# Analysis of Synchronous Generators' Local Mode Eigenvalues in Modern Power Systems
## 4 Local Mode Analysis of Synchronous Generators with AVR  
### Synchronous Generators' Voltage Control Systems  
Synchronous generators operating in a power system require an extensive control system for coordinated operation. There are two fundamental synchronous generator control systems: a primary speed control system (i.e., frequency restoration control system) and a voltage control system. The main difference between these two control systems is in the response speed of the final control element. Turbines that generate the mechanical torque needed in primary speed control systems are very slow-acting, so this control system does not affect synchronous generators' electromechanical oscillations significantly. On the other hand, exciters in modern power plants are almost always static semiconductor devices, which are considerably faster and influence electromechanical oscillations greatly. Voltage control systems can decrease the damping of the local oscillations significantly. New voltage control systems for synchronous generators are often realized with a cascade control system, where the outer control loop regulates the voltage (or reactive power), and the inner loop maintains the static U/Q characteristic. Both control loops could be implemented with proportional-integral (PI) controllers. However, many synchronous generators are still equipped with a simple voltage control system based on a proportional (P)-type voltage controller and static exciter. This control system performs voltage regulation and provides the static U/Q characteristic. Such a voltage control concept is the most transparent for analysis and also the most commonly used in similar studies [11; 12; 13]. Therefore, we also limited this study to the analysis of the influence of this type of voltage control system on the local mode eigenvalues. A voltage control system with a P-controller and static exciter has a significant impact on local oscillations. The controller in this system is usually called an automatic voltage regulator (AVR). The block diagram of the AVR system is shown in Figure 9. This simple control system essentially changes the synchronous generator dynamics. \begin{table}
\begin{tabular}{c c c c c} \hline  & \(\boldsymbol{\sigma_{\min}}\) & \(\boldsymbol{\sigma_{\max}}\) & \(\boldsymbol{\omega_{\min}\ (s^{-1})}\) & \(\boldsymbol{\omega_{\max}\ (s^{-1})}\) \\ \hline hydro-generators & \(-0.3193\) & \(-0.0399\) & \(3.3636\) & \(9.1832\) \\ \hline turbo-generators & \(-0.6361\) & \(-0.0807\) & \(3.6418\) & \(9.4520\) \\ \hline \end{tabular}
\end{table} 
Table 4: The boundaries of real and imaginary parts of local mode eigenvalues of hydro and turbo synchronous generators of different nominal powers in an operating range from \(P\) = 0.0 pu to \(P\) = 1.0 pu and from \(Q\) = 0.0 pu to \(Q\) = 1.0 pu. \begin{table}
\begin{tabular}{c c c c c} \hline  & \(\boldsymbol{\Sigma_{\min}}\) & \(\boldsymbol{\sigma_{\max}}\) & \(\boldsymbol{\omega_{\min}\ (s^{-1})}\) & \(\boldsymbol{\omega_{\max}\ (s^{-1})}\) \\ \hline hydro-generators & \(-0.4558\) & \(-0.0399\) & \(2.8554\) & \(10.8259\) \\ \hline turbo-generators & \(-0.8871\) & \(-0.0807\) & \(3.6418\) & \(10.5455\) \\ \hline \end{tabular}
\end{table} 
Table 3: The boundaries of real and imaginary parts of local mode eigenvalues of hydro and turbo synchronous generators of different nominal powers in an operating range from \(P\) = 0.0 pu to \(P\) = 1.2 pu and from \(Q\) = 0.0 pu to \(Q\) = 1.2 pu. The simplest mathematical model of the regulator in series with the exciter is a first-order lag, described with the transfer function:  
\[G(s)=\frac{E_{\text{FD\Delta}}(s)}{V_{\text{t\Delta,ref}}(s)-V_{\text{t\Delta}}(s )}=\frac{k_{\text{AVR}}}{sT_{\text{AVR}}+1} \tag{5}\]  
where \(V_{\text{t\Delta,ref}}(s)\) represents the reference terminal (stator) voltage, \(V_{\text{t\Delta}}(s)\) represents the actual terminal voltage, and \(E_{\text{t\Delta}}(s)\) represents the output from the exciter, i.e., rotor winding voltage.