# 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
All symbols denote Laplace transforms of time deviation signals where \(s\) is a complex frequency variable and \({}_{\Delta}\) denotes deviations. The symbol \(k_{\text{AVR}}\) represents the product of the voltage controller and exciter gains, and \(T_{\text{AVR}}\) represents the exciter time constant. The time constant \(T_{\text{AVR}}\) depends on the available exciter. For a static exciter, \(T_{\text{AVR}}\) is very short. It is mainly in the range of 2-10 ms. There are many publications that have studied the tuning of the gain \(k_{\text{AVR}}\). The rule for setting \(k_{\text{AVR}}\) was described for the first time in [21]. In [21], it was derived that for the well-damped performance of the voltage control loop, it is desirable to maintain a crossover frequency lower than \(1/(2\ T_{\text{AVR}})\), which would mean that the gain \(k_{\text{AVR}}\) \"should be approximately less than \(T^{\prime}{}_{\text{d0}}/(2T_{\text{AVR}})\)\". Later, in [22], it was shown that gain \(k_{\text{AVR}}\) has a significant impact on the local mode eigenvalue and that the \(k_{\text{AVR}}\) values which are close to the estimated maximal value \(T^{\prime}{}_{\text{d0}}/(2\ T_{\text{AVR}})\) cause instability. Therefore, in this work, the impact of the gain \(k_{\text{AVR}}\) on the local mode eigenvalue was considered first. The H-P model (Equations (1) and (2)), extended with the mathematical model of the AVR system (Equation (5)), is presented with Equations (6) and (7). \[\left[ \begin{array}{c}\dot{\delta}_{\Delta}(t)\\ \dot{\omega}_{\Delta}(t)\\ \dot{E}^{{}^{\prime}}{}_{\text{q\Delta}}(t)\\ \dot{E}_{\text{t\Delta}}(t)\end{array} \right]=\left[ \begin{array}{cccc}0& \omega_{\text{s}}&0&0\\ -\frac{K_{1}}{2H}&-\frac{D}{2H}&-\frac{K_{2}}{2H}&0\\ -\frac{K_{4}}{74\text{d0}}&0&-\frac{1}{T^{\prime}{}_{\text{d0}}K_{3}}&\frac{1 }{T^{\prime}{}_{\text{d0}}}\\ -\frac{k_{\text{AVR}}K_{5}}{T_{\text{AVR}}}&0&-\frac{k_{\text{AVR}}K_{6}}{T_{ \text{AVR}}}&-\frac{1}{T_{\text{AVR}}}\end{array} \right]\left[ \begin{array}[ ]{c}\delta_{\Delta}(t)\\ \omega_{\Delta}(t)\\ E^{\prime}{}_{\text{q\Delta}}(t)\\ E_{\text{fd\Delta}}(t)\end{array} \right]+\left[ \begin{array}{c}0&0\\ 0&0\\ 0&\frac{k_{\text{AVR}}}{T_{\text{AVR}}}\end{array} \right]\left[ \begin{array}[ ]{c}T_{\text{m\Delta}}(t)\\ V_{\text{trefA}}(t)\\ \end{array} \right] \tag{6}\]  
\[\left[ \begin{array}{c}T_{\text{e\Delta}}(t)\\ V_{\text{t\Delta}}(t)\end{array} \right]=\left[ \begin{array}{cccc}K_{1}&0&K_{2 }\\ K_{5}&0&K_{6}\end{array} \right]\left[ \begin{array}{cccc}K_{1}&0&K_{2}&0\\ K_{5}&0&K_{6}&0\end{array} \right]\left[ \begin{array}{c}\delta_{\Delta}(t)\\ \omega_{\Delta}(t)\\ E^{\prime}{}_{\text{q\Delta}}(t)\\ E_{\text{fd\Delta}}(t)\end{array} \right]+\left[ \begin{array}{cccc}0&0\\ 0&0\end{array} \right]\left[ \begin{array}{c}T_{\text{m\Delta}}(t)\\ V_{\text{trefA}}(t)\end{array} \right] \tag{7}\]  
Influence of AVR Gain \(k_{\text{AVR}}\) on Local Mode Eigenvalues of Synchronous Generators with AVR System  
The AVR gain \(k_{\text{AVR}}\) has a significant influence on the synchronous generator dynamics. The eigenvalues of the extended H-P model (Equations (6) and (7)) must be calculated to analyze the local oscillations of the synchronous generator with the AVR system. The obtained characteristic equation is presented with (Equation (8)). Figure 9: Block diagram of the studied AVR system. \[ \begin{array}{l}2K_{3}HT^{\prime}{}_{d0}T_{AVR}A^{4}+\\ \left(K_{3}DT^{\prime}{}_{d0}T_{AVR}+2 HT_{AVR}+2 HK_{3}T^{\prime}{}_{d0}\right) \lambda^{3}+\\ \left(K_{1}K_{3}T^{\prime}{}_{d0}T_{AVR}\omega_{s}+DT_{AVR}+K_{3}DT^{\prime}{}_ {d0}+2K_{3}K_{6}K_{AVR}H+2H\right)\lambda^{2}+\\ \left(K_{3}K_{6}K_{AVR}D+K_{1}K_{3}T^{\prime}{}_{d0}\omega_{s}+D+K_{1}T_{AVR} \omega_{s}-K_{2}K_{3}K_{4}T_{AVR}\omega_{s}\right)\lambda+\\ \left(K_{1}K_{3}K_{6}K_{AVR}\omega_{s}-K_{2}K_{3}K_{5}K_{AVR}\omega_{s}+K_{1} \omega_{s}-K_{2}K_{3}K_{4}\omega_{s}\right)=0\end{array}