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In der Stromversorgung vollzieht sich seit etwa zwanzig Jahren ein grundlegender Wandel von konventioneller Stromerzeugung durch hauptsächlich Kohle- und Atomkraftwerke hin zu erneuerbaren Stromerzeugung durch hauptsächlich Windkraft- und Solaranlagen. Die Hauptunterschiede zwischen diesen beiden Formen der Erzeugung liegt in der geringeren Leistung, die eneuerbare Erzeuger typischerweise gegenüuber konventionellen Erzeugern produzieren und in dem vermehrten Auftreten von Fluktuationen in deren Leistungserzeugung. Aufgrund dieser fundamentalen Unterschiede zwischen diesen beiden Formen der Stromerzeugung stellt dieser Wandel das stabile Funktionieren des Stromnetzes vor grosse Herausforderungen. In dieser Arbeit werden anhand eines einfachen Modells für Stromnetze verschiedene Fragestellungen die Stabilität des Stromnetzes betreffend untersucht. Im ersten Teil der Arbeit wird das Modell im Detail vorgestellt. Anschliessend wird das Modell für das einfachst möglichste Stromnetz untersucht, hier bestehend aus einem Erzeuger und einem Verbraucher. In diesem einfachen Fall lassen sich die mathematischen Gleichungen des Modells analytisch lösen, was Einsichten in die Eigenschaften des Modells erlaubt. Diese Eigenschaften sind auch bei grösseren Netzen vorzufinden. Es wird gezeigt, dass das Modell die wichtigsten Eigenschaften des realen Netzes erfasst, an erster Stelle dass die Entwicklung der Dynamik des Stromnetzes sowohl hin zu einem stabilen Zustand, als auch zu einem instabilen Zustand hin möglich ist, abhängig von dem aktuellen Zustand des Netzes. Im zweiten Teil der Arbeit wird das Phänomen der Dezentralisierung untersucht. Da Erzeuger, die auf erneuerbaren Energieträgern basieren, typischerweise weniger Leistung produzieren koennen als konventionelle Kraftwerke, müssen, um einzelne konventionelle Kraftwerke zu ersetzen, mehrere erneuerbare Erzeuger neu an das bestehende Stromnetz angeschlossen werden. Dies führt zu der sogenannten Dezentralisierung, womit gemeint ist, dass die neu angeschlossen Kraftwerke oftmals weit von der Masse der Verbraucher, zum Beispiel grosse Städte, entfernt sind. Dies hat verschiedene Konsequenzen auf die Stabilität des Stromnetzes, welche in diesem Kapitel im Einzelnen untersucht werden. Zusammenfassend lässt sich sagen, dass die Robustheit des Netzes gegen grosse Störungen fuer dezentrale Netze geringer ist als für zentrale, während die strukturelle Stabilität gegen einzelne Leitungsausfälle zunimmt. Im letzten Teil der Arbeit wird die strukturelle Stablität des Stromnetzes genauer untersucht. Es werden neuartige Verfahren entwickelt, um Vorhersagen zu können, welche Leitungen, falls sie ausfallen sollten, einen systemweiten Stromausfall nach sich ziehen und welche nicht. Es wird gezeigt, dass die Leistung, die eine Leitung transportiert, kein ausreichendes Kriterium ist, um präzise Vorhersagen für das Verhalten des gesamten Netzes im Falle des Ausfalls einer Leitung zu treffen. Abschliessend werden im Detail neue Kriterien entwickelt, die sich für Vorhersagen als geeignet herausstellen.

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The stability of evaporating liquids is examined. The geometries investigated are semi-infinite liquid sheets, bounded liquid sheets, sessile droplets, and funnels. Stability parameters are generated to characterize the stability of evaporating semi-infinite liquid sheets, and bounded liquid sheets. The derivation is made possible by introducing evaporation as the specific heat transfer mechanism at the interface, and using the statistical rate theory expression for evaporation flux so there are no fitting parameters. It is demonstrated that a single parameter can be used to predict the onset criterion instead of two parameters. A linear stability analysis is performed for spherical sessile droplets evaporating on substrates constructed of either insulating or conducting materials. A stability parameter is generated to characterize the stability of sessile droplets evaporating on insulating substrates and conducting substrates. The results indicate that spherical sessile droplets evaporating on insulating substrates are predicted to transition to Marangoni convection. Since there are currently no experimental results to compare the theory with, another analysis is performed for liquids evaporating from funnels, which can be compared with existing experimental observations. A linear stability analysis predicts stable evaporation for funnels constructed of insulating materials, in contrast to the sessile droplet case, and generates a new stability parameter for funnels constructed of conducting materials. The stability parameter is free of fitting variables since the statistical rate theory expression for the evaporation flux is used. The theoretical predictions are found to be consistent with experimental observations for water evaporating from a funnel constructed of poly(methyl methacrylate) (PMMA) and for water and heavy water evaporating from a funnel constructed of stainless steel. A parametric analysis is performed on the new stability parameter for liquids evaporating from funnels constructed of conducting materials, indicating that smaller interfacial temperature discontinuities, higher evaporation rates, and smaller radii correspond to less stable systems. It is also illustrated that calculations using statistical rate theory predict an instability, which is consistent with experimental observations, whereas using the Hertz-Knudsen theory does not predict any instability.

The stability of evaporating liquids is examined. The geometries investigated are semi-infinite liquid sheets, bounded liquid sheets, sessile droplets, and funnels. Stability parameters are generated to characterize the stability of evaporating semi-infinite liquid sheets, and bounded liquid sheets. The derivation is made possible by introducing evaporation as the specific heat transfer mechanism at the interface, and using the statistical rate theory expression for evaporation flux so there are no fitting parameters. It is demonstrated that a single parameter can be used to predict the onset criterion instead of two parameters. A linear stability analysis is performed for spherical sessile droplets evaporating on substrates constructed of either insulating or conducting materials. A stability parameter is generated to characterize the stability of sessile droplets evaporating on insulating substrates and conducting substrates. The results indicate that spherical sessile droplets evaporating on insulating substrates are predicted to transition to Marangoni convection. Since there are currently no experimental results to compare the theory with, another analysis is performed for liquids evaporating from funnels, which can be compared with existing experimental observations. A linear stability analysis predicts stable evaporation for funnels constructed of insulating materials, in contrast to the sessile droplet case, and generates a new stability parameter for funnels constructed of conducting materials. The stability parameter is free of fitting variables since the statistical rate theory expression for the evaporation flux is used. The theoretical predictions are found to be consistent with experimental observations for water evaporating from a funnel constructed of poly(methyl methacrylate) (PMMA) and for water and heavy water evaporating from a funnel constructed of stainless steel. A parametric analysis is performed on the new stability parameter for liquids evaporating from funnels constructed of conducting materials, indicating that smaller interfacial temperature discontinuities, higher evaporation rates, and smaller radii correspond to less stable systems. It is also illustrated that calculations using statistical rate theory predict an instability, which is consistent with experimental observations, whereas using the Hertz-Knudsen theory does not predict any instability.

Drug stability is one of the vital subjects in the pharmaceutical industry. All drug products should be kept stable and protected against any chemical, physical, and microbiological degradation to ensure their efficacy and safety until released for public use. Hence, stability is very important to be estimated or predicted. This work involved studying the stability of three different drug agents using three different mathematical models. These models included both empirical models (linear regression and artificial neural network), and mechanistic (kinetic) models. The stability of each drug in the three cases studied was expressed in terms of concentration, hardness, temperature and humidity. The predicted values obtained from the models were compared to the observed values of drug concentrations obtained experimentally and then evaluated by calculating the mean of squared. Among the models used in this work, the mechanistic model was found to be the most accurate and reliable method of stability testing given the fact that it had the smallest calculated errors. Overall, the accuracy of these mathematical models as indicated by the proximity of their stability measurements to the observed values, led to the assumption that such models can be reliable and time-saving alternatives to the analytical techniques used in practice.

Bargaining theory seeks to answer the question of how to divide a jointly generated surplus between multiple agents. John Nash proposed the Nash Bargaining Solution to answer this question for the special case of two agents. Kleinberg and Tardos extended this idea to network games, and introduced a model they call the Bargaining Game. They search for surplus divisions with a notion of fairness, defined as balanced solutions, that follow the Nash Bargaining Solution for all contracting agents. Unfortunately, many networks exist where no balanced solution can be found, which we call unstable. In this thesis, we explore methods of changing unstable network structures to find fair bargaining solutions. We define the concept of Blocking Sets, introduced by Biro, Kern and Paulusma, and use them to create stability. We show that by removing a blocking set from an unstable network, we can find a balanced bargaining division in polynomial time. This motivates the search for minimal blocking sets. Unfortunately this problem is NP-hard, and hence no known efficient algorithm exists for solving it. To overcome this hardness, we consider the problem when restricted to special graph classes. We introduce a O(1)-factor approximation algorithm for the problem on planar graphs with unit edge weights. We then provide an algorithm to solve the problem optimally in graphs of bounded treewidth, which generalize trees.

The complexity of large industrial engineering systems such as chemical plants has continued to increase over the years. As a result, flexible control systems are required to handle variation in the operating conditions. In the classical approach, first the plant model should be linearized at the nominal operating point and then, a robust controller should be designed for the resulting linear system. However, the performance of a controller designed by this method deteriorates when operation deviates from the nominal point. When the distance between the operating region and the nominal operating point increases, this performance degradation may lead to instability. In the context of traditional linear control, one method to solve this problem is to consider the impact of nonlinearity as “uncertainty” around the nominal model and design a controller such that the desired performance is satisfied for all possible systems in the uncertainty set. As the size of uncertainty increases, conservatism occurs and at some point, it becomes impossible to design a controller that can provide satisfactory performance. One of the methods proposed to overcome the aforementioned shortcomings is the so-called Multiple Model approach. Using Multi-Models, local designs are performed for various operating regions and membership functions or a supervisory switching scheme is used to interpolate or switch among the controllers as the operating point moves among local regions. Since the Multiple Model method is a natural extension of the linear control method, it inherits some benefits of linear control such as simplicity of analysis and implementation. However, all these benefits are valid locally. For example, the multiple model method may be vulnerable when global stability is taken into account. The core objective of this thesis is to develop new tools to study stability of closed-loop nonlinear systems controlled by local controllers in order to improve design of multiple model control systems. For example, one of the aims of this work is to investigate how to determine the region where closed loop system is stable. A secondary objective is to study the effects of the exogenous signals on stability of such systems. / Controls

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