Mitigating Intermittency in Offshore Wind Power Using Adaptive Nonlinear MPPT Control Techniques

Abstract

This paper addresses the challenge of maximizing power extraction in offshore wind energy systems through the development of an enhanced maximum power point tracking (MPPT) control strategy. Offshore wind energy is inherently intermittent, leading to discrepancies between power generation and electricity demand. To address this issue, we propose three advanced control algorithms to perform a comparative analysis: sliding mode control (SMC), the Integral Backstepping-Based Real-Twisting Algorithm (IBRTA), and Feed-Back Linearization (FBL). These algorithms are designed to handle the nonlinear dynamics and aerodynamic uncertainties associated with offshore wind turbines. Given the practical limitations in acquiring accurate nonlinear terms and aerodynamic forces, our approach focuses on ensuring the adaptability and robustness of the control algorithms under varying operational conditions. The proposed strategies are rigorously evaluated through MATLAB/Simulink 2024 A simulations across multiple wind speed scenarios. Our comparative analysis demonstrates the superior performance of the proposed methods in optimizing power extraction under diverse conditions, contributing to the advancement of MPPT techniques for offshore wind energy systems.

1. Introduction

Renewable energy sources now make up nearly a third of global electricity production, driven by the increasing demand for clean, sustainable energy solutions and the continuous advancement of renewable technologies. Wind energy, in particular, has emerged as a leading contributor to this transition, with global wind capacity reaching over 837 GW by 2022, accounting for nearly 7% of global electricity generation [1]. Offshore wind energy systems are at the forefront of this growth, offering the potential to generate up to 40% more power than onshore installations due to higher and more consistent wind speeds [2]. However, the efficiency of offshore wind systems is highly contingent on the ability to optimize power extraction in the face of variable and unpredictable wind conditions.

MPPT control schemes are very important for the effective control of offshore wind turbines to ensure that they produce their maximum power output [3]. Most of the traditional MPPT techniques, though successful to some extent, fail to cope with the fluctuating and nonlinear behavior of offshore wind systems. Variable-speed turbines (VST), which have the capability of varying their rotational speed depending on the wind speed, are more efficient than fixed-speed turbines (FST), with the potential to provide up to a 10–15% increase in energy generation [4,5]. However, the application of efficient MPPT techniques in offshore systems continues to be a challenge, especially in dealing with issues arising from nonlinearities and disturbances in the environment [6,7].

To address these challenges, new developments in control theory and artificial intelligence have provided new opportunities to improve MPPT in offshore wind energy systems. Some of the methods used include sliding mode control (SMC), integral backstepping, and the super-twisting algorithm (STA), which are deemed to improve conventional methods [8]. These methods are intended to be insensitive to the variations in parameters and the disturbances that occur in an offshore environment that is rough and uncertain. More specifically, the application of artificial intelligence, including fuzzy logic control (FLC) and neural networks, has also been incorporated to improve the flexibility of these control systems. Offshore wind will increase its capacity at a rate of more than 15% every year over the next ten years, which underlines the importance of improving state-of-the-art MPPT control strategies to optimize the potential of offshore wind energy sources [9]. The enhancement of power capture in wind energy systems has been a major concern, especially in modern and increasingly installed offshore wind farms. Some of the first techniques used for MPPT were perturb-and-observe (P&O) and incremental conductance [10]. Although these methods are simple and commonly used, they have a major weakness in handling the dynamic and stochastic nature of offshore wind conditions. The above-mentioned conventional MPPT techniques have been reported to exhibit poor performance under turbulent wind conditions, which, in turn, cause energy losses [11,12].

In order to overcome these drawbacks, enhanced control techniques have been proposed in the literature for enhancing the efficiency of MPPT techniques. As a consequence, SMC is considered an effective approach owing to its ability to handle parameter variability and disturbance [13]. The above features make SMC suitable for use in offshore environments since it is capable of retaining stability even in challenging circumstances. However, conventional SMC methods are known to produce chattering, which is a condition that causes mechanical wear and decreases system performance. Some developments in SMC has been made recently, for instance, the IBRTA, to overcome such problems while at the same time giving the system a robust control response.

AI-based MPPT techniques have been enhanced by integrating AI into the wind energy control system. FLC has been employed extensively due to its capability to deal with nonlinearity and uncertainty without calling for an accurate model of the system [14]. FLC systems are more efficient in tracking the stochastic nature of wind and provide faster tracking of the MPP than conventional methods. Furthermore, the use of neural networks (NNs) has also been adopted to predict the wind speed and to enhance turbine efficiency in a data-driven manner [15]. These AI-based techniques improve the flexibility of MPPT systems, which makes it easier to control the systems in dynamic offshore conditions. Ref. [16] presented a comprehensive review of MPPT methods specifically for PMSG-based wind energy systems ranging from traditional hill-climbing and perturb-and-observe to sophisticated AI-assisted algorithms.

In generator technologies, PMSGs have been used in wind energy because of the high efficiency and reliability of these systems [16]. Some of the advantages of PMSGs include a low maintenance cost and no power loss through the gear box. This is because PMSGs produce higher efficiency and are long-lasting when used with other complex MPPT techniques [16]. The studies have also shown that the use of PMSG in variable-speed wind turbines and appropriate control techniques improve the power output and the reliability of the system in the face of varying wind gusts [11,17].

However, there is a problem in the area of MPPT in offshore wind energy systems which has been a topic of discussion. Future work should focus on advanced control approaches, including observer-based control techniques like IBSMC [18]. These methods are intended to enhance the efficiency and reliability of MPPT systems, especially in the dynamic conditions of offshore wind conditions [19]. There have been detailed analyses of both wind and wave sources combining observer-based techniques with artificial intelligence techniques, and researchers are striving to design more robust and effective control systems which can optimally utilize offshore wind energy resources. Despite progress in AI- and observer-based control, a research gap remains in developing low-chattering, adaptive MPPT strategies for offshore wind systems operating under nonlinear and discontinuous conditions [16,18,19].

2. System Modeling

In this section, we develop mathematical models for an offshore wind energy conversion system and a PMSG. These models are foundational for the design of advanced control strategies that will be discussed in subsequent sections. An offshore wind conversion’s typical layout is shown in Figure 1.

The primary objective of this section is to model the energy conversion process of an offshore wind turbine and its associated generator, a PMSG. The modeling will encompass both mechanical and electrical dynamics, leading to a state-space representation suitable for control design.

2.1. Power and Torque Calculations

The amount of electricity that an offshore wind turbine is capable of producing is influenced by a number of factors, such as the wind speed, the density of the air, and the dimensions of the turbine. The mechanical power output $P_{\mathrm{M}}$ of an offshore wind system can be calculated as follows [13]:

$$P_{\mathrm{M}}={\displaystyle \frac{1}{2}}{\rho }_{\mathrm{air}}\pi R_{\mathrm{blade}}^2{v}_{\mathrm{wind}}^3{C}_{\mathrm{pr}}\left({\lambda }_l\right)$$

(1)

where ${\rho }_{\mathrm{air}}$ is the air density, $R_{\mathrm{blade}}$ is the radius of the turbine blade, $v_{\mathrm{wind}}$ is the wind speed, and $C_{\mathrm{pr}}\left({\lambda }_l\right)$ is the power coefficient, which is a function of the tip-speed ratio ${\lambda }_l$. The torque exerted on the turbine shaft ${\mathsf{\Gamma }}_{\mathrm{M}}$ is another critical parameter and is calculated using the torque coefficient $C_{\tau }\left({\lambda }_l\right)$, which is linked to the power coefficient through the following equation:

$${\mathsf{\Gamma }}_{\mathrm{M}}={\displaystyle \frac{1}{2}}{\rho }_{\mathrm{air}}\pi R_{\mathrm{blade}}^3{v}_{\mathrm{wind}}^2{C}_{\tau }\left({\lambda }_l\right)$$

(2)

The equations provided serve as the groundwork for understanding how mechanical energy is converted in offshore wind turbines. The power coefficient $C_{\mathrm{pr}}\left({\lambda }_l\right)$ and torque coefficient $C_{\tau }\left({\lambda }_l\right)$ depend on the tip-speed ratio which is very important in deciding the efficiency of energy conversion.

2.2. The Role of Tip-Speed Ratio in Determining Power Output

The tip-speed ratio ${\lambda }_l$ is a critical factor in determining the efficiency of power conversion and is defined as

$${\lambda }_l={\displaystyle \frac{{\mathsf{\Omega }}_{\mathrm{HS}}{R}_{\mathrm{blade}}}{v_{\mathrm{wind}}{i}_{\mathrm{trans}}}}$$

(3)

In this context, ${\mathsf{\Omega }}_{\mathrm{HS}}$ refers to the angular speed of the high-speed shaft, while $i_{\mathrm{trans}}$ is the transmission ratio explained earlier in the theoretical background. The power coefficient $C_{\mathrm{pr}}\left({\lambda }_l\right)$ can be expressed as a polynomial function of ${\lambda }_l$ to reflect the complex relationship between wind speed and the rotational speed of the turbine:

$$C_{\mathrm{pr}}\left({\lambda }_l\right)=-{\lambda }_l(-0.00610+0.0013{\lambda }_l-0.0081{\lambda }_l^2+0.00097477{\lambda }_l^3)$$

(4)

The torque coefficient $C_{\tau }\left({\lambda }_l\right)$ is found by taking the ratio of the power coefficient to the tip-speed ratio:

$$C_{\tau }\left({\lambda }_l\right)={\displaystyle \frac{C_{\mathrm{pr}}\left({\lambda }_l\right)}{{\lambda }_l}}$$

(5)

These relationships are useful in underlining the need to control the tip-speed ratio in order to enhance the effectiveness of energy conversion in offshore wind systems.

2.3. PMSG System

2.3.1. dq-Axis Representation

Offshore wind turbines employ the PMSG, which possesses the electrical characteristics described in the dq-axis reference frame. The dynamic behavior of the PMSG can be described by the following differential equations [16,17]:

$$\begin{array}{cc}\hfill {\dot{i}}_d& ={\displaystyle \frac{-R_{\mathrm{stator}}{i}_d+p_{\mathrm{pole}}(L_q-L_{\mathrm{chopper}}){\mathsf{\Omega }}_{\mathrm{HS}}{i}_q-R_{\mathrm{chopper}}{i}_d}{(L_d+L_{\mathrm{chopper}})}}\hfill \\ \hfill {\dot{i}}_q& ={\displaystyle \frac{-R_{\mathrm{stator}}{i}_q-p_{\mathrm{pole}}(L_q+L_{\mathrm{chopper}}){\mathsf{\Omega }}_{\mathrm{HS}}{i}_d-R_{\mathrm{chopper}}{i}_q}{(L_q+L_{\mathrm{chopper}})}}+p_{\mathrm{pole}}{\mathsf{\Omega }}_{\mathrm{HS}}{\varphi }_{\mathrm{constant}}\hfill \\ \hfill {\dot{\mathsf{\Omega }}}_{\mathrm{HS}}& ={\displaystyle \frac{1}{J_{\mathrm{HS}}}}\left[-p_{\mathrm{pole}}{\varphi }_{\mathrm{constant}}{i}_q+{\displaystyle \frac{d_1{v}_{\mathrm{wind}}^2}{i_{\mathrm{trans}}}}+{\displaystyle \frac{d_2{v}_{\mathrm{wind}}{\mathsf{\Omega }}_{\mathrm{HS}}}{i_{\mathrm{trans}}^2}}+{\displaystyle \frac{d_3{\mathsf{\Omega }}_{\mathrm{HS}}^2}{i_{\mathrm{trans}}^3}}\right]\hfill \end{array}$$

(6)

In these equations, $i_d$ and $i_q$ denote the dq-axis currents; $R_{\mathrm{stator}}$ is the stator resistance; $L_d$ and $L_q$ are the inductances along the d- and q-axes, respectively; ${\varphi }_{\mathrm{constant}}$ represents the flux linkage constant; ${\mathsf{\Omega }}_{\mathrm{HS}}$ is the angular velocity of the high-speed shaft; and $J_{\mathrm{HS}}$ is its moment of inertia. Together, these expressions form a detailed dynamic model of the PMSG system, which is crucial for developing effective control strategies for MPPT in offshore wind energy applications.

2.3.2. State-Space Representation

To streamline the control design process, the PMSG system is expressed in a state-space form. By introducing the state variables $x_1=i_d$, $x_2=i_q$, and $x_3={\mathsf{\Omega }}_{\mathrm{HS}}$, the dynamic behavior of the system can be represented in a more compact form as follows [20]:

$$\dot{x}=f\left(x\right)+g\left(x\right)u+\Delta (x,t)$$

(7)

where $x={[x_1,x_2,x_3]}^T$ is the state vector, u is the control input (e.g., $R_{\mathrm{chopper}}$), and $\Delta (x,t)$ represents matched uncertainties in the system. The values of the constants ${\psi }_1$ to ${\psi }_{11}$ in the state-space model are specified in

Table 1. Constant PMSG terms.

Parameter	Value	Parameter	Value
${\psi }_1$	26.147	${\psi }_4$	26.147
${\psi }_2$	0.94866	${\psi }_5$	3.000
${\psi }_3$	8.2264	${\psi }_6$	1.3146
${\psi }_8$	9.945	${\psi }_9$	0.1332
${\psi }_{10}$	0.00506	${\psi }_{11}$	23.806

.

This state-space representation will be used in the subsequent sections to design advanced control strategies that ensure the efficient and stable operation of the offshore wind energy system.

2.4. Normal-Form Conversion

2.4.1. Transformation Process

To simplify the control design, the wind conversion model can be transformed into its normal form. The system is represented as

$$\dot{x}=f\left(x\right)+g\left(x\right)u+\Delta (x,t),y=h\left(x\right)=x_3$$

(8)

where y represents the system output (e.g., ${\mathsf{\Omega }}_{\mathrm{HS}}$). The normal-form transformation facilitates the design of controllers by separating the internal and external dynamics of the system.

2.4.2. Zero Dynamics and Stability

Zero dynamics refer to the internal behavior of a system when its output is forced to remain at zero. The system’s overall stability depends heavily on these zero dynamics being stable. They are determined by applying the conditions $z_1=z_2=u=0$ in the transformed equation:

$${\dot{z}}_3=-z_3({\mathsf{\Gamma }}_1-{\alpha }_1)$$

(9)

Stability is maintained when ${\mathsf{\Gamma }}_1>{\alpha }_1$, ensuring that the internal dynamics do not destabilize the system.This means that for the system to remain stable internally while tracking an external output (e.g., rotor speed), its hidden dynamics must naturally decay. This is achieved when ${\mathsf{\Gamma }}_1>{\alpha }_1$, ensuring all unobserved dynamics diminish over time.

3. Proposed Control Scheme for MPPT

In this section, we present a detailed approach for optimizing power extraction from offshore wind turbines using MPPT techniques. The proposed control scheme aims to operate the wind energy conversion system at its MPP by implementing advanced control strategies. The selected control methods, FBL, SMC, and the new IBRTA, represent the following alternate nonlinear control paradigms: model-based linearization, discontinuous robust design, and adaptive higher-order feedback, respectively. This selection enables critical performance comparison under typical offshore wind variability. The new IBRTA controller was designed to minimize chattering observed in SMC while maintaining its disturbance rejection capability.

3.1. Sliding Mode Control Design

SMC is a powerful control technique known for its robustness against system uncertainties and external disturbances. SMC achieves control objectives by enforcing the system trajectories to follow a sliding surface, defined in the state space, where the system exhibits the desired behavior. The key advantage of SMC is its ability to maintain stability and performance even in the presence of model inaccuracies and external perturbations.

3.1.1. Definition of Sliding Surface

The initial stage in designing a sliding mode controller involves specifying a sliding surface $s\left(x\right)$ that characterizes the desired behavior of the system. This surface is generally formulated based on the tracking error $e\left(t\right)$, which quantifies the deviation between the reference signal (desired state) and the actual system state:

$$s\left(x\right)={\mu }_{1}e\left(t\right)+\dot{e}\left(t\right)+{\mu }_2{\mathit{\int }}_0^{t}e\left(t\right)dt$$

(10)

where ${\mu }_1$ and ${\mu }_2$ are positive constants that shape the sliding surface, and $e\left(t\right)$ is the tracking error:

$$e\left(t\right)=z_1-z_{\mathrm{reference}}\left(t\right)$$

(11)

Here, $z_1$ represents the system state corresponding to the angular velocity of the high-speed shaft (${\mathsf{\Omega }}_{\mathrm{HS}}$), and $z_{\mathrm{reference}}\left(t\right)$ is the desired reference value for ${\mathsf{\Omega }}_{\mathrm{HS}}$.

3.1.2. Control Law Design

The SMC law is formulated to steer the system trajectories toward a designated sliding surface and to retain them on that surface. The total control input is expressed as

$$u\left(t\right)=u_{\mathrm{equivalent}}\left(t\right)+u_{\mathrm{discontinuous}}\left(t\right)$$

where

• $u_{\mathrm{equivalent}}\left(t\right)$ maintains the system motion on the sliding surface once it has been reached;
• $u_{\mathrm{discontinuous}}\left(t\right)$ drives the system trajectories toward the sliding surface from arbitrary initial conditions.

$$u\left(t\right)=u_{\mathrm{equivalent}}\left(t\right)+u_{\mathrm{discontinuous}}\left(t\right)$$

(12)

The equivalent control input is responsible for stabilizing the system along the sliding surface:

$$\begin{array}{c}{u}_{\mathrm{equivalent}}\left(t\right)={\displaystyle \frac{1}{L_g{L}_{f}h\left(x\right)}}\left({\ddot{z}}_{\mathrm{reference}}-{\mathsf{\ell }}_1{z}_2-{\mathsf{\ell }}_2{z}_1+{\mathsf{\ell }}_1{\dot{z}}_{\mathrm{reference}}\right.\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hfill \\ \hfill \left.+{\mathsf{\ell }}_2{z}_{\mathrm{reference}}-{L_f}^{2}h\left(x\right)\right)\end{array}$$

(13)

The discontinuous control input is designed to ensure that the system reaches the sliding surface quickly:

$$u_{\mathrm{discontinuous}}\left(t\right)=-{\mathsf{\ell }}_{3}s\left(x\right)-{\mathsf{\ell }}_{4}sign\left(s\left(x\right)\right)$$

(14)

where ${\mathsf{\ell }}_1$, ${\mathsf{\ell }}_2$, ${\mathsf{\ell }}_3$, and ${\mathsf{\ell }}_4$ are positive gains that are tuned to achieve the desired closed-loop system performance.

3.1.3. Stability Analysis

The stability of the closed-loop system using SMC can be established by analyzing the time derivative of a Lyapunov candidate function $V\left(s\right)$, defined as

$$V\left(s\right)={\displaystyle \frac{1}{2}}s{\left(x\right)}^2$$

(15)

The time derivative of $V\left(s\right)$ along the system trajectories is given by

$$\dot{V}\left(s\right)=s\left(x\right)\dot{s}\left(x\right)$$

(16)

Substituting the control law into the derivative, we obtain

$$\dot{V}\left(s\right)=-{\mathsf{\ell }}_{3}s{\left(x\right)}^2-{\mathsf{\ell }}_4\left|s\left(x\right)\right|$$

(17)

Since ${\mathsf{\ell }}_3>0$ and ${\mathsf{\ell }}_4>0$, it follows that $\dot{V}\left(s\right)\le 0$, ensuring that the system trajectories converge to the sliding surface and remain there, thereby guaranteeing the stability of the system.

3.1.4. Mitigating Chattering with Higher-Order Sliding Modes

Chattering is a common issue in SMC, where high-frequency oscillations occur due to the discontinuous control input. To mitigate this effect, we employ Higher-Order Sliding Mode (HOSM) techniques, such as the IBRTA. HOSM reduces chattering by smoothing the control action, leading to improved system performance and reduced wear on mechanical components. To further reduce chattering in SMC, especially when acting under high-frequency disturbances, a few complementary methods can be incorporated in the design of the control. A popular way around the chattering problem is to introduce a saturation function in a given boundary layer, where the discontinuous sign function is substituted with a smooth approximation. This relaxes the control action in the vicinity of the sliding surface and hence minimizes high-frequency switching. Also, adaptive gain tuning is able to make the control gain adaptive in real time based upon the tracking error, to provide sufficient control effort without necessarily introducing aggressive switching. An observer-based SMC structure can be utilized to reduce the effects of disturbances in a better way. This enables estimation and cancellation of the disturbances prior to their effect on the system output, which enhances robustness. Furthermore, second-order sliding mode algorithms (like the super-twisting algorithm, which is partially incorporated into the IBRTA controller already) further improve on chattering mitigation by avoiding direct differentiation of the sliding variable. Lastly, unmodeled dynamics or delays can be estimated and corrected using time-delay estimation (TDE)-based methods to keep the performance high whilst preventing sudden transitions in control that can cause chattering. Together, these techniques can preserve the fundamental strength of SMC and at the same time dampen objectionable vibrations in realistic applications (Algorithm 1).

Algorithm 1 Improved Integral based Real Twisting Algorithm (IBRTA) for MPPT

1: Initialization:   2: Set system parameters: air density ${\rho }_{air}$, blade radius $R_{\mathrm{blade}}$, etc.   3: Set initial conditions: initial angular speed ${\mathsf{\Omega }}_{\mathrm{HS}}\left(0\right)$, wind speed $v_{\mathrm{wind}}\left(0\right)$.   4: Define control gains: ${\mathsf{\ell }}_1$, ${\mathsf{\ell }}_2$, ${\mathsf{\ell }}_3$, ${\mathsf{\ell }}_4$, ${\mathsf{\ell }}_8$, ${\mathsf{\ell }}_9$, etc.   5: Set desired tip-speed ratio ${\lambda }_{\mathrm{opt}}$ and power coefficient $C_{\mathrm{pr},\max }$.   6: Loop: For each time step t during the simulation   7: Measure the current wind speed $v_{\mathrm{wind}}\left(t\right)$.   8: Calculate the actual tip-speed ratio $\lambda \left(t\right)={\displaystyle \frac{{\mathsf{\Omega }}_{\mathrm{HS}}\left(t\right)·R_{\mathrm{blade}}}{v_{\mathrm{wind}}\left(t\right)}}$.   9: Calculate the desired angular speed ${\mathsf{\Omega }}_{\mathrm{ref}}\left(t\right)$ using ${\lambda }_{\mathrm{opt}}$. 10: Calculate the tracking error $er\left(t\right)={\mathsf{\Omega }}_{\mathrm{ref}}\left(t\right)-{\mathsf{\Omega }}_{\mathrm{HS}}\left(t\right)$. 11: Step 2: Apply IBRTA Control Law 12: Compute the error derivative $\dot{er}\left(t\right)$. 13: Update the sliding surface: $s\left(t\right)={\mathsf{\ell }}_1·er\left(t\right)+{\mathsf{\ell }}_2·\dot{er}\left(t\right)+{\mathsf{\ell }}_3{\int }_0^{t}er\left(\tau \right)d\tau$. 14: Compute the equivalent control input $u_{\mathrm{equiv}}\left(t\right)$. 15: Compute the discontinuous control input $u_{\mathrm{discon}}\left(t\right)=-{\mathsf{\ell }}_9·s\left(t\right)-{\mathsf{\ell }}_{10}·\mathrm{sign}\left(s\left(t\right)\right)$. 16: Apply the total control input $u\left(t\right)=u_{\mathrm{equiv}}\left(t\right)+u_{\mathrm{discon}}\left(t\right)$. 17: Step 3: Update system state 18: Apply the control input $u\left(t\right)$ to the system. 19: Step 4: Check for convergence 20: If the tracking error $\left|er\right(t\left)\right|$ is within the acceptable range, proceed with normal operation. 21: End Loop 22: Continue until the end of the simulation time $T_{\mathrm{sim}}$. 23: Output: Final values of angular speed, power output, and other performance metrics.

3.2. Design of MPPT Control Strategy Based on IBRTA

The IBRTA is an advanced control strategy designed to overcome the limitations of traditional SMC, particularly the chattering phenomenon. IBRTA combines the robustness of SMC with the smooth control action of backstepping and twisting algorithms, resulting in enhanced tracking performance and reduced chattering.

3.2.1. Design of the IBRTA Control Law

The IBRTA control law is composed of an ideal control input $u_{\mathrm{ideal}}\left(t\right)$ and a discontinuous control input $u_{\mathrm{discontinuous}}\left(t\right)$:

$$u\left(t\right)=u_{\mathrm{ideal}}\left(t\right)+u_{\mathrm{discontinuous}}\left(t\right)$$

(18)

The ideal control input is designed using a backstepping approach:

$$u_{\mathrm{ideal}}\left(t\right)=-{\mathsf{\ell }}_{5}e{r}_1-{\mathsf{\ell }}_{6}e{r}_2$$

(19)

where ${\mathsf{\ell }}_5$ and ${\mathsf{\ell }}_6$ are positive gains, and $e{r}_1=e\left(t\right)$ and $e{r}_2=\dot{e}\left(t\right)$ are the tracking errors. The discontinuous control input is designed to ensure robustness against disturbances and uncertainties:

$$u_{\mathrm{discontinuous}}\left(t\right)={\displaystyle \frac{-1}{L_g{L}_{f}h\left(x\right)}}\left({L_f}^{2}h\left(x\right)+{\mathsf{\ell }}_8{\dot{er}}_1+{\mathsf{\ell }}_9{s}_1+{\mathsf{\ell }}_{10}sign\left(s_1\right)\right)$$

(20)

where ${\mathsf{\ell }}_8$, ${\mathsf{\ell }}_9$, and ${\mathsf{\ell }}_{10}$ are positive gains that are tuned to optimize the control performance.

3.2.2. Lyapunov Stability and Convergence Analysis

The stability of the IBRTA control law is analyzed using a composite Lyapunov function $V_2(e{r}_1,s_1)$, defined as

$$V_2(e{r}_1,s_1)={\displaystyle \frac{1}{2}}\left(e{r}_1^2+s_1^2\right)$$

(21)

Taking the time derivative of $V_2$ and substituting the control law, we obtain

$${\dot{V}}_2=-{\mathsf{\ell }}_{8}e{r}_1^2-{\mathsf{\ell }}_9{s}_1^2-{\mathsf{\ell }}_{10}\left|s_1\right|$$

(22)

Since all gains are positive, ${\dot{V}}_2\le 0$, ensuring that the system is globally asymptotically stable and that the tracking error converges to zero.

3.2.3. Control Gains

The control gains used in the SMC and IBRTA are provided in

Table 2. Parameters used in control algorithms.

Control Method	Parameter	Value
SMC	Gain 1, ${\mathsf{\ell }}_1$	103
	Gain 2, ${\mathsf{\ell }}_2$	2000
	Gain 3, ${\mathsf{\ell }}_3$	0.01
	Gain 4, ${\mathsf{\ell }}_4$	50
IBRTA	Gain 1, ${\mathsf{\ell }}_8$	0.1
	Gain 2, ${\mathsf{\ell }}_9$	100
	Gain 3, ${\mathsf{\ell }}_{10}$	0.001
	Gain 4, ${\mathsf{\ell }}_{12}$	2
	Gain 5, ${\mathsf{\ell }}_{11}$	700

. These gains are tuned to achieve the desired closed-loop performance, ensuring robustness and stability under varying wind conditions. The control gains in Table 2 were initially selected based on physical system constraints and then fine-tuned through iterative simulation under stochastic and deterministic wind scenarios to ensure convergence, robustness and chattering suppression. The parameterized using real offshore turbine characteristics (

Table 3. Key parameters of system components.

Component	Parameter	Value
Wind Turbine	Air Density, ${\rho }_{air}$	1.2500 kg/m3
	Radius of Blades, $R_{blade}$	2.5000 m
	Optimal Tip-Speed Ratio, ${\lambda }_{lopt}$	7.000
	Transmission Gear Ratio, $i_{trans}$	7.000
	Maximum Power Coefficient, $C_{p{r}_{max}}$	0.476
	Mean Wind Velocity, $v_{wind}$	7.000 m/s
PMSG	Stator Resistance, $R_{stator}$	3.300 $\mathsf{\Omega }$
	Inductance of Load, $L_{chopper}$	0.00800 H
	Flux Linkage Constant, ${\varphi }_{constant}$	438.200 mWb
	Number of Pole Pairs, $p_{pole}$	3

), ensuring credible simulation outcomes

3.2.4. Inverter Model and Power Conversion

A VSI unit serves as a power electronic component to control the output voltage and current. The model of an inverter behaves as follows:

$$V_{d}c=V_{i}n-I_{d}c{R}_f-L_f\ast \left({\displaystyle \frac{I_{d}c}{I_{d}t}}\right)$$

(23)

where $V_{d}c$ is the DC-link voltage, $V_{i}n$ is the input voltage from the generator, $I_{d}c$ is the DC-link current, and $R_f-L_f$ are the filter resistance and inductance.

4. Simulation Results and Discussion

This section presents the simulation results of the proposed control strategies, evaluating their performance under two distinct scenarios: a stochastic wind speed profile and a deterministic offshore wind speed profile. The performance of the IBRTA, SMC, and FBL controllers is compared to assess their effectiveness in MPPT under varying wind conditions.

4.1. Stochastic Wind Speed Profile

In this scenario, we assess the robustness and adaptability of the control strategies in the presence of a stochastic wind speed profile. This profile is characterized by random variations in wind speed over time, representing real-world fluctuations that offshore wind turbines typically encounter. The simulations are conducted over a 100-s duration to observe the controllers’ ability to track the MPP and maintain optimal performance. Figure 2 presents the angular speed tracking performance of the three controllers. The FBL controller exhibits a noticeable steady-state error and oscillatory behavior, which indicates suboptimal performance in maintaining the desired speed. The SMC controller, while reducing the steady-state error compared to FBL, introduces chattering, as is evident from the oscillations around the reference speed. In contrast, the IBRTA controller significantly outperforms both SMC and FBL, displaying minimal steady-state errors and faster convergence, as highlighted in the zoomed-in section of Figure 2. The stochastic wind profiles used are based on a turbulence model validated in prior studies [13].

Further analysis is provided in Figure 3 and Figure 4, where the IBRTA controller demonstrates superior performance in maintaining the tip-speed ratio (TSR) and turbine power coefficient ($C_{pr}$) within optimal ranges. This indicates that the IBRTA controller is more effective in achieving MPPT under stochastic wind conditions compared to SMC and FBL. Additionally, Figure 5 shows that the IBRTA and SMC both outperform FBL in maintaining the mechanical power of the turbine shaft within the optimal range, confirming the elimination of chattering by the IBRTA strategy.

The results from this scenario suggest that the IBRTA controller offers significant improvements in steady-state accuracy and response time over the SMC and FBL controllers, making it the preferred choice for handling the unpredictability of offshore wind conditions.

4.2. Deterministic Offshore Wind Speed Profile

In this case study, the controllers’ performance is evaluated using a deterministic offshore wind speed profile characterized by abrupt and significant fluctuations in wind velocity.

This scenario aims at assessing the controllers’ performance in adjusting the rotor speed as a result of changes in wind speeds while at the same time ensuring maximum power production. As seen from Figure 6, all three controllers are able to track the deterministic wind speed profile. The SMC and FBL controllers have oscillations and sudden changes in their output, suggesting that they have some difficulty in managing the variation in wind speed.

The IBRTA controller, however, shows better performance and stability compared to the other controller, exhibiting less oscillation and better tracking of the desired speed. The effectiveness of the IBRTA controller is also emphasized in Figure 7 and Figure 8, where it has a better and smooth power coefficient ($C_{pr}$), and TSR in the face of the deterministic wind profile. This indicates that the proposed IBRTA controller is capable of MPPT despite sudden variations in wind speed. Figure 9 and Figure 10 show the power generated and power of the turbine respectively.

4.3. Discussion and Recommendations

From the simulation results of both scenarios, it can be observed that the proposed IBRTA control system outperforms the SMC and FBL controllers in terms of system stability, speed tracking, and power output. The IBRTA controller’s steady-state error reduction, chattering reduction, and optimal performance under both stochastic and deterministic wind conditions makes it the best strategy for MPPT in offshore wind energy systems. As for its application, power engineers are especially encouraged to employ the IBRTA controller when wind conditions are transient or volatile. The proposed IBRTA strategy offers several benefits in terms of energy yield optimization and offshore wind energy system reliability due to its flexibility and resilience.The IBRTA method shows important strengths against MPPT methods based on machine learning (ML), including the predictive features of the data-driven models. However, the IBRTA approach is superior regarding robustness and reliability, as shown in Figure 9 and Figure 10. ML approaches often exhibit good performance in conditions that they know or are trained under, but when tested in an untrained setting, they may underperform, necessitating retraining and large amounts of data. Also, they may be black boxes, which can impede explainability and responsiveness in real-time. MPPT methods based on adaptive control have superior parameter tuning capabilities and are capable of withstanding moderate changes in operating conditions. However, they tend to have limits when highly nonlinear time-varying or discontinuous dynamics are involved, as is the case in offshore wind systems. In the proposed IBRTA controller, which combines the backstepping methodology with robust sliding mode logic, a good compromise between deterministic robustness and smooth adaptive behavior is obtained. It guarantees fast convergence speed, good disturbance rejection, and minimal chattering without the need for any data-driven training. These features enable IBRTA to be a more high-performing, confident, and robust solution to maximum power point tracking, particularly in severe and varying environments like those faced in offshore renewable energy systems. Although controller design and initial validation were carried out with MATLAB/Simulink, a tool that is commonly employed in research studies due to its ease of use and flexibility, we recognize that industry-level simulation packages such as OpenFAST or Bladed provide higher fidelity for end-to-end system analysis. The present work concentrates on early-stage validation, with follow-up work expected to include the use of OpenFAST to make the simulations more robust and realistic. Our quantitative analysis shows that IBRTA achieves a 68 % reduction in chattering (as measured by the RMS error in high-frequency components) compared to SMC, validating its smoother control output and reduced actuator stress.

5. Conclusions

This research proposes a novel system for offshore PMSG-based wind energy conversion systems. A model that was initially based on three states is reduced to a two-state normal form with emphasis on output control. For the purpose of improving the control strategy, SMC is used, which allows for appropriate wind speed control across a range of conditions, including normal and deterministic offshore wind conditions. Therefore, the proposed MPPT strategy, especially the IBRTA, is evaluated using MATLAB/Simulink. The IBRTA strategy appears to be the most effective strategy among all of the discussed strategies, as it has potential to offer enhanced efficiency and reliability for offshore wind energy conversion.

Future work could explore how the application of artificial intelligence and machine learning might be combined with the IBRTA approach to better address the highly dynamic and nonlinear characteristics of offshore wind flow. Although not analyzed in the current study, such integration could offer self-tuning and predictive capabilities that further enhance the robustness and adaptability of MPPT control. For their real-world implementation, a pilot experiment of the proposed MPPT control methods must be developed on the basis of factors such as real-time compatibility with embedded processors (e.g., FPGA or DSP), the integration of SCADA systems into offshore wind farms without any downtime, and simulation against actual-site wind/load conditions. Safety-centric boundaries such as over-speed protection, hardware redundancy, and protection relays have to be implemented. In addition, performance should be evaluated using quantitative parameters like power capture efficiency, response time, mechanical loading, and control stability.