OpenFAST model with full DOFs (24 DOFs) whereas it gives accurate results when applied to the reduced model. Indeed, the backstepping control is designed on available model (single-mass model); however, when it is implemented on the OpenFAST platform with a complex set of wind turbine dynamics, it may struggle with uncertain dynamics. The OpenFAST platform includes a detailed model of the mechanical system, including dynamics of the rotor, blades, drive train, and tower. The backstepping control algorithm is designed on a simplified model of the wind turbine system (5) that does not include the complete dynamics and behaviors. This fact has significant impact on the performance of the wind turbine control system and therefore must be accounted for in the design and evaluation of control strategies. The control strategies developed in the sequel, based on an adaptive sliding mode theory and requiring very few knowledge of the model, allow to mitigate the effects of uncertain dynamics and improve the overall performance of the wind turbine system.

### 3.3. Recalls on super-twisting-based control design

Designing a controller for a FOWT based on a single-mass model (5), without considering the precise and complex system dynamics – including aerodynamics, hydrodynamics, and control part dynamics
<figure><img src="image_6.png" /><figcaption>Fig. 3. Top. Backstepping technique on single-mass model. Rotor speed (rpm) versus time (sec). Bottom. Backstepping technique on OpenFAST model. Rotor speed (rpm) versus time (sec).</figcaption></figure>

– may not yield the desired performance. For accurate and efficient results based on the designed control, it is crucial to consider the complete and complex dynamics of an FOWT. However, mathematically expressing such dynamics can be intricate. Therefore, it is essential to utilize a comprehensive simulation tool that thoroughly captures the dynamics of an FOWT. In this context, we employ the OpenFAST platform to model and simulate the complete FOWT dynamics. This aids us in achieving an effective and reliable response to the proposed control strategy. However, due to the complex dynamics that evolve within OpenFAST, it is advisable to pursue a control design that necessitates minimal knowledge of the system’s model. In our case, only the system’s relative degree should be available. This strategy avoids dependence on a specific model. To implement this approach, we utilize measurements from OpenFAST, such as the measured rotor speed. To pursue this goal, in this section, an STW-based control with different adaptation laws is designed to verify system robustness against uncertainties and variations in system parameters. Implementing this approach enables us to study the system’s overall behavior in a more realistic situation. Take the following uncertain system (Shtessel et al., 2012)<fnref n="1" />

$$\dot{s} = a(\cdot) + b(\cdot)u \qquad (27)$$

with a as an unknown but bounded function and b also as an unknown bounded positive one. In the context of SMC theory, s is the sliding variable defined such that, when s = 0, the control objectives will be fulfilled. The objective of the control u is to guarantee s = 0 in a finite time even if a is not known. Furthermore, when second-order sliding mode controllers (as STW) are used, the control guarantees in a finite time s = $\dot{s}$ = 0.

**Assumption 1.** The function $a(\cdot)$ is unknown and bounded in the operating domain that can be written as Shtessel et al. (2012)

$$a(\cdot) = \delta_a(\cdot) + a_n(\cdot) \qquad (28)$$

with $\delta_a$ the uncertain (unknown) part such that $\delta a_m \le \delta_a \le \delta a_M$ and $a_n$ the nominal (known) part. Similarly, within the operational domain, the function $b(\cdot)$ represents an unknown, bounded, and positive constant function, which can be expressed as follows

$$b(\cdot) = \delta_b(\cdot) + b_n(\cdot) \qquad (29)$$

with $\delta_b(\cdot)$ the unknown part, while $b_n(\cdot) > 0$ denotes the nominal part of b, such that Shtessel et al. (2012)

$$\frac{|\delta_b(\cdot)|}{b_n(\cdot)} \ll 1. \qquad (30)$$

Considering the following control law

$$u = \frac{1}{b_n} (-a_n + w) \qquad (31)$$

with w being defined in the sequel, the closed-loop system dynamics (27) rearranges as

$$\begin{aligned} \dot{s} &= \delta_a + a_n + (\delta_b + b_n) \cdot u \\ &= \delta_a - \frac{a_n}{b_n} \delta_b + \left(1 + \frac{\delta_b}{b_n}\right) w. \end{aligned} \qquad (32)$$

Taking $\delta = \delta_a - \frac{a_n}{b_n} \delta_b + \frac{\delta_b}{b_n} w$, then (32) can be written as

$$\dot{s} = w + \delta(\cdot) \qquad (33)$$

Notice that, given Assumption 1, the term s is bounded. Now, the objective is to propose a control input w that forces s to 0 in a finite time in spite of the uncertain term $\delta$. Based on Levant (1993) and Shtessel et al. (2012), if we consider the nonlinear system (33) assuming that the control input is denoted by w and $\delta(t)$ represents an unknown perturbation, the super-twisting controller (Levant, 1993) is designed to operate without requiring precise knowledge of the system’s dynamic model. If there exist positive constants $\delta_M$ and $\Delta$, such that $|\delta(t)| < \delta_M$ and $|\dot{\delta}(t)| < \Delta$ for all $t \ge 0$, then it is possible to find gains $k_1$ and $k_2$ such that the controller defined as

$$w = -k_1|s|^{\frac{1}{2}} \text{sign}(s) + v \qquad (34)$$

$$\dot{v} = -k_2 \text{sign}(s)$$

ensures the convergence of s and $\dot{s}$ towards 0 within a finite time. Conditions on $k_1$ and $k_2$ are given in Shtessel et al. (2012), Taleb et al. (2021) and Mirzaei et al. (2022); the values of $k_1$ and $k_2$ are depending on $\Delta$. In practical applications, the bounds of uncertainties and perturbations (finally, $\delta_a, \delta_b, \delta_M$, and $\Delta$) are difficult to determine. They are often over-estimated which degrades the control performance by introducing chattering, large magnitudes, saturations, and so on. One solution to overcome these defects is the use of adaptive gain approaches (Shtessel et al., 2014). These approaches allow for an increase in the gains when accuracy is not sufficient and reduce them

<footnote n="1">¹ The Eq. (27) means that the relative degree of s equals 1 (Isidori, 1985).</footnote>
when control objectives are achieved. Moreover, the adaptive gain approach does not require the determination of the bounds $\delta_M$ and especially $\Delta$. Only the relative degree (Isidori, 1985), which must be equal to 1, needs to be known: it means that s-dynamics reads as (27). In this study, three adaptive versions of STW algorithms are adopted, differing in the definition of their adaptation laws. The first one is the adaptive super-twisting (ASTW) controller proposed in Shtessel et al. (2012), requiring a set of parameters to be tuned. The second one is an ASTW variant with a reduced set of parameters, developed in Taleb et al. (2021) and called “adaptive super-twisting with a reduced set of parameters” (ASTR), in which only two controller parameters need to be tuned. The third one is the “self-tuning adaptive super-twisting” (STAS) (Mirzaei et al., 2022), offering a single tuning parameter to ensure system stability. All three algorithms discussed in this study offer the advantage of reducing the tuning effort for the controller gains, as the gains are automatically adjusted online. Below, the definitions of these algorithms are provided. These protocols globally follow the same idea: when the targeted accuracy is not reached, it could be due to too small gains, so, gains are increasing. On the other hand, when the targeted accuracy is reached, the gains are reduced.

*   **ASTW** (Shtessel et al., 2012). The controller gains $k_1$ and $k_2$ of (34) are defined as

$$ \dot{k}_1 = \begin{cases} \alpha \operatorname{sign}(|s| - \epsilon) & \text{if } k_1 > k_{1m} \\ \eta & \text{if } k_1 < k_{1m} \end{cases} \quad \text{and} \quad k_2 = \beta k_1 \quad (35) $$

with $\eta$, $\epsilon$, $\beta$, $\alpha$ and $k_{1m}$ as positive constants. The initial value of $k_1$ exceeds the constant $k_{1m}$, i.e., $k_1(0) > k_{1m}$. The protocol (35) automatically increases the gains $k_1$ and $k_2$ when the targeted accuracy $\epsilon$ is not achieved and reduces the gains otherwise. The parameter $k_{1m}$ is used to set a lower limit for the gains, preventing them from going negative, the parameter $\eta$ defining the gain dynamics in such case. The most important parameter is the targeted accuracy $\epsilon$ which can significantly impact the gain values, potentially causing them to become very large or consistently increasing. Lastly, the gain dynamics are determined by $\alpha$ which can be a limitation when dealing with perturbations and their dynamics, especially if they are unknown.

*   **ASTR** (Taleb et al., 2021). The controller gains $k_1$ and $k_2$ are adjusted dynamically based on the given protocol

$$ \dot{k}_1 = \begin{cases} \frac{\alpha}{|\hat{s}|+\epsilon} & \text{if } |s| > \epsilon \\ -k_1 & \text{if } |s| \le \epsilon \end{cases} \quad \text{and} \quad \dot{k}_2 = \begin{cases} \frac{\alpha}{2|s|^{\frac{1}{2}}} & \text{if } |s| > \epsilon \\ -k_2 & \text{if } |s| \le \epsilon \end{cases} \quad (36) $$

with $\alpha$ and $\epsilon$ as positive constants and $\hat{s}$ denoting the estimation of first-time derivative of $s$. Thanks to (36), the gains $k_1$ and $k_2$ increase to ensure convergence of the sliding variable to the origin within a finite time, as long as the sliding variable $s$ remains outside the interval $[-\epsilon, \epsilon]$, $\epsilon$ being the targeted accuracy. When $s$ enters the interval $[-\epsilon, \epsilon]$, both $\dot{k}_1$ and $\dot{k}_2$ become negative, making the gains decreasing. Additionally, to modify the convergence speed towards $s = \dot{s} = 0$, the parameter $\alpha$ should be adjusted. Notice that ASTR has only 2 tuning parameters whereas ASTW needs 6 parameters. The parameter $\alpha$ in (36) has to be fixed. However, this parameter has to be tuned w.r.t perturbations that are not known. This drawback is removed thanks to the next protocol, in which $\alpha$ is time-varying allowing the adaptation of $(k_1, k_2)$-rates.

*   **STAS** (Mirzaei et al., 2022). Under significant perturbations, the convergence time typically grows, necessitating an increase in convergence speed to maintain high performances. This implies that the reaching speed is heavily impacted by the intensity of the applied perturbations. To address this issue, a self-adjusting

approach featuring a time-varying rate has been introduced. The gains $k_1$ and $k_2$ are adapted according to the following protocol

$$ \dot{k}_1 = \begin{cases} \frac{a(t)}{|\hat{s}|+\epsilon} & \text{if } |s| > \epsilon \\ -k_1 & \text{if } |s| \le \epsilon \end{cases} \quad \text{&} \quad \dot{k}_2 = \begin{cases} \frac{a(t)}{2|s|^{\frac{1}{2}}} & \text{if } |s| > \epsilon \\ -k_2 & \text{if } |s| \le \epsilon \end{cases} \quad (37) $$

with any positive initial conditions $k_1(0)$ and $k_2(0)$ and a time-varying $\alpha(t)$ and the self-tuning dynamics obtained as

$$ \begin{cases} \alpha(t) = |k_1| |s|^{\frac{1}{2}} + |\hat{s}| + |w| \\ \dot{w} = k_2 \operatorname{sgn}(s), \quad w(0) = 0 \end{cases} \quad (38) $$

Here, $\epsilon$ is the single parameter related to target accuracy, acting as a positive design element (for further details, see Mirzaei et al. (2022)).

### 3.4. Application of STW-based control approach to the FOWT

Based on the provided dynamics for a FOWT (7), as discussed in Zhang and Plestan (2021a), these dynamics can be considered as a class of nonlinear systems. This nonlinear representation provides several advantages in terms of control design and adaptation across different operational points without requiring individual parameter tuning. The representation reads as follows

$$ \dot{x} = f(x, t) + g(t) \cdot u \quad (39) $$

with the control input, denoted as $u$, being the generator torque $T_g$. However, due to the connection to a PMSG, direct control over $T_g$ is not feasible. To apply control over $T_g$, the stator voltages for design purposes should be utilized. To this end, the dynamics of a FOWT equipped with a PMSG is taken as (10) and shown in the following nonlinear uncertain system affine in $U$

$$ \dot{x} = F(x, t) + \Delta F(M, t) + G(t) \cdot U \quad (40) $$

with $x = [\phi \ \dot{\phi} \ \omega \ i_d \ i_q]^T$, $F$ and $G$ unknown and bounded functions, $\Delta F$ as an uncertain part and $U = [v_d, v_q]^T$.

**Assumption 2.** As the wind and perturbations are bounded in real-world conditions, in the operating domain, the function $F(x, t)$ is unknown and bounded, and also $G(t)$ is a bounded function with a constant and unknown value that incorporates the system inertia, as described in Section 2.

Besides, the dynamics of the platform pitch angle rate are supposed to be unknown; it yields

$$ \dot{\phi} = f_{\phi}(\cdot) \quad (41) $$

with $f_{\phi}(\cdot)$ supposed to be bounded in the operating domain. Additionally, the first derivative (acceleration of platform pitch) and the second derivative (jerk of platform pitch) of the platform pitch rate are considered to be unknown and bounded.

Based on the prior discussions, the control inputs are $v_q$ and $v_d$ (see Section 2). Meanwhile, the system output consists of the rotor speed error, $\sigma_\omega$, and the $d$-axis current error, $\sigma_{i_d}$. The main objective is to design controllers that ensure both $\omega$ (rotor speed) and $i_d$ ($d$-axis current) track their respective references. Considering the rotational speed error $\sigma_\omega$ defined as (12), the system admits a relative degree equal to two, given that the control input $v_q$ appears at its second time derivative. That means that an STW-based control cannot be directly applied as it can only be applied to a system with a relative degree equal to one. So, to apply an STW-based control, the sliding variable is defined as

$$ s_{\omega} = \delta_{\omega} + \gamma \sigma_{\omega} \quad (42) $$

with $\gamma > 0$. Notice that once $s_\omega$, then $\sigma_\omega \to 0$.
### 3.4.1. Rotor speed control design

The sliding variable (42) read as

$$ 
\begin{aligned}
s_{\omega} &= \dot{\sigma}_{\omega} + \gamma \sigma_{\omega} \\
&= \dot{\omega} - \dot{\omega}^* + \gamma(\omega - \omega^*) \\
&= \frac{1}{J_t}(T_a - T_g - K_t \omega) - \frac{\lambda^*}{R}\dot{\nu} + k\ddot{\phi} + \gamma(\omega - \omega^*)
\end{aligned} 
\quad (43) $$

The time derivative of $s_{\omega}$ reads as

$$ 
\begin{aligned}
\dot{s}_{\omega} &= \ddot{\sigma}_{\omega} + \gamma \dot{\sigma}_{\omega} \\
&= \ddot{\omega} - \ddot{\omega}^* + \gamma(\dot{\omega} - \dot{\omega}^*) \\
&= \frac{1}{J_t}(\dot{T}_a - \dot{T}_g - K_t \dot{\omega}) - \frac{\lambda^*}{R}\dot{\nu} + k\phi^{(3)} \\
&\quad + \gamma\left(\frac{1}{J_t}(T_a - T_g - K_t \omega) - \frac{\lambda^*}{R}\dot{\nu} + k\ddot{\phi}\right)
\end{aligned} 
\quad (44) $$

By recalling that in the considered PMSG, losses are assumed to be insignificant ($T_g \approx T_{em}$), and supposing that $L_d = L_q$ (i.e., $T_{em} = p\phi_f i_q$), one gets

$$ 
\begin{aligned}
\dot{s}_{\omega} &= \frac{1}{J_t}(\dot{T}_a - K_t \dot{\omega}) - \frac{\lambda^*}{R}\dot{\nu} + k\phi^{(3)} \\
&\quad + \gamma\left(\frac{1}{J_t}(T_a - T_g - K_t \omega) - \frac{\lambda^*}{R}\dot{\nu} + k\ddot{\phi}\right) - \frac{1}{J_t}p\phi_f i_q
\end{aligned} 
\quad (45) $$

The above equation can be written in the form (27)

$$ 
\begin{aligned}
\dot{s}_{\omega} &= \frac{1}{J_t}(\dot{T}_a - K_t \dot{\omega}) - \frac{\lambda^*}{R}\dot{\nu} + k\phi^{(3)} + \gamma\left(\frac{1}{J_t}(T_a - T_g - K_t \omega) - \frac{\lambda^*}{R}\dot{\nu} + k\ddot{\phi}\right) \\
&\quad - \frac{1}{J_t}p\phi_f\left(-\frac{R_s}{L_q}i_q - p\frac{L_d}{L_q}\omega i_d - p\frac{1}{L_q}\phi_f\omega\right) - \frac{1}{L_q J_t}p\phi_f v_q \\
&= \underbrace{\frac{1}{J_t}(\dot{T}_a - K_t \dot{\omega}) - \frac{\lambda^*}{R}\dot{\nu} + k\phi^{(3)} + \gamma\left(\frac{1}{J_t}(T_a - T_g - K_t \omega) - \frac{\lambda^*}{R}\dot{\nu} + k\ddot{\phi}\right)}_{a_{\omega_1}} \\
&\quad - \underbrace{\frac{1}{J_t}p\phi_f\left(-\frac{R_s}{L_q}i_q - p\frac{L_d}{L_q}\omega i_d - p\frac{1}{L_q}\phi_f\omega\right)}_{a_{\omega_2}} \\
&\quad - \underbrace{\frac{1}{L_q J_t}p\phi_f v_q}_{b_{\omega}}.
\end{aligned} 
\quad (46) $$

Consequently, it can be rewritten in the following form

$$ \dot{s}_{\omega} = a_{\omega}(\cdot) + b_{\omega}(\cdot)v_q \quad (47) $$

with $a_{\omega} = a_{\omega_1} + a_{\omega_2} = a_{\omega_n} + \delta a_{\omega}$, where $a_{\omega_n}$ is the known (nominal) part and $\delta a_{\omega}$ is the unknown (uncertain) part. Similar notations for ($b_{\omega} = b_{\omega_n} + \delta b_{\omega}$). For example, the inductor $L_q$ can be expressed as $L_q = L_{q_n} + \delta L_q$, where $L_{q_n}$ represents its nominal value and $\delta L_q$ is the corresponding uncertainty. The nominal parts of (47) read as

$$ a_{\omega_n} = \frac{1}{J_{t_n}} p \phi_f J_n \left( - \frac{R_{s_n}}{L_{q_n}} i_q - p \frac{L_{d_n}}{L_{q_n}} \omega i_d - p \frac{1}{L_{q_n}} \phi_f \omega \right) \quad (48) $$

and

$$ b_{\omega_n} = \frac{-p}{L_{q_n} \cdot J_{t_n}} \phi_f J_n \quad (49) $$

the values of $R_{s_n}$, $L_{d_n}$, $L_{q_n}$, $J_{t_n}$ and $\phi_f$ are given by the manufacturer (see Table 2). Then, as (31), the control input $v_q$ is defined as,<fnref n="2" />

$$ v_q = \frac{1}{b_{\omega_n}} (-a_{\omega_n} + w_q) \quad (50) $$

The term $w_q$ is taken based on (34). Then, one gets

$$ 
\begin{aligned}
v_q = & -R_{s_n} i_q - p L_{d_n} \omega i_d - p \phi_f n \omega \\
& + \frac{L_{q_n} \cdot J_{t_n}}{p \phi_f n} \left( k_{\omega_1} |s_{\omega}|^{\frac{1}{2}} \operatorname{sign}(s_{\omega}) + k_{\omega_2} \int_0^t \operatorname{sign}(s_{\omega}) d\tau \right).
\end{aligned} 
\quad (51) $$

The gains $k_{\omega_1}$ and $k_{\omega_2}$ are the proposed gains which will be tuned through the successive adaptive laws (35)–(37). Then, the sliding variables $s_{\omega}$ and its time derivative $\dot{s}_{\omega}$ converge to a vicinity of 0 ensuring $\omega \to \omega^*$ as shown by simulations in the sequel.

### 3.4.2. Direct current control designs

In this case, the sliding variable is defined as

$$ s_{i_d} = \sigma_{i_d} = i_d - i_d^* \quad (52) $$

which gives

$$ \dot{s}_{i_d} = -\frac{R_s}{L_d} i_d + p \frac{L_q}{L_d} \omega i_q + \frac{1}{L_d} v_d. \quad (53) $$

The relative degree of $s_{i_d}$ with respect to $v_d$ equals 1 and $s_{i_d}$-dynamics can be written as (27)

$$ \dot{s}_{i_d} = a_{i_d}(\cdot) + b_{i_d}(\cdot)v_d. \quad (54) $$

As previously detailed, both terms $a_{i_d}$ and $b_{i_d}$ can be expressed as known nominal parts and uncertain ones. Consequently, one has $a_{i_d} = a_{i_{d_n}} + \delta a_{i_d}$ and $b_{i_d} = b_{i_{d_n}} + \delta b_{i_d}$, with

$$ a_{i_{d_n}} = -\frac{R_{s_n}}{L_{d_n}} i_d + p \frac{L_{q_n}}{L_{d_n}} \omega i_q, \quad b_{i_{d_n}} = \frac{1}{L_{d_n}} \quad (55) $$

Finally, the control input $v_d$ that forces $\sigma_{i_d} \to 0$ is defined as

$$ 
\begin{aligned}
v_d &= \frac{1}{b_{i_{d_n}}} (-a_{i_{d_n}} + w_d) \\
&= R_{s_n} i_d - p L_{q_n} \omega i_q - L_{d_n} k_{i_{d1}} |s_{i_d}|^{\frac{1}{2}} \operatorname{sign}(s_{i_d}) - L_{d_n} k_{i_{d2}} \int_0^t \operatorname{sign}(s_{i_d}) d\tau
\end{aligned} 
\quad (56) $$

The term $w_d$ is taken based on (34). The gains $k_{i_{d1}}$ and $k_{i_{d2}}$ are obtained from the adaptive laws (35)–(37). It is important to recall that the adaptive gain super-twisting-based control method does not require precise knowledge of system uncertainty bounds. In the next section, a comparative analysis of the controllers will evaluate the performance and suitability for different criteria.

**Remark 2.** The implemented STW-based controllers ensure system stability even if there are parameter changes given that they are designed to manage system uncertainties and disturbances. This feature makes these control techniques superior in delivering consistent performance for evolving systems.

## 4. Simulations and analysis

### 4.1. Simulation set-up

This section is devoted to the performance analysis of the NREL's 5MW OC3 FOWT modeled via OpenFAST, equipped with a PMSG, under the influence of various control strategies. The mechanical/ electrical parameters of the considered turbine and generator are specified in Tables 1 and 2. The control strategies are executed thanks to Matlab/SIMULINK. Although the control strategy is tailored for a simplified model, all 24 degrees of freedom are enabled in OpenFAST. Simulations are conducted over 1200 s, using the ODE1 (Euler) integration algorithm with a fixed time step of 0.0125 s. In the following part, a comparison of several aspects is drawn among the proposed controllers. Here, for the super-twisting-based control, the three different adaptation laws are considered as

<footnote n="2">² It is important to note that, in contrast to what is outlined in (31) $b_1$ holds a negative value, and it is imperative not to mistake or disregard this aspect.</footnote>
<figure><img src="image_7.png" /><figcaption>Fig. 4. Wind velocity (top, in ms⁻¹) and wave elevation (bottom, in meters) as functions of time (in seconds).</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 5. Main variables of the FOWT versus time (s).</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 6. Performance criteria for 5MW FOWT.</figcaption></figure>

* **ASTW (Shtessel et al., 2012)**: this controller enhances tuning over the constant gain STW. The control inputs are given by (50) and (56) and (35) for gain adaptation.

* **ASTR (Taleb et al., 2021)**: this controller minimizes tuning effort. The user adjusts three parameters (α for gain rate, ε for targeted accuracy, and τ as low-pass filter time constant). Control inputs and gain adaptation are defined by (50), (56), and (36).

* **STAS (Mirzaei et al., 2022)**: this controller can adapt the rate of gains and requires the tuning of only two parameters: ε for accuracy and τ for low-pass filter time constant. The controller is based on (50), (56), and (37).

All simulations in this paper are carried out under the same conditions, using the wind and wave profiles as depicted in Fig. 4. The stochastic wind speed profile is generated with a mean value of 7 m/s and a turbulence intensity of 10%. Moreover, an irregular wave with a significant height of 3.25 m and a peak spectral period of 9.7 s is considered. The parameters for the proposed controllers are displayed
<figure><img src="image_10.png" /><figcaption>Fig. 7. Motions indicators for 5MW FOWT.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 8. Normalized DEL values of structure loads.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 9. Generated power and rotor speed tracking with constant wind speed (7 m/s).</figcaption></figure>
<table><caption>**Table 1**
FOWT parameters (Jonkman et al., 2009).</caption><thead><tr><th>Parameters</th><th>Value</th></tr></thead><tbody><tr><td>Rotor, hub diameter</td><td>126 m, 3 m</td></tr><tr><td>Hub height</td><td>90 m</td></tr><tr><td>Optimal power coefficient (C<sub>p</sub><sup>*</sup>)</td><td>0.482</td></tr><tr><td>Optimal TSR (λ<sup>*</sup>)</td><td>7.55</td></tr><tr><td>Wind speed (mean value)</td><td>7 m/s (stochastic)</td></tr><tr><td>Wind turbulence intensity</td><td>10%</td></tr><tr><td>Wave height with irregular pattern</td><td>3.25 m</td></tr><tr><td>Wave peak spectral period</td><td>9.7 s</td></tr><tr><td>Specific density of air</td><td>1.22 kg/m<sup>3</sup></td></tr><tr><td>Rotor inertia</td><td>38.759.228 kg m<sup>2</sup></td></tr><tr><td>External damping</td><td>0.0085 kg m<sup>2</sup> s<sup>-1</sup></td></tr></tbody></table>

<table><caption>**Table 2**
PMSG parameters (Zhang and Plestan, 2021a).</caption><thead><tr><th>Parameters</th><th>Value</th></tr></thead><tbody><tr><td>d-axis inductance (L<sub>d</sub>)</td><td>14.29 mH</td></tr><tr><td>q-axis inductance (L<sub>q</sub>)</td><td>14.29 mH</td></tr><tr><td>Stator resistance (R<sub>s</sub>)</td><td>1.2 ohm</td></tr><tr><td>Viscous friction coefficient (f<sub>v</sub>)</td><td>0.0085 kg m<sup>2</sup> s<sup>-1</sup></td></tr><tr><td>Generator flux (φ<sub>f</sub>)</td><td>8.6 Wb</td></tr><tr><td>Pole pair (p)</td><td>5</td></tr></tbody></table>

<table><caption>**Table 3**
Tuning parameters for controllers.</caption><thead><tr><th>Controller</th><th>Parameters</th></tr></thead><tbody><tr><td>ASTW</td><td>k<sub>1m</sub> = 0.05, α = 0.01, ε = 0.001, β = 0.02, η = 0.05</td></tr><tr><td>ASTR</td><td>α = 10<sup>-6</sup>, ε = 10<sup>-3</sup></td></tr><tr><td>STAS</td><td>ε = 10<sup>-3</sup></td></tr><tr><td>Backstepping</td><td>k<sub>ω</sub> = 150, k<sub>q</sub> = 5000, k<sub>d</sub> = 50</td></tr></tbody></table>

in Table 3. All these parameters have been fine-tuned to achieve the best performance. The initial conditions for the gains are set as $k_{ω1}(0) = 5$, $k_{ω2}(0) = 0.01$, $k_{id1}(0) = 5$, and $k_{id2}(0) = 0.01$.

Additionally, the values of $\dot{s}_{\omega}$ and $\dot{s}_{id}$ are estimated using a low-pass filter as follows

$$ \dot{s}_{\omega} = \left( \frac{s}{1 + \tau s} \right) s_{\omega}, \quad \tau = 0.01 \text{ sec} \quad (57) $$

$$ \dot{s}_{id} = \left( \frac{s}{1 + \tau s} \right) s_{id}, \quad \tau = 0.01 \text{ sec} $$

## 4.2. Results and analysis

In Fig. 5, the results obtained with each STW-based controller are depicted. Detailed comparisons are shown in Figs. 6–8. To assess the effectiveness of the proposed STW-based controllers and the backstepping approach the mean power is evaluated. It allows determine which approach yields the higher power generation. Additionally, the effectiveness of each control is made by comparing the power coefficient error defined as $\frac{100|C_p - C_p^*|}{C_p^*}$. This assessment plays a crucial role in maintaining the optimal TSR. A lower power coefficient error indicates superior control performance in achieving MPPT. Based on a comparison as illustrated in Fig. 6, it is evident that STAS outperforms the other methods, yielding 12 kW, 55 kW, and 64.6 kW more power than ASTW, ASTR, and backstepping, respectively. In addition, the mean power for baseline control is 1.06 MW which is lower than other controllers. The power coefficient error, defined as the difference between the actual value of the power coefficient $C_p$ and its reference value $C_p^*$, serves as an indicator of how closely the generated power adheres to its MPPT target. A smaller power coefficient error indicates a higher level of efficiency and control performance. This criterion can also be used to assess the reliability of an approach. As illustrated in Fig. 6, the STAS demonstrates enhanced control performance compared to other approaches. It exhibits a 3.54% power coefficient error,

while the ASTR, ASTW, and backstepping controllers have errors of 4.69%, 3.93%, and 10.34%, respectively. This confirms that the STAS is more efficient in terms of control performance. Therefore, reducing the power coefficient error is critical to achieving optimal performance and maximizing the energy output of FOWTs in Region II. Data normalization is used to analyze and assess possible damages to the structure. The data is normalized using the baseline generator-torque control strategy (Jonkman et al., 2009). Normalized indicators are in Figs. 7–8, aiming for the smallest values. This process ensures that all considered variables contribute equally. It is also decided not to analyze the fatigue reduction features of the backstepping control due to its unsatisfactory performance in the first part. Fig. 7 reveals that STW-based controls are generally superior, except in roll motion and the rate of roll motion where the generator-torque control is better. Fig. 8 displays the damage equivalent load (DEL) criteria to evaluate tower base (TB) moments, blade root (BR) moments, fair-lead force (FF), and anchor force (AF). In light of the results in the top side of Fig. 8, the baseline generator-torque control strategy excels in managing side-to-side TB, whereas STAS and ASTW demonstrate greater efficiency in addressing BR flap-wise moments and mooring lines ($FF_1-FF_3$ and $AF_1-AF_3$), respectively. Specifically, from the bottom results of Fig. 8, concerning the forces acting on mooring lines, ASTW outperforms other approaches, imposing less force.

**Remark 3.** Wind turbines often face variable wind conditions, leading to frequent transient states. An effective response to these changes is vital for control systems, contributing to their performance. For constant wind conditions, as in Fig. 9, all STW-based controllers perform efficiently, generating powers with minor oscillations around the optimal power due to irregular wave effects on the turbine platform.

In conclusion, a comparative analysis is useful for choosing suitable control strategies in low-wind scenarios. Fig. 3 shows that a backstepping control may not be ideal on the OpenFAST platform due to its reliance on a complete system model. Conversely, the implemented STW-based controls achieve satisfactory rotor speed tracking with just a rotor speed signal feedback from the OpenFAST model. This strategy accomplishes control objectives and results in MPPT even without precise system dynamics knowledge.

## 5. Conclusion

In conclusion, this study has provided valuable insights into the implementation of nonlinear control strategies for FOWTs with PMSG in Region II. Our goal was to optimize power generation while minimizing fatigue loads. The adaptive form of the STW controls successfully manages the complex dynamics inherent in the FOWTs, requiring minimal system model knowledge. The limitations of the backstepping controller when applied to the OpenFAST platform were highlighted. Through comparative analysis, the efficacy of the STW-based approaches is confirmed. Lastly, results obtained from testing on the OpenFAST simulator reveal the ability of the applied controls in power production under various operating conditions, highlighting the potential for more robust and reliable real-world performance. The next steps consist of applying this class of controllers to a real experimental system. Another research axis consists of developing Lidar-based control decoupled with adaptive STW.

### CRediT authorship contribution statement

**Ehsan Aslmostafa:** Writing – original draft, Validation.
**Mohamed Assaad Hamida:** Supervision, Methodology, Conceptualization.
**Franck Plestan:** Writing – review & editing, Supervision, Project administration.
Declaration of competing interest

The authors declare the following financial interests/personal rela-
tionships which may be considered as potential competing interests:
Ehsan Aslmostafa reports financial support was provided by Ecole
Centrale de Nantes. Franck Plestan reports a relationship with Ecole
Centrale de Nantes that includes: board membership. If there are other
authors, they declare that they have no known competing financial in-
terests or personal relationships that could have appeared to influence
the work reported in this paper.

Data availability

No data was used for the research described in the article.

Acknowledgments

This work has been partially supported by Région Pays de la Loire
and is a part of ANR CREATIF project (No. ANR-20-CE05-0039).

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