## Appendix A  
Model of the dual fed induction generator (DFIG) connected to a power system (PS) through a long transmission line [10]  
\[R_{1}i_{d1}+L_{d1}\frac{d}{dt}\big{(}i_{d1}\big{)}+L_{d21}\frac{d}{dt}\big{(}i_{ d2}\big{)}+L_{1h}\frac{d}{dt}\big{(}i_{d3}\big{)}-\omega_{1}\big{(}L_{q1}i_{q1}+L_{q21}i_{ q2}+L_{1h}i_{q3}\big{)}-u_{ex}=0 \tag{11}\]  
\[R_{1}i_{q1}+L_{q1}\frac{d}{dt}\big{(}i_{q1}\big{)}+L_{q21}\frac{d}{dt}\big{(}i_ {q2}\big{)}+L_{1h}\frac{d}{dt}\big{(}i_{q3}\big{)}+\omega_{1}\big{(}L_{d1}i_{d1 }+L_{d21}i_{d2}+L_{1h}i_{d3}\big{)}=0 \tag{12}\]  
\[-(R_{2}+R_{C})i_{d2}-L_{d12}\frac{d}{dt}\big{(}i_{d1}\big{)}-L_{d2}\frac{d}{dt }\big{(}i_{d2}\big{)}-L_{1h}\frac{d}{dt}\big{(}i_{d3}\big{)}+\omega_{1}\big{(}L _{q12}i_{q1}+L_{q2}i_{q2}+L_{1h}i_{q3}\big{)}=0 \tag{13}\]  
\[-(R_{2}+R_{C})i_{q2}-L_{q12}\frac{d}{dt}\big{(}i_{q1}\big{)}-L_{q2}\frac{d}{dt }\big{(}i_{q2}\big{)}-L_{1h}\frac{d}{dt}\big{(}i_{q3}\big{)}-\omega_{1}\big{(} L_{412}i_{d1}+L_{d2}i_{d2}+L_{1h}i_{d3}\big{)}=0 \tag{14}\]  
\[R_{3}i_{d3}+L_{1h}\frac{d}{dt}\big{(}i_{d1}\big{)}+L_{1h}\frac{d}{dt}\big{(} i_{d2}\big{)}+L_{d3}\frac{d}{dt}\big{(}i_{d3}\big{)}-(\omega_{1}-\omega)\Big{(}L _{1h}i_{q1}+L_{1h}i_{q2}+L_{q3}i_{q3}\Big{)}=0 \tag{15}\]  
\[R_{3}i_{q3}+L_{1h}\frac{d}{dt}\big{(}i_{q1}\big{)}+L_{1h}\frac{d}{dt}\big{(} i_{q2}\big{)}+L_{q3}\frac{d}{dt}\big{(}i_{q3}\big{)}+(\omega_{1}-\omega)\big{(}L _{1h}i_{d1}+L_{1h}i_{d2}+L_{d3}i_{d3}\big{)}=0 \tag{16}\]  
\[\frac{d\omega}{dt}=\Big{(}p\cdot L_{1h}\cdot\big{(}i_{q1}\cdot i_{d3}+i_{d3} \cdot i_{q2}-i_{d1}\cdot i_{q3}-i_{q3}\cdot i_{d2}\big{)}+T_{m}\Big{)}\cdot p/J \tag{17}\]  
\[i_{d}=G_{1}V_{d}-B_{1}V_{q}-G_{2}V_{b}\sin(\delta)+B_{2}V_{b}\cos(\delta) \tag{18}\]  
\[i_{q}=G_{1}V_{q}+B_{1}V_{d}-G_{2}V_{b}\cos(\delta)-B_{2}V_{b}\sin(\delta) \tag{19}\]  
\[i_{d}=i_{d2}\text{ and }i_{q}=i_{q2} \tag{20}\]  
\[R_{1}i_{q1}+L_{q1}\frac{d}{dt}\big{(}i_{q1}\big{)}+L_{q21}\frac{d}{dt}\big{(}i_ {q2}\big{)}+L_{1h}\frac{d}{dt}\big{(}i_{q3}\big{)}+\omega_{1}\big{(}L_{d1}i_{d1 }+L_{d21}i_{d2}+L_{1h}i_{d3}\big{)}=0 \tag{12}\]0  
\[R_{1}i_{q1}+L_{q1}\frac{d}{dt}\big{(}i_{q1}\big{)}+L_{q21}\frac{d}{dt}\big{(}i_ {q2}\big{)}+L_{1h}\frac{d}{dt}\big{(}i_{q3}\big{)}+\omega_{1}\big{(}L_{d1}i_{d1 }+L_{d21}i_{d2}+L_{1h}i_{d3}\big{)}=0 \tag{12}\]1  
\[R_{1}i_{q1}+L_{q1}\frac{d}{dt}\big{(}i_{q1}\big{)}+L_{q21}\frac{d}{dt}\big{(}i_ {q2}\big{)}+L_{1h}\frac{d}{dt}\big{(}i_{q3}\big{)}+\omega_{1}\big{(}L_{d1}i_{d1 }+L_{d21}i_{d2}+L_{1h}i_{d3}\big{)}=0 \tag{12}\]2  
\[R_{1}i_{q1}+L_{q1}\frac{d}{dt}\big{(}i_{q1}\big{)}+L_{q21}\frac{d}{dt}\big{(}i_ {q2}\big{)}+L_{1h}\frac{d}{dt}\big{(}i_{q3}\big{)}+\omega_{1}\big{(}L_{d1}i_{d1 }+L_{d21}i_{d2}+L_{1h}i_{d3}\big{)}=0 \tag{12}\]3  
\[R_{1}i_{q1}+L_{q1}\frac{d}{dt}\big{(}i_{q1}\big{)}+L_{q21}\frac{d}{dt}\big{(}i_ {q2}\big{)}+L_{1h}\frac{d}{dt}\big{(}i_{q3}\big{)}+\omega_{1}\big{(}L_{d1}i_{d1 }+L_{d21}i_{d2}+L_{1h}i_{d3}\big{)}=0 \tag{12}\]4  
\[R_{1}i_{q1}+L_{q1}\frac{d}{dt}\big{(}i_{q1}\big{)}+L_{q21}\frac{d}{dt}\big{(}i_ {q2}\big{)}+L_{1h}\frac{d}{dt}\big{(}i_{q3}\big{)}+\omega_{1}\big{(}L_{d1}i_{d1 }+L_{d21}i_{d2}+L_{1h}i_{d3}\big{)}=0 \tag{12}\]5  
The following notations were used in equations: \(\omega\) is the rotation speed; \(\omega_{1}\) is the synchronous speed; \(T_{m}\) is the mechanical torque; \(i_{d1}\), \(i_{d2}\), \(i_{d3}\), \(i_{q1}\), \(i_{q2}\), \(i_{q3}\) are the currents projections on \(d\)-\(q\) axis, for each of the three windings: stator excitation, stator load and rotor; \(J\) is the inertia moment; \(u_{ex}\) is the excitation voltage; \(R_{1}\) is the stator excitation winding resistance; \(R_{2}\) is the stator load winding resistance; \(R_{3}\) is the rotor winding resistance; \(L_{d1}\), \(L_{q1}\), are the \(d\)-\(q\) axis inductance projections of the stator excitation winding; \(L_{d2}\), \(L_{q2}\) are the \(d\)-\(q\) axis inductance projections of the rotor winding; \(L_{d3}\), \(L_{q3}\) are the \(d\)-\(q\) axis inductance projections of the rotor winding; \(L_{d12}\), \(L_{d21}\), \(L_{q12}\), \(L_{q21}\), \(L_{1h}\) are the leakage/mutual inductances; \(p\) is the number of pole pairs; \(Y_{1}=G_{1}+jB_{1}=Y_{L}+Y_{C}\) is the transmission line admittance seen at the generator terminals (\(G_{1}\), \(B_{1}\), -conductance, susceptance)--see Figure 11; \(Y_{2}=G_{2}+jB_{2}\) is thetransfer admittance between the generator and the power grid (\(G_{2}\), \(B_{2}\)- conductance, susceptance)--see Figure 11; \(R_{L}\),\(R_{C}\) are the line and consumer resistances; \(X_{L}\), \(X_{C}\) are the line and consumer reactances; \(i_{d}\) and \(i_{q}\) are the terminals current on \(d\)-\(q\) axis; \(V_{d}\) and \(V_{q}\) are voltage projections of stator load winding on \(d\)-\(q\) axis (and also projections of \(V_{t}\) generator terminals voltage on \(d\)-\(q\) axis); \(V_{t}=\sqrt{V_{d}^{2}+V_{q}^{2}}\) is the the generator terminal voltage (effective value).