A Hybrid Dynamic Nonlinear Controller for Variable Speed Wind Turbine in Low Wind Velocity Regime

POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 22 |NUMBER: 2 |2024 |JUNE A Hybrid Dynamic Nonlinear Controller For Variable Speed Wind Turbine In Low Wind Velocity Regime EL Kabira EL MJABBER1, Abdellatif KHAMLICHI1 1Team 3M, Department of Industrial and Civil Sciences and Technologies, National School of Applied Sciences, University Abdelmalek Essaadi, Street Sebta, Tetouan, Morocco kabira.mjabb[email protected], akhamlic[email protected] DOI: 10.15598/aeee.v22i2.5335 Article history: Received Jul 27, 2023; Revised Dec 17, 2023; Accepted Apr 22, 2024; Published Jun 30, 2024. This is an open access article under the BY-CC license. Abstract. The purpose of this paper is to propose a new control approach to be applied to a variable rotor speed wind turbine at a low wind velocity zone. The aim is to reduce dynamic mechanical loads and optimize energy production by acting on the generator torque through a new hybrid adaptive controller. This combines two well-known nonlinear methods: nonlinear control based on Radial Basis Function Neural Networks used to estimate the nonlinear part of the wind turbine system and Integral Sliding Mode Control to tackle system uncertainties. Lyapunov’s approach has been applied to assess the stability of this new controller. Then, simulations were carried out using the Matlab/Simulink software package. The obtained results demonstrated the superior performance of the hybrid controller compared to each controller taken alone. Keywords Nonlinear control, integral sliding mode control, radial basis function neural network controller, variable speed wind turbine control, wind energy. 1. Introduction Wind energy is a renewable clean energy that has become competitive nowadays due to various technological advances that have been achieved these recent years. Optimization of energy capture from wind still suffers from obstacles such as the stochastic nature of wind speed. These cause continuous fluctuations that impede tracking the optimal rotor speed without causing variations of mechanical loads acting on key parts of the wind turbine installation, such as torque in the transmission line. So, the controllers are required to ensure a safe and efficient conversion of kinetic energy contained in wind into electricity while dealing with the presence of these sources of disturbances. Otherwise, these may deteriorate energy quality and reduce the life of equipment due to accelerated fatigue [1]. Numerous approaches to controlling wind turbines have been proposed in the literature, including both linear and nonlinear methods [2–5]. Since wind turbines are nonlinear systems, conventional linear controllers are known to lack robustness and can only remain stable within a limited operating range [3] and [6]. Consequently, there is a growing interest in employing nonlinear control laws, which can better adapt to variations in system parameters and environmental conditions. As a result, several nonlinear control strategies have been introduced. Notably, the works cited in [6] and [7] represent significant approaches in the literature. New perspectives have emerged recently in the field of nonlinear control with the apparition of intelligent adaptive controllers. The best-known methods in this field are controllers based on neural networks [8], fuzzy logic [9], and Radial Basis Function Neural Networks (RBF-NN) which are mostly used to adjust Proportional Integral (PI) gains and to perform dynamic neural approximation [10–12]. The Sliding Mode Control (SMC) is particularly suitable for wind turbines because of its robust be- ©2024 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 115 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 22 |NUMBER: 2 |2024 |JUNE havior under uncertainties that can affect the system [13–15]. The researchers have used SMC-based controllers to maximize the capture of wind energy while reducing load fluctuations on the drive train to limit the fatigue loads [1] and [16]. SMC is recognized for its drawback of frequent switching between various control values, leading to a chattering phenomenon that induces increased mechanical stress on system components. To alleviate this issue, researchers have introduced several techniques to mitigate chattering effects [17]. Capitalizing on the inherent robustness of SMC, a derivative technique, the Super Twisting Algorithm (STA), has been amalgamated with other approaches. Authors in [18] utilized STA to alleviate the chattering problem. The integration of STA with SMC, denoted as STA-SMC, employing Space Vector Modulation (SVM), has been employed by [19] for the control of the Double Fed Induction Generator (DFIG). The effectiveness of SMC-based Artificial Neural Networks (ANN) has been demonstrated in various studies. For instance, [20] applied an SMC-based ANN to regulate the active and reactive power of DFIG. Additionally, the STA-based ANN proposed by [21] has been utilized to control a floating wind turbine in Region III. The existing literature extensively covers various control strategies for wind turbine systems aimed at optimizing energy production and reducing mechanical loads. However, there is a clear need for further research into integrating nonlinear control methods, specifically combining RBF-NN for nonlinear system estimation and ISMC for managing system uncertainties. While individual studies have examined these methods separately, there is limited research on this joint application of wind turbine control, especially in maximizing energy by considering the mechanical part modeled by two masses. Additionally, there is a lack of thorough analysis regarding the proposed hybrid adaptive controller using the Lyapunov approach. Addressing these research gaps is crucial for advancing our understanding of effective control strategies in wind energy systems and enhancing their performance under dynamic conditions. Contribution of this paper: This paper focuses on a variable-speed wind turbine operating at low wind speeds (below-rated speed) and presents the following contributions: •An intelligent hybrid nonlinear controller: The paper proposed a controller that combines the use of RBF-NN to estimate the uncertain variables in the wind turbine system and Integral Sliding Mode Control ISMC protocol to track the optimal rotor speed. This hybrid controller aims to address the chattering phenomenon typically associated with conventional SMC-based controllers while compensating for the lack of an accurate wind turbine model through RBF-NN training. The goal is to synergistically leverage the strengths of both controllers to enhance the robust behavior of the ISMC strategy. •Maximizing wind energy extraction: The proposed control strategy aims to maximize the extraction of wind energy by following the optimal Tip Speed Ratio (TSR). By optimizing the rotor speed based on TSR, the controller seeks to achieve optimal energy capture from the wind. •Minimizing mechanical load fluctuations: Another objective of the proposed controller is to minimize fluctuations that affect mechanical loads to enhance the lifetime of the wind turbine system. By regulating the electromagnetic torque, the controller aims to minimize control errors while tracking the optimal rotor speed. In summary, the paper introduces an intelligent hybrid nonlinear controller that combines RBF-NN and ISMC techniques. The controller aims to improve the robustness of the ISMC strategy, maximize mechanical load fluctuations to enhance the longevity of the wind turbine system . 2. Wind turbine modeling Wind turbines with horizontal rotor axes are nowadays widely used [22]. Often for control purposes, a simplified mechanical model of such a complex system is considered [23,24]. The aerodynamic power extracted by a wind turbine can be expressed as: Pa=1 2ρπR2Cpv3,(1) where ρdenotes air density in (kg.m−3), Rrepresents the radius of the wind turbine in (m), vis the average wind speed in (m.s−1) at the rotor level, and Cp the power coefficient. This last is a characteristic factor specific to a particular wind turbine. It represents its ability to extract energy from the kinetic energy present in the wind. This coefficient varies depending on the design of the wind turbine blades and the operating conditions. It can be determined through experimental testing or estimated using numerical simulations and computational models. For the specific wind turbine chosen in this study, Cpcan be approximated using the following equation [25]: Cp(λ, β) = c1(c2χ−c3β−c4)e−c5χ+c6λ χ=1 λ+ 0.08β−0.035 β3+ 1,(2) ©2024 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 116 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 22 |NUMBER: 2 |2024 |JUNE in which the constants c1, c2, c3, c4, c5, c6depend on the given wind turbine, βis the pitching angle of the wind turbine blades in (◦) and λis the tip speed ratio. This last parameter is defined as the ratio of the linear velocity at the tip of the blades divided by the wind speed. It is written as: λ=ωt vR, (3) where ωtis the rotational speed at which the rotor turns actually. This speed is typically measured and expressed in revolutions per minute (rpm). Fig. 1 presents a three-dimensional representation of the power coefficient for the specific wind turbine considered in this work. This figure illustrates that the power coefficient reaches a maximum value for an optimal pitch β=βopt and an optimal tip-speed ratio λ=λopt = 7.5. The aerodynamic torque is obtained by dividing the aerodynamic power, as defined in Eq. (1), by the rotor speed ωt: Ta=Pa ωt =ρπR2 2 Cp(λ, β)v3 ωt .(4) In general, the drive train of a wind turbine consists of several components, including a generator, a gearbox, a rotor, a high-speed shaft and a low-speed shaft. This mechanical system can be represented roughly by a rigid one degree of freedom system in which the whole inertia of turning parts is integrated. However, a refined model that gives more insight into the transmission line dynamics has been introduced by taking into account the flexibility of the primary shaft. The resulting wind turbine model takes then the form of an equivalent discrete two-mass model representing the inertia of the turbine rotor and the power train. The schematic representation of this two degrees of freedom system is depicted in Fig. 2. Fig. 1: 3D representation of the power coefficient as function of blades pitch angle βand tip speed ratio λ. In Fig. 2, Kls, Kr, Kgdenote the external damping, Bls is the low-speed shaft stiffness, Tls and Ths are respectively the low and high-speed shaft torques, Tem represents the electromagnetic torque, ωgis the generator speed, and Jrand Jgare the rotor and generator inertia respectively. Fig. 2: Equivalent wind turbine model with two masses. The dynamic equation of the inertia in the rotor side Jrin (kg.m2) expresses the evolution of the rotor speed ωtunder the action of the aerodynamic torque Tain (Nm), the torque exerted on the gearbox extremity linked to slow speed shaft, denoted Tls, and a viscous frictional torque Krωt. This equation takes the form [22]: ˙ωt=Ta Jr −Tls Jr −Kr Jr ωt.(5) The dynamic of the inertia on the generator side Jg in (kg.m2) describes the time variation of generator shaft speed resulting from the application of the torque Ths on the gearbox side, as well as the torque Tem and a viscous frictional torque Kgωg.This equation writes [12]: ˙ωg=Ths Jg −Kg Jg ωg−Tem Jg .(6) The torques Tls and Ths vary inversely with the speeds ωtand ωgsince: ng=Tls Ths =ωg ωt =θg θt ,(7) where ngis the gearbox ratio The elastic behavior of the slow-speed shaft, coupled with the absence of external loading between its extremities [26], allows for the torque it experiences on the gearbox side Tls to be related to the elastic and frictional effects by: Tls =Bls (θt−θls) + Kls (ωt−ωls).(8) Using ωls =ωg/ngand Ths =Tls/ng, and substituting (5) and (6) into the expression of the time deriva- ©2024 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 117 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 22 |NUMBER: 2 |2024 |JUNE tive of Eq. (8), one gets: ˙ Tls =Bls −KlsKr Jrωt+1 ngKlsKg Jg −Blsωg +Kls Ta Jr +Tem ngJg−KlsTls Jr+n2 gJg n2 gJrJg!.(9) The global state system, Eq. (10), which describes the dynamics of the two masses wind turbine model, is deduced from (5), (6), and (9). 3. Wind turbine control 3.1. Design of integral sliding mode control ISMC SMC is well suited for controlling nonlinear and uncertain equilibrium trajectories and enhance control stability [27]: S(t) = δ e +de dt +GiZe(t)dt, (11) where δrepresents positive gain and Giis the integral gain. The control error (tracking error) is defined by the optimum rotor speed ωopt and rotor speed ωtas: e=ωopt −ωt.(12) The optimum rotor speed expression can be obtained from Eq. (3) as follows: ωopt =λoptv R.(13) The control input denoted ushould ensure local attractiveness to surface Sin its vicinity, i.e., the system trajectory (11) must be directed towards it and intersect it. For this purpose, a sufficient stability condition of S(x, t)=0, called the attractiveness condition, has to be satisfied by the controller. In the context of the direct Lyapunov method, an unbounded function V(S) called the Lyapunov function should be exhibited with V(0) = 0 and V(∞) = ∞. Its time derivative dV dt provides information on the stability of the system, such that if dV dt <0for S= 0 then the system is asymptotically stable. One of the simplest Lyapunov functions that can be proposed in the context of ISMC is the classical quadratic systems. ISMC enhances SMC by integrating integral action, providing improved tracking accuracy and disturbance rejection compared to standard SMC. In the context of modeling wind turbines with a two-mass system and enhancing control stability, incorporating integral action into the expression of the sliding surface is essential. This refined surface formulation aims to better describe function: V(S) = 1 2S(x, t)2.(14) The function V(S)is positive. Then, it is sufficient to impose its time derivative to be negative to ensure convergence of S towards zero. This condition writes dV (S) dt =S(x, t)dS(x, t) dt ⩽0.(15) In this study, the Lyapunov function is chosen to be defined by Eq. (14) to ensure the stability of the system through condition (15). Forcing the quantity S(x, t)2 to decrease all the time serves to force the trajectory of the system towards the sliding surface. In this case it is of course assumed that the switching frequency is infinite. The ideal sliding regime is obtained by using the time derivative of Eq. (11) and subsituting it into Eq. (15) under the following form: Sδdωopt dt −dωt dt +d2e dt2⩽0.(16) ISMC control law: Considering the time derivative of the sliding surface as defined by Eq. (11) along with Eq. (12) gives: dS dt =δde dt +d2e dt +Gie =δdωopt dt −dωt dt +d2e dt +Gie. (17) Using now equations (5), (6) and (7), the low speed shaft torque can be put under the following form: Tls =ngJg˙ωg+Kgngωg+ngTem.(18) Using Eqs. (6) and (18), the rotor dynamics under the action of Tem is obtained as: dωt dt =Ta Jr −Kr Jr ωt−ngJg Jr dωg dt −ngKg Jr ωg−ng Jr Tem.(19) To apply ISMC, Eq. (19) must be rewritten under the general nonlinear control form as: dωt dt =f(ωt, ωg) + gTem,(20) with f(ωt, ωg) = Ta Jr−Kr Jrωt−ngJg Jr dωg d t −ngKg Jrωgand g=−ng Jr. ©2024 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 118 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 22 |NUMBER: 2 |2024 |JUNE    ˙ωt ˙ωg ˙ Tls   =   −Kr/Jr0−1/Jr 0−Kg/Jg1/[ngJg] Bls −KlsKr Jr 1 ngKlsKg Jg−Bls−Kls Jr+n2 gJg n2 gJgJr     ωt ωg Tls   +   1/Jr 0 Kls/Jr   Ta+   0 −1/Jg Kls/[ngJg]   Tem. (10) The control input uconsists of two terms. The equivalent term ueq =Tem is utilized to control system variations, meaning that it provides the tracking reference for the system. On the other hand, the robustness term uris used to force the system to vary on the sliding surface. u=ueq +ur.(21) By subsituting ˙ωtobtained from Eq. (20) into Eq. (17) with imposing ˙ S= 0, the expression of ueq is then found as follows: ueq =1 g−f(ωt, ωg) + 1 δ de2 dt +G δe+ωopt v dv dt . (22) To minimize the interference phenomenon, the hyperbolic tangent function (tanh) is used to obtain a smooth control law under the form: ur=ηtanh(S),(23) where the constant ηrepresents the switching gain. By using Eqs. (21), (22) and (23), the global law of the control is deduced as: u=1 g−f(ωt, ωg) + 1 δ de2 dt +G δe+ωopt v dv dt  +ηtanh (24) 3.2. Design of nonlinear controller RBF-NN RBF-NN offers several advantages such as the fact that its local behavior is very intuitive. RBF-NN admits a single hidden layer which makes it easier to use and faster than conventional multilayer perceptron structure. RBF-NN has gained interest for its universal faculty of computation, estimation and control of uncertain systems. RBF, Fig. 3, consists of three layers, an input layer that contains the input variables xi, a hidden layer that is activated by radial basis function ψjand an output layer f. Taking into account the approximate network error, the output of an RBF-NN is written as [28]: f(ωt) = WTΨ(e) + ν, (25) where Ψ(e)=[ψ1, ψ2, ..., ψp]is a radial function vector, WTis the vector of optimal weights, and vis the uncertainty that can be assumed to be bounded by νmax:|ν|< νmax. Fig. 3: RBF network architecture. The optimal weighs are solutions to an optimization problem which has the form [29]: W= arg min hsup  ˆ f(X)−f(X)i.(26) To produce the output of the hidden layer, the following nonlinear activation function is used [29]: ψj(e) = exp −∥e−Cj∥2 2σ2 j!,(27) where Cjrepresents the center of Gaussian function ψj and σjthe width of each ψj. RBF-NN Control law: To extract the control law u=Tem, the second order dynamic is imposed on the tracking error as stated by Eq. (12): d2e(t) dt +k1 de dt +k2e= 0,(28) with k1and k2being two positive scalars, chosen so that s2+k1s+k2polynomial is Hurwitz. In order to ensure stability of the equilibrium, de dt is substituted by ©2024 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 119 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 22 |NUMBER: 2 |2024 |JUNE the time derivative of the right-hand side of Eq. (12), which leads to: d2e(t) dt +k1dωopt dt −dωt dt +k2e= 0.(29) Replacing the dωt dt in Eq. (29) with the expression from Eq. (20), the control law becomes as follows:        ueq =1 g−f(ωt, ωg) + 1 k1 de2 dt +k2 k1 e+ωopt v dv dt  f(ωt, ωg) = Ta Jr −Kr Jr ωt−ngJg Jr ˙ωg−ngKg Jr ωg . (30) Applying the control law of the ISMC as given by Eq. (24), and imposing the dynamic to the error of speed to be that given in Eq. (28), the global ISMC control law writes:              u=1 g−f(ωt, ωg) + 1 k1 de2 dt +k2 k1 e+ωopt v dv dt  +ηtanh(S) f(ωt, ωg) = Ta Jr −Kr Jr ωt−ngJg Jr dωg dt −ngKg Jr ωg . (31) As f(ωt, ωg)is highly nonlinear which may affect the control reliability, estimating this function by an RBFNN allows for a more accurate representation of the system. The estimation is considered under the form: ˆ f(e) = ˆ WTΨ(e).(32) The RBFNN control law developed of the twomasses wind system is given by:      ˆu=1 g−ˆ f(e) + 1 k1 d2e dt +k2 k1 e+ωopt v dv dt  ˆ f(e) = ˆ WTψ(e) .(33) The weights ˆ Wcan be updated using the following rule: ˙ ˆW=−ςˆ f e +χ|e|ˆ W, (34) where ς > 0and χ > 0. 3.3. Hybrid control ISMC-RBF: The ISMC controller, as described in Eq. (30), often exhibits a chattering phenomenon, which can have detrimental effects on the power quality and the mechanical and electrical components of the wind turbine. Given the nonlinear nature of wind turbine systems, Fig. 4: Block scheme of the hybrid proposed controller ISMCRBF. the accuracy of uncertain components is crucial. Integrating an RBF-NN representation of these nonlinearities is expected to be beneficial in mitigating this ISMC perturbations. Our proposed method aims to address the chattering phenomenon while retaining the advantages of the sliding control mode approach. The ISMP component introduces integral action, effectively eliminating steadystate errors and reducing the impact of high-frequency oscillations. Simultaneously, the incorporation of RBF helps to smooth the control signal transitions, providing a more continuous and stable response. Together, these elements synergistically address the chattering phenomenon inherent in the traditional sliding mode techniques. This yields a new hybrid controller that combines both ISMC and RBF-NN methods which constitutes the main contribution of this work, (see Fig. 4). Using the estimation f(ωt, ωg)as given in Eq. (33), instead of the explicit expression given in Eq. (31), the hybrid control law writes as follows:          u=1 g−f(ωt, ωg) + 1 k1 de2 dt +k2 k1 e+ωopt v dv dt  +ηtanh(S) ˆ f(e) = ˆ WTψ(e) . (35) In addition to system nonlinearities, a disturbance d is assumed to disrupt the dynamics of the wind turbine. The system dynamic equation in terms of the rotor speed, Eq. (20), is then modified according to: dωt/dt =f(ωt) + gu +d, (36) in which the disturbance is considered to be bounded such that |d|⩽D. ©2024 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 120 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 22 |NUMBER: 2 |2024 |JUNE Stability study is essential for any system control. In this work, considering the dynamic and non-linear nature of the proposed controller presented by Eq. (35), Lyapunov’s theorem was chosen to ensure the stability of the learning algorithm, as follows: dˆ W.dt =−(1/γ)eΨ (e), γ > 0,(37) Proof of stability for the hybrid controller ISMC-RBF Take the Lyapunov function to be: V=1 2γ¯ WT¯ W+1 2e2,(38) with ¯ W=W−ˆ Wand γ > 0. Taking the time derivative of Eq. (38), one gets: dV dt =1 2γd¯ W dt T ¯ W+1 2γ¯ WTd¯ W dt +ede dt =γ¯ WTdW dt −d¯ W dt +ede dt .(39) The stationary property of is derived from its state of optimality and can be expressed as follows: dW dt = 0.(40) Substituting Eq. (40) into Eq. (39), yields: dV dt =−γ¯ WT dˆ W dt !+ede dt .(41) By using Eqs. (15) and (36), Eq. (41) can be simplified as: dV dt =−γ¯ WT dˆ W dt !+edωopt dt −f−gu −d. (42) By substituting Eq. (36) into Eq. (42), the derivative of the Lyapunov function can be expressed as follows: dV dt =e−f+ˆ f−1 k1de2 dt −k2 k1 e−ηg tanh(S)−d −γ¯ WT dˆ W dt !.(43) The optimization error can be denoted by: ¯ f=f−ˆ f. (44) By utilizing equations Eqs. (25) and (32), it is possible to express this error in the following manner: ¯ f=¯ WΨ + ν. (45) Substituting (45) into (42), one obtains after some obvious algebraic calculation: dV dt =−e1 k1de2 dt +k2 k1 e+ηg tanh(S) + ν+d ¯ WT"−γ dˆ W dt !−eΨ#.(46) Considering the dynamics of the weight estimates according to the learning algorithm given by Eq. (37), then the Lyapunov function derivative given in Eq. (46) becomes: dV dt =−1 δede2 dt −eν+d+k2 k1 e+ηg tanh(S). (47) We can now easily see that for v, d →0the condition of stability to be met has the obvious following form: dV dt =−1 k1ede2 dt −k2 k1 e2−eηg tanh(S)⩽0.(48) 4. Results and discussion In this study, the performance of the developed control approach is evaluated through numerical simulations conducted on the Controls Advanced Research Turbine (CART) [3]. CART has been specifically designed to facilitate the investigation and research of control tests for large-scale turbines. This variable-speed wind turbine is equipped with a flexible hub, variable pitch mechanism, and two blades, each driven independently by its electromechanical system, allowing for individual pitching control. The nominal power output of CART is 600 kW. The other characteristics of this turbine are presented in Appendix A. To assess efficiency of the hybrid controller, various parameters are taken into consideration across different mean wind speeds. These include: •Tracking of optimal rotor speed: This involves minimizing tracking errors to ensure the rotor speed closely follows the desired trajectory. •Maximum electromagnetic torque (Max (Tem)): This parameter determines the maximum torque exerted on the turbine generator, providing insights into mechanical loading. •Maximum electrical power (Max (Pe)): This represents the highest level of power output generated by the wind turbine electrical system. •Standard Deviation (STD) of electrical power: The STD indicates the variability or fluctuation in electrical power output, providing information on the stability of power generation. ©2024 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 121 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 22 |NUMBER: 2 |2024 |JUNE Fig. 5: Wind velocity profile with mean value = 8 m.s−1. Fig. 6: The time response of the rotor speed. •Efficiency (ηel): Efficiency measures how effectively the controller converts available wind energy into usable electrical power, typically represented as the ratio of electrical power output to the available wind power. ηel(%) = tf R ti Pedt tf R ti Pa,optdt .(49) Fig. 7: The time response of the tracking error. Fig. 8: The time response of the electromagnetic torque. Fig. 9: The time response of the electrical power. where tfand tiare the considered limits of the time interval. Peis the actual power extracted and Pa,opt is the optimal aerodynamic power which is given by: Pa,opt = 0.5ρπR2Cp,maxv3.(50) Assessing these parameters across different mean wind speeds provides valuable insights into the hybrid controller performance and its ability to adapt to varying environmental conditions. The variations in wind speed are modeled to range between 7 and 9 m.s−1, representing 50% of the standard of the deviation wind spectrum. Fig. 5 illustrates the wind data used in the simulation, with a mean wind speed of 8 m.s−1. A comparison among three control techniques, RBFNN, ISMC, and ISMC-RBF, is conducted to ascertain the most effective method and highlight the advantages of combining different control techniques. The analysis of results from the three controllers, as depicted in Fig. 6, indicates that ISMC alone struggles to accurately track the optimal rotor speed. However, the ©2024 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 122 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 22 |NUMBER: 2 |2024 |JUNE Tab. 1: Benchmarking performance of the 3 controllers with mean speed= 7m/s. 7 m/s ηel Max (Pe) Max (Tem) STD (Tem) ISMC 90.2% 350 1.97 0.245 NN-RBF 93% 296 1.50 0.120 ISMCRBF 95.5% 290 1.21 0.022 hybrid technique ISMC-RBF yields satisfactory results compared to the other two methods. Moreover, the presence of disturbance does not affect the rotor speed controlled by the ISMC-RBF controller, which continues to precisely follow the optimal speed ωopt. The simulation of tracking error e=ωt−ωopt, as illustrated in Fig. 7, confirms these observations. The controlled speed provided by ISMC-RBF fluctuates around the optimal speed with the least error, whereas both RBFNN and ISMC controllers presented higher variations in tracking error. The results of the electromagnetic torque Tem simulation, as depicted in Fig. 8, demonstrate satisfactory performance for both controllers, ISMC-RBF and RBF-NN, with smooth variations observed. To underscore the significance of ISMC-RBF, the STD was considered about the fluctuations of the electromagnetic torque. A lower standard deviation indicates a more stable and consistent torque output from the generator, suggesting that the controller effectively adapts to changing wind conditions and maintains a steady power output. The findings presented in Table 2 indicate that the STD obtained by ISMC-RBF (0.22 kN.m) is lower than that of RBF-NN (0.224 kN.m) and ISMC (0.474 kN.m). Additionally, Table 2 demonstrates that the proposed ISMC-RBF achieves a maximum electromagnetic torque value of only 1.21 kN.m, whereas ISMC reaches 3.88 kN.m. Consequently, the proposed ISMC-RBF controller demonstrates high performance in rejecting fluctuations, thereby minimizing mechanical loads that could potentially cause fatigue. Electrical power efficiency (ηel)is a crucial factor to consider, representing the ratio of electrical power output to the available wind power. A higher efficiency value suggests that the controller effectively converts a larger portion of wind energy into usable electrical power. This indicates a well-designed and optimized control system. Referring to the temporal variations depicted in Fig. 9 and the performance results shown in Table 3, it is evident that the proposed ISMC-RBF controller delivers the most stable electrical power among all three considered controllers. The efficiency reaches up to 95%. Consequently, the ISMC-RBF controller enables higher power extraction compared to the reference controllers. Tab. 2: Benchmarking performance of the 3 controllers with mean speed= 8m/s. 8 m/s ηel Max (Pe) Max (Tem) STD (Tem) ISMC 85% 385 3.88 0.474 NN-RBF 93% 296 1.50 0.227 ISMCRBF 95.5% 290 1.21 0.220 Tab. 3: Benchmarking performance of the 3 controllers with mean speed= 9m/s. 9 m/s ηel Max (Pe) Max (Tem) STD (Tem) ISMC 74% 610 4.10 0.510 NN-RBF 78.8 % 454 3.99 0.254 ISMCRBF 79% 450 3.80 0.240 Tables 1, 2, and 3 present the results for wind speeds of 7, 8, and 9 m.s−1, respectively. It is observed from the tables that the performances of all methods decrease with increasing the mean wind speed. However, the proposed ISMC-RBF method consistently outperforms the other two methods across all wind speeds. 5. Conclusion A new hybrid controller ISMC-RBF was developed in this work. Its performances were compared to those of the classical RBF-NN and ISMC in the low wind speed regime where the control consists of extracting maximum energy from wind. Adding RBF to ISMC enabled to mitigate the main drawback of this last which is known as chattering phenomenon, and exhibited due to the inherent imposed fast dynamics of the controller. Using the RBF-NN to estimate the nonlinear dynamics of the two-mass model of a wind turbine has enabled to enhance the behavior of ISMC controller and succeeded in limiting these perturbations. This was achieved through ensuring Lyapunov like condition of stability of the hybrid controller while the nonlinear part of system dynamics was actualized through a RBF based learning algorithm. The obtained results have shown that the ISMC-RBF gives the best tracking of the reference wind speed. It yields better extraction of the electrical energy while minimizing fluctuations of mechanical loads. Author Contributions The two authors have worked together on the theoretical formalism. They have performed the analytic ©2024 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 123