With the incorporation of renewables and new technologies, the modern power grid is facing new challenges that can be partially resolved with mathematics. Since the basic mathematical structure of the traditional power grid is retained in the modern grid, analysis of the fundamental mathematical equations is crucial. Specifically, the governing equations are a large algebraic-differential system defined on a weighted graph, with the dynamics modelled by a coupled system of oscillators and the algebraic constraints arising from the Kirchoff/Ohm's law. In this talk, we derive the spatial multilevel/multiscale features of the power grid, which can be used to construct accurate coarse-grain models for fast real-time simulations, fast parameter analysis, and fast multigrid solvers for a variety of power grid problems. These features are obtained by exploiting a synchrony condition that describes a stable power network. In particular, this condition is based on the weighted graph Laplacian given by the so-called admittance matrix, which reflects the physical properties of the transmission lines. Nodes or DOFS for the coarse model are selected to ensure that the resulting coarse graph Laplacian is an accurate approximation to the graph Laplacian of the fine-grain model. To achieve this coarse node selection, state-of-the-art multigrid techniques will be used.
Unfortunately, the selection of the coarse nodes and a multigrid coarsening of the weighted Laplacian matrix are not sufficient to produce a stable coarse model. To obtain a stable system, other physical parameters of the power grid must be carefully aggregated using the intergrid operators constructed in the coarsening of the Laplacian, and other system structures of the fine-grain model must be preserved. In this talk, the multigrid coarse-graining algorithm and the procedures to generate a stable coarse-grain model are described. Numerical examples will be provided to demonstrate the potential of these coarsening methods.
BIO: Barry Lee obtained his Ph.D. in 1996 from the Department of Applied Mathematics at University of Colorado, under the supervision of Professors Steve McCormick and Tom Manteuffel. During the past 8 years, he has been affiliated with the Department of Mathematics at Southern Methodist University. Prior to joining SMU, he was an applied/computational mathematician at the National Oceanic and Atmospheric Administration, Lawrence Livermore National Lab, and Pacific Northwest National Lab. His research focuses on numerical algorithmic development and scientific computing for large-scale industrial and laboratory applications. His current research interests include the development of efficient algorithms for general coupled systems of elliptic partial differential equations, large systems of algebraic-differential equations (electric power grid networks), the Boltzmann transport equation (neutron/photon transport), Maxwell equations (fusion), and uncertainty quantification. Central to his research is the development of schemes that give optimal computational efficiency on serial and large-scale parallel computer platforms. Thus, an essential component of his research is computational linear algebra, particularly scalable multigrid and multilevel methods.