Quantification of nonlinear modal interactions using modal signal energies

Increased system stress has brought about the need for a better understanding of nonlinearities in power system dynamic behavior. The relationship between system stress and the nonlinearity of the differential equations should be quantified. The similarity transformation used to determine the Jordan canonical form of a linear system provides a view of the system in a ‘modal’ state space where the linear parts are decoupled and easily accounted for. The nonlinear system can also be transformed to the modal state space using the linear variable transformation. In the transformed state space the effects of nonlinearities and linearities are clearly separated and are expressed in terms of their effects on the system modes. Thus the nonlinear interactions between the modes are viewed directly. The effects of the nonlinearities on the modes are measured by considering the signal energy associated with the nonlinearities in the Jordan-form state space. The concept of nonlinear signal energy provides a numerical value to gauge the influence of nonlinearities on a given mode. Application to the IEEE 50-generator test system shows the relationship between interarea modes and nonlinear interactions between low- and high-frequency modes.