## 5 MPPT Control Design for PMSG-WECS
In this article, the FDTSMC offers a finite-time rotor speed tracking error convergence of the PMSG-WECS for ensuring the MPPT. This control strategy is developed using a function augmented nonlinear sliding surface that guarantees the tracking error convergence to the equilibrium point in a finite time. Moreover, the high-frequency chattering is completely eliminated by eliminating the reaching phase. The input-output form for FDTSMC can be described as follows:  
\[ \begin{split}\dot{z}_{1}&=z_{2}\\ \dot{z}_{2}&=\widehat{L}_{f}^{2}h(\mathbf{x})+ \widehat{L}_{g}L_{f}h(\mathbf{x})u\\ \dot{z}_{3}&=\widehat{L}_{f}^{3}h(\mathbf{x})+ \widehat{L}_{g}L_{f}^{2}h(\mathbf{x})\dot{u}\end{split}  \tag{17}\]  
where \(\widehat{L}_{f}^{3}h(\mathbf{x})\) and \(\widehat{L}_{g}L_{f}^{2}h(\mathbf{x})\) have been expressed in Equations (A1) and (A2), in Appendix A, respectively. The nonlinear sliding surface for the proposed FDTSMC can be expressed as follows:  
\[s_{3}=\dot{\sigma}+c_{2}\sigma+\beta_{3}\sigma^{q_{3}/p_{3}} \tag{18}\]  
where \(c_{2}\), \(\beta_{3}\), \(p_{3}\), \(q_{3}>0\). Note that both \(p_{3}\) and \(q_{3}\) must be odd integers, such that (\(p_{3}>q_{3}\)). The term \(\sigma\) in Equation (18) denotes a sliding mode surface that can be expressed as follows:  
\[\sigma=\dot{e}+c_{3}e=(z_{2}-\dot{z}_{ref})+c_{3}(z_{1}-z_{ref}) \tag{19}\]  
Now, the successive differentiation of Equation (19) yields:  
\[ \begin{split}\dot{\sigma}&=(\dot{z}_{2}-\dot{z}_{ ref})+c_{3}(\dot{z}_{1}-\dot{z}_{ref})=(z_{3}-\ddot{z}_{ref})+c_{3}(z_{2}-\dot{z}_{ ref})\\ \dot{\sigma}&=(\dot{z}_{3}-\dddot{z}_{ref})+c_{3}( \dot{z}_{2}-\dot{z}_{ref})=\widehat{L}_{f}^{3}h(\mathbf{x})+\widehat{L}_{g}L_ {f}^{2}h(\mathbf{x})\dot{u}-\dddot{z}_{ref}+c_{3}(\dot{e})\end{split}  \tag{20}\]  
Next, the FDTSMC sliding surface expressed in Equation (18) is differentiated and simplified using Equation (20) in order to derive the equivalent controller as follows:  
\[\dot{s_{3}}=\widehat{L}_{f}^{3}h(\mathbf{x})+\widehat{L}_{g}L_{f}^{2}h( \mathbf{x})\dot{u}-\dddot{z}_{ref}+c_{3}\dot{e}+c_{2}\dot{\sigma}+\beta_{3} \frac{q_{3}}{p_{3}}\sigma^{\left(\frac{q_{3}}{p_{3}}-1\right)}\dot{\sigma} \tag{21}\]  
The equivalent controller, \(\dot{u}_{eq}D\), can be obtained by setting Equation (21) to zero as follows:  
\[\dot{u}_{eq}D=\frac{-1}{\widehat{L}_{g}L_{f}^{2}h(\mathbf{x})}\Big{[}\widehat {L}_{f}^{3}h(\mathbf{x})-\dddot{z}_{ref}+c_{3}\dot{e}+c_{2}\dot{\sigma}+\beta_ {3}\frac{q_{3}}{p_{3}}\sigma^{\left(\frac{q_{3}}{p_{3}}-1\right)}\dot{\sigma} \Big{]} \tag{22}\]While, the discontinuous controller, \(\dot{u}_{discD}\), can be given as follows:  
\[\dot{u}_{discD}=-\mathcal{K}_{1}s_{3}-\mathcal{K}_{2}\,\text{sign}(s_{3}) \tag{23}\]  
where \(\mathcal{K}_{1}\), \(\mathcal{K}_{2}>0\) are the switching controller gains. Finally, the FDTSMC-based MPPT law is obtained by combining Equations (22) and (23), and passing through an integrator (low-pass filter) as follows:  
\[ \begin{split}& u_{FDTSMC}=\int\Big{[}\underbrace{\frac{-1}{ \mathcal{\dot{\mathcal{L}}}_{S}L_{f}^{2}h(\mathbf{x})}\Big{\{}\widehat{L}_{f}^ {3}h(\mathbf{x})-\vec{z}_{ref}+c_{3}\dot{v}+c_{2}\dot{v}+\beta_{3}\frac{q_{3}} {p_{3}}\sigma^{\left(\frac{q_{3}}{p_{3}}-1\right)}\dot{v}\Big{\}}}_{\dot{u}_{ engD}}\\ &\underbrace{-\mathcal{K}_{1}s_{3}-\mathcal{K}_{2}\,\text{sign}( s_{3})}_{\dot{u}_{discD}}\Big{]}.dt\end{split}  \tag{24}\]  
The computational flow chart for implementation of the proposed MPPT strategy is depicted in Figure 4, while the closed-loop control system for PMSG-WECS is given in Figure 5.  
Figure 4: Computational flow chart for implementation of the proposed MPPT strategy.