Analyzing dynamic performance of power systems over parameter space using the method of normal forms of vector fields

Today\u27s power systems have become more and more stressed due to the high utilization of available facilities. The complex dynamic behavior of large stressed power systems following disturbances can not be fully explained with present tools, such as linear eigen-analysis tools and nonlinear time-domain simulation methods. This research work applies a nonlinear analytical tool, the method of normal forms of vector fields, to help understand the complex transient oscillations in stressed power systems;The method of normal forms is a well-known mathematical tool to study systems of differential equations. The basic idea is to simplify the dynamical system by a sequence of nonlinear coordinate transformations. If there is no resonance in the system, then the nonlinear vector field can be turned into a linear one by the transformations. Previous work applied the second-order normal form transformation under non-resonance condition to power system dynamical equations. The nonlinear interaction among the fundamental modes was investigated. Based on these efforts, this work extends the application of normal forms to evaluate the dynamic performance of power systems taking into account changing operation conditions;As the resonance and near-resonance could occur in parameter space, a new normal form transformation under second order resonance condition is derived. The analysis shows that the high nonlinearity resulting from the resonance and near-resonance among poorly damped oscillatory modes and control modes is detrimental to the system performance. An approach to determine the resonance and near-resonance regions in parameter space is developed. The modes contributing to the detrimental behavior associated with the near-resonance region are identified by a procedure based on certain modal interaction indices. The state variables showing detrimental behavior are then determined using nonlinear participation factors. The accuracy of the prediction is verified by conducting nonlinear time-domain simulation. In order to compare the effect of nonlinear modal interaction quantitatively under different operating conditions, a new index in the state space of machine variables is developed. The nonlinear modal interaction together with the linear modal characteristics accounts for the dynamic performance of the system over a range of operating conditions. The method and procedures are tested and validated on a sample test system

Formula manipulation in the bond graph modelling and simulation of large mechanical systems

A multibond graph element for a general single moving body is derived. A multibody system can easily be described as an interconnection of these elements. 3-D mechanical systems usually contain dependent inertias having both differential and integral causality. A method is described for the transformation of inertias with differential causality to an integral form, using formula manipulation. The program also helps to find experimentally the optimal choice for the generalized coordinates. The resulting explicit differential equation may be solved using a standard integration routine or simulation program