The analysis of the PMSG-WECS involves a wide array of constants and parameters. These include critical factors such as the mechanical power output of the turbine (P _M ), wind speed expressed in meters per second (V _wind), air density (ρ), and area swept by the turbine blades (A). Additionally, the conversion of the wind turbine rotor power is governed by the wind turbine rotor power conversion coefficient (Γ_wind) and the blade radius is represented by R t . In this context, variables of significant importance include the tip speed ratio (λ), blade pitch angle (β), and optimal tip speed ratio (λ _opt). The analysis also considers the maximum power coefficient ( C p max ) and other parameters relevant to the PMSG-WECS.

Turning to the modeling of the PMSG itself, a meticulous examination of the electrical properties was conducted. This includes variables such as the stator resistance ( R s ) , permanent magnet flux variables (ω _m and ϕ _m ), rotor inductances along the d-axis and q-axis (L _d and L _q ), and rotor inductance (L), particularly for non-salient PMSG configurations. Furthermore, the analysis explores variables, including the currents along the d-axis (i _d ) and q-axis (i _q ), as well as the high-speed shaft speed (ω _h ). Various constants (d ₁, d ₂, d ₃) are also incorporated into the modeling process of the PMSG-WECS system. The system introduces a set of state variables, specifically x ₁, x ₂, and x ₃, which effectively represent the state of the PMSG-WECS system. These state variables evolve dynamically over time, guided by the state vector field (f(x)). Within this framework, we also consider a matrix of smooth functions (g(x)) and output function (h(x)). Here, the Lie derivatives ( L f and L g ) play a significant role in investigating the dynamics of the system.

Control strategies played an integral role in the analysis. A control input (u) is introduced into the system, and the design incorporates constants such as α z , β z , and γ z ), which are of paramount importance in the realm of sliding mode control. Elements, such as the sliding surface ( σ z ) and tracking error ( e z ) , are fundamental components of the control framework. The derivative of the sliding surface ( σ ̇ z ) provides valuable insights into the behavior of the system. The design also incorporates control inputs for strong reachability factors (u _q(equ)) and discrete control (u _q(dis)) along with pertinent control constants ( K c and K d ). Throughout the analysis, specific indexes are judiciously applied to signify optimal values (opt) and variables associated with sliding mode control ( z ) to enhance clarity and facilitate a deeper understanding of the subject matter.

The following set of mathematical equations describes a wind turbine with fixed-pitch configuration Zafran et al. (2020); Khan et al. (2021): P M = 1 2 ρ A V wind 3 = Γ wind ω t (1) where P _M represents the turbine output mechanical power, V _wind is the wind speed in m/s, ρ denotes the air density, and A is the area swept by the blades. The parameter Γ _wind is defined as: Γ wind = 0.5 ρ π R t 3 V w 2 C T λ (2) The dynamic modelling of the variable-speed standalone PMSG-WECS involves key parameters such as the blade radius R t in meters, wind turbine rotor power conversion coefficient C _T (λ), and tip-speed ratio λ = R t ω t V wind . Additionally, the air density ρ (usually 1.25 kg/m³ at 100°C at sea level), power coefficient C _p , blade pitch angle β, and the optimal tip-speed ratio λ _opt = 7 play vital roles in maximizing power extraction under varying wind speeds. To achieve optimal power extraction under varying wind conditions, the blade pitch angle β is set to 0, resulting in a maximum power coefficient C _pmax = 0.47 at the corresponding tip speed ratio λ _nom or λ _opt . The mechanical power produced by the wind turbine at different wind speeds is illustrated in Figure 2. At each wind speed, a distinct maximum power point P _mopt is evident, which collectively defines the optimal regime characteristic (ORC). The ORC designates a specific operating region in which the WECS efficiently extracts the maximum energy, even under varying wind conditions. The MPPT control strategy ensures that the wind turbine operates within the designated ORC despite fluctuations in wind speed. The MPPT control scheme continuously adjusts the turbine operation, aligns it with the ORC, and maximizes the power extraction efficiency.

For the variable speed wind turbine reported in Munteanu et al. (2009), C _p can be determined by the numerical following approximation while considering β = 0⁰: C p λ = 0 + 0.0061 λ − 0.0013 λ 2 + 0.0081 λ 3 − 0.000974 λ 4 + 0.0000654 λ 5 + 0.00000130 λ 6 − 0.000000454 λ 7 (3) The VSWT is optimized to track the maximum value of C _p , denoted as C p max , which occurs at a specific tip speed ratio λ _opt . By maintaining the tip-speed ratio at its optimal value λ _opt , VSWT can effectively extract the maximum wind power Chand et al. (2023); Majout et al. (2022); Zafran et al. (2020).

In this study, the critical wind turbine parameters include the air density ρ (set to 1.25 kg/m³), gears ratio or transmission ratio i (equal to 7), blade radius R t (measuring 2.5 m), maximum power coefficient C p max (equal to 0.47), optimal tip-speed ratio λ _opt (valued at 7), and high-speed shaft inertia J h (measured as 0.0552 kg ⋅m²) Chand et al. (2023); Majout et al. (2022); Khan et al. (2022, 2021); Ullah A. et al. (2020); Alam et al. (2022).

Having discussed the essential rotor blade parameters, the next section briefly explores them.

A simplified dq model of the three-phase PMSG in the dq-reference frame, neglecting the zero components, is utilized for the dynamical modelling of the variable speed standalone WECS. The dq-axis voltages represent output variables. The following set of differential equations describes the PMSG model. v d = R a i d + L d d d t i d − L q i q v q = R a i q + L q d d t i q + L d i d + ϕ m (4) where v _d and v _q represent the voltages along the dq-axes, R s is the stator resistance, ω _m is the permanent magnet flux, and ϕ _m is the maximum magnetic flux. The rotor inductances L _d and L _q are assumed to be equal (i.e., L _d = L _q = L) for a non-salient PMSG. The PMSG-WECS system has three states: i _d , i _q , and ω _h . To design a controller, the PMSG equations are reformulated as nonlinear dynamic equations as follows: L d + L c h x ̇ 1 = − R s x 1 + p L q − L c h x 2 x 3 − R ini x 1 L q + L c h x ̇ 2 = p L q + L c h ϕ m x 3 − R s x 2 − p L d + L c h x 1 x 3 − R ini x 2 J h x ̇ 3 = d 1 v w 2 i + d 2 v w x 3 i 3 + d 3 x 3 2 i 3 − p ϕ m x 2 (5) where p denotes the number of pole pairs, J h represents the moment of inertia of the PMSG high-speed shaft, and R ini and L _ch refer to the resistance and inductance of the load, respectively.

The system, characterized by its input vector u = R ini ∈ R m , output vector y = h ( x ) ∈ R l , and state vector x ∈ R n , can be presented in a general form as follows: x ̇ = f x + g x u y = h x (6) The nonlinear smooth state vector field f ( x ) ∈ R n with a matrix of smooth functions g ( x ) ∈ R n × m can be expressed as: f x = − R s x 1 + p L q − L c h x 2 x 3 L d + L c h − R s x 2 − p L d + L c h x 1 x 3 L q + L c h + p ϕ m x 3 d 1 v w 2 i + d 2 v w x 3 i 3 + d 3 x 3 2 i 3 − p ϕ m x 2 1 J h ; g x = − x 1 L d + L c h − x 2 L q + L c h 0 ; x = i d i q ω h (7) Now, by utilizing Equations 6, 7, we can represent the PMSG-WECS model in state-space form as follows: x ̇ 1 x ̇ 2 x ̇ 3 = − k 1 x 1 − k 2 x 2 x 3 − l 1 x 2 − l 2 x 1 x 3 + l 3 x 3 − m 1 − m 2 x 3 − m 3 x 3 2 − m 4 x 2 + − k 3 x 1 − l 4 x 2 0 u y = h x = 0 0 1 x 1 x 2 x 3 (8) where k 1 = R s ( L d + L c h ) , k 2 = − p ( L q − L c h ) ( L d + L c h ) , k 3 = 1 ( L d + L c h ) , l 1 = R s ( L q + L c h ) , l 2 = p ( L d + L c h ) ( L q + L c h ) , l ₃ = pϕ _m , l 4 = 1 ( L q + L c h ) , m 1 = − d 1 v w 2 i J h , m 2 = − d 2 v w i 3 J h , m 3 = − d 3 i 3 J h and m 4 = p ϕ m J h .

To determine the relative degree r of the considered system, one can ensure the satisfaction of the following equation is satisfied: L g L f r − 1 h x ≠ 0 for r = 1,2,3 , … (9) where L g and L f are the Lie derivatives of the nonlinear functions g(x) and f(x) with respect to the system state vector x, respectively, and h(x) is a output function of the observer state vector x.

By implementing (Eq. 9), the following expressions are obtained. L g h x = 0 L f h x = − m 1 − m 2 x 3 − m 3 x 3 2 − m 4 x 2 L g L f h x = l 4 m 4 x 2 ≠ 0 (10) This confirms that the studied system had a relative degree of 2. Subsequently, we transform the system into an input-output form in the following subsection and design the control input u accordingly.

Prior to expressing the system in a standard canonical form (normal form or input-output form), a coordinate transformation for the system is required. This transformation is performed as follows: z = z 1 , L f h x , L f 2 h x , … , L f r − 1 h x (11) where z 1 = x 3 z 2 = L f h x = − m 1 − m 2 x 3 − m 3 x 3 2 − m 4 x 2 z 3 = L f 2 h x = x 1 x 2 (12) and the inverse coordinates transform are x 1 = z 3 m 1 − z 2 − m 2 z 1 − m 3 z 1 2 / m 4 x 2 = m 1 − z 2 − m 2 z 1 − m 3 z 1 2 / m 4 x 3 = z 1 (13) Now, the system dynamics in the z -domain can be written as z ̇ 1 = x ̇ 3 = z 2 z ̇ 2 = L f 2 h x + L g L f h x u (14) where L f 2 h ( x ) = − m 4 f 2 − ( m 2 + 2 m 3 x 3 ) f 3 and L g L f h ( x ) = l 4 m 4 x 2 . Hence, the linearized model of the system in the z domain is z ̇ 1 z ̇ 2 = 0 1 0 0 z 1 z 2 + 0 1 u y = 1 0 z 1 z 2 (15) Note that the control input u will be designed with the help of a fast integral terminal SMC scheme, and we check the stability of the system’s third internal dynamic state in the following subsection.

After all the calculations and transformation into a z -coordinate system, we achieved the third internal dynamic state of the system, that is z ̇ 3 = m 4 m 1 − k 1 z 3 m 1 m 4 − k 2 z 1 m 1 m 4 − k 3 z 3 m 1 u m 4 + z 3 m 4 m 1 l 1 m 1 m 4 + l 2 m 1 z 3 z 1 m 4 − l 3 z 1 + l 4 m 1 u m 4 (16) To analyze the stability of the zero-dynamic state, we set the following variables to zero: z 1 = z 2 = u = 0 . By simplifying Equation 16, we get z ̇ 3 = − z 3 k 1 − l 1 (17) where k ₁ > l ₁, so z ̇ 3 = − k 1 z 3 (18)

The zero-dynamic state remains stable for all k ₁ > l ₁, where k ₁ is a positive integer.

In the next step, we will design a control algorithm to adjust the duty cycle and maximize power extraction from the wind turbine system.

The first step in the control design is to precisely track the reference speed. To accomplish this, we define the tracking error as e z = z 1 − z d (19)

where z _d represents the desired or reference value for the variable z. By taking the double derivative of the error, we obtain e ̇ z = z 2 − z ̇ d e ̈ z = z ̇ 2 − z ̈ d (20) The finite-time integral terminal sliding surface is defined as σ z = e ̇ z + α z e z + β z ∫ 0 t e z γ z sign e z d τ (21) Here, α z and β z are positive design constants, and γ z = q z p z , where q z and p z are positive odd integers. The first derivative of the sliding surface is expressed as σ ̇ z = e ̈ z + α z e ̇ z + β z e z γ z sign e z σ ̇ z = z ̇ 2 − z ̈ d + α z z 2 − z ̇ d + β z z 1 − z d γ z sign z 1 − z d (22) By utilizing the values of z ̇ 1 and z ̇ 2 , Eq. 23 can be derived as σ ̇ z = L f 2 h x + L g L f h x u q + Δ − z ̈ d + α z z 2 − z ̇ d + β z z 1 − z d γ z sign z 1 − z d (23) Assuming σ ̇ z = 0 , the control input u _q(equ) can be calculated as follows: u q e q u = 1 L g L f h x − L f 2 h x + z ̈ d − α z z 2 − z ̇ d − β z z 1 − z d γ z sign z 1 − z d (24) The strong reachability factor is defined as u q d i s = − K c σ z − K d sign σ z (25) Hence, the overall control input becomes u z = 1 L g L f h x − L f 2 h x + z ̈ d − α z z 2 − z ̇ d − β z z 1 − z d γ z sign z 1 − z d − K c σ z − K d sign σ z (26)

To demonstrate the stability of the PMSG-WECS under the FITSMC, we use a Lyapunov-based stability analysis and use the following Lyapunov function to prove the enforcement of the sliding mode: Λ z = 1 2 σ z 2 (27) Taking the derivative of Λ z , we have that Λ ̇ z = σ z σ ̇ z (28) By substituting the value of σ ̇ z from Equation 23 and simplifying it, we obtain Λ ̇ z = σ z L f 2 h x + L g L f h x u z + Δ − z ̈ d + α z z 2 − z ̇ d + β z z 1 − z d γ z sign z 1 − z d (29) After substituting the controller u _(q) value and simplifying it, we obtain Λ ̇ z = − L g L f h x K c σ z 2 − L g L f h x K d sign σ z σ z + Δ σ z (30) Because sign ( σ z ) σ z = σ z , Equation 30 becomes Λ ̇ z ≤ − L g L f h x K c σ z 2 − L g L f h x K d σ z + Δ σ z (31) Λ ̇ z ≤ − L g L f h x K c σ z 2 − σ z L g L f h x K d − Δ (32) The uncertainty term η z is defined as follows: L g L f h x K d − Δ ≥ η z (33) We can rewrite inequality (Eq. 32) as: Λ ̇ z ≤ − L g L f h x K c σ z 2 − σ z η z (34) To solve the Lyapunov function for the sliding surface, we have σ z = 2 Λ z (35) By substituting σ z with 2 Λ z in Eq. 34, we have Λ ̇ z ≤ − L g L f h x K c 2 Λ z − η z 2 Λ z Λ ̇ z + L g L f h x K c 2 Λ z + η z 2 Λ z ≤ 0 Λ ̇ z + K ̄ Λ z + η ̄ z Λ z ≤ 0 (36) where K ̄ c = L g L f h ( x ) K c 2 and η ̄ z = 2 η z . T f ≤ 1 2 K ̄ c ln K ̄ c Λ z σ z 0 + η ̄ z η ̄ z (37) The above inequality signifies rapid finite-time convergence, meaning that, as time progresses, Λ z tends to zero at a fast rate. Consequently, the sliding surface converged to zero, indicating that the control system achieved its desired state quickly. This robust convergence behavior ensures successful enforcement of the sliding mode, providing a reliable and efficient control mechanism for the WECS.