MULTI-OBJECTIVE OPTIMIZATION OF UNIDIRECTIONAL NON-ISOLATED DC/DC CONVERTERS by Andrija Stupar A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Edward S. Rogers Department of Electrical and Computer Engineering University of Toronto © Copyright 2017 by Andrija Stupar Abstract Multi-Objective Optimization of Unidirectional Non-Isolated DC/DC Converters Andrija Stupar Doctor of Philosophy Graduate Department of Edward S. Rogers Department of Electrical and Computer Engineering University of Toronto 2017 Engineers have to fulfill multiple requirements and strive towards often competing goals while designing power electronic systems. This can be an analytically complex and computationally intensive task since the relationship between the parameters of a system’s design space is not always obvious. Furthermore, a number of possible solutions to a particular problem may exist. To find an optimal system, many different possible designs must be evaluated. Literature on power electronics optimization focuses on the modeling and design of partic- ular converters, with little thought given to the mathematical formulation of the optimization problem. Therefore, converter optimization has generally been a slow process, with exhaus- tive search, the execution time of which is exponential in the number of design variables, the prevalent approach. In this thesis, geometric programming (GP), a type of convex optimization, the execution time of which is polynomial in the number of design variables, is proposed and demonstrated as an efficient and comprehensive framework for the multi-objective optimization of non-isolated unidirectional DC/DC converters. A GP model of multilevel flying capacitor step-down convert- ers is developed and experimentally verified on a 15-to-3.3 V, 9.9 W discrete prototype, with sets of loss-volume Pareto optimal designs generated in under one minute. It is also demon- strated how the GP model can be used to determine the sensitivity of the optimized designs to the design parameters. Furthermore, a general procedure for deriving GP models is presented and demonstrated on the example of inductors for higher power applications. ii Finally, using the example of the seven-switch flying capacitor converter, it is shown how in cases where a converter-level GP model is not possible, component-level GP models can be coupled with a circuit simulator to perform efficiency optimization quickly. This hybrid approach is used to design a wide input and output voltage range, 2 A output current converter IC. The results show that GP is of great value to both researchers and practicing engineers. The methods demonstrated for deriving GP models of power converters are extendable to other converter topologies, allowing for the creation of a rigorous mathematical framework for the optimization of power electronics that guarantees globally optimum designs generated quickly. iii When sorrows come, they come not single spies But in battalions. — King Claudius in Shakespeare’s Hamlet (Act IV, Scene 5) For at the moment of the final division, the final miniaturization of matter, suddenly the whole cosmos opened up. — Thomas Mann, Der Zauberberg iv To Tanja, for sticking with me to the end Acknowledgements This doctoral degree has been a long, and often turbulent, eight years in the making and now that it is finally over there is a long list of people to thank, some of whom will be invariably missed in these few brief paragraphs - to them I wholeheartedly apologize in advance. I be- gan work towards a PhD degree at the Swiss Federal Institute of Technology (ETH) in Zurich, Switzerland, in February 2009, where I spent slightly more than four years as a full-time stu- dent and employee at the Power Electronic Systems Laboratory (PES), and a subsequent two as an “external” PhD student while working to develop a spin-off company, Gecko-Simulations AG, that my colleagues from PES and I had co-founded in 2012. Although my work at PES/ETH has found its way into many refereed publications, conference and workshop presentations, and Gecko-Simulations products, little of it is present directly in the text of this thesis. However, the work I did at PES has greatly informed this thesis, and has been a basis for the work pre- sented therein without which, I feel, this thesis would not be possible, and would certainly not have the form that it does. PES at ETH was and continues to be at the forefront of research into the optimization of power converters, and I was privileged not only to have taken part of that work myself, but also to have been able to absorb the knowledge and skills generated by my colleagues and supervisors there, both during and before my time. I was able to learn the state-of-the-art when it comes to optimizing power electronics, and how to improve it – which is what, as I hope the reader will in the end conclude, this thesis is about. I would therefore like to thank first and foremost my former supervisor at ETH Zurich and the Director of PES, Prof. Dr. Johann W. Kolar for giving me the opportunity to work and learn in a challenging and world-class academic environment, for all that I have learned under his supervision, and for accommodating the various arrangements over the years. I would also like to thank the post- docs under whose supervision I worked at PES: Dr. Uwe Drofenik, under whom I got started v and who provided some important words of encouragement; Dr. Dominik Bortis, who was of immense practical help in the early years; Dr. Florian Krismer, who was always ready to offer help and advice; and finally, Dr. Thomas Friedli, who was probably simultaneously the kindest, most committed and most knowledgeable person I have had the pleasure of working with. I would also like to thank my fellow PhD students whom I collaborated most closely with at PES and at Gecko-Simulations AG, Dr. Ivana Kovaˇcevi´c-Badstübner, Dr. Andreas Müsing, and Dr. Jonas Mühlethaler, from whom I learned and with whom I built a lot. They provided great pro- fessional and personal support during my time at ETH Zurich. I would also like to thank other colleagues from PES I have the opportunity to work with and know: Dr. Uwe Badstübner, who was especially helpful, Daniel Christen, Dr. David Boillat, Dr. Arda Tüysüz, Prof. Dr. Jürgen Biela, Dr. Toke Andersen, Dr. Matthias Kasper, Dr. Johann Miniböck, Dr. Cristoph Marxgut, Dr. Yannick Lobsiger, Dr. Hirofumi Uemura, Florian Vancu, Dr. Benjamin Wrzecionko, Dr. Thiago Soeiro, Dr. Christof Zwyssig, Dr. Thomas Baumgartner, staff members Monica Kohn-Müller and Peter Seitz, and others. Unfortunately it was not possible for my to bring my degree to completion at ETH Zürich, and a solution was found in my “transferring back”, in 2015, to my alma mater, the Univer- sity of Toronto, to which I transferred my course credits and research from ETH. Therefore most importantly, I would like to thank most of all, my PhD co-supervisors at the University of Toronto, Prof. Aleksandar Prodi´c and Asst. Prof. Joshua A. Taylor, for rescuing my academic career when it seemed at its lowest – and breaking – point and for giving me another chance. Moreover, I owe my start in power electronics research to Prof. Prodi´c, who invited my to work at his lab as an undergraduate student back in the summer of 2004. It was in his lab and under his supervision that I completed my Master’s degree in 2008. My return to the Laboratory for Power Management and Integrated Switch-Mode Power Supplies in the Energy Systems Group at the Faculty of Applied Science and Engineering of the University of Toronto was thus a home- coming - made successful not only by Prof. Prodi´c’s support and steady supply of challenging research projects, but also by support, guidance, and key insights provided by Prof. Taylor, whose insistence on using convex optimization re-oriented my research into a much more fruit- ful direction. I would like to stress that although greatly informed by my previous work, the major results of thesis and its major contribution – the use of geometric programming for the vi multi-objective optimization of power converters – have been developed only and fully during my time at the University of Toronto. I would once more like to thank my two co-supervisors Profs. Taylor and Prodi´c for all the support and guidance they have provided over the past two years, without which the – I believe – significant results of this thesis would not be possible. I would also like to thank my colleagues in Prof. Prodi´c’s lab for all I’ve learned from them and for the support they have provided. Specifically I’d like to thank Timothy McRae and Nenad Vukadinovi´c for their help with developing posynomial models of ML-FC converters, as well as for their help in all sorts of work-related matters and their friendship, and Tom Moiannou with whom I worked on the 7SFC converter project. I would also like to thank Dr. S.M. Ahsanuzzaman, Samuel Carvalho, Michael Halamicek, and others. Furthermore I would like to thank, from the University of Toronto, Assoc. Prof. Stark Draper, whose course on convex optimization I wish I had taken at least four years earlier, and where the use of geometric programming for power electronics got started as a course project, and Prof. Jason Anderson for being members of my PhD qualifying exam committee; Prof. Reza Iravani for being a member of my PhD thesis proposal and then departmental defense committee; Prof. Peter Lehn for being a member of my qualifying exam, thesis proposal, and final defense committees; Assoc.