## 2 Mathematical Modeling of PMSG-WECS
### Wind Turbine Mathematical Modeling  
The mechanical power, \(P_{m}\), captured by the VSWT is expressed as follows [16]:  
Figure 1: Schematic of the overall variable speed permanent magnet synchronous generator–wind energy conversion system (PMSG–WECS). \[P_{m}=\frac{1}{2}\rho\pi{R_{t}}^{2}v_{w}^{3}C_{p}(\lambda,\beta) \tag{1}\]  
where \(\rho\) denotes the air density (generally, its value is \(1.25\,\mathrm{kg/m^{3}}\), at \(10\,\mathrm{\SIUnitSymbolCelsius}\), at sea level), \(R_{t}\) stands for the turbine blade radius (m), \(v_{w}\) represents the wind speed (m/s), and \(C_{p}\) indicates the wind turbine rotor power conversion coefficient that defines the wind turbine rotor aerodynamic efficiency. Generally, \(C_{p}\) varies with both the tip speed ratio, \(\lambda\), and the blade pitch angle, \(\beta\). However, \(\beta\) is assumed to be a constant (\(\beta=0\,\mathrm{\SIUnitSymbolDegree}\)) in this article. \(C_{p}\) is determined by numerical approximation; such as in [17], for \(\beta=0\,\mathrm{\SIUnitSymbolDegree}\), it is given as follows:  
\[ \begin{split}& C_{p}(\lambda)=0.0061\lambda-0.0013\lambda^{2}+0.0 081\lambda^{3}-9.7477\times 10^{-4}\lambda^{4}-6.5416\times 10^{-5}\lambda^{5} \\ &+1.3027\times 10^{-6}\lambda^{6}-4.54\times 10^{-7}\lambda^{7} \end{split}  \tag{2}\]  
\(C_{p}\) has a unique maximum value (say, \(C_{p_{max}}\)) at a particular \(\lambda\), called \(\lambda_{opt}\). Thus, a VSWT tracks the \(C_{p_{max}}\), while maintaining the tip speed ratio at its optimal value, \(\lambda_{opt}\), for the maximum wind power extraction. The ratio of the peripheral speed of the wind turbine blades to the wind speed indicates the tip speed ratio, as follows:  
\[\lambda=\frac{\Omega_{\ell}R_{t}}{v_{w}} \tag{3}\]  
where \(\Omega_{\ell}\) represents the rotational speed of the wind turbine blades (rad/s, i.e., rotational speed of the low-speed shaft). The wind turbine mechanical torque (aerodynamic torque), \(T_{m}\), can be obtained from Equations (1) and (3), as follows:  
\[T_{m}=\frac{P_{m}}{\Omega_{\ell}}=\frac{1}{2}\rho\pi R_{t}^{3}v_{w}^{2}C_{t}(\lambda) \tag{4}\]  
where \(C_{t}(\lambda)=C_{p}(\lambda)/\lambda\) represents the wind turbine torque coefficient. It can be found from a second-order polynomial expression, as a function of the tip speed ratio, as follows:  
\[C_{t}(\lambda)=\alpha_{0}+\alpha_{1}\lambda+\alpha_{2}\lambda^{2} \tag{5}\]  
Now, substituting Equations (3) and (5) into Equation (4), \(T_{m}\) can also be expressed as follows:  
\[T_{m}=d_{1}v_{w}^{2}+\frac{d_{2}v_{w}\Omega_{h}}{g_{r}}+\frac{d_{3}\Omega_{h} ^{2}}{g_{r}^{2}} \tag{6}\]  
where \(\Omega_{h}\) is rotational speed of the high speed shaft (PMSG rotor) and \(g_{r}=\Omega_{h}/\Omega_{\ell}\) is the gear ratio (or transmission ratio). Moreover,  
\[ \begin{cases}d_{1}=0.5\pi\rho R_{t}^{3}\alpha_{0},\ d_{2}=0.5\pi\rho R_{t}^{4} \alpha_{1},\ d_{3}=0.5\pi\rho R_{t}^{5}\alpha_{2}\\ \alpha_{0}=0.1253,\ \alpha_{1}=-0.0047,\ \alpha_{2}=-0.0005\end{cases} \]  
Knowing \(C_{p_{max}}\) and \(\lambda_{opt}\), the optimal mechanical power, \(P_{m_{opt}}\), generated by the VSWT can be worked out from Equation (1) by substituting Equation (3) into it as follows:  
\[P_{m_{opt}}=\frac{\rho\pi{R_{t}}^{5}\Omega_{\ell}^{3}C_{p_{max}}(\lambda)}{2 \lambda_{opt}^{3}} \tag{7}\]  
Figure 2 demonstrates the mechanical power output of the VSWT under different wind speeds. There exists a unique maximum power point, \(P_{m_{opt}}\), at each wind speed. All these maximum power points constitute an optimal regime characteristic (ORC). The ORC indicates the operating region  
where the maximum possible energy can be extracted from the fluctuating wind. The primary goal of the MPPT control strategy is to continuously operate the VSWT on the ORC, despite fluctuations in the wind speed.