Power distribution systems continue to evolve to accommodate the advancements in the field of microgrids and renewable energy resources. The future grids will be highly connected and will require increased reliability of the network. To this effect, low-voltage distribution systems with meshed or networked topology can be utilized. Currently, the use of low-voltage heavily-meshed distribution systems is restricted to urban areas with high load density that require increased reliability of power. A reason for this is the high cost of construction of such systems and complex topology which creates additional challenges. The direction of power flow in such systems is not unidirectional, which makes the power flow analysis difficult. Complicated network analysis techniques are required to determine the fault currents and protection settings in the network. Due to the aforementioned reasons, there is limited work analyzing the effectiveness of existing power flow algorithms to solve complex meshed systems. In this thesis, the robustness of two power flow algorithms is compared using an index called static stability breakdown margin parameter of circuit elements. For this study, a low-voltage heavily-meshed distribution test system is also proposed. Additionally, a study is conducted to show how reliable the meshed test system is against any fault in the system. The steady-state voltage stability of the test system is observed during the event of a fault. The stability margin parameter is then used to determine the vulnerable components in the system which need to be strengthened to increase the stability and voltage profile of the system. / Master of Science / Distribution systems carry electricity from the transmission system and deliver it to the customers. Distribution systems mainly operate using two topologies for their feeders: Radial and Meshed. The majority of customers are served using radial distribution systems, as in the radial feeders power flows in one direction (i.e. from substation to the end-user). They are simple in design and operation and are constructed at a moderate cost. However, if there is a fault along the main feeder, there will be an interruption of power to the end-use customer. On the other hand, meshed distribution systems involve multiple paths of power flow between all the points in the network. If a fault occurs along the feeder, the power flow is rerouted to the other available paths. Thus, Heavily Due to their complex topology, meshed systems are expensive to construct and deploy. The power flow analysis of these systems poses many challenges. Because of these reasons, their use is mainly restricted to urban areas with high load density which require very high reliability. The future grid is becoming increasingly complex and evolving to a meshed distribution topology has its own advantages. However, as presently the use of meshed systems is sparse, the work done on evaluating the stability of these systems is minimal. As a result of which, this thesis focuses on determining the optimal power flow solvers for these complex systems, analyzing their stability under abnormal operating conditions, and suggesting methods to reinforce the vulnerabilities in the system.

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When a power system is subjected to a disturbance, the power flow changes, leading to deviations in the synchronous generator rotor angles. The rotor angle deviations propagate as electromechanical waves (EMWs) throughout the power system. These waves became observable since the development of synchrophasor measurement instruments. The speed of EMW propagation is hundreds of miles per second, much less than the electromagnetic wave propagation speed, which is the speed of light. Recently, with the development of renewable energy resources and a growth in using HVDC and FACTS devices, these waves are propagating slower, and their impacts are more considerable and complicated. The protection system needs a control system that can take suitable action based on local measurements to overcome the results of power system faults. Therefore, the dynamic behavior of power systems should be properly observed. The EMW propagation in the literature was studied using assumptions such as constant voltage throughout the entire power system and zero resistances and equal series reactances for the transmission lines. Although these assumptions help simplify the power system study model, the model cannot capture the entire power system's dynamic behaviors, since these assumptions are unrealistic. This research will develop an accurate model for EMW propagation when the system is facing a disturbance using a continuum model. The model includes a novel inertia distribution. It also investigates the impacts of voltage changes in the power system on EMW behaviors and when these impacts are negligible. Furthermore, the impacts of the internal reactances of synchronous generators and the resistances of transmission lines on EMW propagation are explored. / Doctor of Philosophy / Power systems, essential for electricity supply, undergo disturbances causing changes in power flow and synchronous generator behavior. These disturbances create electromechanical waves (EMWs) that influence system dynamics. Recent advancements, including renewable energy integration and new technologies, alter EMW behavior, posing challenges for control and protection systems. Existing studies simplify models, limiting their accuracy. This research aims to develop a realistic EMW propagation model considering factors like novel inertia distribution, voltage changes, and internal generator properties. This work addresses the evolving power landscape, enhancing our understanding of power system dynamics for improved control and reliability.

We study the nonlinear Schrödinger (NLS) equation on star graphs with the Neumann- Kirchhoff (NK) boundary conditions at the vertex. We analyze the stability of standing wave solutions of the NLS equation by using different techniques. We consider a half-soliton state of the NLS equation, and by using normal forms, we prove it is nonlinearly unstable due to small perturbations that grow slowly in time. Moreover, under certain constraints on parameters of the generalized NK conditions, we show the existence of a family of shifted states, which are parametrized by a translational parameter. We obtain the spectral stability/instability result for shifted states by using the Sturm theory for counting the Morse indices of the shifted states. For the spectrally stable shifted states, we show that the momentum of the NLS equation is not conserved which results in the irreversible drift of the family of shifted states towards the vertex of the star graph. As a result, the spectrally stable shifted states are nonlinearly unstable. We also study the NLS equation on star graphs with a delta-interaction at the vertex. The presence of the interaction modifies the NK boundary conditions by adding an extra parameter. Depending on the value of the parameter, the NLS equation admits symmetric and asymmetric standing waves with either monotonic or non-monotonic structure on each edge. By using the Sturm theory approach, we prove the orbital instability of the standing waves. / Thesis / Doctor of Philosophy (PhD)

Turbulent swirling flows are important in many applications including gas turbines, furnaces and cyclone dust separators among others. Although the mean flow fields have been relatively well studied, a complete understanding of the flow field including its dynamics has not been achieved. The work contained in this dissertation attempts to shed further light on the behavior of turbulent swirling flows, especially focused on the dynamic behavior of a turbulent swirling flow encountering a sudden expansion. Experiments were performed in a new isothermal turbulent swirling flow test facility. Two geometrical nozzle configurations were studied. The \cb\ nozzle configuration exhibits a cylindrical \cb\ in the center of the nozzle. The free vortex nozzle configuration is obtained when the cylindrical \cb\ is removed. Detailed laser velocimeter measurements were performed to map out the flow field near the sudden expansion of the 2.9" (ID) nozzle leading to the 7.4" (ID) downstream section. In addition to presenting detailed flow profiles for both nozzle and downstream flow fields, representative frequency spectra of the flow dynamics are presented. Along with the flow time histories and histograms, the wide variety of dynamic behavior was thus described in great detail. The dynamics observed in the experiment can be classified into three main categories: coherent and large scale motion, intermittent motion and coherent periodic motion. Free vortex geometry flows, in the parameter space of the experiments (Swirl number = 0 - 0.21), exhibited mostly coherent and large scale motion. The spectra in these cases were broadband with very light concentration of spectral energy observed in some specific cases. Center--body geometry flows exhibited all three categories of flows as swirl strength was increased from zero. Flows with little or no swirl exhibited broad--band spectra similar to those for the free vortex geometry. Intermediate swirl levels resulted in a large amount of low frequency energy which, with the aid of the time histories, was identified as a large scale intermittence associated with radial movement of the annular jet as it enters the sudden expansion. Large swirl levels resulted in high magnitude coherent oscillations concentrated largely just downstream of the sudden expansion. Linear stability analysis was used to help in the interpretation of the observed dynamics. Although, as implemented here (using the parallel flow assumption), the analysis was not successful in quantitatively matching the experimentally observed dynamics, significant insight into the physical mechanisms of the observed dynamics was obtained from the analysis. Specifically, the coherent oscillations observed for larger swirl levels were able to be described in terms of the interaction between the inner and outer shear layers of the flow field. / Ph. D.