In this paper, two nonlinear control approaches using optimal fractional high-order fast terminal sliding mode (FHOFTSM) control are proposed to maximize the captured wind energy and minimize the mechanical loads in the variable speed wind turbines (VSWTs). The optimal condition of the system which uses the presented approaches is achieved by creating a compromise between maximizing the energy extraction and minimizing the control input. So, the mechanical stress on the drive-train is reduced by minimizing the control input. To achieve this compromise, the behavior of the wind turbine system should be precisely modeled. Fractional calculus theory is an effective method for modeling the VSWT, which provides a more accurate description of the system performance compared with integer-order modeling. Based on the fractional-order modeling of the system, the optimal approach is designed by calculating the error dynamics of the system and defining two types of performance indexes, integer-order and fractional-order performance indexes. Hence, the design procedure of an optimal FHOFTSM control for each performance index requires two-stages process. At first phase, an optimal controller is presented for the system nominal error to minimize the quadratic performance index. Then a switching controller is offered in the second phase that is obtained by defining the fractional high-order sliding manifold and fractional nonsingular fast terminal sliding manifold to overcome the unknown disturbance. Hence, the order of the controllers is smaller than the second order. The optimal strategy is applied to minimize the control input, whereas the FHOFTSM method is used to obtain the maximum wind power extraction, reduce mechanical loads, attenuate the chattering, and achieve fast finite-time convergence. The closed-loop stability of the control system is approved by the fractional Lyapunov direct method and the Mittag-Leffler stability theorem. To illustrate the effectiveness of the proposed approaches, the proposed controllers are compared with some already existing controllers. The performance of controllers is verified through three scenarios, i.e., random variation of wind speed, step change of wind speed, and robustness against parameter uncertainties, in which the simulation results confirm the effectiveness of the proposed controllers.

在本文中，提出了两种使用最优分数高阶快速终端滑模（FHOFTSM）控制的非线性控制方法，以最大化捕获的风能并最小化变速风力涡轮机（VSWT）中的机械负载。使用所提出方法的系统的最佳条件是通过在最大化能量提取和最小化控制输入之间建立折衷来实现的。因此，通过最小化控制输入来降低传动系统上的机械应力。为了实现这种折衷，应该对风力涡轮机系统的行为进行精确建模。分数阶微积分理论是一种有效的 VSWT 建模方法，与整数阶建模相比，它可以更准确地描述系统性能。在对系统进行分数阶建模的基础上，通过计算系统的误差动态并定义整数阶和分数阶两种性能指标来设计最优方法。因此，每个性能指标的最优 FHOFTSM 控制的设计过程需要两个阶段的过程。在第一阶段，针对系统标称误差提出了一个最优控制器，以最小化二次性能指标。然后在第二阶段提供切换控制器，通过定义分数高阶滑动流形和分数非奇异快速终端滑动流形来克服未知干扰。因此，控制器的阶数小于二阶。应用最优策略来最小化控制输入，而 FHOFTSM 方法用于获得最大的风能提取，减少机械负载，衰减颤振，并实现快速的有限时间收敛。控制系统的闭环稳定性通过分数阶 Lyapunov 直接法和 Mittag-Leffler 稳定性定理得到认可。为了说明所提出方法的有效性，将所提出的控制器与一些已经存在的控制器进行了比较。通过风速随机变化、风速阶跃变化和对参数不确定性的鲁棒性三个场景验证控制器的性能，其中仿真结果证实了所提出控制器的有效性。并实现快速的有限时间收敛。控制系统的闭环稳定性通过分数阶 Lyapunov 直接法和 Mittag-Leffler 稳定性定理得到认可。为了说明所提出方法的有效性，将所提出的控制器与一些已经存在的控制器进行了比较。通过风速随机变化、风速阶跃变化和对参数不确定性的鲁棒性三个场景验证控制器的性能，其中仿真结果证实了所提出控制器的有效性。并实现快速的有限时间收敛。控制系统的闭环稳定性通过分数阶 Lyapunov 直接法和 Mittag-Leffler 稳定性定理得到认可。为了说明所提出方法的有效性，将所提出的控制器与一些已经存在的控制器进行了比较。通过风速随机变化、风速阶跃变化和对参数不确定性的鲁棒性三个场景验证控制器的性能，其中仿真结果证实了所提出控制器的有效性。建议的控制器与一些已经存在的控制器进行了比较。通过风速随机变化、风速阶跃变化和对参数不确定性的鲁棒性三个场景验证控制器的性能，其中仿真结果证实了所提出控制器的有效性。建议的控制器与一些已经存在的控制器进行了比较。通过风速随机变化、风速阶跃变化和对参数不确定性的鲁棒性三个场景验证控制器的性能，其中仿真结果证实了所提出控制器的有效性。