## 3 Modelling, Analysis Tools and Experimental Support for Stability Analysis
### Nonlinear Analytic State Model  
#### 3.1.1 State Model Derivation  
The analytic derivation for the generator and for the motor(s) is based on a [PERSON]'s model, which depends on the topology, which may be either single- (Figure 1b) or multi-motor (Figure 1c). Each electric machine (generator or motor) is classically modelled by a [PERSON]'s (\(d_{\beta}\)) model that links the \(q\) axis with the electromotive force (emf): \(E_{x}=E_{xq}\) as illustrated in Figure 3 in which a generator and a motor are respectively represented in the (\(d_{g}\),\(d_{\beta}\)) and (\(d_{m}\),\(d_{m}\)) rotating reference frames. In the case of a direct AC coupling of both machines (Figure 1b), the emfs can be phase shifted by the angle \(\delta\). Let us note that this derivation may be generalized for a multi-motor topology, in which each electric motor (\(M_{x}\)) would have its own reference frame (\(d_{mx}\),\(d_{mx}\)) with a \(\delta_{x}\) phase shift with the generator. In order to represent the direct coupling between the generator and the motor(s) and to generalize the model derivation whatever the number of motors, it was chosen to represent all machines in the generator reference frame (\(d_{g}\),\(d_{\beta}\)). In the rest of the paper, only a single-motor topology will be considered in order to simplify the stability analysis which remains complex even with that simplification. However, most of the theoretical consideration proposed in the next section remain relevant for multi-motor topology. The Park's model of a generator (1) and single-motor (2) association can be derived, where bold type is used to highlight vector or matrix variables as opposed to scalar quantities:  
\[ \begin{split}\mathbf{V}_{dq0}^{g}=-\big{(}R_{g}*\mathbf{I}_{3}\big{)}\cdot \mathbf{R}_{dq0}^{g}+\frac{d}{dt}\Big{(}\mathbf{\Phi}_{6 dq0}^{g}\Big{)}+\omega_{g}* \begin{pmatrix}0&-1&0\\ 1&0&0\\ 0&0&0\end{pmatrix}\cdot\mathbf{\Phi}_{6 dq0}^{g}\\ \mathbf{\Phi}_{6 dq0}^{g}=-L_{g}\cdot\mathbf{R}_{dq0}^{g}+\phi_{g}*\begin{pmatrix}1\\ 0\\ 0\end{pmatrix}\end{split}  \tag{1}\]  
Figure 3: [PERSON]’s reference frames: (**a**) generator voltage and current vectors in the generator reference frame (\(d_{g}\),\(d_{\beta}\)); (**b**) motor voltage and current vectors in the motor reference frame (\(d_{m}\),\(d_{m}\)); (**c**) generator and motor voltage and current vectors with phase shift in the generator reference frame (\(d_{g}\),\(d_{\beta}\)). \[ \begin{array}{c}\mathbf{V}_{dq\mathbf{0}}^{g}=(R_{m}*\mathbf{I_{3}})\cdot I_{dq\mathbf{0}}^{g}+ \frac{d}{dt}\left(\mathbf{\Phi}_{m}^{g}\mathbf{\dot{q}}_{d\mathbf{0}}\right)+\omega_{g}* \begin{pmatrix}0&-1&0\\ 1&0&0\\ 0&0&0\end{pmatrix}\cdot\mathbf{\Phi}_{m}^{g}\mathbf{\dot{q}}_{d\mathbf{0}}\\ \\ \mathbf{\Phi}_{m}^{g}\mathbf{\dot{q}}_{d\mathbf{0}}=L_{m}\cdot\mathbf{I}_{dq\mathbf{0}}^{g}+\phi_{ q_{m}}*\begin{pmatrix}\cos(\delta)\\ -\sin(\delta)\\ 0\end{pmatrix}\\ \\ \delta=\theta_{g}-\theta_{m};\;\frac{d}{dt}\delta=\frac{d}{dt}\big{(}\theta_ {g}-\theta_{m}\big{)}=\omega_{g}-\omega_{m}\end{array}  \tag{2}\]  
where \(\mathbf{I_{3}}\) is the third order identity matrix, \(R_{g}\), \(R_{m}\), \(L_{g}\), \(L_{m}\), \(\theta_{q_{g}}\) and \(\phi_{q_{m}}\) are, respectively, the generator and motor resistances, inductances and rotor (magnet) fluxes. \(\theta_{g}\), \(\theta_{m}\), \(\omega_{g}\) and \(\omega_{m}\), respectively, denote the generator and motor electric positions and angular speeds, \(\delta\) being the electric angle that represents the phase shift between both machine frames. When both machines are directly connected through an AC bus, the equality of stator voltages (\(\mathbf{V}_{dq\mathbf{0}}^{g}\)) leads to the final analytic model of the association that can be derived as a non-linear state model:  
\[