Continuum modeling of electromechanical dynamics in large-scale power systems

In this paper, we propose a continuum model for real power systems that tend to be highly irregular in terms of their geographical topology and the power injections, loads, and shunt elements at the bus locations. The continuum model presented here therefore relaxes the isotropy and homogeneity constraints assumed in our prior work. The network, with its transmission lines, generators, and loads, are treated as a continuum in spatial coordinates. We are consequently able to model the system as a pair of nonlinear partial differential equations (PDEs). The first PDE is the continuum equivalent of the load flow equations of the power system and is a boundary value problem. The second equation is the continuum equivalent of the swing equations of the power system. The parameters of these equations are functions of spatial coordinates and the network topology is embedded in them. The computational effort needed to solve the PDEs depends on the uniformity in the parameter distributions. A systematic approach of smoothening the parameter distributions is also proposed. While a continuum system with these smooth parameter distributions looses some of its ability to accurately model the detailed behavior of the power system, the global behavior of the system remains preserved. Furthermore, the electromechanical wave propagation behavior observed in actual power systems is readily recognized from the PDE model. A theoretical analysis of the continuum model as well as test simulations show that disturbances in the system´s phase angles propagate through the continuum system with velocities much slower than the speed of light and exhibit dispersion phenomena.