# Analysis of Synchronous Generators' Local Mode Eigenvalues in Modern Power Systems
## 5 Experimental Analysis of the Laboratory-Size Synchronous Generator
Local oscillations were analyzed throughout the entire operating range. The desired studied steady-state operating point was settled using a power amplifier and induction machine. To provoke the local oscillations, we generated a step change of the torque of the induction machine. Currents and voltages were measured, and generated powers were calculated in all phases. In the computed instantaneous 3-phase power, the local oscillation was seen clearly. It had the same time course as the step response of the second-order lag. To determine the local mode eigenvalues, we identified a second-order dynamic system with the same response as local oscillations. The second-order transfer functions were identified at different active and reactive power. The MATLAB Identification Toolbox was used to identify the transfer functions at different operating points. The poles were calculated from the c transfer functions. They corresponded to the eigenvalues of the local oscillations. Figures 21 and 22 show the time responses of the measured instantaneous 3-phase power of the laboratory synchronous generator in the vicinity of two distinctive operating points. Figure 21 presents the local oscillation in the vicinity of the operating point \(P=12\) kW, \(Q=0\) kVAr, where the oscillation was well-damped. The figure also shows the step response of the identified transfer function used to determine the eigenvalues of the local oscillation. The measured response and the simulated response of the identified model for the least-damped operating point \(P=0\) kW, \(Q=12\) kVAr are presented in Figure 22. In both cases, the simulated responses of the identified models correspond to the measured responses in the time interval after the occurrence of the step change of mechanical torque. Figure 21: Time response of the instantaneous three-phase power of the laboratory synchronous generator \(S_{\mathrm{n}}=12\) kV in the vicinity of the operating point \(P=12\) kW, \(Q=0\) kVAr on the step change of mechanical torque corresponding to mechanical power change from 11.5 kW to 13.5 kW compared with the identified local oscillation. Measurements and identification were carried out at operating points in the entire working area. Table 10 presents the real and imaginary parts of the local mode eigenvalues, the undamped natural frequencies, and the damping ratios at measured operating points. The used base values were \(P_{\text{base}}\) = 12 kW and \(Q_{\text{base}}\) = 12 kVAr. Experimental results confirmed the main conclusions of the correlation between the operating point and the frequency and damping of local oscillations that were obtained by the numerical analysis. As expected, the values of the oscillation frequencies of the laboratory generator did not correspond to the numerically calculated limit values of power plants' synchronous generators. The numerically obtained boundaries of the eigenvalues' regions presented in Tables 14-16 are valid for synchronous generators with nominal power between \(S_{\text{n}}\) = 1 MVA and \(S_{\text{n}}\) = 1000 MVA, and they deviate significantly from the eigenvalues of the laboratory synchronous generator with nominal apparent power \(S_{\text{n}}\) = 15 kVA. Figure 23 presents the identified local mode eigenvalues (Table 10) together with the eigenvalue loci obtained with numerical analysis of the H-P models of synchronous generators with the parameters in Tables 6, 7 and 9. Active power was changed in the range of 0.0 pu to 1.2 pu at constant reactive power. The measurements were made at active power values \(P\) = 0.0 pu, \(P\) = 0.4 pu, \(P\) = 0.8 pu, \(P\) = 1.0 pu, and \(P\) = 1.2 pu, all at reactive power values Q = 0.0 pu, Q = 0.5 pu, and Q = 1.0 pu. The individual curves correspond to a constant reactive power. Local mode eigenvalues at nominal power \(P\) = 1 pu are marked on the figures with a square.