This thesis proposes a novel out-of-step protection technique using the state-plane representation of the generator speed and power angle. The critical clearing angle is computed using the principle that the total energy of the system at the instant the fault is cleared should be equal to the maximum potential energy of the system. The critical clearing time corresponding to this value of critical clearing angle is obtained directly using the time calibration of the relative speed versus power angle solution curve. The simultaneous calculation of the critical clearing angle and the time makes the proposed state plane approach much faster than the two-blinder scheme, Equal Area Criterion (EAC) method, rate of change of impedance method, the Swing Center Voltage (SCV) technique, transient energy calculation method, and the frequency deviation calculation from voltage signal method discussed in the literature. The proposed state plane prediction scheme is used to detect the rst swing out-of-step condition in single machine in nite bus (SMIB) system as well as larger power system con gurations (two-area and IEEE 39-bus test systems) using system wide information. A coherency analysis is performed in a multi-machine system to find out the two critical groups of generators. The critical generator groups are then represented with a SMIB equivalent system, and the state plane algorithm is applied to the reduced equivalent. Electromagnetic transient simulations are carried out using PSCAD/EMTDCTM to test the proposed algorithm in the above discussed test systems. The simulation studies show that the proposed method is computationally e cient, and accurate even for the larger power systems. The technique also does not require any o ine studies. This thesis also proposes another out-of-step protection technique using generator state deviations to detect multi-swing instability conditions in power system. It uses wide-area measurements of generator electrical power and speed deviations as inputs to the proposed scheme to detect instability. This technique is not as fast as the state plane approach but can predict multi-swing instability conditions in power system. The state plane method and state deviation method are used together to nd rst swing and multi-swing instability conditions. Two-area power system con guration is used to demonstrate multi-swing instability prediction. Di erent power swing conditions such as stable, rst swing unstable and multi-swing unstable scenarios are created and the proposed techniques are tested to verify their performance. The proposed techniques are also compared with the conventional two blinder technique. A facility for hardware-in-the-loop testing of the relays using a digital simulator is available in the Power System Laboratory at the University of Saskatchewan. An out-of-step relay module is developed in a digital signal processing board (ADSP BF533TM from Analog Devices Inc.) and a closed loop test is performed using the real time digital simulator (RTDSTM). The simulator mimics the power system behaviour in real time, and the analog time signals from simulator can be communicated to the relay module. The relay can also feed back the signals to the simulator which can be used to operate the circuit breaker elements in the power system. The SMIB and two area systems are used to test the relay in real time. The relay prototypes for both of the proposed techniques are developed in this thesis. The hardware-in-the-loop implementation and testing show that the calculation times required for the proposed methods are small, and the state plane method especially can predict instability condition much faster than all other methods in current literature.

The future voltage regulator module (VRM) challenges of high bandwidth control with fast transient response, high current output, simple implementation, and efficient 48V solution are tackled in this dissertation. With the push for control bandwidth to meet design specifications for microprocessor VRM with larger and faster load transients, control can be saturated and lost for a significant period of time during transient. During this time, undesirable transient responses such as large undershoot and ringback occurs. Due to the loss of control, the existing tools to study the dynamic behavior of the system, such as small signal model, are insufficient to analyze the behavior of the system during this time. In order to have a better understanding of the system dynamic performance, the operation the VRM is analyzed in the state-plane for a clear visual understanding of the steady-state and transient behaviors. Using the state-plane, a simplified state-plane trajectory control is proposed for constant on-time (COT) control to achieve the best transient possible for applications with adaptive voltage positioning (AVP). When the COT control is lost during a load step-up transient, the state-plane trajectory control will extend on-time to provide the a near optimal transient response. By observing the COT control law in the state-plane, a simplified state-plane trajectory control with analog implementation is proposed to achieve the best transient possible with smooth transitions in and out of the steady-state COT control. The concept of the simplified state-plane trajectory control is then extended to multiphase COT. For multiphase operation, additional operating behavior, such as phase overlapping during transient and interleaving during steady-state, need to be taken into consideration to design the desired state-plane trajectory control. A simple state-plane trajectory control with improved Ton extension is proposed and verified using multiphase COT control. After tackling the state-plane trajectory control for current mode COT, the idea is then extended to V2 COT. V2 COT is a more advanced current mode control which requires a more advanced state-plane trajectory control to COT. By calculating the intersection of the extended on-stage trajectory during transient and the ideal off trajectory in the form of a current limiting wall, a near optimal transient response can be achieved. For V2 COT with state-plane trajectory control, implementations using inductor vs. capacitor current, effect of component tolerance, and effect of IC delay are studied. The proposed state-plane trajectory control is then extended to enhanced V2 COT. Aside from tackling existing VRM challenges, the future datacenter 48V VRM challenge of a high efficiency, high power density solution to meet the VRM specifications is studied. The sigma converter is proposed for the 48V VRM solution due to exhibition of high efficiency and high-power density from hardware evaluation. An accurate model for the sigma converter is derived using the new modeling approach of modularizing the small signal components. Using the proposed model, the sigma converter is shown to naturally have very low output impedance, making the sigma converter suitable for microprocessor applications. The sigma converter is designed and optimized to achieve AVP and very fast transient response using both voltage-mode and current-mode controls. / Doctor of Philosophy / Microprocessors, such as central processing unit (CPU) and graphics processing unit (GPU) are the basis of today's electronics. In the recent decades, the demand for more powerful and faster data processing lead to a significant increase in power consumption by these microprocessors. Even with the introduction of multi-core processors and adaptive voltage positioning (AVP) to reduce the average power provided by the power supplies, the microprocessor can still draw a large amount of instantaneous power in a short period of time. With the microprocessors demanding high amount of current at fast slew-rate, the challenges for the next generation of microprocessor power supply, or voltage regulator modules (VRM), are fast response speed to ensure proper operation of the microprocessors, and high efficiency VRM to minimize the overall system power consumption. The challenge of a VRM with fast response speed is tackled first. To meet the AVP and transient requirements of microprocessor, the VR need to utilize high-bandwidth control methods. Of the control methods used by the industry, high control bandwidth can be easily achieved using constant on-time (COT) control. With the ever-increasing output current level and transient slew-rate requirements, COT control can saturate and lose its steady-state control for a period of time during load step-up transient. During this time, the system will operate with a fixed frequency control until COT control is recovered. Although the method is widely used in the industry, the method is too slow to meet the transient requirements. Many state-of-art methods have been proposed to resolve the load step-up transient issue of COT. However, of the methods proposed, it is difficult to optimize the transient improvement while having a simple analog implementation to ensure a fast response for the wide operating range and aggressive transient conditions observed in microprocessor VRM application. In this dissertation, COT control is studied using the state-plane to provide a clear visual understanding of the transient behavior of the control. Using the state-plane, a state-plane trajectory control is proposed to achieve near optimal load step-up transient response. The concept is then extended to multiphase VRM, which is typically used for high current applications. The state-plane trajectory control concept is then further extended to V2 COT control for VRM without AVP, such as those used by GPU and smartphone CPU. For the proposed state-plane trajectory controls, hardware implementation, evaluation, and experimental results are provided. After tackling the challenge of a VRM with fast response speed, the challenge of an efficient VRM is then tackled. In recent years, a significant amount of research has been put into studying VRM for a power delivery architecture which uses a 48V bus instead of the 12V bus. By using the 48V bus, less redundancy in the power delivery path can greatly increase the overall system efficiency if the VRM stage retains its efficiency. However, the increase in input voltage for the VRM provides an additional challenge to maintain high efficiency for the VRM stage itself. To maintain good efficiency, it is difficult to increase converter switching frequency beyond 300kHz. This limitation on switching frequency will limit the ability to achieve high bandwidth design and fast transient requirements. A 48V VRM using a different topology, the sigma converter, has demonstrated high-efficiency and high-power density, but the converter behavior and control methodology for VRM application is unclear. In this dissertation, the modeling and control of the sigma converter are studied using the proposed small-signal model. By evaluating the proposed small-signal model, the sigma converter can naturally have very low output impedance, making it an ideal candidate for 48V VRM. Then, the design guideline of the sigma converter with current-mode control is provided. With the work discussed in this dissertation, further study of the sigma converter with COT and state-plane trajectory control can be conducted in the future.

Due to modularity and easy scalability, modular multilevel converters (MMCs) are deemed the most suitable for high-voltage and medium-voltage power conversion applications. However, large module capacitors are usually required in MMCs to store large circulating power of line-frequency and its harmonics that flow through the capacitors. Even though several methods for minimizing the circulating power have been proposed in the literature, there is still the need for a systematic and simplified approach of addressing these control strategies and evaluating their efficacy. Moreover, the generally accepted feedback control architecture for the MMC is complicated, derived through a rigorous mathematical analysis, and therefore, not easy to intuitively comprehend. Recently, a method of modeling of the MMC based on state-plane analysis and coordinate transformation, is proposed in the literature. Based on the state-plane analysis, two kinds of circulating power in the MMC are identified that are orthogonal to each other. This means these two circulating power can be controlled individually without affecting each other. To control these circulating power, in the literature, a decoupled equivalent circuit model is developed through the coordinate transformation which clearly suggests a means for minimizing these circulating power. Further extending this work, in this thesis, the existing control concepts for reducing the circulating power are unveiled in a systematic and simplified manner utilizing the decoupled equivalent circuit model. A graphical visualization of circulating power using the state-planes is provided for each control strategy to readily compare its efficacy. Moreover, the generally accepted control architecture of the MMC is presented in an intuitive and simplified way using the decoupled circuit model. The important physics related to control implementation, originally hidden behind the complicated mathematics, is explained in detail. / Master of Science / A power converter is an electrical device that converts electrical energy from one form to another in order to be compatible with the load demand. A typical power converter consists of semiconductor switches, inductor, capacitor etc. These power converters are required in a wide range of applications: automotive and traction, motor drives, renewable energy conversion, energy storage, aircraft, power generation, transmission, and distribution, to name a few. Many of these applications are continuously increasing their power capacity to handle the escalating demands of energy that exist due to rising population numbers, industrialization, urbanization etc. Consequently, it has been a responsibility of power electronics engineers and researchers to develop power converters that can handle high voltages and high currents. Multilevel power converters have been the key-enabling developments that can withstand high-voltages while using traditional low-voltage semiconductor switches. Several multilevel converters such as the neutral point clamped converter, flying capacitor converter, cascaded H-bridge converter, modular multilevel converter (MMC) etc. have been developed and commercialized in the last two decades. Among them, the MMC is a widely accepted topology for medium- and high-voltage power conversion applications. In an MMC, several modules are stacked together in series, and each module consists of semiconductor switches and a capacitor. The series connection of the modules enables the MMC to handle high-voltage power conversion using low-voltage traditional semiconductor switches. The voltage rating of an MMC can be easily scaled-up by simply increasing the number of modules in each arm. Moreover, since several identical modules are connected in each arm, the structure of the MMC is highly modular which helps greatly in manufacturing and design. Nonetheless, in MMCs, generally large circulating power flow to the capacitor in each module, which leads to significant voltage ripples. To suppress these voltage ripples, a large capacitor is required in each module, leading to large size and weight of the converter. In the literature, several control strategies have been proposed to minimize the circulating power. However, there is still the need for a systematic and simplified approach of addressing these control strategies and evaluating their efficacy. Moreover, the generally accepted feedback control architecture for the MMC is complicated, derived through a rigorous mathematical analysis, and therefore, not easy to intuitively comprehend. Recently, a decoupled equivalent circuit model has been developed in the literature. This model clearly explains the process of power flow in the MMC between input and output and the nature of the circulating power. The equivalent circuit model provides the circulating power, that are orthogonal to each other, meaning they can be controlled individually without affecting each other. Moreover, the equivalent circuit model clearly suggests a means for minimize the circulating power by providing two "ideal" control laws. Further extending this work, in this thesis, the existing control concepts for reducing the circulating power are unveiled in a systematic and simplified manner utilizing the decoupled equivalent circuit model. Moreover, the generally accepted control architecture of the MMC is presented in an intuitive and simplified way via the decoupled circuit model. The important physics related to control implementation, originally hidden behind the complicated mathematics, is explained in detail.

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.