## 3 Proposed Model  
The proposed FTC scheme is applied to suppress the overshoot DC voltage under the perturbation condition of the system. A balanced short-circuit (SC) fault is applied to test the validity of the proposed FTC scheme. The short circuit is applied in the premeditated system at GSC.  
The conditions and standard measurements are arranged and observed in the SC, and then the FTC algorithm values are tuned according to the acceptable system values. The FTC algorithm performance is appreciated under the balanced SC depicted in Figure 1. It reduces the SC and holds it within the minimum value. This response is favorable for system protection. Figure 2 shows the DFIG testing block diagram, which consists of the power grid; four busses, bus-1 to bus-4; two transformers, T-1 and T-2; the transmission line, T-line; and the fault point at grid side converters.  
Figure 1: Proposed scheme for the fault tolerant control (FTC) controller. PAC, pitch angle control; MPPT, maximum power point tracking; RSC, rotor side converter.  
The DFIG model designed and presented in Figure 2 is a state-space based model. In the alpha-beta (\(\alpha\)-\(\beta\)) stationary reference frame, the DFIG model can be represented in the state-space form as follows:  
\[\begin{array}{l}\dot{x}(t)=Ax(t)+Bu(t)\\ y(t)=Cx(t)\end{array} \tag{7}\]  
where \(x=[I_{\alpha s}I_{\beta s}I_{\alpha r}i_{\beta r}]^{T}\) is the rotor and stator current component; and \(u\) (t) is the system input, which can be written as \(x=[u_{\alpha s}u_{\beta s}u_{\alpha r}u_{\beta r}]^{T}\), which is the rotor and the stator voltage components. Hence, the state matrix \(A\) and the matrix input \(B\) are defined as follows in Equation (8):  
\[A=\left(\begin{array}{cc}A_{11}&A_{12}\\ A_{21}&A_{22}\end{array}\right)B=\left(\begin{array}{cc}\frac{L_{m}}{ \sigma L_{L}L_{r}}I_{2}&-\frac{1}{\sigma L_{s}}I_{2}\\ -\frac{1}{\sigma L_{s}}I_{2}&\frac{L_{m}}{\sigma L_{L}L_{r}}I_{2}\end{array} \right)\mathrm{K}=\left(\begin{array}{cc}0&-1\\ 1&0\end{array}\right) \tag{8}\]  
where  
\[A_{11}=-(\frac{R_{\mathrm{s}}}{\sigma L_{\mathrm{s}}}I_{2}+(\omega_{s}+\frac {\mathrm{L}_{m}^{2}\mathrm{P}\Omega_{mech}}{\sigma L_{\mathrm{s}}L_{r}}) \mathrm{K}) \tag{9}\]  
\[A_{12}=(\frac{L_{m}R_{r}}{\sigma L_{\mathrm{s}}L_{r}}I_{2}-\frac{\mathrm{L}_{ m}\mathrm{P}\Omega_{mech}}{\sigma L_{\mathrm{s}}}\mathrm{K}) \tag{10}\]  
\[A_{21}=(\frac{L_{m}R_{\mathrm{s}}}{\sigma L_{\mathrm{s}}L_{r}}I_{2}+\frac{ \mathrm{L}_{m}\mathrm{P}\Omega_{mech}}{\sigma L_{r}}\mathrm{K}) \tag{11}\]  
\[A_{22}=-(\frac{1}{\sigma L_{r}}I_{2}+(\omega_{s}-\frac{\mathrm{P}\Omega_{mech }}{\sigma})\mathrm{K}) \tag{12}\]  
\[\mathrm{Calculating}\ \sigma\ \mathrm{as}\ \sigma=1-L_{m}^{2}/(\mathrm{L_{s}L_{r}}) \tag{13}\]  
The rotor and stator resistances are expressed by \(R_{\mathrm{s}}\) and \(R_{\mathrm{r}}\), respectively. The current is denoted by \(I\), and \(\omega_{\mathrm{s}}\) represents the synchronous speed. The \(\Omega_{\mathrm{mech}}\) represents the mechanical rotational speed. The inductance and magnetization inductance of the rotor and stator are represented by \(L_{\mathrm{r}}\), \(L_{\mathrm{s}}\), and \(L_{\mathrm{m}}\), respectively.