## 4 Stability Analysis of the Powertrain
### Stability Analysis with Respect to the Generator Speed  
The first analysis relates to the original parametric configuration of the system in accordance with the experimental bench. Here the dynamic (bandwidth) of the speed control for the gas turbine--generator association is set to 1 Hz for the IP controller bandwidth with a damping coefficient equal to 1. On the propeller side, a linear torque-speed characteristic is applied: \(T_{prop}\)=k.\(\Omega\); with k = 0.1 This choice is made for this preliminary study in order to avoid a nonlinearity at the input of the system, which could bring a possible additional complexity to the understanding of the instability phenomenon. This choice is possible as the evaluated speed range is quite low thus making it possible to approximate the propeller load with an equivalent linear load without making too much of an error. This analysis is set with respect to an evolution of the generator speed. Based on the linearized state model, the root locus is analytically derived with operating speed which varies step by step (every 10 rpm) between various steady-state points, here between 200 and 800 rpm for the generator mechanical speed. On that root locus, blue crosses denote the generator speed steady states while purple values are related to the system damping and black values on the right are related to the system angular frequency. Only the dominant poles close to the imaginary axis are represented here. On the transient time simulation (Matlab Simulink) and on the experimental test rig, the protocol follows a similar approach as the system evolves between different stages of generator speed, ranging from 0 to 800 rpm. These stages are spaced 10 rpm apart, and are maintained for 5 s.  
The root locus of Figure 7 shows that, inside a speed range between 260 and 620 rpm for the generator mechanical speed, the real parts of the poles are positive, indicating the unstable domain of the system. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Generator** & **Motor** \\ \hline Cyclic inductance (mH) & 6.5 & 2.7 \\ Magnetic flux of the magnets (mWb) & 286.5 & 162.1 \\ Inertia (kg.m\({}^{2}\)) & 0.0097 & 0.0062 \\ Inertia of the associated asynchronous machines acting as a gas turbine (generator) or propeller (motor) (kg.m\({}^{2}\)) & 0.0113 & 0.024 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Original parameters of the system. Figure 7: Small-signal model-based root locus with respect to the generator speed (blue crosses) with a bandwidth of 1 Hz for the generator speed control. Dashed lines illustrate iso-values of system damping (linear dashed lines), and iso-values of system angular frequency (horizontal dashed lines), the red line indicate the limit between stable and unstable zone: the ordinates axe. The time simulation result of Figure 8 also highlights a phenomenon of instability similar to the one observed from the root locus analysis. It is indeed possible to find a generator speed range during which the AC powertrain becomes oscillating (phase of \"quasi-instability\"), these non-divergent oscillations being maintained in steady state. On the contrary, before and after this speed range, the behavior of the system remains perfectly stable. Moreover, the speed range of \"quasi-instability\" is emphasized by the simulation tool between 290 and 580 rpm, which is quite consistent with the unstable range obtained with the root locus (between 260 and 620 rpm). This last comparison shows the concordance of the root locus analysis based on the analytic linearized state model and of the time simulation of the whole system, without any linearization. The experimental results of the Figure 9 also underline the same phenomenon of \"quasi-instability\". The experimental test bench has an oscillating speed range between 300 and 560 rpm, while its behavior is stable outside this interval. This speed range is also close to the previous ranges given by the theoretical means of study.