## 5 MPPT Control Design for PMSG-WECS  
The main goal of the MPPT control design is to maximize the power output of the PMSG-WECS. This goal is accomplished by proposing an FDTSMC-based MPPT control strategy. The conventional TSMC strategy may not be able to guarantee the same convergence speed for the system states located distant from the equilibrium point, O. However, it can offer a finite-time convergence for the system states located near the equilibrium point. The conventional TSMC concept was first introduced in [23], for which the nonlinear sliding surface can be expressed as follows:  
\[s_{1}=\dot{e}+\beta_{1}e^{\frac{q_{1}}{p_{1}}}=0 \tag{14}\]  
where \(e\) represents the tracking error and \(\beta_{1}\), \(p_{1}\), \(q_{1}>0\). Note that both \(p_{1}\) and \(q_{1}\) must be odd integers, such that \((p_{1}>q_{1})\). It can easily be verified that for any given initial tracking error, \(e(0)\
eq 0\), the dynamics expressed in Equation (14) will converge to \(e=0\) in a finite time, as determined by the reaching time, \(t^{s_{1}}=[p_{1}/\{\beta_{1}\,(p_{1}-q_{1})\}]|e(0)|^{(p_{1}-q_{1})/p_{1}}\), obtained from the analytical solution of Equation (14). The equilibrium point O is termed as the terminal attractor. The reaching time can be tuned by proper selection of the parameters \(\beta_{1}\), \(p_{1}\) and \(q_{1}\). The inclusion of the nonlinear term \(e^{q_{1}/p_{1}}\) in Equation (14) improves the convergence towards the equilibrium point such that the closer the \(e^{q_{1}/p_{1}}\) is to the equilibrium point, the faster the convergence will be, thus offering a finite-time convergence. However, for the system states located distant from the equilibrium point, the conventional TSMC strategy does not outperform its conventional SMC counterpart (where \(p_{1}=q_{1}\)) because the \(e^{q_{1}/p_{1}}\) term now tends to retard the convergence towards the equilibrium point. To counteract this problem, a solution has been proposed in [24] that is known as the FTSMC, for which the nonlinear sliding surface can be expressed as follows:  
\[s_{2}=\dot{e}+c_{1}e+\beta_{2}e^{q_{2}/p_{2}}=0 \tag{15}\]  
where \(c_{1}\), \(\beta_{2}\), \(p_{2}\), \(q_{2}>0\). Note that both \(p_{2}\) and \(q_{2}\) must be odd integers, such that \((p_{2}>q_{2})\). Thus, Equation (15) gives \(\dot{e}=-c_{1}e-\beta_{2}e^{q_{2}/p_{2}}\). By properly choosing \(p_{2}\) and \(q_{2}\), for a given initial tracking error \(e(0)\
eq 0\), the dynamics expressed in Equation (15) will converge to \(e=0\) in a finite time. Physically, it can be interpreted as: when \(e\) is located far away from the equilibrium point, the approximate dynamics can then be expressed as: \(\dot{e}=-c_{1}e\), which offers a fast and finite-time convergence. However, when \(e\) is located close to the equilibrium point, the approximate dynamics then become: \(e=-\beta_{2}e^{q_{2}/p_{2}}\), which is the terminal attractor offering fast and finite-time convergence. Figure 3: ANFIS network structure for PMSG–WECS Lie-derivatives estimation. More precisely, the analytical solution of Equation (15) gives the exact time to reach the equilibrium point (reaching time) as follows:  
\[t^{p_{2}}=\frac{p_{2}}{c_{1}\left(p_{2}-q_{2}\right)}\Big{[}\ln\{c_{1}e(0)^{(p_{2 }-q_{2})/p_{2}}+\beta_{2}\}-\ln\beta_{2}\Big{]} \tag{16}\]  
_FDTSMC-Based MPPT Control Design for PMSG-WECS_  
The FTSMC renders the advantages of fast dynamic response, high steady-state tracking accuracy and finite-time convergence. However, chattering of the control signal due to the switching action of the discontinuous control signal, and sensitivity to high-frequency noise, still remain as the main disadvantages of the FTSMC. To counteract these issues, a novel strategy has been proposed in [25], namely the global FDTSMC. In the stated strategy, the time derivative of the control input has been taken as the control variable for the augmented system such that the control input appears as chattering-free due to an integrator (acting as a low-pass filter) placed next to the system.