This manuscript reports the first step towards building a robust and efficient model reduction methodology to capture transient dynamics in a transmission level electric power system. Such dynamics is normally modeled on seconds-to-tens-of-seconds time scales by the so-called swing equations, which are ordinary differential equations defined on a spatially discrete model of the power grid. Following Seymlyen (1974) and Thorpe, Seyler, and Phadke (1999), we suggest to map the swing equations onto a linear, inhomogeneous Partial Differential Equation (PDE) of parabolic type in two space and one time dimensions with time-independent coefficients and properly defined boundary conditions. We illustrate our method on the synchronous transmission grid of continental Europe. We show that, when properly coarse-grained, i.e., with the PDE coefficients and source terms extracted from a spatial convolution procedure of the respective discrete coefficients in the swing equations, the resulting PDE reproduces faithfully and efficiently the original swing dynamics. We finally discuss future extensions of this work, where the presented PDEbased modeling will initialize a physics-informed machine learning approach for real-time modeling, n − 1 feasibility assessment and transient stability analysis of power systems.
We investigate coherent oscillations in large scale transmission power grids, where large groups of generators respond in unison to a distant disturbance. Such long wavelength coherent phenomena are known as inter-area oscillations. Their existence in networks of weakly connected areas is well captured by singular perturbation theory. However, they are also observed in strongly connected networks without time-scale separation, where applying singular perturbation theory is not justified. We show that the occurrence of these oscillations is actually generic. Applying matrix perturbation theory, we show that, because these modes have the lowest oscillation frequencies of the system, they are only moderately sensitive to increased network connectivity between well chosen, initially weakly connected areas, and that their general structure remains the same, regardless of the strength of the inter-area coupling. This is qualitatively understood by bringing together the standard singular perturbation theory and Courant's nodal domain theorem.INDEX TERMS Slow coherency; transmission power systems; matrix perturbation theory; inter-area oscillations.
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