## 3 Modelling, Analysis Tools and Experimental Support for Stability Analysis
### Nonlinear Analytic State Model
#### 3.1.1 State Model Derivation
\begin{array}{c}\frac{d}{dt}\begin{pmatrix}I_{d\mathbf{0}}^{g}\\ I_{q\mathbf{0}}^{g}\\ \delta\end{pmatrix}=\begin{pmatrix}-\frac{\left(R_{g}+R_{m}\right)}{\left(L_{g} +L_{m}\right)}&\omega_{g}&0\\ -\omega_{g}&-\frac{\left(R_{g}+R_{m}\right)}{\left(L_{g}+L_{m}\right)}&0\\ 0&0&0\end{pmatrix}\begin{pmatrix}I_{d\mathbf{0}}^{g}\\ I_{q\mathbf{0}}^{g}\\ \delta\end{pmatrix}+\begin{pmatrix}-\frac{\phi_{q_{m}}\sin(\delta)}{L_{g}+L_{m} }&0\\ -\frac{\phi_{q_{m}}\cos(\delta)}{L_{g}+L_{m}}&\frac{\phi_{q_{g}}}{L_{g}+L_{m}}\\ -1&1\end{pmatrix}\begin{pmatrix}\omega_{m}\\ \omega_{g}\end{pmatrix}\end{array}  \tag{4}\]  
where \(I_{d}^{g}\), \(I_{q}^{g}\) are the stator currents that have become common to both generator and motor after the direct AC connexion. The variables in red illustrate the model non linearities inside this state matrix form. The torques are also derived for both machines by taking account of the total flux derivation for the generator (\(\mathbf{\Phi}_{\mathbf{\dot{g}dq\mathbf{0}}^{g}}^{g}\)) and the motor (\(\mathbf{\Phi}_{\mathbf{m}d\mathbf{0}}^{g}\)), respectively described in Equations (1) and (2):  
\[T_{em_{g}}=\frac{3p_{g}}{2}\Big{(}\Phi_{\mathbf{\dot{g}}_{d}}^{g}\cdot I_{q}^{g}- \Phi_{\mathbf{\dot{g}}_{d}}^{g}\cdot I_{d}^{g}\Big{)}=\frac{3p_{g}}{2}\phi_{q_{g}} \cdot I_{q} \tag{5}\]  
\[T_{em_{m}}=\frac{3p_{m}}{2}\Big{(}\Phi_{m_{d}}^{g}\cdot I_{q}^{g}-\Phi_{m_{d} }^{g}\cdot I_{d}^{g}\Big{)}==\frac{3p_{m}}{2}\phi_{q_{m}}\cdot\big{(}I_{d}sin (\delta)+I_{q}cos(\delta)\big{)} \tag{6}\]  
where \(p_{g}\) and \(p_{m}\) are the number of pole pairs, respectively, for the generator and motor. In the previous equations and for the rest of the derivation, the currents in the generator reference frame are simply denoted as the Park's currents (\(I_{d}^{g}=I_{d}\), \(I_{q}^{g}=I_{q}\)). The mechanical equations at both (generator and motor) sides have to be added to complete the state model:  
\[ \begin{array}{c}\frac{d}{dt}\omega_{g}=\frac{p_{g}}{I_{qq_{g}}}\Big{(}T_{TAG }-T_{em_{g}}\Big{)}-\frac{f_{g}}{I_{eq_{g}}}\omega_{g}\;\text{with}\;J_{eq_{g }}=J_{g}+J_{TAG}\;\text{and}\;\omega_{g}=p_{g}\cdot\Omega_{g}\\ \\ \frac{d}{dt}\omega_{m}=\frac{p_{m}}{I_{qm}}\big{(}T_{em_{m}}-T_{prop_{m}} \big{)}-\frac{f_{m}}{I_{qm}}\omega_{m}\;\text{with}\;J_{eq_{m}}=J_{m}+J_{prop}\\ \text{and}\;\omega_{m}=p_{m}\cdot\Omega_{m}\end{array}  \tag{8}\]  
where \(T_{TAG}\) and \(T_{em_{g}}\) are, respectively, the gas turbine and the generator electromagnetic torques. \(T_{prop}\) and \(T_{em_{m}}\) are, respectively, the propeller (load) and the motor electromagnetic torques. \(J_{g}\), \(J_{TAG}\), \(J_{m}\),\(J_{prop}\) are, respectively, the generator, gas turbine, motor and propeller moment of inertia. The two previous equations also describe the link between electric angular speeds \(\omega_{x}\) and rotation mechanical speeds \(\Omega_{x}\). The complete 5 th order analytic state model with gas turbine and propeller torques as inputs is then obtained:\[\frac{d}{dt}\left( \begin{array}{c}I_{d}\\ I_{q}\\ \delta\\ \omega_{m}\\ \omega_{g}\end{array} \right)=\left( \begin{array}{cccc}-\frac{R_{g}+R_{m}}{I_{g }+L_{m}}&\omega_{g}&0-\frac{\phi_{m}\sin(\delta)}{I_{g}+L_{m}}&0\\ -\omega_{g}&-\frac{R_{g}+R_{m}}{I_{g}+L_{m}}&0-\frac{\phi_{m}\cos(\delta)}{I_{g }+L_{m}}&\frac{\phi_{g}}{I_{g}+L_{m}}\\ 0&0&-1&1\\ \frac{3p_{m}^{2}}{2I_{\omega_{m}}}\phi_{\omega_{m}}\sin(\delta)\frac{3p_{m}^{2 }}{2I_{\omega_{m}}}\phi_{\omega_{m}}\cos(\delta)0&\frac{-f_{m}}{I_{\omega_{m}} }&0\\ 0&-\frac{3p_{g}^{2}}{2I_{\omega_{m}}}\phi_{\omega_{g}}&0&\frac{-f_{g}}{I_{ \omega_{g}}}\\ 0&-\frac{3p_{g}^{2}}{2I_{\omega_{g}}}\phi_{\omega_{g}}&0&\frac{-f_{g}}{I_{ \omega_{g}}}\\ \end{array} \right)\left( \begin{array}{c}I_{d}\\ I_{q}\\ \delta\\ \omega_{m}\\ \omega_{g}\end{array} \right)+\left( \begin{array}{c}0&0\\ 0&0\\ 0&0\\ 0&0\\ 0&0\\ 0&0\\ 0&-\frac{p_{m}}{I_{\omega_{m}}}\\ \end{array} \right) \tag{9}\]  
Actually, the gas turbine is classically controlled to drive its rotation speed.