# Analysis of Synchronous Generators' Local Mode Eigenvalues in Modern Power Systems
## 2 Mathematical Model of a Synchronous Generator Connected to an Infinite Bus  
Due to the extreme importance of synchronous generators, it is expected that the activities in the field of synchronous generator modeling have been, and still are, very intensive. There is a distinct interest in adequate static and dynamic models of synchronous generators. These models accelerate the design and the construction of synchronous generators, facilitate the analysis and simulation of synchronous generators' operation, and enable the systematic synthesis of their control systems [8]. There are many models of different levels of complexity that are suitable for various qualitative and quantitative analyses of synchronous generators. The most fundamental model of a synchronous generator is the non-linear seventh-order dq model (NLM), where the magnetic coupling of stator, field, and damper windings is a function of the position and saturation of a rotor. For the analysis of electromechanical oscillations of a synchronous generator connected to a strong network, the generator model is supplemented with the simple lumped-parameters model of the transmission line between a synchronous generator and an infinite bus. This model certainly represents the best description of the dynamic and static behavior of a synchronous generator. The disadvantage of this model is the need to know a large number of the parameters of the synchronous generator. These parameters are mainly unknown and challenging to measure. Therefore, the use of this model is inappropriate in cases where a model-based quantitative analysis of a real synchronous generator is intended. For the purposes of small-signal electromechanical oscillation analysis, the use of linearized synchronous generator models is satisfactory. They describe the synchronous generator's dynamics in the proximity of the selected equilibrium point. Linearized models are characterized by the dependence of their parameters on the equilibrium point from which they are derived. By changing the synchronous generator's operating point (i.e., equilibrium point), the parameters of the linearized models also change. Linearized models enable easy evaluation of eigenvalue analysis, which offers good insight into the frequency and damping of electromechanical oscillations. Linearized models of different levels are known and used. Among many linearized models of different orders, the Heffron-Phillips model (H-P model) is the most popular. A detailed description of the assumptions regarding the development of this model, comparison of the H-P model with NLM, and the research of applicability of the H-P model in an entire operating range can be found in [8]. The H-P model has two inputs and three state-space variables. The inputs are mechanical torque deviation \(T_{\text{m}\Delta}(t)\) and rotor winding voltage deviation \(E_{\text{fd}\Delta}(t)\). The state-space variables are rotor angle deviation \(\delta_{\Delta}(t)\), rotor speed deviation \(\omega_{\Delta}(t)\), and voltage behind transient reactance deviation \(E^{\prime}{}_{\text{q}\Delta}(t)\). Additional outputs are electric torque deviation \(T_{\text{e}\Delta}(t)\) and terminal stator voltage deviation \(V_{\text{t}\Delta}(t)\). All the inputs and the state-space variables denote the deviations (subscript \({}_{\Delta}\)) from the equilibrium state. The minimum set of the parameters and the operating point values of the synchronous generator, transmission line, and infinite bus necessary to calculate the H-P model's parameters are presented in Table 1.  
The parameters \(L_{\text{d}}\) and \(L_{\text{q}}\) in Table 1 are d- and q-axes inductances; \(L_{\text{d}}\)' is d-axis transient inductance; \(H\) is an inertia constant; \(R_{\text{e}}\) and \(L_{\text{e}}\) are transmission line resistance and inductance; \(D\) is a damping coefficient representing total lumped damping effects; and \(T_{\text{d}0}\) is an open circuit time constant of a direct axis.