Robust Adaptive Super Twisting Algorithm Sliding Mode Control of a Wind System Based on the PMSG Generator

Abstract

In the field of optimizing wind system control approaches and enhancing the quality of electricity generated on the grid, this research makes a fresh addition. The Sliding Mode Control (SMC) technique produces some fairly intriguing outcomes, but it has a severe flaw in the oscillations (phenomenon of reluctance: chattering) that diminish the system’s efficiency. In this paper, an AST (adaptive super twisting) approach is proposed to control the wind energy conversion system of the permanent magnet synchronous generator (PMSG), which is connected to the electrical system via two converters (grid-side and machine-side) and a capacitor serves as a DC link between them. This research seeks to regulate the generator and grid-side converter to monitor the wind rate reference given by the MPPT technique in order to eliminate the occurrence of the chattering phenomenon. With the help of this approach, precision and stability flaws will be resolved, and the wind system will perform significantly better in terms of productivity. To evaluate the performance of each control in terms of reference tracking, response time, stability, and the quality of the signal sent to the network under different wind conditions, a detailed description of the individual controls is given, preceded by a simulation in the Matlab/Simulink environment. The simulation study validates the control method and demonstrates that the AST control based on the Lyapunov stability theory provides excellent THD and power factor results. This work is completed by a comparative analysis of the other commands to identify the effect on the PMSG wind energy conversion system.

1. Introduction

The need for all forms of energy is expanding. The world’s population is projected to grow by almost two billion people over the next twenty years. As a result of growing living standards, it is anticipated that between 2021 and 2040, energy consumption would [1] increase by 50% (by the International Energy Agency) (Figure 1).

Renewable energy sources [2,3] have the potential to meet a large part of the world’s current energy demand [4,5]. They can increase market diversity in the energy supply sector, as well as secure long-term, sustainable energy sources.

Figure 1. Energy consumption between 2021 and 2040 [6].

Figure 1. Energy consumption between 2021 and 2040 [6].

As well as lowering local and global emissions, they can also offer commercially attractive choices to meet specific energy service needs (particularly in developing countries and rural areas), provide new job opportunities, and create prospects for local equipment production.

Since 2020, the use of renewable energy sources like wind [6] and solar PV has grown quickly. When economies were weakening under the weight of COVID-19’s blockades, electric vehicles [7,8] set new sales records. The future energy economy will be more electric, efficient, networked, and environmentally friendly than the old one. Its rise is the result of a virtuous circle of policy action and technological innovation [9], and lower costs are currently sustaining its speed. Solar photovoltaic and wind power are now the least expensive sources of production of new electricity in the majority of markets. Clean energy [10] technology is rapidly emerging as a major new source of investment and job creation, as well as a vibrant field for international collaboration and rivalry.

Wind energy is a common source of fresh power production [11] and an important participant in the world energy market. The technical maturity and speed of deployment of wind power are recognized as leading energy technologies, as is the fact that there is no realistic maximum limit to the percentage of wind that can be connected to the electricity grid [12].

The total solar power absorbed by the planet is estimated to be around ($1.8\times {10}^{11}$ MW), and only 2% ($3.6\times {10}^9$ MW) of total solar input is transformed into wind power. Moreover, within 1000 m of the earth’s surface, over 35% of wind energy is lost. As a result, there is a total of 1.26 ${10}^9$ MW of wind energy that can be converted [13] into other sources of energy. This number is 20 times the current pace of global energy uses, suggesting that wind energy could hypothetically supply all of the world’s energy requirements.

A wind energy conversion system (WECS) [14,15] uses wind energy to generate mechanical energy, which is then sent to an electrical generator to generate electricity. The connectivity of a WECS is depicted in Figure 2. A permanent magnet synchronous generator [16,17] (PMSG) is employed in wind turbines due to the following benefits over other generators:

• Permanent magnet poles provide a high energy density.
• High efficiency, reliability.
• Self-excitation characteristics.
• Lightweight and simplicity of reactive power management.

WECSs are extremely nonlinear systems with a significant quantity of disturbances and uncertainty, according to the control literature [18]. As a result, current WECSs necessitate high-performance control approaches such as Backstepping control [19], fractional-order proportional-derivative control [20], and predictive control [21,22,23].

In furtherance of the control methods outlined previously mentioned, sliding mode control (SMC) is a prominent approach for sophisticated WECS control. SMC is a popular nonlinear resilient method that, in comparison to other sophisticated control systems, has a comparatively straightforward creation and execution procedure. The major drawback of this approach is the high-frequency oscillations of the state around the sliding commutator, known as “chattering”. This is why, in recent years, most SMC studies [24,25,26] using higher-order sliding mode (HOSM) [26] have been focused on strategies for chattering reduction.

The best-known control used by HOSM is the super-twisting method (ST) [27], which is increasing in popularity due to its ease of execution and efficacy.

The wind energy is received by the generator from the wind turbine. To extract the greatest power of the WECS, the spinning speed of the generator is regulated by a modulation converter. The generator’s power is sent to the grid through a generator-side converter (MSC) [28] and a grid-side inverter (GSC).

To manage active and reactive power to the grid, an adaptive SMC super-twist control (AST-SMC) is used.

The article is organized as follows: Section 1 is an introduction and provides a literature review of some renewable WECS studies on adaptive control [29,30]; Section 2 presents a description of our work and problem formulation; Section 3 introduces the modeling of a wind energy [31,32] conversion system; in Section 4, the sliding mode control [33] and the adaptive super twisting [34] of SMC are discussed with their application to the PMSG in view speed control; and Section 5 presents the performance results of the SMC and AST controls, followed by an analysis and comparison with other controls (

Table 1. Explanation of the paper’s acronyms.

Acronym	Version with More Information
PMSG	Permanent Magnet Synchronous Generator
WECS	Wind Energy Conversion System
SMC	Sliding Mode Control
AST	Adaptive Super Twisting
MSC	Machine-Side Converter
GSC	Grid-Side Converter
HOSMC	Higher Order Sliding Mode Control
MPPT	Maximum Power Point Tracking

).

2. Literature Review

Previous renewable WECS studies on adaptive control include:

Many researchers are constantly working on more sophisticated control algorithms that take into account the problems associated with nonlinear machine models in order to produce reliable electrical machine controls that resist the failures of traditional controls. Predictive control, adaptive Backstepping control, fuzzy logic control, sliding mode control (SMC), and others are examples of these control methods.

In this short review, certain control algorithms employed in the wind energy conversion system are published:

Wenping Cao et al. (2020) [26] present a new adaptive-model-based approach for estimating rotor position and speed as an observer. The suggested solution is based on model predictive control, which eliminates the requirement for speed and position sensors while simultaneously improving the performance of the model-referenced adaptive control system.

Jian Chen et al. (2019) [27]: The authors compare two controls based on a doubly-fed induction generator (DFIG-WT) used to convert wind energy: one focusing on vector control (VC) controllers and the other on a perturbation observer-based multiloop adaptive control (POMAC). POMAC outperforms VC in a variety of scenarios, including changeable wind speed situations, according to the simulation and experiment results.

Manuel Lara et al. (2021) [28] design a wind turbine adaptive control structure based on the pitch variable using multi-objective optimization, with the objectives to maintain the generator speed at its nominal value, minimize fluctuations, and maintain the generated power constant.

Saadatmand, M., Gharehpetian, G. B. (2021) [35] proposed a low-frequency oscillation damping system based on a fractional-order proportional-derivative controller for a (eolien- photovoltaic) park connected to synchronous generators. Finally, the suggested method’s performance is assessed under various operating situations of a reference intelligent system.

El Mourabit Youness et al. (2019) [36] improve the performance of the conversion system by studying the Backstepping control experimental validation for a permanent magnet synchronous generator (PMSG) wind turbine using the dSPACE DS1104 control board and the Matlab-Simulink environment in both static and dynamic operating modes.

Behnaz Babaghorbani et al. (2021) [37] propose a nonlinear strategy for predictive control of permanent magnet synchronous generator (PMSG) wind turbine systems based on the Lyapunov model. Based on the results, the proposed technique provides stability and performance and covers the DC link voltage during faults.

Tummala S. L. V. Ayyarao (2019) [38] develops a new vector control for a DFIG system that is resistant to external perturbations. For this purpose, the internal current loop is replaced by an adaptive sliding mode controller. Moreover, to prevent chattering and ensure limited time convergence, the control gains are selected based on a positive semi-definite barrier function.

Kanasottu Anil Naik et al. (2020) [39] construct and develop a DFIG interval type-2 fuzzy-PI controlled rotor side converter, which was used to estimate the gains of the PI controller when the system operating environment varied. As a result, an adaptive structure has been developed which is vital for regulating the DFIG’s rotor-side converter. The proposed technique’s performance has been studied for a variety of DFIG operating scenarios, including critical failures, power drops, and wind speed variations.

This brief literature review discusses recent WECS controls that are essentially based on PMSG and DFIG. The objective of this paper is to implement and validate the nonlinear adaptive super twisting sliding mode control for a high-power PMSG-based WECS. A detailed evaluation will be presented in this article, along with a comparison of the results with other research, to demonstrate this control’s usefulness.

3. The Paper Contribution

According to studies and the wind turbine industry, the most frequent generators used in variable-speed WECSs are PMSGs and doubly-fed induction generators (DFIGs). PMSGs are widely employed due to their ability to provide high torques. Compared to other types of generators, it produces less noise. At the same time, being small and lightweight, DFIGs [40] are also based on WECS and are a suitable solution for them.

The generator used in this work is the PMSG-based WECS. The main result of this study is the development of a robust continuous control approach based on sliding mode control with a super twist operation to reduce the impact of chattering effect on the control variable, with the aim of guaranteeing and increasing the stability and robustness of our system. The approach will then be called: Robust Adaptive Super Twisting Algorithm Sliding Mode Control (AST).

The originality of this work resides in the fact it introduces an innovative control topology that differs from most PMSG wind turbine systems based on sliding mode control. In the face of outside disturbances, the controller can maintain constant transient performance, minimize the load resulting from electrical power generation, and improve the quality of the electrical power supplied.

4. Problem Formulation

Due to system restrictions and parametric uncertainty, actual hardware attributes and behavior may differ from the model. Therefore, a controller must be robust to external disturbances to guarantee the robustness of the controller.

In this work, an adaptive super twisting sliding mode control is used to obtain good transient performance throughout the system’s operating range.

5. Wind Energy Conversion System Modeling

Creating an adaptive nonlinear super-torque sliding mode for PMSG-based WECS requires careful machine modeling. Therefore, analytical modeling of synchronous generators is a necessary step in order to obtain better results. Subsequently, the analytical model must be very similar to the real machine model.

The general model of the wind energy conversion system (Figure 2) consists of the following elements: the wind turbine connected to the PMSG using maximum power point tracking controllers [41]; and the machine-side converter (MSC) which transmits the energy extracted by the wind turbine to the grid-side converter (GSC), transmitting only active power to the grid.

5.1. Wind Turbine Model

A wind energy conversion system is a sophisticated device that converts wind energy to rotational energy and, eventually, electrical energy. Wind power (P_wind) is presented by [42,43]:

$$P_{wind}=\frac{1}{2}\times \rho \times \pi \times R^2\times v^3$$

(1)

The aerodynamic power captured by the wind turbine is [6,7]

$$P_{tur}=\frac{1}{2}\times C_p\left(\lambda ,\beta \right)\times \rho \times \pi \times R^2\times v^3$$

(2)

The mechanical torque of the wind turbine is [10,12]

$$T_m=\frac{P_{tur}}{\omega }=\frac{1}{2}\times C_p\left(\lambda ,\beta \right)\times \rho \times \pi \times R^3\times v^2$$

(3)

The power coefficient C_p or the aerodynamic efficiency of the wind turbine can be described as [44,45,46,47]

$$C_p\left(\lambda ,\beta \right)=\left(0.44-0.167\beta \right)sin \left(\frac{\pi \left(\lambda -3\right)}{15-0.3\beta }\right)-0.00184\left(\lambda -3\right)\beta$$

(4)

The Cp is as shown in Figure 3 has a theoretical maximum value of 0.59 known as the Betz limit: it depends on the pitch angle (β) and the tip speed ratio, known as the Tip Speed Ratio (TSR), indicated by (λ) [44,48]:

$$\mathrm{TSR}=\lambda =\frac{\omega R}{V}$$

(5)

5.2. PMSG Modeling

The PMSG equivalent model in the d-q reference system is seen in Figure 4.

A PMSG’s stator (d-q) voltages are written as

$$V_{ds}=R_s{i}_{ds}+\dot{\mathsf{\Psi }}{}_{sd}-{\omega }_e{\mathsf{\Psi }}_{qs}$$

(6)

$$V_{qs}=R_s{i}_{ds}+\dot{\mathsf{\Psi }}{}_{qs}+{\omega }_e{\mathsf{\Psi }}_{ds}$$

(7)

with:

$${\mathsf{\Psi }}_{ds}=L_d\times i_{ds}+{\mathsf{\Psi }}_f$$

(8)

$${\mathsf{\Psi }}_{qs}=L_q\times i_{qs}$$

(9)

The following are the equations:

$$V_{ds}=R_s{i}_{ds}+L_d\frac{d{i}_{ds}}{dt}-{\omega }_e{L}_q\times i_{qs}$$

(10)

$$V_{qs}=R_s{i}_{qs}+L_q\frac{d{i}_{qs}}{dt}+{\omega }_e{L}_d\times i_{ds}+{\omega }_e{\mathsf{\Psi }}_f$$

(11)

The PMSG’s electromagnetic torque is characterized as

$$T_{em}=\frac{-3}{2} p\left[{\mathsf{\Psi }}_{ds}\times i_{qs}-{\mathsf{\Psi }}_{qs}\times i_{ds}\right]$$

(12)

$$T_{em}=\frac{-3}{2} p\left[(L_d-L_q)i_{ds}{i}_{qs}+{\mathsf{\Psi }}_f\times i_{qs}\right]$$

(13)

$$T_{em-ref}=\frac{-3}{2} p\times {\mathsf{\Psi }}_f\times i_{qs}$$

(14)

(For a machine with plain poles [9,11] $(L_d=L_q$)).

Powers of active and reactive expression

$$P_{gen}=T_{em}\times \Omega =\frac{3}{2}·\left[V_{ds}\times i_{ds}+V_{qs}\times i_{qs}\right]$$

(15)

$$Q_{gen}=\frac{3}{2}·\left[V_{ds}\times i_{ds}-V_{qs}\times i_{qs}\right]$$

(16)

5.3. DC-Link Modeling

The DC-link is the link that connects the generator to the power grid. The voltage equation for a dynamic DC-link [35] can be written as

$$C_{dc}=\frac{d{V}_{dc}}{dt}=\frac{d{P}_1}{V_{dc}}-\frac{d{P}_2}{V_{dc}}$$

(17)

$$\frac{C_{dc}}{2}\frac{d{\left(V_{dc}\right)}^2}{dt}=\frac{C_{dc}}{2}s{\left(V_{dc}\right)}^2=P_1-P_2$$

(18)

with $P_1=V_s I_{sd}$ being the generator’s power output and $P_2=V_g I_{gd}$ being the grid-side converter’s power output.

5.4. GSC Modeling

GSC’s mathematical model can be summarized as follows [49,50]:

$$L_g\frac{d{i}_{gd}}{dt}=V_{gd}-R_g{i}_{gd}+{\omega }_g{L}_g{i}_{gq}-u_{gd}$$

(19)

$$L_g\frac{d{i}_{gq}}{dt}=V_{gq}-R_g{i}_{gq}-{\omega }_g{L}_g{i}_{gd}-u_{gq}$$

(20)

where ${\omega }_g$, $L_g$, and $R_g$ signify the power grid’s angular frequency, grid filter inductance, and grid filter resistance, respectively; $V_{gd}$, $V_{gq}$, $u_{gd}$, and $u_{gq}$ denote the grid and GSC output dq voltages; and $i_{gd}$ and $i_{gq}$ denote the d–q currents injected from the GSC into the distribution grid (

Table 2. Experimental System (Turbine+ PMSG) Parameters.

Symbol	Parameter	Value
$N_p$	Pole pairs	75
$R_s$	Nominal stator resistance	6.25 × 10−3 Ω
$Lsd$	d axis inductance	4.229 × 10−3 H
Lsq	q axis inductance	4.229 × 10−3 H
J	Mechanical inertia moment	10,000 N·m
R	Rotor radius	55 m
$\rho$	Air density	1.25 kg/${\mathrm{m}}^3$
$\omega$	The rotor angular speed	rad/s
$P_r$	Rated power	1.5 kw
C	DC-link nominal voltage	V
$\varnothing f$	Generator flux	11.1464 Wb
Λopt	Tip-speed ratio	8

).

6. Sliding Mode Control

Sliding mode control was developed following innovative work in the former Soviet Union in the 1960s. It is a specific type of variable-structure system with one decision rule and several feedback control rules.

The design of an SMC involves two actions: (the first) the creation of a stable sliding surface to achieve the desired dynamics; and (the second) the creation of a control law to ensure that states reach the selected sliding surface in a finite time and remain there.

The sliding surface [49,51]:

$$S\left(x,t\right)={\left(\frac{d}{dt}+\lambda \right)}^{n-1} e\left(t\right)$$

(21)

• 𝜆 is a positive coefficient.
• n is the order of the system.
• $e\left(t\right)$ is the error in the output state with: $e\left(t\right)=x_{ref}\left(t\right)-x\left(t\right).$

The convergence condition is specified by the Lyapunov equation:

$$S\left(x\right)\times \dot{S}\left(x\right)<0$$

(22)

Two components make up the controller structure $u_{eq}\left(t\right)$ and $u_N\left(t\right)$ with

$$u\left(t\right)=u_{eq}\left(t\right)+u_N\left(t\right)$$

(23)

• $u_{eq}\left(t\right)$, the equivalent command, is calculated based on the system behavior S(x) = 0 to precise linearization of the system.
• $u_N\left(t\right)$ is utilized to verify the convergence condition of Lyapunov with $u_N\left(t\right)=K\times sgn\left(S\left(x\right)\right)$, where:

  $$sgn\left(S\left(x\right)\right)=\left\{\begin{array}{c}1 if S\left(x\right)<0 \\ 0 if S\left(x\right)=0\\ -1 if S\left(x\right)>0\end{array}\right.$$

  (24)

6.1. State-of-the-Art SMC Implementation for PMSG Control

In order to reduce or almost eliminate chattering, several researchers proposed numerous novel SMC configurations. The image below depicts increased SMC control structures that were developed to enhance traditional SMC Figure 5. The higher-order sliding mode control method (HOSMC) [50,52] has been developed in the literature by (Bartolini et al., 1998; Levant, 1993). The principal purpose of this control is to retain the advantages of the original strategy by acting on the higher-order time derivatives of the system’s deviation from the constraint rather than on the first derivative, as is the case in conventional sliding modes.

To achieve fast, finite-time convergence of the state, terminal sliding mode control (TSMC) [37] uses terminal sliding surfaces where fractional power is introduced. The rate of convergence accelerates close to the equilibrium point, making this controller the best choice for high-precision control. The two-phase control approach in TSM has been used to overcome the problem by ensuring that the system state is non-singular before applying finite-time convergence control.

Many works have merged SMC to remove the chattering problem, such as artificial intelligence [38], adaptive control technique, sliding mode predictive control [39], and reach law SMC [40]. The main advantages of these techniques are their independence from specific prior knowledge of the system dynamics and their ability to suppress errors caused by parameter variation.

To realize robust control for uncertain systems, Backstepping control [29] and sliding mode control (SMC) are integrated to provide the Backstepping-SMC command.

The traditional SMC controller is built for a single operating point, as is common knowledge. Fuzzy SMC [41] is created to address the issues with fixed parameters SMC because the SMC is frequently not calibrated properly. It should be emphasized that conventional SMC does not work well when there are load torque disturbances. The observer SMC [42] (DOSMC) is developed to put an end to this. It has a viewer that provides a tool to eliminate the effects of load torque.

6.2. Higher Order Sliding Mode Control

Higher-order sliding modes [26] are designed to reduce the chattering problem while retaining finite-time convergence properties; they also improve asymptotic accuracy.

In this approach, the discontinuous term no longer appears directly in the synthesized control but in one of its higher derivatives, thus reducing chattering.

The aims are to force the system to evolve on the surface and on its (r − 1) first derivative and to keep the sliding sum at zero.

$$S=\dot{S}=\cdots =S^{r-1}=0$$

(25)

6.3. Sliding Mode Control Using the Super-Twisting Method (ST-SMC)

The super twisting technique [44] is a good solution for second-order slip modes, as it only requires knowledge of the surface S and not of the other derivatives. It is a continuous regulator that ensures all of the first-order SMC control qualities for the system with matching limited uncertainties and disturbances. The super twisting algorithm’s phase plane trajectory is represented in Figure 6.

It is composed of a discontinued part and a $u_2$ continuous part $u_1$:

$$u=u_1\left(t\right)+u_2\left(t\right)$$

(26)

$${\dot{u}}_1=\left\{\begin{array}{c}-u if \left|u\right|>1\\ -W sgn\left(S\right) if \left|u\right|\le 1\end{array}\right.$$

$${\dot{u}}_2=\left\{\begin{array}{c}-{\beta }_1{\left|S_0\right|}^{\rho 1} sgn\left(S\right) if \left|u\right|>S_0\\ -{\beta }_1{\left|S\right|}^{\rho 1} sgn\left(S\right) if \left|u\right|\le S_0\end{array}\right.$$

The finite-time convergence conditions

$$\left\{\begin{array}{c}\alpha >\frac{\varphi }{{\Gamma }_m}\\ {\beta }_{\mathrm{1,2}} \ge \frac{4\varphi {\Gamma }_M(w+\varphi )}{{{\Gamma }_m}^2{\Gamma }_m(w-\varphi )}\\ 0<\rho 1\le 0.5\end{array}\right.$$

(27)

The gains of the super twisting sliding mode controller are

$${\Gamma }_M=0.967 ; {\Gamma }_m=0.76$$

These are positive terms that are determined by the second derivative of the sliding surface S, with ($\varphi$) being the disturbance [52].

$$\ddot{S}=\varphi \left(x,t\right)+\Gamma \left(x,t\right)\dot{u}$$

(28)

$${\Gamma }_m<\Gamma \le {\Gamma }_M$$

(29)

The simplified control law is

$$\left\{\begin{array}{c}u=-{\beta }_1{\left|s\right|}^{\frac{1}{2}} sgn\left(S\right)+u_1\\ {\dot{u}}_1={\alpha }_{i}sgn\left(S\right)\end{array}\right.$$

(30)

6.4. Application of the AST to the PMSG

The organization of sliding mode control order two using AST is presented in (Figure 7). The internal circuit controls the currents ($i_{ds},i_{qs}$), while the external circuit controls the speed ${\omega }_m$.

We used the following three sliding surfaces:

$$S_1={\omega }_{mref}-{\omega }_m S_2=i_{qref}-i_{qs} S_3=i_{dref}-i_{ds}$$

(31)

The super twisting control law is

$$u=-{\beta }_i{\left|s\right|}^{\frac{1}{2}} sgn\left(S\right)+u_i \dot{u_i}={\alpha }_{i}sgn\left(S\right) i=1,2,3$$

(32)

• Generator rotation speed regulation

The equation of the electromagnetic couple is defined by [49]:

$$T_{em-ref}=\frac{-3}{2} \times {\mathsf{\Psi }}_f\times i_{sqref}$$

(33)

$$i_{sqref}=\frac{2}{3\times p\times {\mathsf{\Psi }}_f} T_{em-ref}$$

(34)

we have: $T_{em-ref}=u_1+u_2$

$$\dot{u_1}=-{\alpha }_{1}sgn\left(S_{{\omega }_m}\right) u_2=-{\beta }_1{\left|S_{{\omega }_m}\right|}^{\frac{1}{2}} sgn\left(S_{{\omega }_m}\right)$$

(35)

The current equation becomes:

$$i_{sqref}=\frac{2}{3\times p\times {\mathsf{\Psi }}_f} T_{em-ref}=\frac{2}{3\times p\times {\mathsf{\Psi }}_f}-({\beta }_1{\left|S_{{\omega }_m}\right|}^{\frac{1}{2}} sgn(S_{{\omega }_m})+{\alpha }_{1}sgn(S_{{\omega }_m}))$$

(36)

6.5. Regulation of Stator Currents

We will adjust the components of the currents $i_{ds}$ and $i_{qs}$ to their references $i_{dref}, i_{qref}$.

$$\left\{\begin{array}{c}{S}_{I_{ds}}=I_{dref}-I_{ds}\\ S_{I_{qs}}=I_{qref}-I_{qs}\end{array}\right.$$

(37)

The control voltages for the d and q axes are defined as follows:

$$V_{dsref}={\mu }_1+{\mu }_2-Fe{m}_d ; V_{dqref}=W_1+W_2+Fe{m}_q$$

(38)

$$W_2=-{\beta }_2{\left|S_{I_{ds}}\right|}^{\frac{1}{2}} sgn\left(S_{I_{ds}}\right) ; \dot{W_1}=-{\alpha }_{2}sgn\left(S_{I_{ds}}\right); Fe{m}_d=p\times {\omega }_m\times L_q\times I_{qs}$$

(39)

$$W_2=-{\beta }_3{\left|S_{I_{qs}}\right|}^{\frac{1}{2}} sgn\left(S_{I_{qs}}\right); \dot{W_1}=-{\alpha }_{3}sgn\left(S_{I_{qs}}\right); Fe{m}_q=p\times {\omega }_m\times \left(L_d\times I_{ds}+{\mathsf{\Psi }}_f\right)$$

(40)

$Fe{m}_d$ and $Fe{m}_q$ are compensation terms.

The control voltages become:

$$V_{dsref}={\displaystyle \int }-{\alpha }_{2}sgn\left(S_{I_{ds}}\right)dt-{\beta }_2{\left|S_{I_{ds}}\right|}^{\frac{1}{2}} sgn\left(S_{I_{ds}}\right)-p\times {\omega }_m\times L_q\times I_{qs}$$

(41)

$$V_{dqref}={\displaystyle \int }-{\alpha }_{3}sgn\left(S_{I_{qs}}\right)dt-{\beta }_3{\left|S_{I_{qs}}\right|}^{\frac{1}{2}} sgn\left(S_{I_{qs}}\right)+p\times {\omega }_m\times \left(L_d\times I_{ds}+{\mathsf{\Psi }}_f\right)$$

(42)

To verify that the sliding surface converges to zero in limited time, the gains can be chosen as follows:

$$\left\{\begin{array}{c}{\alpha }_i>\frac{\varphi }{{\Gamma }_m}\\ {\beta }_i \ge \frac{4\varphi {\Gamma }_m(w+\varphi )}{{{\Gamma }_m}^2{\Gamma }_m(w-\varphi )}\\ {\rho }_i=0.5\end{array}\right.$$

(43)

The control voltages (for the GSC) for the d and q axes are defined as follows:

$$V_{gd}=R_g{i}_{gd}-{\omega }_g{L}_g{i}_{gq}+u_{gd}+L_g\frac{d{i}_{gd}}{dt}$$

(44)

$$V_{gq}=R_g{i}_{gq}+{\omega }_g{L}_g{i}_{gd}+u_{gq}+L_g\frac{d{i}_{gq}}{dt}$$

(45)

6.6. Adaptive Super Twisting Algorithm (AST)

To alleviate the chattering problem, high-order sliding mode control [46] is an efficient solution. A control degree with «d» affects «d» derivatives. This feature helps to reduce unwanted chattering while maintaining the robustness of the SMC technique. The development of an «r» degree controller, on the other hand, necessitates knowledge of the successive sliding surfaces. The super twisting algorithm is an exception, as it simply requires information on the sliding surfaces ‘S’.

The adaptive and standard ST algorithms differ in that the controller gains $\mathrm{and} {\alpha }_i \mathrm{and} {\beta }_i$ are calculated automatically. Moreover, rather than being forced to zero, the sliding variables are forced to a specific neighborhood ($𝜇$) of the sliding surfaces.

The adaptive super twisting technique is presented to increase the WT-PMSG system’s performance and efficacy.

To solve this difficulty, this research develops a new gain adaptive:

$${\alpha }_1=2{\epsilon }_1{\beta }_1$$

$$\dot{{\beta }_1}={\omega }_{1}sign(\left|S_1\right|-{\mu }_1);if {\beta }_1>{\beta }_{1m} {\eta }_1;if{\beta }_1\le {\beta }_{1m}$$

(46)

$${\alpha }_2=2{\epsilon }_2{\beta }_3$$

$$\dot{{\beta }_2}={\omega }_{2}sign(\left|S_2\right|-{\mu }_2);if {\beta }_2>{\beta }_{2m} {\eta }_2;if{\beta }_2\le {\beta }_{2m}$$

(47)

$${\alpha }_3=2{\epsilon }_3{\beta }_3$$

$$\dot{{\beta }_3}={\omega }_{3}sign(\left|S_3\right|-{\mu }_3);if {\beta }_3>{\beta }_{3m} {\eta }_3;if{\beta }_3\le {\beta }_{3m}$$

(48)

To ensure system convergence (using the Lyapunov theory), all controller gains $\omega , \eta , \epsilon , \mu , {\beta }_m$ should be positive, and they are calibrated to accomplish the appropriate reaction time and predefined sliding variable neighborhood (

Table 3. Several methods used for SMC control.

	Methods	Researchers
	Higher order-SMC	Ozer, H. O., Hacioglu, Y., & Yagiz, N. (2017) [26]
	Second order-SMC	Aschemann, H., Haus, B., & Mercorelli [36]
	Terminal-SMC	Rajendiran, S., & Lakshmi, [37] Lochan, K., Roy, B. K., & Subudhi, B. [43]
	Integral-SMC	Singh, P. P., & Roy, B. K. [45]
Sliding Mode Control	Backstepping-SMC	Majout, B., El Alami, H., Salime, H., Zine Laabidine, N. [17]
	Direct power control-SMC	Ouchen, S., Benbouzid, M., Blaabjerg, F., Betka, A., & Steinhart, H. [44]
	Fuzzy logic-SMC	Charfeddine, S.; Boudjemline, A.; Ben Aoun, S.; Jerbi, H.; Kchaou, M.; Alshammari, O.; Elleuch, Z.; Abbassi, [41]
	Artificial neural network-SMC	Hamid Chojaa, Aziz Derouich, Seif Eddine Chehaidia, Othmane Zamzoum, Mohammed Taoussi, Hasnae Elouatouat [38]
	Observer-SMC	Ali, N., Ur Rehman, A., Alam, W., & Maqsood, H. [42]
	Predictive sliding mode	Sun, X., Cao, J., Lei, G., Guo, Y., & Zhu, J. [39]
	Reaching Law-SMC	Chen, X., Li, Y., Ma, H., Tang, H., & Xie, Y. [40]

).

The values of the variable gains utilized during the tests are shown in

Table 4. Gains for the control algorithm’s controllers.

Parameter	Value	Parameter	Value	Parameter	Value	Parameter	Value	Parameter	Value
${\omega }_1$	50 × 105	${\mu }_1$	0.15	${\beta }_{1m}$	50	${\eta }_1$	80	${\epsilon }_1$	1
${\omega }_2$	60 × 105	${\mu }_2$	0.15	${\beta }_{2m}$	66	${\eta }_2$	98	${\epsilon }_2$	1
${\omega }_3$	58 × 105	${\mu }_3$	0.15	${\beta }_{3m}$	62	${\eta }_3$	100	${\epsilon }_3$	1

. For the numerical validation, ${\omega }_i$ (with i = 1, 2, 3) is selected to account for the requirements suggested in Equations (44)–(46) and to ensure $S=\dot{S}=0$ at all times; each ${\epsilon }_i$ is proposed to be equivalent to one. The adaptive super twisting algorithm thus ensures the continuity of $\dot{{\beta }_1},\dot{{\beta }_2},\dot{{\beta }_3}$ that converges to zero in finite time. Furthermore, each $W_i$ will be compensated for all t $>$ 0.

Some ideas from [49] are used for this stability analysis.

7. Results and Discussion

In this part, detailed simulation studies using a 1.5-kW WECS system are provided. The simulations were carried out using MATLAB/Simulink 2020a. Table 2 displays the nominal characteristics of the wind turbine and PMSG system. The wind speed profile (Figure 8) depicts a turbulent wind speed fluctuation of 2.7 to 8.3 m/s with a grid frequency of 50 Hz.

The SMC and AST methods, respectively, are used to control the generator to demonstrate reference tracking performance and controller resilience.

As shown in Figure 9 and Figure 10, respectively, the mechanical speed deviations follow its references. It is affected by both turbine design and wind speed, and the mechanical power shape is comparable to that of the wind.

Figure 11 and Figure 12 show active and reactive power. Figure 11 shows the active power, shaped like the wind profile, with negative values since the machine is in generator mode. The reactive power amount shown in Figure 13 is kept at a highly acceptable level in comparison to the immense power of the employed generator. The relatively small reactive power figure indicates that the installation has a unity power factor.

Oscillations are visible in the shape of the active and reactive power control. Unexpected departures that are rapidly directed toward zero are observed (Figure 13 and Figure 14). These transitions occur when the wind turbine switches between partial and full load regions; this is also because the disturbance terms and their variations over time are higher than the maximum level at which the controls have been configured.

Figure 12 shows a response time of around 0.025 s using the SMC control and 0.023 s with AST control: AST control provides faster response.

The three-phase injected currents ig-abc are illustrated in Figure 15 and Figure 16. In SMC, some undesirable distortion arises in their morphologies (Figure 17), whereas the injected current waveform acquires a more optimal sinusoidal shape (Figure 18) using AST control. This guarantees that the currents injected into the grid are of high quality and that they are in phase with the grid voltages.

However, a harmonic analysis of the grid current was carried out to examine the impact of these two controls on the quality of the signal delivered to the grid, as shown in Figure 19. We notice that compared with the result produced by the total harmonic distortion (THD) of the SMC (THD = 3.01%; Figure 17 right), the THD achieved by the suggested AST control (Figure 17 left) was significantly reduced (THD = 1.24%).

Figure 20 shows that the power factor is equal to 0.984 with a changing wind profile using the SMC simulation. However, with the AST approach, this rate reaches 0.999. These observed values explain the high quality and performance of the electricity generated.

Figure 21 depicts the progression of the DC bus voltage in both controls. According to these results, the adaptive approach provides greater $V_{DC}$ voltage regulation and stays stable despite wind variations.

Compared with the first-order sliding mode controller [24], which exhibits undesirable chatter, the adaptive super twisting control offers flawless performance, and the problem is mitigated. The evolution of PMSG stator currents [27] demonstrates that the super twisting controller can track changes in wind speed better than the SMC approach.

An evaluation table is used to compare the results with some recently published research to validate the AST control.

Table 5. Comparison of the suggested approach to those used in previous articles.

Publication Paper	Control Methods	Response Time (s)	Power Factor	Overshot (%)	Performance
[47]	Direct Control	1	0.97	12	Low
[48]	Foc	1	-	-	Low
[29]	Backstepping	0.029	0.99	-	Robust
[51]	Fractional SMC	-	0.98	-	Robust
[17]	MPC	0.05	0.98	1.17	Medium
Proposed SMC	SMC	0.025	0.98	1.12	Robust
Proposal Adaptive Control	AST	0.023	0.99	0	High

compares some of the results, which generally attest to the quality of the proposed control.

8. Conclusions

The present research proposes two types of nonlinear controllers for use in wind energy conversion systems: first-order SMC sliding mode control and super twisting controllers [49] (second-order SMC) [36]. The theory of the two applied algorithms is presented with an application to the PMSG wind energy conversion system [17] for robust power control. MPPT control is also used in this study to achieve the highest power output under rapidly changing wind profiles.

Using a variable wind profile, a simulation test of both controllers was carried out in MATLAB/Simulink software. The results show that the first-order SMC controller exhibits undesirable chattering due to its discontinuous Vn term, which has an impact on system performance. Compared with the first order, the adaptive super twisting control exhibits extremely low oscillations, indicating a reduction in the chattering phenomenon. Simulation results indicate that a wind energy conversion system operates more efficiently, both statically and dynamically, when using the AST control approach.

Future work will focus on the experimental validation of super twisting control using dSPACE DS1104 to verify and validate the effectiveness of the PMSG-based wind energy conversion system.