# Analysis of Synchronous Generators' Local Mode Eigenvalues in Modern Power Systems
## 2 Mathematical Model of a Synchronous Generator Connected to an Infinite Bus
The operating point values \(P\) and \(Q\) in Table 1 are active and reactive power at a machine terminal, respectively; \(\omega_{\text{s}}\) is electric synchronous speed; and \(V_{\infty}\) is infinite bus voltage. From Table 1 can be seen the reduction of the parameters' request of the H-P model compared to the NLM [8]. For calculation of the H-P model parameters, there is no need to know the synchronous generator's d-axis sub-transient inductances; d- and q-axes leakage inductances; or stator, field, d-axis damping, and q-axis damping windings resistances. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters of Synchronous Generator** & \(L_{\text{d}}\) (H), \(L_{\text{q}}\) (H), \(L_{\text{d}}\)\({}^{\prime}\) (H), \(H\) (s), \(R_{\text{e}}\) (\(\Omega\)), \(L_{\text{e}}\) (H), \(D\) (pu), \\  & \(T_{\text{d}0^{\prime}}\)(s) \\ \hline Operating point & \(P\) (W), \(Q\) (Var), \(\omega_{\text{s}}\) (rad s\({}^{-1}\)), \(V_{\infty}\) (V) \\ quantities & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Data of synchronous generator and network for H–P model parameters calculation. From the data in Table 1, the auxiliary operating point variables' values are calculated through synchronous machine phasor equations. The auxiliary operating point variables values in Table 2 below must be determined before calculating the H-P model parameters. The variables in Table 2: \(\delta(t)\) is the electric rotor angle; \(V_{\text{d}}(t)\) and \(V_{\text{q}}(t)\) are stator terminal d- and q-axis voltages, respectively; \(I_{\text{d}}(t)\) and \(I_{\text{q}}(t)\) are the stator d- and q-axis currents, respectively; and \(E\), \(E_{\text{qa}}\), and \(E^{\prime}_{\text{q}}\) are standardized phasor diagram EMFs. Index \({}_{\text{op}}\) means an operating point value. The H-P model is presented in a state-space form by (1) and (2) [8]. The linearization parameters \(K_{1}\) to \(K_{6}\) of the H-P model are calculated based on the values in Tables 1 and 2. All variables in the H-P model are normalized except electric rotor angle \(\delta(t)\). \[\left[ \begin{array}{c}\dot{\delta}_{\text{A}}(t)\\ \dot{\omega}_{\text{A}}(t)\\ \dot{E^{\prime}}_{\text{qA}}(t)\end{array} \right]=\left[ \begin{array}{ccc}0& \omega_{\text{s}}&0\\ -\frac{K_{1}}{2H}&-\frac{D}{2H}&-\frac{K_{3}}{2H}\\ -\frac{K_{4}}{7a}&0&-\frac{K_{1}}{K_{3}T_{\text{u}0}}\end{array} \right] \left[ \begin{array}{c}\delta_{\text{A}}(t)\\ \omega_{\text{A}}(t)\\ E^{\prime}_{\text{qA}}(t)\end{array} \right]+\left[ \begin{array}{c}0&0\\ \frac{1}{2H}&0\\ 0&\frac{1}{T_{\text{u}0}}\end{array} \right]\left[ \begin{array}{c}T_{\text{ ma}}(t)\\ E_{\text{fdA}}(t)\end{array} \right] \tag{1}\]  
\[\left[ \begin{array}{c}T_{\text{eA}}(t)\\ V_{\text{tA}}(t)\end{array} \right]=\left[ \begin{array}{ccc}K_{1}&0&K_{2}\\ K_{5}&0&K_{6}\end{array} \right]\left[ \begin{array}{c}\delta_{\text{A}}(t)\\ \omega_{\text{A}}(t)\\ E^{\prime}_{\text{qA}}(t)\end{array} \right]+\left[ \begin{array}{c}0&0\\ 0&0\end{array} \right]\left[ \begin{array}{c}T_{\text{ma}}(t)\\ E_{\text{fdA}}(t)\end{array} \right] \tag{2}\]  
The accuracy of the H-P model was investigated in [8]. Simulations with both NLM and H-P models were made and compared with experiments. The simulations and experiments were made systematically for different operating points in the entire operating range, as well as different dynamical changes of inputs (mechanical torque and excitation voltage). The high accordance of the H-P model is evident from the obtained results. Agreement in the first amplitude, the frequency, and the damping of the transient response is very high. The conclusion of [8] was that the H-P model describes synchronous generators' electromechanical phenomena in the entire operating range in the case of the small input's perturbations, which could be used for a detailed analysis of local oscillations.