We first provide a detailed background of a geometric projection methodology developed by Professor Roswitha Marz at Humboldt University in Berlin for showing uniqueness and existence of solutions for ordinary differential-algebraic equations (DAEs). Because of the geometric and operator-theoretic aspects of this particular method, it can be extended to the case of infinite-dimensional abstract DAEs. For example, partial differential equations (PDEs) are often formulated as abstract Cauchy or evolution problems which we label abstract ordinary differential equations or AODE. Using this abstract formulation, existence and uniqueness of the Cauchy problem has been studied. Similarly, we look at an AODE system with operator constraint equations to formulate an abstract differential-algebraic equation or ADAE problem. Existence and uniqueness of solutions is shown under certain conditions on the operators for both index-1 and index-2 abstract DAEs. These existence and uniqueness results are then applied to some index-1 DAEs in the area of thermodynamic modeling of a chemical vapor deposition reactor and to a structural dynamics problem. The application for the structural dynamics problem, in particular, provides a detailed construction of the model and development of the DAE framework. Existence and uniqueness are primarily demonstrated using a semigroup approach. Finally, an exploration of some issues which arise from discretizing the abstract DAE are discussed. / Ph. D.

Diese Arbeit beschäftigt sich mit verschiedenen mathematischen Fragestellungen hinsichtlich der Modellierung, Analysis und numerischen Simulation von Gasnetzen. Hierbei liegt der Fokus auf der mathematischen Handhabung von partiellen differential-algebraischen Gleichungen, die mit algebraischen Gleichungen gekoppelt sind. Diese bieten einen einfachen Zugang hinsichtlich der Modellierung von dynamischen Strukturen auf Netzen Somit sind sie insbesondere für Gasnetze geeignet, denen im Zuge der steigenden Bedeutung von erneuerbaren Energien ein gestiegenes Interesse seitens der Öffentlichkeit, Politik und Wissenschaft entgegen gebracht wird. Wir führen zunächst die gängigsten Elemente, die in Gasnetzen benötigt werden ein und formulieren zwei PDAE-Klassen für solche Netze: Eine für reine Rohrnetze, und eine, die zusätzliche Elemente wie Verdichter und Widerstände beinhaltet. Des Weiteren untersuchen wir die Sensitivität der Lösung der Rohrnetz-PDAE hinsichtlich Störungen. Dabei berücksichtigen wir Störungen, die nicht nur den dynamischen Teil der PDAE beeinflussen, sondern auch Störungen in den algebraischen Gleichungen und weisen Stabilitätseigenschaften für die Lösung der PDAE nach. Darüber hinaus beschäftigen wir uns mit einer neu entwickelten, an die Netztopologie angepassten Ortsdiskretisierung, welche die Stabilitätseigenschaften der PDAE auf DAE Systeme überträgt. Des Weiteren zeigen wir, wie sich die Gasnetz-DAE zu einer gewöhnlichen Differentialgleichung, welche die inhärente Dynamik der DAE widerspiegelt entkoppeln lässt. Dieses entkoppelte System kann darüber hinaus direkt aus den Topologie- und Elementinformationen des Netzes aufgestellt werden. Abschließend demonstrieren wir die Ergebnisse an Benchmark-Gasnetzen. Dabei vergleichen wir sowohl die entkoppelte Differentialgleichung mit dem ursprünglichen DAE System, zeigen aber auch, welche Vorteile die an die Netztopologie angepasste Ortsdiskretisierung gegenüber existierenden Verfahren besitzt. / This thesis addresses several aspects regarding modelling, analysis and numerical simulation of gas networks. Hereby, our focus lies on (partial) differential-algebraic equations, thus systems of partial and ordinary differential equations which are coupled by algebraic equations. These coupled systems allow an easy approach towards the modelling of dynamic structures on networks. Therefore, they are well suited for gas networks, which have gained a rise of attention in society, politics and science due to the focus towards renewable energies. We give an introduction towards gas network modelling that includes the most common elements that also appear in real gas networks and present two PDAE systems: One for pipe networks and one that includes additional elements like resistors and compressors. Furthermore, we investigate the impact of perturbations onto the pipe network PDAE, where we explicitly allow perturbations to affect the system in the differential as well as in the algebraic components. We conclude that the solution of the PDAE possesses stability properties. In addition, this thesis introduces a new spatial discretisation that is adapted to the net- work topology. This topology-adapted semi-discretisation results in a DAE which possesses the same perturbation behaviour as the space continuous PDAE. Furthermore, we present a topology based decoupling procedure that allows to reformulate the DAE as an ordinary differential equation (ODE), which represents the inherent dynamics of the DAE system. This ODE, together with a decoupled set of algebraic equations, can be derived from the topology and element information directly. We conclude by demonstrating the established results for several benchmark networks. This includes a comparison of numerical solutions for the decoupled ODE and the DAE system. In addition we present the advantages of the topology-adapted spatial discretisation over existing well established methods.

Indiana University-Purdue University Indianapolis (IUPUI) / A gradient free function optimization technique, namely particle swarm optimization (PSO) algorithm, is utilized in parameter identification of the electrochemical model of a Lithium-Ion battery having a LiCoO2 chemistry. Battery electrochemical model parameters are subject to change under severe or abusive operating conditions resulting in, for example, Navy over-discharged battery, 24-hr over-discharged battery, and over-charged battery. It is important for a battery management system to have these parameters changes fully captured in a bank of battery models that can be used to monitor battery conditions in real time. In this work, PSO methodology has been used to identify four electrochemical model parameters that exhibit significant variations under severe operating conditions. The identified battery models were validated by comparing the model output voltage with the experimental output voltage for the stated operating conditions. These identified conditions of the battery were then used to monitor condition of the battery that can aid the battery management system (BMS) in improving overall performance. An adaptive estimation technique, namely multiple model adaptive estimation (MMAE) method, was implemented for this purpose. In this estimation algorithm, all the identified models were simulated for a battery current input profile extracted from the hybrid pulse power characterization (HPPC) cycle simulation of a hybrid electric vehicle (HEV). A partial differential algebraic equation (PDAE) observer was utilized to obtain the estimated voltage, which was used to generate the residuals. Analysis of these residuals through MMAE provided the probability of matching the current battery operating condition to that of one of the identified models. Simulation results show that the proposed model based method offered an accurate and effective fault diagnosis of the battery conditions. This type of fault diagnosis, which is based on the models capturing true physics of the battery electrochemistry, can lead to a more accurate and robust battery fault diagnosis and help BMS take appropriate steps to prevent battery operation in any of the stated severe or abusive conditions.