A Growth Curve Model (GCM) is a multivariate linear model used for analyzing longitudinal data with short to moderate time series. It is a special case of Generalized Multivariate Analysis of Variance (GMANOVA) models. Analysis using the GCM involves comparison of mean growths among different groups. The classical GCM, however, possesses some limitations including distributional assumptions, assumption of identical degree of polynomials for all groups and it requires larger sample size than the number of time points. In this thesis, we relax some of the assumptions of the traditional GCM and develop appropriate inferential tools for its analysis, with the aim of reducing bias, improving precision and to gain increased power as well as overcome limitations of high-dimensionality. Existing methods for estimating the parameters of the GCM assume that the underlying distribution for the error terms is multivariate normal. In practical problems, however, we often come across skewed data and hence estimation techniques developed under the normality assumption may not be optimal. Simulation studies conducted in this thesis, in fact, show that existing methods are sensitive to the presence of skewness in the data, where estimators are associated with increased bias and mean square error (MSE), when the normality assumption is violated. Methods appropriate for skewed distributions are, therefore, required. In this thesis, we relax the distributional assumption of the GCM and provide estimators for the mean and covariance matrices of the GCM under multivariate skew normal (MSN) distribution. An estimator for the additional skewness parameter of the MSN distribution is also provided. The estimators are derived using the expectation maximization (EM) algorithm and extensive simulations are performed to examine the performance of the estimators. Comparisons with existing estimators show that our estimators perform better than existing estimators, when the underlying distribution is multivariate skew normal. Illustration using real data set is also provided, wherein Triglyceride levels from the Framingham Heart Study is modelled over time. The GCM assumes equal degree of polynomial for each group. Therefore, when groups means follow different shapes of polynomials, the GCM fails to accommodate this difference in one model. We consider an extension of the GCM, wherein mean responses from different groups can have different shapes, represented by polynomials of different degree. Such a model is referred to as Extended Growth Curve Model (EGCM). We extend our work on GCM to EGCM, and develop estimators for the mean and covariance matrices under MSN errors. We adopted the Restricted Expectation Maximization (REM) algorithm, which is based on the multivariate Newton-Raphson (NR) method and Lagrangian optimization. However, the multivariate NR method and hence, the existing REM algorithm are applicable to vector parameters and the parameters of interest in this study are matrices. We, therefore, extended the NR approach to matrix parameters, which consequently allowed us to extend the REM algorithm to matrix parameters. The performance of the proposed estimators were examined using extensive simulations and a motivating real data example was provided to illustrate the application of the proposed estimators. Finally, this thesis deals with high-dimensional application of GCM. Existing methods for a GCM are developed under the assumption of ‘small p large n’ (n >> p) and are not appropriate for analyzing high-dimensional longitudinal data, due to singularity of the sample covariance matrix. In a previous work, we used Moore-Penrose generalized inverse to overcome this challenge. However, the method has some limitations around near singularity, when p~n. In this thesis, a Bayesian framework was used to derive a test for testing the linear hypothesis on the mean parameter of the GCM, which is applicable in high-dimensional situations. Extensive simulations are performed to investigate the performance of the test statistic and establish optimality characteristics. Results show that this test performs well, under different conditions, including the near singularity zone. Sensitivity of the test to mis-specification of the parameters of the prior distribution are also examined empirically. A numerical example is provided to illustrate the usefulness of the proposed method in practical situations. / Thesis / Doctor of Philosophy (PhD)

The modern day power system is witnessing a tremendous change. There has been a rapid rise in the distributed generation, along with this the deregulation has resulted in a more complex system. The power demand is on a rise, the generation and trans-mission infrastructure hasn't yet adapted to this growing demand. The economic and operational constraints have forced the system to be operated close to its design limits, making the system vulnerable to disturbances and possible grid failure. This makes the study of voltage stability of the system important more than ever. Generally, voltage stability studies are carried on a single phase equivalent system assuming that the system is perfectly balanced. However, the three phase power system is not always in balanced state. There are a number of untransposed lines, single phase and double phase lines. This thesis deals with three phase voltage stability analysis, in particular the voltage stability index known as L-Index. The equivalent single phase analysis for voltage stability fails to work in case of any unbalance in the system or in presence of asymmetrical contingency. Moreover, as the system operators are giving importance to synchrophasor measurements, PMUs are being installed throughout the system. Hence, the three phase voltages can be obtained, making three phase analysis easier. To study the effect of unbalanced system on voltage stability a three phase L-Index based on traditional L-Index has been proposed. The proposed index takes into consideration the unbalance resulting due to untransposed transmission lines and unbalanced loads in the system. This index can handle any unbalance in the system and is much more realistic. To obtain bus voltages during unbalanced operation of the system a three phase decoupled Newton Raphson load ow was used. Reactive power distribution in a system can be altered using generators voltage set-ting, transformers OLTC settings and SVC settings. All these settings are usually in balanced mode i.e. all the phases have the same setting. Based on this reactive power optimization using LP technique on an equivalent single phase system is proposed. This method takes into account generator voltage settings, OLTC settings of transformers and SVC settings. The optimal settings so obtained are applied to corresponding three phase system. The effectiveness of the optimal settings during unbalanced scenario is studied. This method ensures better voltage pro les and decrease in power loss. Case studies of the proposed methods are carried on 12 bus and 24 bus EHV systems of southern Indian grid and a modified IEEE 30 bus system. Both balanced and unbalanced systems are studied and the results are compared.

Der Vortrag stellt das neue, unendlich reibungsbehaftete Kontaktmodell der FEM-Berechnungssoftware Pro/ENGINEER Mechanica in der Version Wildfire 4.0 von PTC vor. Dabei werden sowohl die Grundlagen des reibungsfreien Kontaktes als auch die Theorie des unendlich reibungsbehafteten Kontaktmodells behandelt sowie die Grundlagen der zur numerischen Lösung in der Software verwendeten Penalty- und Newton-Raphson-Methode erläutert. Als Anwendungsbeispiel für das reibungsfreie Kontaktmodell wird ein Zylinderrollenlager vollständig mit sämtlichen Wälzkontakten für verschiedene Lager- und Einbauspiele berechnet, die Ergebnisse umfassend dargestellt sowie eine analytische Gegenrechnung nach der Hertzschen Theorie ausgeführt, die sehr gute Übereinstimmung mit der numerischen Simulation zeigt. Für das reibungsbehaftete Kontaktmodell wird exemplarisch eine geschrumpfte Welle-Nabe-Verbindung unter Torsion berechnet. Diese wird einer analytischen Lösung sowie verschiedenen 2D-Idealisierungen (ebener Spannungszustand, ebener Dehnungszustand, 2D-Axialsymmetrie) gegenübergestellt.

Der Vortrag stellt das neue, unendlich reibungsbehaftete Kontaktmodell der FEM-Berechnungssoftware Pro/ENGINEER Mechanica in der Version Wildfire 4.0 von PTC vor. Dabei werden sowohl die Grundlagen des reibungsfreien Kontaktes als auch die Theorie des unendlich reibungsbehafteten Kontaktmodells behandelt sowie die Grundlagen der zur numerischen Lösung in der Software verwendeten Penalty- und Newton-Raphson-Methode erläutert. Als Anwendungsbeispiel für das reibungsfreie Kontaktmodell wird ein Zylinderrollenlager vollständig mit sämtlichen Wälzkontakten für verschiedene Lager- und Einbauspiele berechnet, die Ergebnisse umfassend dargestellt sowie eine analytische Gegenrechnung nach der Hertzschen Theorie ausgeführt, die sehr gute Übereinstimmung mit der numerischen Simulation zeigt. Für das reibungsbehaftete Kontaktmodell wird exemplarisch eine geschrumpfte Welle-Nabe-Verbindung unter Torsion berechnet. Diese wird einer analytischen Lösung sowie verschiedenen 2D-Idealisierungen (ebener Spannungszustand, ebener Dehnungszustand, 2D-Axialsymmetrie) gegenübergestellt.