## 3 Existence of Synchronous Motion
### The Stability of Synchronous Solutions  
Equations (18)-(21) after linearization around \(\overline{\pi}_{10}\), \(\overline{\pi}_{20}\), \(\overline{\pi}_{30}\) and \(\omega_{\rm m0}\), are respectively,  
\[k\biggl{(}\frac{3}{2}\dot{\overline{\pi}}_{1}+\dot{\overline{\pi}}_{2}+\frac{ 1}{2}\dot{\overline{\pi}}_{3}\biggr{)}=-\sum_{i=1}^{3}T_{\rm u}\biggl{[}\frac{ \partial(\chi_{\rm f1}+\chi_{\rm a1})}{\partial\overline{\pi}_{i}}\biggr{]}_ {0}\Delta\alpha_{i} \tag{37}\]  
\[k_{\rm R1}\biggl{(}-\frac{1}{2}\dot{\overline{\pi}}_{1}+\dot{\overline{\pi}}_{2 }+\frac{1}{2}\dot{\overline{\pi}}_{3}\biggr{)}=-\sum_{i=1}^{3}T_{\rm u}\biggl{[} \frac{\partial(\chi_{\rm f2}+\chi_{\rm a2})}{\partial\overline{\pi}_{i}} \biggr{]}_{0}\Delta\alpha_{i} \tag{38}\]\[k_{\text{R2}}\bigg{(}-\frac{1}{2}\ddot{\overline{\pi}}_{1}-\ddot{\overline{\pi}}_{2 }+\frac{1}{2}\ddot{\overline{\pi}}_{3}\bigg{)}=-\sum_{i=1}^{3}T_{\text{u}}\bigg{[} \frac{\partial(x_{\text{R3}}+\chi_{\text{A3}})}{\partial\ddot{\overline{\pi}}_{i }}\bigg{]}_{0}\Delta a_{i} \tag{39}\]  
\[k_{\text{R3}}\bigg{(}-\frac{1}{2}\ddot{\overline{\pi}}_{1}-\ddot{\overline{\pi} }_{2}-\frac{3}{2}\ddot{\overline{\pi}}_{3}\bigg{)}=-\sum_{i=1}^{3}T_{\text{u}} \bigg{[}\frac{\partial(x_{\text{R4}}+\chi_{\text{A4}})}{\partial\ddot{\overline {\pi}}_{i}}\bigg{]}_{0}\Delta a_{i} \tag{40}\]  
where \([\bullet]_{0}\) denotes the values for \(\overline{\pi}_{1}=\overline{\pi}_{10}\), \(\overline{\pi}_{2}=\overline{\pi}_{20}\), \(\overline{\pi}_{3}=\overline{\pi}_{30}\); \(\Delta a_{i}=\overline{\pi}_{i}-\ddot{\overline{\pi}}_{i0}\), \(i=1,\,2,\,3\); \(k=k_{\text{e01}}+f_{1}\); \(k_{Ri}=\Big{[}\frac{\mathrm{d}(f_{Ri}\eta_{i}\overline{\pi}_{2}^{T}\overline{ \pi}_{1}(2+\eta_{i}W_{\text{e01}}))}{\mathrm{d}\omega_{\text{m}}}\Big{]}_{ \omega_{\text{m}}=\omega_{\text{m}}},\,i=1,\,2,\,3\).  
Calculating the differences of Equations (37) and (38), Equations (38) and (39), Equations (39) and (40), and writing the results into a matrix form, then we have  
\[\mathbf{E}\Delta\dot{\alpha}=\mathbf{D}\Delta\alpha \tag{41}\]  
with \(\mathbf{E}=\left[e_{ij}\right]_{3\times 3}\) and \(\mathbf{D}=\left[d_{ij}\right]_{3\times 3}\). The explicit expressions for \(e_{ij}\) and \(d_{ij}\) are provided in Appendix C.  
Rewriting Equation (41), we obtain  
\[\Delta\dot{\alpha}=\mathbf{C}\Delta\alpha,\,\,\mathbf{C}=\mathbf{E}^{-1} \mathbf{D} \tag{42}\]  
Solving the determinant equation \(\det(\lambda\mathbf{I}-\mathbf{C})=0\), we can deduce the characteristic equation for the eigenvalue \(\lambda\) as follows:  
\[\lambda^{3}+a_{1}\lambda^{2}+a_{2}\lambda+a_{3}=0 \tag{43}\]  
By means of Routh-Hurwitz principle, it can be determined that when Equation (44) is satisfied, the zero solutions of Equation (42) are stable.  
\[a_{1}>0,\,\,a_{3}>0,\,\,a_{1}a_{2}>a_{3} \tag{44}\]  
That is to say, Equation (44) is the stability criterion of synchronous operation. Simplifying Equation (44), we have  
\[H_{1}>0,\,\,H_{2}>0,\,\,H_{3}>0 \tag{45}\]  
Here, \(H_{i}\) is referred to as the stability coefficient. The greater the stability coefficient \(H_{i}\) is, the stronger the stability is. Explicit expression for \(H_{i}\) is provided in Appendix C.