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Thesis (MSc (Physics))--University of Stellenbosch, 2006. / The aim of this thesis is the semi-classical implementation of Wegner’s flow equations and comparison with the well-established Wentzel-Kramers-Brillouin method. We do this by converting operators, in particular the Hamiltonian, into scalar functions, while an isomorphism with the operator product is maintained by the introduction of the Moyal product. A flow equation in terms of these scalar functions is set up and then approximated by expanding it to first order in ~. We apply this method to two potentials, namely the quartic anharmonic oscillator and the symmetric double-well potential. Results obtained via the flow equations are then compared with those obtained from the WKB method.

Thesis (MSc (Physics))--University of Stellenbosch, 2005. / The goal of this thesis is the development and implementation of a non-perturbative solution method for Wegner’s flow equations. We show that a parameterization of the flowing Hamiltonian in terms of a scalar function allows the flow equation to be rewritten as a nonlinear partial differential equation. The implementation is non-perturbative in that the derivation of the PDE is based on an expansion controlled by the size of the system rather than the coupling constant. We apply this method to the Lipkin model and obtain very accurate results for the spectrum, expectation values and eigenstates for all values of the coupling and in the thermodynamic limit. New aspects of the phase structure, made apparent by this non-perturbative treatment, are also investigated. The Dicke model is treated using a two-step diagonalization procedure which illustrates how an effective Hamiltonian may be constructed and subsequently solved within this framework.

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abstract: Nowadays, the widespread introduction of distributed generators (DGs) brings great challenges to the design, planning, and reliable operation of the power system. Therefore, assessing the capability of a distribution network to accommodate renewable power generations is urgent and necessary. In this respect, the concept of hosting capacity (HC) is generally accepted by engineers to evaluate the reliability and sustainability of the system with high penetration of DGs. For HC calculation, existing research provides simulation-based methods which are not able to find global optimal. Others use OPF (optimal power flow) based methods where too many constraints prevent them from obtaining the solution exactly. They also can not get global optimal solution. Due to this situation, I proposed a new methodology to overcome the shortcomings. First, I start with an optimization problem formulation and provide a flexible objective function to satisfy different requirements. Power flow equations are the basic rule and I transfer them from the commonly used polar coordinate to the rectangular coordinate. Due to the operation criteria, several constraints are incrementally added. I aim to preserve convexity as much as possible so that I can obtain optimal solution. Second, I provide the geometric view of the convex problem model. The process to find global optimal can be visualized clearly. Then, I implement segmental optimization tool to speed up the computation. A large network is able to be divided into segments and calculated in parallel computing where the results stay the same. Finally, the robustness of my methodology is demonstrated by doing extensive simulations regarding IEEE distribution networks (e.g. 8-bus, 16-bus, 32-bus, 64-bus, 128-bus). Thus, it shows that the proposed method is verified to calculate accurate hosting capacity and ensure to get global optimal solution. / Dissertation/Thesis / Masters Thesis Electrical Engineering 2018

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