Ph.D. Defense by William Johns (Mathematical Sciences, MSU)

04/16/2021  3:00pm  WebEx Meeting

Abstract:

This research began during a summer internship at the National Renewable Energy Lab (NREL).  One long term research goal of the Complex Systems Simulation & Optimization  (CSSO) group at NREL is to develop technology that identifies the location of a fault in an electrical grid using data from sparse detectors for electro-magnetic traveling waves launched from the fault event.  Such technology relies on having accurate electro-magnetic transients models that are amenable to fast simulation.  Multi-input Multi-output (MIMO) rational approximations are widely used for model order reduction of time invariant linear dynamical systems including electro-magnetic transients models. We have developed methodology for extending the Adaptive Antoulas Anderson (AAA) algorithm from single input approximations to these MIMO problems. For this application it is necessary that all of the approximation poles be stable. We introduce a new methodology (smiAAA) for enforcing this stability constraint in MIMO AAA approximations, and examples show that the algorithm produces approximations with smaller error and fewer poles than existing approaches. For applications where uniformly bounded errors are desirable, we extend the ideas of single input Barycentric Lawson optimization to MIMO problems. We compare this new algorithm with the industry standard Vector Fitting algorithm and the newly developed RKFIT algorithm. Both of these algorithms require the user to input the number of poles to be used in the approximations; the locations of the poles are then optimized, and the best approximation found is returned.  With this approach, the user is not guaranteed any particular final accuracy in the approximation, and one must rerun the algorithm with additional poles if insufficient accuracy is achieved. In contrast, the smiAAA algorithm has been designed to prioritize the accuracy of the approximation.  The algorithm allows the user to specify a desired accuracy, and it automatically detects the number of poles required to achieve that accuracy, guaranteeing that the algorithm never needs to be rerun. The presentation will focus on contrasting the performance of these algorithms on two test problems that highlight the advantages and disadvantages inherent to each numerical approach.