# Analysis of Synchronous Generators' Local Mode Eigenvalues in Modern Power Systems
## 6 Conclusions  
Oscillations and stability are fundamental power system problems. Oscillations cause fatigue in rotor shafts, reduce rotor life, induce power quality concerns, provoke undesirable tripping of generation units, etc. Knowledge of local oscillation modes of synchronous generators is crucial for stability analysis and instability prediction. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \(P=0.0\) pu & \(P=0.4\) pu & \(P=0.8\) pu & \(P=1.0\) pu & \(P=1.2\) pu \\ \hline \multirow{4}{*}{\(Q=0.0\) pu} & \(\sigma=-1.75\) & \(\sigma=-4.50\) & \(\sigma=-7.16\) & \(\sigma=-7.70\) & \(\sigma=-8.06\) \\ \cline{2-6}  & \(\omega=22.85\) & \(\omega=27.33\) & \(\omega=34.15\) & \(\omega=37.80\) & \(\omega=41.34\) \\ \cline{2-6}  & \(\omega_{0}=22.92\) & \(\omega_{0}=27.70\) & \(\omega_{0}=34.89\) & \(\omega_{0}=38.57\) & \(\omega_{0}=42.12\) \\ \cline{2-6}  & \(\zeta=0.0764\) & \(\zeta=0.1625\) & \(\zeta=0.2053\) & \(\zeta=0.1996\) & \(\zeta=0.1915\) \\ \hline \multirow{4}{*}{\(Q=0.5\) pu} & \(\sigma=-1.63\) & \(\sigma=-2.51\) & \(\sigma=-4.18\) & \(\sigma=-5.15\) & \(\sigma=-5.93\) \\ \cline{2-6}  & \(\omega=29.22\) & \(\omega=31.22\) & \(\omega=33.47\) & \(\omega=36.43\) & \(\omega=40.06\) \\ \cline{2-6}  & \(\omega_{0}=29.26\) & \(\omega_{0}=31.32\) & \(\omega_{0}=33.73\) & \(\omega_{0}=36.79\) & \(\omega_{0}=40.49\) \\ \cline{2-6}  & \(\zeta=0.0558\) & \(\zeta=0.0801\) & \(\zeta=0.1241\) & \(\zeta=0.1400\) & \(\zeta=0.1465\) \\ \hline \multirow{4}{*}{\(Q=1.0\) pu} & \(\sigma=-1.10\) & \(\sigma=-2.39\) & \(\sigma=-2.89\) & \(\sigma=-3.32\) & \(\sigma=-3.91\) \\ \cline{2-6}  & \(\omega=34.25\) & \(\omega=36.83\) & \(\omega=37.16\) & \(\omega=37.69\) & \(\omega=39.12\) \\ \cline{2-6}  & \(\omega_{0}=34.27\) & \(\omega_{0}=36.91\) & \(\omega_{0}=37.27\) & \(\omega_{0}=37.84\) & \(\omega_{0}=39.32\) \\ \cline{2-6}  & \(\zeta=0.0321\) & \(\zeta=0.0649\) & \(\zeta=0.0776\) & \(\zeta=0.0877\) & \(\zeta=0.0996\) \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Real and imaginary parts of the eigenvalues, the undamped natural frequencies, and the damping ratios of the local oscillations of the laboratory synchronous generator at different operating points. Figure 23: Eigenvalue loci of the local mode of the laboratory synchronous generator \(P_{\rm n}=12\) kW obtained with numerical analysis of an H–P model and with identification of the measured time responses of the oscillations conducted with step changes of the mechanical torque. Local oscillations are influenced by the construction of a synchronous generator, its connection to the network and parameters, and the structure of the voltage control system and operating point. In this paper, we investigated more than 100 synchronous generators in the nominal power range from 1 MVA to 1000 MVA operating in present-day power systems. We tried to find out in what range the frequencies and dampings of local oscillations are and how individual factors affect the oscillations. Since the frequency and damping of oscillation are directly related to the eigenvalues of the mathematical model of the synchronous generator connected to the network, we carried out the study by means of eigenvalue analysis. The following conclusions were obtained:  
* Differences in the construction of the generators (slow, salient pole hydro-type synchronous generators or fast cylindrical rotor turbo-type synchronous generators) do not have a recognizable effect on the frequency and damping of local oscillations. * It is also not possible to draw conclusions about the influence of size (within the considered size range) on the oscillation of generators. * The variations of the impedance of the transmission line have a small effect on the oscillations. * The changing of the operating point of the synchronous generator has a very large influence on the frequency and damping of oscillations.