Using phase-space localized basis functions to obtain vibrational energies of molecules

For decades scientists have attempted to use ideas of classical mechanics to choose basis functions for calculating spectra. The hope is that a classically-motivated basis set will be small because it covers only the dynamically important part of phase space. One popular idea is to use phase space localized (PSL) basis functions. This thesis improves on previous efforts to use PSL functions and examines the usefulness of these improvements. Because the overlap matrix, in the matrix eigenvalue problem obtained by using PSL functions with the variational method, is not an identity, it is costly to use iterative methods to solve the matrix eigenvalue problem. We show that it is possible to circumvent the orthogonality (overlap) problem and use iterative eigensolvers. We also present an altered method of calculating the matrix elements that improves the performance of the PSL basis functions, and also a new method which more efficiently chooses which PSL functions to include. These improvements are applied to a variety of single well molecules. We conclude that for single minimum molecules, the PSL functions are inferior to other basis functions. However, the ideas developed here can be applied to other types of basis functions, and PSL functions may be useful for multi-well systems.

ADVANCED ISLANDING DETECTION METHODS FOR SINGLE PHASE GRID CONNECTED PV INVERTERS

Due to the growing concerns associated with fossil fuels, emphasis has been placed on clean and sustainable energy generation. This has resulted in the increase in Photovoltaics (PV) units being integrated into the utility system. The integration of PV units has raised some concerns for utility power systems, including the consequences of failing to detect islanding. Numerous methods for islanding detection have been introduced in literature. They can be categorized into local methods and remote methods. The local methods are categorically divided into passive and active methods. Active methods generally have smaller Non-Detection Zone (NDZ) but the injecting disturbances will slightly degrade the power quality and reliability of the power system. Slip Mode Frequency Shift Islanding Detection Method (SMS IDM) is an active method that uses positive feedback for islanding detection. In this method, the phase angle of the converter is controlled to have a sinusoidal function of the deviation of the Point of Common Coupling (PCC) voltage frequency from the nominal grid frequency. This method has a non-detection zone which means it fails to detect islanding for specific local load conditions. If the SMS IDM employs a different function other than the sinusoidal function for drifting the phase angle of the inverter, its non-detection zone could be smaller. In addition, Advanced Slip Mode Frequency Shift Islanding Detection Method (Advanced SMS IDM), which has been introduced in this thesis, eliminates the non-detection zone of the SMS IDM. In this method the parameters of SMS IDM change based on the local load impedance value. Moreover, the stability of the system is investigated by developing the dynamical equations of the system for two operation modes; grid connected and islanded mode. It is mathematically proven that for some loading conditions the nominal frequency is an unstable point and the operation frequency slides to another stable point, while for other loading conditions the nominal frequency is the only stable point of the system upon islanding occurring. Simulation and experimental results show the accuracy of the proposed methods in detection of islanding and verify the validity of the mathematical analysis.