## 3 Simulation Model
### VSWG Model  
A VSWG model made up of a three-phase current source [17], as shown in Figure 7, was used in this work. The current source model has sufficient accuracy in the analysis of the power system with a great number of WGs, which cannot be simulated by using the detailed generator model. In this work, the wind farm is composed of 100 VSWGs each rated at 5 MW. The VSWG parameters [17] are listed in Table 2.  
Figure 6: Control model of the PV system. Figure 7: \(Cont.\)  
The wind turbine power output, \(P_{w}\), can be calculated as follows [18]:  
\[P_{w}=\frac{1}{2}\rho\pi R^{2}V_{w}^{3}C_{p}(\lambda,\ \beta) \tag{3}\]  
\[C_{p}(\lambda,\ \beta)=0.5176\left(\frac{116}{\lambda_{i}}-0.4\beta-5\right)e^{-21/ \lambda_{i}}\ +0.0068\lambda \tag{4}\]  
\[\frac{1}{\lambda_{i}}=\frac{1}{\lambda-0.08\beta}\ -\ \frac{0.035}{\beta^{3}+1} \tag{5}\]  
\[\lambda=\frac{\omega_{r}R}{V_{w}} \tag{6}\]  
where \(\rho\) is the air density, \(R\) is the radius of rotor blade, \(V_{w}\) is the wind speed, \(C_{p}\) is the performance coefficient, \(\lambda\) is tip speed ratio and \(\beta\) is blade pitch angle. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{3}{c}{**VSWG**} \\ \hline \multicolumn{3}{c}{**Wind Turbine**} & \multicolumn{3}{c}{**PMSG**} \\ \hline \(R\) & (m) & 60 & \(P_{s}\) & (MW) & 5 \\ \(V_{w}\) & (m/s) & 8.8 & \(V_{s}\) & (kV) & 1 \\ \(\omega_{r}\) & (rad/s) & 1.2 & \(R_{s}\) & (\(\Omega\)) & 0.004 \\ \(C_{p\ opt}\) & & 0.48 & \(L_{sd}\) & (mH) & 1.528 \\ \(\lambda_{opt}\) & & 8.2 & \(L_{sq}\) & (mH) & 1.21 \\ \(\rho\) & (kg/m\({}^{3}\)) & 1.225 & \(\psi_{m}\) & (Wb) & 11.255 \\ \(I_{t}\) & (kgm\({}^{2}\)) & \(49.5\times 10^{6}\) & \(C_{dc}\) & (\(\mu\)F) & 2500 \\  & & & \(V_{dc}\) & (kV) & 1.75 \\  & & & \(R_{c}\) & (\(\Omega\)) & 0.45 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: VSWG parameters. Figure 7: Control model of the VSWG: (**a**) overall; (**b**) stator side controller; (**c**) grid side controller; (**d**) protection circuit of DC-link. From Equation (3), the maximum power point, which changes according to the rotor speed, is calculated as follows [18]:  
\[P_{mptt}=\frac{1}{2}\rho\pi R^{2}\bigg{(}\frac{\omega_{r}R}{\lambda_{opt}}\bigg{)} ^{3}C_{p\ opt} \tag{7}\]  
where \(\lambda_{opt}\) and \(C_{p\ opt}\) are the optimal values of the tip speed ratio and the power coefficient, respectively. The wind turbine parameters are also listed in Table 2. In this case study, we assumed that \(V_{w}\) is kept constant at 8.8 (m/s) at which the active power reference of the VSWG is 0.45 pu (VSWG capacity base) within the simulation period. The equation of motion of the drive train is expressed as follows:  
\[T_{m}-T_{e}=J_{t}\frac{\mathrm{d}\omega_{r}}{\mathrm{dt}} \tag{8}\]  
where \(T_{m},\ T_{e},\ J_{t}\) and \(\omega_{r}\) are mechanical torque, electrical torque, moment of inertia and rotor speed of the wind turbine, respectively. The dynamic model of the PMSG in the d-q rotating reference frame is expressed as follows:  
\[V_{sd}=R_{s}I_{sd}+L_{sd}\frac{\mathrm{d}I_{sd}}{\mathrm{dt}}-\omega_{e}L_{sq} I_{sq} \tag{9}\]  
\[V_{sq}=R_{s}I_{sq}+L_{sq}\frac{\mathrm{d}I_{sq}}{\mathrm{dt}}+\omega_{e}L_{sd} I_{sd}+\omega_{e}\psi_{m} \tag{10}\]  
where \(V_{sd}\) and \(V_{sq}\) are stator voltages, \(R_{s}\) is the stator winding resistance, \(I_{sd}\) and \(I_{sq}\) are stator currents, \(\omega_{e}\) is the rotor speed, \(L_{sd}\) and \(L_{sq}\) are the synchronous inductances of stator winding and \(\psi_{m}\) is the flux of permanent magnet. Active and reactive power of the PMSG are expressed as follows:  
\[P_{s}=V_{sd}I_{sd}+V_{sq}I_{sq} \tag{11}\]  
\[Q_{s}=V_{sq}I_{sd}-V_{sd}I_{sq} \tag{12}\]  
The electrical torque of the PMSG can be expressed as follows;  
\[T_{e}=p\Big{[}\psi_{m}I_{sq}+\big{(}L_{sd}-L_{sq}\big{)}I_{sd}I_{sq}\Big{)} \tag{13}\]  
where \(p\) is the number of pole pairs. The dynamic behavior of DC capacitor voltage \(V_{dc}\) is expressed as follows:  
Figure 7:0  
\[P_{rc}=