Developing and analysis of a quadratic approximation of a nonlinear two-area power system model in the state space

We present the results of mathematical modeling of a two-area four-generator electric power system (EPS) in the state space, with accounting the second order terms. Software for construction of quadratic approximations of the EPS was developed. The model is based on a 6-th order nonlinear synchronous generator model that integrates the electromechanical equations of the rotor motion, the excitation winding electromotive force (EMF) dynamics, and the dynamics of the three damping windings. Automatic voltage regulators and system stabilizers are not considered. For construction of the quadratic approximation, the Carleman linearization method is used, which is widely used for representation of nonlinear dynamic systems models in bilinear form. A detailed analysis of the dynamics matrix spectrum of the obtained quadratic approximation is presented. The results of numerical experiments are presented to demonstrate that in comparison with the traditionally used linearized model the developed quadratic approximation gives qualitatively new information about the transient processes of the initial EPS model, arising both at the deviation of some state variables and at the disturbance through system inputs. The resulting model can be used for analytical studies of the nonlinear behavior of the EPS near its operating point, in particular, for the analysis of interarea oscillations and parametric resonances. It should be noted that the method for constructing a quadratic approximation (Carleman linearization) applied in this work can be extended to EPS of larger sizes, and the developed software allows upgrading for taking into account terms of the third and higher orders of smallness.