Transformers are very important parts of any electrical network, and their size increase so does their price. Protecting these important devices is a daunting task due to the wide variety of operating conditions. This thesis develops a new protection scheme based on state estimation.The foundation upon which our protection scheme is built is the modeling of the single phase transformer system of equations. The transformer equations are composed of polynomial and differential equations and this system of equations involving the transformer's electrical quantities are modeled into a system of equations such that highest degree of each of the system's equations is quadratic―in a process named Quadratization and then integrated using a technique called Quadratic integration to give a set of algebraic companion equations that can be solved numerically to determine the health of the transformer.

The major achievements of this work are based on two categories: (A) introduction of an advanced simulation technique in both time domain and frequency domain, and (B) realistic and reliable models for converters applicable to analysis of alternative transmission systems. The proposed modeling-methodology using a combination of model quadratization and quadratic integration (QMQI) is demonstrated as a more robust, stable, and accurate method than previous modeling methodologies for power system analyses. The quadratic-integration method is free of artificial numerical-oscillations exhibited by trapezoidal integration (which is the most popularly used method in power system analyses). Artificial numerical oscillations can be the direct reason for switching malfunction of switching systems. However, the quadratic-integration method has a natural characteristic to eliminate fictitious oscillations with great simulation accuracy. Also, model quadratization permits nonlinear equations to be solved without simplification or approximation, leading to realistic models of nonlinearities. Therefore, the QMQI method is suitable for simulations of network systems with nonlinear components and switching subsystems. Realistic and reliable converter models by the application of the QMQI method can be used for advanced designs and optimization studies for alternative transmission systems; they can also be used to perform a comprehensive evaluation of the technical performance and economics of alternative transmission systems. For example, the converters can be used for comprehensive methodology for determining the optimal topology, kV-levels, etc. of alternative transmission systems for wind farms, for given distances of wind farms from major power grid substations. In this case, a comprehensive evaluation may help make more-informed decisions for the type of transmission (HVAC, HVDC, and LFAC) for wind farms.