Dan Gordon

Dan Gordon, Rachel Gordon

Linear systems with large differences between coefficients ("discontinuous coefficients") arise in many cases in which partial differential equations(PDEs) model physical phenomena involving heterogeneous media. The standard approach to solving such problems is to use domain decomposition techniques, with domain boundaries conforming to the boundaries between the different media. This approach can be difficult to implement when the geometry of the domain boundaries is complicated or the grid is unstructured. This work examines the simple preconditioning technique of scaling the equations by dividing each equation by the Lp-norm of its coefficients. This preconditioning is called geometric scaling (GS). It has long been known that diagonal scaling can be useful in improving convergence, but there is no study on the general usefulness of this approach for discontinuous coefficients. GS was tested on several nonsymmetric linear systems with discontinuous coefficients derived from convection-diffusion elliptic PDEs with small to moderate convection terms. It is shown that GS improved the convergence properties of restarted GMRES and Bi-CGSTAB, with and without the ILUT preconditioner. GS was also shown to improve the distribution of the eigenvalues by reducing their concentration around the origin very significantly.

Carleton Coffrin, Dan Gordon, Paul Scott

In recent years the power systems research community has seen an explosion of work applying operations research techniques to challenging power network optimization problems. Regardless of the application under consideration, all of these works rely on power system test cases for evaluation and validation. However, many of the well established power system test cases were developed as far back as the 1960s with the aim of testing AC power flow algorithms. It is unclear if these power flow test cases are suitable for power system optimization studies. This report surveys all of the publicly available AC transmission system test cases, to the best of our knowledge, and assess their suitability for optimization tasks. It finds that many of the traditional test cases are missing key network operation constraints, such as line thermal limits and generator capability curves. To incorporate these missing constraints, data driven models are developed from a variety of publicly available data sources. The resulting extended test cases form a compressive archive, NESTA, for the evaluation and validation of power system optimization algorithms.