Applied Science, Faculty of
Electrical and Computer Engineering, Department of
DSpace
UBCV
Degioanni, Franco
Doctor of Philosophy - PhD
University of British Columbia
The ever-growing global energy demand accentuates the importance of integrating renewable energy sources into the grid. Beyond fulfilling escalating energy needs, this integration holds the potential to reduce the dependency on fossil fuels. In this context, power electronics systems assume a key role in the efficient utilization of renewable energy. Particularly, the three-phase pulse-width-modulated (PWM) converter serves as the bridge that facilitates the seamless interaction between the grid, distributed re- sources, and loads. However, as power electronics systems increase in complexity, challenges emerge in terms of modeling, control design, and implementation. Overcoming these challenges necessitates ad- vancements in control design to enhance dynamic performance and optimize the efficiency of integrating renewable energy. This thesis aims to improve the understanding of the dynamic characteristics of three-phase PWM converters, and to develop novel tools for modeling, analyzing and improving their dynamic performance in grid-connected applications. By employing concepts such as normalization, state plane representation, and geometric analysis, a comprehensive large-signal model for the three-phase converter is developed. This model offers an intuitive graphical interpretation of the system’s dynamic behavior by illustrating its evolution in the state plane. Initially, the application of this framework facilitates the identification and characterization of the theoretical limits of dynamic performance. Consequently, it serves as a point of reference for control design engineers to conduct an objective assessment of the converter’s dynamic performance. Furthermore, the introduced analysis enables the development of high-performance control methods, even for challenging scenarios such as controlling active loads and bidirectional operation with wide operating range requirements. These control techniques ensure consistent large-signal behavior, fast transient responses, and low implementation requirements. In this manner, the thesis contributes to advancing power electronics modeling and control through the enhancement of the dynamic performance of three-phase converters.
Fast Dynamic Transient Solutions for Three-Phase PWM ConvertersbyFranco DegioanniIng., Universidad Nacional de Co´rdoba, 2014M.A.Sc., The University of British Columbia, 2018A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Electrical and Computer Engineering)The University of British Columbia(Vancouver)August 2023© Franco Degioanni, 2023The following individuals certify that they have read, and recommend to the Faculty of Graduate andPostdoctoral Studies for acceptance, the thesis entitled:Fast Dynamic Transient Solutions for Three-Phase PWM Converterssubmitted by Franco Degioanni in partial fulfillment of the requirements forthe degree of Doctor of Philosophyin Electrical and Computer EngineeringExamining Committee:Dr. Martin Ordonez, Professor, Electrical and Computer Engineering, UBCSupervisorDr. Emanuel Serban, Adjunct Professor, Electrical and Computer Engineering, UBCSupervisory Committee MemberDr. Alireza Nojeh, Professor, Electrical and Computer Engineering, UBCUniversity ExaminerDr. Ryozo Nagamune, Professor, Mechanical Engineering, UBCUniversity ExaminerAdditional Supervisory Committee Members:Dr. Jose´ Martı´, Professor, Electrical and Computer Engineering, UBCSupervisory Committee MemberiiAbstractThe ever-growing global energy demand accentuates the importance of integrating renewable energysources into the grid. Beyond fulfilling escalating energy needs, this integration holds the potential toreduce the dependency on fossil fuels. In this context, power electronics systems assume a key role inthe efficient utilization of renewable energy. Particularly, the three-phase pulse-width-modulated (PWM)converter serves as the bridge that facilitates the seamless interaction between the grid, distributed re-sources, and loads. However, as power electronics systems increase in complexity, challenges emerge interms of modeling, control design, and implementation. Overcoming these challenges necessitates ad-vancements in control design to enhance dynamic performance and optimize the efficiency of integratingrenewable energy.This thesis aims to improve the understanding of the dynamic characteristics of three-phase PWMconverters, and to develop novel tools for modeling, analyzing and improving their dynamic performancein grid-connected applications. By employing concepts such as normalization, state plane representation,and geometric analysis, a comprehensive large-signal model for the three-phase converter is developed.This model offers an intuitive graphical interpretation of the system’s dynamic behavior by illustratingits evolution in the state plane. Initially, the application of this framework facilitates the identificationand characterization of the theoretical limits of dynamic performance. Consequently, it serves as a pointof reference for control design engineers to conduct an objective assessment of the converter’s dynamicperformance. Furthermore, the introduced analysis enables the development of high-performance controlmethods, even for challenging scenarios such as controlling active loads and bidirectional operation withwide operating range requirements. These control techniques ensure consistent large-signal behavior,fast transient responses, and low implementation requirements. In this manner, the thesis contributes toadvancing power electronics modeling and control through the enhancement of the dynamic performanceof three-phase converters.iiiLay SummaryWith the ongoing increase in global energy demand, the integration of renewable energy sources into thegrid becomes indispensable. Power electronics systems, including the three-phase pulse-width-modulated(PWM) converter, play a critical role in harnessing the potential of renewable energy sources and makingtheir use more efficient and reliable. However, conventional control methods are no longer adequate toaddress the dynamic performance challenges posed by the integration of advanced and complex systemsinto the grid. This thesis studies the dynamic behavior of three-phase PWM converters and developsinnovative modeling and control techniques to enhance their dynamic performance. A comprehensivelarge-signal model of the converter is presented, offering a complete characterization of the system dy-namics and serving as a foundation for developing different high-performance control solutions. Thesesolutions provide exceptional dynamic performance under varying operating conditions, enabling theefficient and reliable integration of renewable energy into the grid.ivPrefaceThis work is based on research performed at the Department of Electrical and Computer Engineering ofthe University of British Columbia by Franco Degioanni, under the supervision of Dr. Martin Ordonez.Chapters 1, 2 and 7 contain a modified portion of text from all the below-listed publications.Portions of Chapter 3 have been published and presented at the IEEE Applied Power ElectronicsConference (APEC) and IEEE Transactions on Industrial Electronics [1, 2]:• F. Degioanni, I. G. Zurbriggen and M. Ordonez, ”Minimum Recovery Time Transients for Three-Phase Converters in the Synchronous Reference Frame,” in IEEE Transactions on Power Electron-ics, vol. 37, no. 2, pp. 1579-1591, Feb. 2022.• F. Degioanni, I. G. Zurbriggen and M. Ordonez, ”Characterization of the Minimum Recovery TimeTransients for Three-Phase PWM Rectifiers,” 2021 IEEE Applied Power Electronics Conferenceand Exposition (APEC), 2021, pp. 2503-2509.Portions of Chapter 4 have been published and presented at the IEEE Applied Power Electronics Con-ference (APEC) and an extended version is under review on IEEE Transactions on Power Electronics [3]:• F. Degioanni, I. G. Zurbriggen and M. Ordonez, ”Improved Control for Three-Phase Rectifierswith Constant Power Loads,” 2023 IEEE Applied Power Electronics Conference and Exposition(APEC), 2023, pp. 2775-2779.Portions of Chapter 5 have been published and presented at the IEEE Applied Power ElectronicsConference (APEC) and IEEE Transactions on Power Electronics [4, 5]:• F. Degioanni, I. G. Zurbriggen and M. Ordonez, ”Enhanced DC-Link Voltage Dynamics for Grid-Connected Converters,” in IEEE Transactions on Industrial Electronics, vol. 69, no. 11, pp. 10787-10796, Nov. 2022.v• F. Degioanni, I. G. Zurbriggen and M. Ordonez, ”Fast and Reliable Geometric-Based Controller forThree-Phase PWM Rectifiers,” 2020 IEEE Applied Power Electronics Conference and Exposition(APEC), 2020, pp. 1891-1896.Portions of Chapter 6 have been published and presented at the IEEE Applied Power ElectronicsConference (APEC) [6]:• F. Degioanni, I. G. Zurbriggen and M. Ordonez, ”Universal Controllers for PWM Converters: aNormalized Approach,” 2019 IEEE Applied Power Electronics Conference and Exposition (APEC),pp. 1770-1775.As the first author of the above-mentioned publications, the author of this thesis developed the the-oretical concepts and wrote the manuscripts, receiving advice and technical support from Dr. MartinOrdonez, and developed simulation and experimental platforms, receiving contributions from Dr. Or-donez’s research team.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.1 Conventional Modeling and Control of Three-Phase PWM Converters . . . . . . 51.2.2 Control of Three-Phase PWM Converters for Wide Operating Range Applications 71.2.3 Control of Three-Phase PWM Converters with Active Loads . . . . . . . . . . . 81.2.4 Minimum Recovery Time Transients for Three-Phase Converters . . . . . . . . 101.3 Contributions of the Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Normalized Decoupled Model of the Three-Phase Converter in the Synchronous Refer-ence Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.1 Syncrhonous Reference Frame Model of the Three-Phase Converter . . . . . . . . . . . 17vii2.1.1 Decoupled Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.1.2 Normalized Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 Steady-State Operating Points Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3 Dynamic Transient Analysis and State Plane Representation . . . . . . . . . . . . . . . 222.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Theoretical Minimum Recovery Time Solutions of the Three-Phase Converter in the Syn-chronous Reference Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.1 Time Optimal Control Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 Three-Phase Converter Saturation Trajectories . . . . . . . . . . . . . . . . . . . . . . . 283.3 Minimum Recovery Time Solutions for the Three-Phase Converter . . . . . . . . . . . . 293.4 Characterization of the Minimum Recovery Time Solutions . . . . . . . . . . . . . . . . 303.4.1 Start-Up Transient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.4.2 DC-bus Current Step-Up Transient . . . . . . . . . . . . . . . . . . . . . . . . . 323.4.3 DC-bus Current Step-Down Transient . . . . . . . . . . . . . . . . . . . . . . . 343.5 Minimum Time Solutions Considering Different System Constraints . . . . . . . . . . . 353.5.1 Effect of the Modulation Technique in the Minimum Time Solutions . . . . . . . 363.5.2 Effect of the Protection Limits in the Minimum Time Solutions . . . . . . . . . 373.6 Simulation and Experimental Validation of the Minimum Recovery Time Transients . . . 383.6.1 Hardware-in-the-Loop Simulation Results . . . . . . . . . . . . . . . . . . . . . 393.6.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 State Plane Control of Three-Phase PWM Converters with Active Loads . . . . . . . . . 474.1 Constant Power Load Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.2 Analysis of the Negative Effects of a CPL connected to the DC-bus of the Three-PhaseConverter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.3 Control of the Three-Phase Converter with Active Loads Connected to the DC-bus . . . 514.3.1 State Plane Control Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.3.2 Controller Operation with a Constant Power Load . . . . . . . . . . . . . . . . . 524.4 State Plane Controller Performance Evaluation with a Constant Power Load . . . . . . . 54viii4.5 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 Circular Trajectory Control for Three-Phase PWM Converters . . . . . . . . . . . . . . . 625.1 Circular Trajectory Control Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.2 Complete Circular Trajectory Control Law . . . . . . . . . . . . . . . . . . . . . . . . . 645.3 Lyapunov Stability Analysis of the Circular Trajectory Controller . . . . . . . . . . . . . 675.4 Transient Analysis of the Circular Trajectory Controller . . . . . . . . . . . . . . . . . . 695.4.1 DC-Bus Current Step-Up Transient . . . . . . . . . . . . . . . . . . . . . . . . 695.4.2 DC-Bus Current Step-Down Transient . . . . . . . . . . . . . . . . . . . . . . . 725.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.5.1 Dynamic Performance Analysis of the Circular Trajectory Controller . . . . . . 745.5.2 Dynamic Performance Comparison with the Dual-Loop PI Controller . . . . . . 775.5.3 Parameter Variation Simulation Analysis . . . . . . . . . . . . . . . . . . . . . 795.6 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.6.1 Experimental Dynamic Performance Evaluation . . . . . . . . . . . . . . . . . . 805.6.2 Grid Voltage Step Transient . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.6.3 Performance Analysis for Distorted Grid Operation . . . . . . . . . . . . . . . . 845.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866 Normalization of PWM Converters and Universal Control Design . . . . . . . . . . . . . 876.1 Normalization Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886.2 Universal Linear Controllers in the Normalized Domain . . . . . . . . . . . . . . . . . . 896.3 Application Example: Normalized Voltage-Mode Synchronous Buck Converter . . . . . 916.3.1 Normalized Dynamic Model of the Buck Converter . . . . . . . . . . . . . . . . 916.3.2 Universal Voltage-Mode Controller for Synchronous Buck Converters . . . . . . 926.4 Simulation and Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 936.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 967 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 987.1 Conclusions and Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98ix7.1.1 Large-Signal Modeling of Three-phase Converters in the Synchronous ReferenceFrame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 997.1.2 Theoretical Limits of the Dynamic Performance for the Three-Phase PWM Con-verter in the Synchronous Reference Frame . . . . . . . . . . . . . . . . . . . . 997.1.3 Performance Improvements of the Three-Phase PWM Converter with Active LoadsControl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1007.1.4 Performance Improvements of the Three-Phase Converter in Wide OperatingRange Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1007.1.5 Normalized Modeling and Universal Controller Design for PWM Converters . . 1017.1.6 Specific Academic Contributions . . . . . . . . . . . . . . . . . . . . . . . . . 1017.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105A Linear Control of Three-Phase Converters in the Synchronous Reference Frame . . . . . 115A.1 Current-loop Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115A.2 Voltage-loop Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117B Experimental Hardware Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120xList of TablesTable 3.1 Parameters of the three converters used in simulation and experiments . . . . . . . . 38Table 3.2 Normalized minimum recovery parameters . . . . . . . . . . . . . . . . . . . . . . . 39Table 4.1 System parameters used in experiments . . . . . . . . . . . . . . . . . . . . . . . . . 56Table 5.1 System parameters used in experiments . . . . . . . . . . . . . . . . . . . . . . . . . 80Table 6.1 Universal voltage-mode compensator coefficients . . . . . . . . . . . . . . . . . . . 92Table 6.2 Parameters of the two buck converters used in simulation and experiments . . . . . . 94xiList of FiguresFigure 1.1 High-level representation of a modern grid. The power electronic interface suppliesmuch more complex loads than in the past. . . . . . . . . . . . . . . . . . . . . . . 3Figure 2.1 Schematic of the three-phase PWM converter topology. . . . . . . . . . . . . . . . . 17Figure 2.2 Conceptual representation of the dynamic model of the three-phase converter in thesynchronous reference frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Figure 2.3 Conceptual representation of the natural trajectories of the three-phase converter inthe synchronous reference frame. The sinusoidal waveforms of the state variables inthe time domain are represented as circles in the state plane with the center, radiusand angular speed determined by md . . . . . . . . . . . . . . . . . . . . . . . . . . 21Figure 2.4 State plane representation of several natural trajectories for different md values andsame initial conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Figure 2.5 State plane representation of several natural trajectories for different DC-bus currentvalues and same initial conditions with constant md . . . . . . . . . . . . . . . . . . 24Figure 2.6 Experimental natural circular trajectories of the three-phase converter in the syn-chronous reference frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Figure 3.1 Natural saturation trajectories λsat+ and λsat− in both time domain and state planerepresentation for the same set of initial conditions. . . . . . . . . . . . . . . . . . . 27Figure 3.2 Conceptual representation of the theoretical minimum time solutions. (a) The stateplane is divided into two regions according to the saturation trajectories. (b) Twominimum time responses in the state plane. (c) Tow minimum time responses in thetime domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29xiiFigure 3.3 Theoretical minimum recovery time start-up transient in state plane and time repre-sentation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Figure 3.4 Theoretical minimum recovery time transient for a DC-bus current step-up in stateplane and time representation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Figure 3.5 Theoretical minimum recovery time transient for a DC-bus current step-down in stateplane and time representation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Figure 3.6 Theoretical minimum recovery times for a DC-bus current step transient for differentinitial values: (a) Step-up. (b) Step-down. . . . . . . . . . . . . . . . . . . . . . . . 35Figure 3.7 Theoretical minimum recovery times for a DC-bus current step with different satura-tion limits for different initial values: (a) Step-up. (b) Step-down. . . . . . . . . . . 36Figure 3.8 Theoretical minimum recovery time transient for a DC-bus current step-up with d-axis current limit in state plane and time representation. . . . . . . . . . . . . . . . . 37Figure 3.9 HIL results of the minimum recovery time transients: (a) Start-up transient for Con-verter I. (b) Start-up transient for Converter II. (c) Start-up transient for Converter III.(d) DC-bus current step transient for Converter I. (e) DC-bus current step transientfor Converter II. (f) DC-bus current step transient for Converter III. . . . . . . . . . 40Figure 3.10 HIL results of minimum recovery time transient for a DC-bus voltage reference step. 41Figure 3.11 Minimum recovery time transients for a DC-bus current step with nonideal grid con-ditions: (a) Grid voltages with harmonic distortion. (b) Grid voltages with 15% im-balance. (c) Grid voltages with harmonic distortion and imbalance. . . . . . . . . . 42Figure 3.12 Experimental minimum recovery time transient for a DC-bus current step. The ob-tained results closely match the theoretical response. . . . . . . . . . . . . . . . . . 43Figure 3.13 Experimental minimum recovery time transients: (a) Start-up. (b) DC-bus currentstep-up. (c) DC-bus current step-down. . . . . . . . . . . . . . . . . . . . . . . . . 44Figure 3.14 The experimental transients for two different dual-loop PI controller designs can becompared with the theoretical minimum time solutions to evaluate their performance. 45Figure 4.1 CPL model and characteristics: (a) CPL model. (b) Linearized equivalent circuitmodel of a CPL around one equilibrium point. (c) Voltage-current characteristics ofa CPL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48xiiiFigure 4.2 Root-locus of the closed-loop linear control of the normalized three-phase converterfor different CPL conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Figure 4.3 Transient responses of the three-phase rectifier under different CPL steps. Althoughthe system is stable around all loading conditions, the CPL characteristics may leadto the DC-bus collapse during transients. . . . . . . . . . . . . . . . . . . . . . . . 50Figure 4.4 Conceptual representation of the proposed controller in the state plane. Although theCPL trajectories deviate from the ideal ones, the transient can be solved by using thecontrol surfaces σsat− and σsat+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Figure 4.5 Simulation results of the state plane controller for the normalized three-phase rec-tifier. Although the CPL trajectories deviate from the ideal case, the transients aresolved by the proposed control law. . . . . . . . . . . . . . . . . . . . . . . . . . . 54Figure 4.6 Transient responses of the state plane controller under different CPL steps. The DC-bus shows improved dynamic responses over the entire operating range, even whenthe largest power step is applied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Figure 4.7 Experimental results of state plane controller for the three-phase rectifier loaded witha CPL in both time and state plane representation. . . . . . . . . . . . . . . . . . . . 56Figure 4.8 Experimental results for the conventional dual-loop PI controller under small CPLsteps. The system is stable under small transients around different operating points. . 57Figure 4.9 Experimental results for a CPL step from no load (0 kW) to full load (2 kW). (a)Conventional dual-loop PI controller. (b) Proposed state plane controller. The con-ventional controller cannot compensate for the negative effects of the CPL whichleads to the collapse of the DC-bus voltage. On the other hand, the proposed methodcan solve the large CPL transient. . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Figure 4.10 Experimental CPL step-up/down transient from 0.5 to 2 kW (∆Pcpl = 1.5 kW). (a)Proposed state plane controller. (b) Conventional dual-loop PI controller. . . . . . . 59Figure 4.11 Experimental CPL step-up/down transient from 1 to 2 kW (∆Pcpl = 1 kW). (a) Pro-posed state plane controller. (b) Conventional dual-loop PI controller. . . . . . . . . 60Figure 5.1 Conceptual representation of the circular control approach in state plane and timedomain for four different sets of initial conditions and one target point. . . . . . . . . 63xivFigure 5.2 Conceptual graphical representation of the circular trajectory controller: (a) The con-trol state is selected according to the position of the operating point in the state plane.(b) Ideal circular trajectory controller transient responses for four different initialstates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Figure 5.3 Conceptual flow chart implementation for the circular trajectory controller. . . . . . 66Figure 5.4 General block diagram of the complete circular trajectory controller. . . . . . . . . . 67Figure 5.5 Theoretical circular trajectory controller transient for a DC-bus current step-up instate plane and time representation. . . . . . . . . . . . . . . . . . . . . . . . . . . 70Figure 5.6 Theoretical circular trajectory controller transient for a DC-bus current step-down instate plane and time representation. . . . . . . . . . . . . . . . . . . . . . . . . . . 72Figure 5.7 Simulation results for start-up and DC-current step transients in rectifier operationfor the circular trajectory controller in state plane and time domain representation. . 74Figure 5.8 Simulation results for start-up and DC-current step transients in rectifier operationfor the circular trajectory controller in state plane and time domain representation. . 75Figure 5.9 Simulation results for several DC-bus current steps in rectifier operation for the pro-posed controller (blue) and the conventional dual-loop PI controller (red). . . . . . . 76Figure 5.10 Simulation results for several DC-bus current steps in inverter operation for the pro-posed controller (blue) and the conventional dual-loop PI controller (red). . . . . . . 77Figure 5.11 Circular trajectory controller simulation results for an LC parameter variation of±20%: (a) Start-up and rectifier operation. (b) Inverter operation. . . . . . . . . . . 79Figure 5.12 Experimental start-up transient: (a) Circular trajectory controller. (b) Conventionaldual-loop PI controller. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81Figure 5.13 Experimental DC-bus current step up/down transients for rectifier operation: (a) Cir-cular trajectory controller. (b) Conventional dual-loop PI controller. . . . . . . . . . 82Figure 5.14 Experimental DC-bus current step-up/down transients for inverter operation: (a) Cir-cular trajectory controller. (b) Conventional dual-loop PI controller. . . . . . . . . . 83Figure 5.15 Experimental results for a grid voltage step transient for the circular trajectory con-troller. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84Figure 5.16 Experimental capture for a current step transient for the circular trajectory controllerunder nonideal grid conditions: (a) Rectifier operation. (b) Inverter operation. . . . . 85xvFigure 6.1 Conceptual representation of the normalization procedure. This method enables thederivation of unified models and controller designs that are not dependent on theparticular converter’s parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Figure 6.2 Synchronous PWM buck converter schematic diagram. . . . . . . . . . . . . . . . . 91Figure 6.3 Bode plot of the voltage loop for the normalized buck converter with the normalizedcompensator Gc(s) for different normalized switching frequencies. . . . . . . . . . . 94Figure 6.4 Step response for two different buck converters. The responses of the two buck con-verter are identical to the normalized response . . . . . . . . . . . . . . . . . . . . . 95Figure 6.5 Experimental voltage loop response for Buck I and II using the same normalizedcompensator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Figure 6.6 Experimental load step response for Buck I and II using the same normalized com-pensator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97Figure A.1 General block diagram of the conventional dual-loop controller implemented in thesynchronous reference frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116Figure A.2 Simplified block diagram of the current-loop for Technical Optimum approach. . . . 117Figure A.3 Bode plot of the voltage plant transfer function under a wide range of operating points. 118Figure B.1 Experimental hardware platform of the three-phase PWM converter. . . . . . . . . . 120xviAcknowledgmentsI would like to acknowledge the invaluable contributions of those who have supported and guided methroughout my journey to completing this thesis. First and foremost, my sincere appreciation goes tomy supervisor, Dr. Martin Ordonez. His consistent guidance, encouragement, and support have beeninvaluable throughout these years.I would also like to extend my special thanks to the members of my supervisory committee: Dr.Emanuel Serban, Dr. William Dunford, and Dr. Jose´ Martı´. Their insightful feedback and valuablesuggestions have significantly enriched the content of this thesis.I am deeply thankful to The University of British Columbia and the Department of Electrical andComputer Engineering for fostering an enriching academic environment and providing the necessaryresources during my time as a PhD student.I wish to extend my acknowledgement to my esteemed lab mates and mentors, including Nacho,Daniel, Abbas, Pancho, Ion, Gabriel, Jorge, German, and the many others whose engaging interactions,insightful discussions, and camaraderie have been an irreplaceable aspect of my academic experience.A heartfelt special thanks goes to my family, whose unwavering support and belief in my abilities havebeen my constant motivation. And last but not least, I am profoundly grateful to my life partner, Sofi, forher continuous encouragement, understanding, and patience throughout this challenging journey.Without the collective support of these individuals and many others, this thesis would not have beenpossible. Thank you all for your contributions, guidance, and belief in me.xviiChapter 1Introduction1.1 MotivationAs the world’s energy requirements continue to rise, the importance of alternative energy sources suchas wind, solar, marine, and biofuels as a means to replace fossil fuels and meet these growing demandscannot be overstated. According to forecasts, global power demand is projected to increase by 62% by2050, with an estimated half of the world’s electricity being generated by wind and solar [7]. Despite thenumerous benefits of renewable energy, it is important to note that these sources tend to be less dense,more distributed, and intermittent in comparison to the conventional, centralized fossil-based productionmethods currently in use. This presents new challenges in terms of controlling, monitoring, transporting,and optimizing energy production and distribution. To fully replace existing technology and displacefossil fuels, it is essential the development of new technology to address these challenges and effectivelyPortions of this chapter have been published in:• F. Degioanni, I. G. Zurbriggen and M. Ordonez, ”Enhanced DC-Link Voltage Dynamics for Grid-Connected Converters,”in IEEE Transactions on Industrial Electronics, vol. 69, no. 11, pp. 10787-10796, Nov. 2022.• F. Degioanni, I. G. Zurbriggen and M. Ordonez, ”Minimum Recovery Time Transients for Three-Phase Converters inthe Synchronous Reference Frame,” in IEEE Transactions on Power Electronics, vol. 37, no. 2, pp. 1579-1591, Feb.2022.• F. Degioanni, I. G. Zurbriggen and M. Ordonez, ”Fast and Reliable Geometric-Based Controller for Three-Phase PWMRectifiers,” 2020 IEEE Applied Power Electronics Conference and Exposition (APEC), 2020, pp. 1891-1896.• F. Degioanni, I. G. Zurbriggen and M. Ordonez, ”Characterization of the Minimum Recovery Time Transients for Three-Phase PWM Rectifiers,” 2021 IEEE Applied Power Electronics Conference and Exposition (APEC), 2021, pp. 2503-2509.• F. Degioanni, I. G. Zurbriggen and M. Ordonez, ”Improved Control for Three-Phase Rectifiers with Constant PowerLoads,” 2023 IEEE Applied Power Electronics Conference and Exposition (APEC), 2023, pp. 2775-2779.• F. Degioanni, I. G. Zurbriggen and M. Ordonez, ”Universal Controllers for PWM Converters: a Normalized Approach,”2019 IEEE Applied Power Electronics Conference and Exposition (APEC), pp. 1770-1775.1harness the potential of renewable energy sources.Power electronics systems play an important role in the development and integration of distributedenergy resources (DERs). They are indispensable for power processing and energy transfer from thesource to the end user. The impact of power electronics can be observed not only in large-scale industrialsystems, but also in smaller systems that have a significant impact on individuals’ daily lives, such asdomestic appliances, modern lighting systems, portable devices, battery chargers, and electric vehicles(EVs). As technology continues to advance, the range of applications for power electronics expands,bringing new challenges for modern energy conversion systems. This underlines the importance of powerelectronics in the implementation and optimization of renewable energy systems, as they are necessaryfor effectively harnessing the potential of these sources and addressing the challenges presented by theirdistributed and intermittent nature.Switching power converters are the foundation of any power electronics conversion system. Thesefundamental components are designed to convert electrical energy from one form to another and areparticularly useful in applications where high efficiency and precise power control are critical. Among thedifferent types of converters, the three-phase pulse-width-modulated (PWM) converter is widely used ina large number of applications that involve the conversion and power exchange from alternating currents(AC) to direct currents (DC), such as wind power [8, 9], solar power [10, 11], motor drives [12, 13],and high voltage DC transmission lines [14, 15], among others [16–18]. The use of three-phase PWMconverters is particularly relevant in modern grid applications due to their numerous benefits, such as highpower factor, low current harmonic distortion and tight DC-bus voltage regulation [19, 20]. As shownin Fig. 1.1, the ability of three-phase converters to support bidirectional power flow is crucial for thesuccessful integration of renewable energy sources and energy storage systems into the grid infrastructure,enabling seamless energy flow between the grid and DERs [21, 22].In the dynamic landscape of evolving technology, the integration of closed-loop controllers becomesan essential requisite for the seamless operation of three-phase converters, ensuring tight regulation ofvoltages and currents. These controllers, which are considered part of the backbone of the power con-verter, have a significant impact on the performance of the entire system. As power electronics systemsbecome more complex, new challenges arise that require higher demands on the control design and im-plementation. Advancements in controller design can result in improved efficiency, reduction in size, andcost savings for the entire power conversion system. Considering the rapidly shifting energy landscape2Consumer ElectronicsDistributed GenerationHybrid SystemsMore Complex SystemClosed-LoopControllerThree-Phase ConverterPower ElectronicsInterfaceElectric GridFigure 1.1: High-level representation of a modern grid. The power electronic interface suppliesmuch more complex loads than in the past.and the necessity of stringent sustainability goals, the study of closed-loop controllers for three-phaseconverters becomes crucial. Consequently, this area of research requires in-depth analysis and compre-hensive design methodologies to fully harness its potential.The dynamic performance of power converters is crucial for evaluating the effectiveness of the controlsystem in maintaining desired steady-state values in the presence of transient disturbances. Metrics likerecovery time, which denotes the duration for a converter to restore stable operation after a disturbance,and the deviation of state variables during transients, such as voltage and current fluctuations, serve as keyindicators for assessing the converter’s dynamic response. Poor dynamic performance can lead to largerecovery times and significant voltage and current deviations, potentially causing system failures andreduced efficiency. In particular, conventional control techniques for three-phase PWM converters, oftendesigned based on linear controllers using small-signal models around a steady operating point, may notfully address the dynamic performance issues that arise during the implementation of these converters inmodern applications. These applications, including bidirectional three-phase converters integrated intothe grid, present challenging control scenarios due to their wide range of operating conditions. As aresult, the development of novel control techniques is essential to enhance the dynamic performance ofthe system, aligning with the needs of contemporary energy conversion systems and ensuring efficient,3stable, fast, and reliable operation.A dynamic model serves as a representation of the system, allowing for the prediction of its behaviorunder various operating conditions. Novel modeling techniques have the potential to enhance system sta-bility by providing more accurate representations of underlying dynamics. Therefore, the ability to modelthe three-phase PWM converter in a simplified, yet accurate and comprehensive manner is vital for thecontrol design process. These improved models capture intricate behaviors, thus enabling better controldesign and analysis. Consequently, this capacity facilitates identifying and mitigating potential sourcesof instability, resulting in more robust and reliable power electronic systems. To meet the demands ofmodern power electronic systems, including rapid transient response and a broad operating range, theadoption of new modeling approaches becomes necessary to support the development of advanced, high-performance control techniques.In summary, the development of new models and advanced controllers for three-phase PWM con-verters is an area of research that holds great potential for contributing to the field of power electronics.Due to the complexity and rapid advancements in technology, the capability to enhance the transientperformance, operating range, and cost-effectiveness of three-phase converters is critical for meeting thedemands of modern systems, such as the integration of renewable energy and energy storage.The main goal of this dissertation is to thoroughly investigate the dynamic behavior of three-phasePWM converters and to propose innovative modeling and control techniques that have the capabilityto significantly enhance the dynamic performance, stability, and operating range of the system. Thisresearch explores concepts such as normalization, state plane representation, and geometric analysis toderive simple yet detailed and comprehensive dynamic models of the three-phase PWM converter. Thesemodels serve as an effective tool for understanding the converter’s behavior and serve as a foundation forthe development of advanced control solutions that overcome the limitations of traditional techniques.This work introduces a detailed and comprehensive large-signal model of the three-phase converter,providing a general representation of the system dynamics. This model enables the derivation and charac-terization of the limits of the dynamic performance of three-phase converters, which not only provides apowerful tool for benchmarking the dynamic performance of the converter, but also a fast control solutionfor challenging scenarios such as the connection of active loads to the DC-bus. Additionally, this model isused to design advanced control methods to improve the large-signal transient responses of applicationswith high control requirements and wide operating ranges. The proposed model and control techniques4significantly enhance the dynamic performance, stability, and operating range of the system.1.2 Literature ReviewThe field of power electronics has seen significant research on the modeling and control of three-phasePWM converters over the past several decades. Researchers have explored various techniques to enhancethe dynamic performance of the converter for various applications. Among the methods employed, small-signal models and linear controllers have been widely adopted as the conventional solution to control thethree-phase converter. However, these techniques have limitations when addressing new and advancedcontrol challenges [23–25]. The purpose of this literature review is to provide an overview of the currentstate of the art by analyzing the most recent developments and advancements in this field.1.2.1 Conventional Modeling and Control of Three-Phase PWM ConvertersPower electronics converters are nonlinear systems by nature, and thus, a variety of modeling approacheshave been proposed to study their dynamic behavior [26–30]. The multiple combinations of switchstates in a PWM converter result in a periodically repeated sequence of possible configurations during itsoperation time interval, also known as the switching period [31]. By using a discrete switching model,each configuration can be mathematically described by a set of state equations, corresponding to differentelectrical configurations throughout the converter’s operation cycle.However, in most power converter control applications, it is the low-frequency behavior that is relevantfor designing the controller [32]. To capture this low-frequency behavior while neglecting high-frequencyvariations due to circuit switching, the average model is commonly used [33, 34]. This approach is naturalas power converters employ filters to limit the ripple of various variables, resulting in a continuous-timemodel that is easier to handle with classical analysis and control techniques. The average model of apower electronic converter is, in general, a nonlinear model. Therefore, if state-space analysis or linearcontrol design is desired, a small-signal representation of the converter in the state-space or frequencydomain is needed [35]. These models assume small perturbations around the steady-state operating point,providing a linear representation of the converter’s dynamic behavior [36, 37].The aforementioned approaches have been also widely adopted for modeling and control design ofthree-phase grid-connected converters [38–40]. However, the dynamic model of these systems showsan additional challenge due to the presence of three-phase sinusoidal voltages and currents on the ACside of the converter [41, 42]. The time-varying nature of these variables results in a lack of time-5invariant steady-state operating points, which can make it difficult to apply averaging and linearizationtechniques [43]. This issue is typically addressed by modeling the converter in a different reference frame,such as the synchronous reference frame (SRF), also known as the dq frame [44, 45]. The SRF model ofthe converter is obtained by rotating the natural reference frame at the same speed as the grid voltage orcurrent vector. This rotation allows for the definition of the AC variables into two vector components, dand q, representing the direct and quadrature axis, respectively. As a result, during steady-state operation,the AC currents and voltages can be represented as DC variables in the SRF, leading to time-invariantsteady-state solutions [46, 47]. This enables the application of linear modeling and control techniques,such as small-signal models, which are widely adopted due to their simplicity [48, 49].The control objectives for three-phase converters in grid-connected applications typically includeachieving phase currents with low harmonic distortion and unity power factor on the AC side, as wellas maintaining tight voltage regulation on the DC-bus [50, 51]. Poor dynamic characteristics can resultin the uncontrollability of the system, making stable operation and reliable dynamic response essen-tial [52, 53]. The conventional solution to meet these requirements is the implementation of dual-loopcontrol architecture, comprising an inner current loop and an outer voltage loop [54, 55].The goal of the inner current loop is to provide fast and accurate grid current regulation. Variousapproaches have been developed to achieve this, such as hysteresis control, model predictive control(MPC) [56, 57], and vector-oriented control (VOC) [58–61]. While hysteresis and MPC techniques canenable faster current dynamics, VOC techniques are typically more widely adopted due to their lowercomputational cost and constant switching frequency operation [56, 57]. Therefore, the implementationof Proportional-integral (PI) regulators in the synchronous reference frame is widely adopted to sepa-rately regulate the d and q-axis currents, simplifying the control design [59–61]. On the other hand, theouter voltage loop provides the reference for the current controller and regulates the DC-bus voltage toits reference value by controlling the converter’s power flow. The dynamics of the DC-bus are mainlydetermined by the voltage controller, which is typically realized by a PI compensator [62]. To achievedifferent transient performance objectives, various tuning methodologies have been proposed to calculateand tune the proper compensator coefficients [63–65]. Hence, DC-bus voltage control becomes a keyfactor in the overall performance of the converter [66].61.2.2 Control of Three-Phase PWM Converters for Wide Operating Range ApplicationsUnfortunately, linear-based controllers may struggle to provide the desired level of performance androbustness in wide operating range applications, such as for three-phase converters with bidirectionalpower flow capabilities [67]. These systems are required to not only maintain tight voltage regulation onthe DC-bus but also mitigate the effects of external disturbances in both rectifier and inverter operatingmodes [68, 69]. The limitations of linear controllers in this scenario result from the fact that the modelsused to design the PI parameters are based on linear system assumptions, which may not hold when thesystem operates over a wide range of conditions. The dynamic model of the three-phase converter in theSRF presents non-minimum phase characteristics when operating in rectifier mode [70–73]. The exis-tence of a right-half-plane-zero (RHPZ) in the control-to-output transfer functions restricts the maximumachievable closed-loop bandwidth of the system, limiting the converter’s dynamic response. Furthermore,the dynamic model shows a right-half-plane-pole (RHPP) in its voltage control loop dynamics for inverteroperation [74, 75]. The behavior of the converter in both modes influences the location of the RHPZ andRHPP, causing stability problems as different operating conditions are encountered. This can result ininadequate dynamic responses and suboptimal performance as the modes of operation change [76].To overcome the limitations of conventional methods and improve the DC-bus voltage dynamics inthree-phase converters, various approaches have been proposed. One such approach is the implementationof an adaptive PI controller in the outer-voltage loop to achieve improved performance under differentconditions [77]. However, this technique neglects the nonlinear nature of the DC-link dynamics. Toaddress this, boundary controllers such as Sliding-Mode Control (SMC) and Natural Switching Surfaces(NSS) methods have been proposed to enhance the DC-bus voltage dynamic regulation [78–83]. Thesecontrollers have been shown to exhibit excellent dynamic performance under large transients and dif-ferent operating conditions. However, SMC is known to have chattering issues and may result in largeovershoot/undershoot values prior to reaching the sliding surface. To mitigate these effects, super twistingcontrol algorithms or higher-order sliding surfaces have been implemented, but at the cost of increasedcomputational burden and complexity of controller design [84, 85]. On the other hand, the NSS methodpresented in [83] eliminates the need for a dual-loop architecture and provides fast DC-link dynamics.However, this approach utilizes a table for selecting switching states, resulting in a variable switching fre-quency that is limited to half of the sampling frequency. This makes output filter design challenging andhigher sampling rates are required to minimize chattering issues in steady-state. Robust control methods7such as H-infinity or Mu-synthesis have also been proposed, which are able to provide stable and robustperformance in the presence of disturbances and uncertainties [86, 87]. However, these methods require agood understanding of the system dynamics and high computational effort. Furthermore, in recent years,intelligent control methods such as artificial neural networks (ANNs) and fuzzy logic (FL) have beenproposed for three-phase converters [88–91]. These methods have the advantage of being able to handlenonlinearities and uncertainties in the system, as well as achieving good dynamic performance and lowharmonic distortion. But, the main limitation of these methods is the difficulty in designing and tuningthe controllers, which can be time-consuming and computationally intensive.Despite the various advanced control methods proposed in the literature, there remains a significantgap in the ability to effectively and efficiently provide excellent dynamic performance in bidirectionalthree-phase converters over a wide operating range. This thesis aims to address this gap by proposing anovel control method that addresses the nonlinearities and uncertainties present in these systems whileachieving fast and reliable dynamic responses in both rectifier and inverter modes. The proposed methodalso maintains a low computation burden and implementation requirements, making it a robust and prac-tical solution for bidirectional three-phase converter control.1.2.3 Control of Three-Phase PWM Converters with Active LoadsThe control of three-phase converters faces some particular challenges when active loads are presentin the system [92]. These loads, which are power electronics converters themselves, tightly regulatetheir output voltages and/or currents. In many applications, multi-converter power architectures are used,where the three-phase converter acts as the first stage, supplying energy from the grid to the DC-bus [93–97]. Another set of power converters then transfers the energy from the DC-bus to one or multiple loads.The presence of active loads in the DC-bus can cause Constant Power Load (CPL) behavior, leading tostability issues [98, 99]. CPLs present negative incremental resistance, meaning that if the input voltagedrops, the input current must increase to maintain constant power delivery to the load [100]. This canimpact the DC-bus voltage dynamics, leading to voltage oscillations, and even system failure under heavyloading conditions or during transients [101]. The negative impact on the DC-bus dynamics and stabilityof three-phase converters must be considered in the control design to ensure stable operation and minimizevoltage and current fluctuations under sudden changes in the loading conditions.The negative incremental resistance exhibited by a CPL reduces the effective damping of the system,which may result in unstable equilibrium points within the rated power of the three-phase converter [102].8To mitigate these negative effects, various stabilization techniques are commonly employed. These tech-niques can be divided into two main categories: active and passive [103–106]. Passive solutions, suchas the implementation of extra RC and RL filters or increasing the DC-bus capacitance, are attractivefor their simplicity, but they come at the cost of increased system size and more expensive implemen-tation [103, 104]. In contrast, active methods modify the control signal to emulate passive elements,offering less expensive solutions, but at the cost of increased complexity [105, 106]. While these strate-gies may provide stable operation under different equilibrium points, they may become ineffective underexternal disturbances, resulting in deteriorated transient responses [107, 108].In instances of power mismatch on the DC-bus, which can cause abrupt changes in current and volt-age, the nonlinear nature of the CPL can result in substantial fluctuations in the DC-bus current. If thesevariations are not addressed in a timely manner, system failure may occur [109]. A fast dynamics con-troller can mitigate these negative effects by enabling rapid current injection in the DC-bus [110, 111].However, conventional linear approaches often restrict the maximum achievable closed-loop bandwidth,limiting the dynamic response of the three-phase converter under sudden changes in the DC-bus. More-over, due to their small-signal nature, they may not be able to provide stable operation during large CPLtransients, leading to a collapse of the DC-bus voltage.In order to address the limitations of linear controllers and improve DC-bus regulation with CPLs, avariety of approaches have been proposed. Feedback linearization methods aim to cancel the nonlineardynamics of the system by selecting a proper feedback path, allowing for the use of conventional designmethods after linearization [112, 113]. However, these approaches assume constant power operatingconditions and do not account for the effects of external disturbances on dynamic performance. Animproved loop cancellation technique was proposed for a three-phase rectifier feeding a DC-microgrid,which eliminates the impact of CPLs on the DC-bus [114, 115]. While this approach achieves stablesteady-state operation within the rated power range, it necessitates the computation of the DC-bus voltagederivative in the feedback loop, which can increase noise levels in the control loop. Furthermore, due tothe nonlinear nature of the CPL dynamics, nonlinear approaches such as SMC or passivity control mayimprove the dynamic performance of the system [78–81, 116, 117]. However, these approaches generallystudy the three-phase converter feeding a passive load, and only a few have explored the effects of aCPL in the DC-bus [82, 118]. Recently, artificial intelligence methods have been also used to mitigatethe effects of CPLs over the entire operating range, however, this approach increases the computational9burden and controller design complexity [119].While various approaches have been proposed to mitigate the negative effects of CPLs on the dynamicstability of three-phase converters, there is still room for improvement. Further research is needed todevelop new methods that can effectively control three-phase converters with active loads connected tothe DC-bus.1.2.4 Minimum Recovery Time Transients for Three-Phase ConvertersThe dynamic behavior of a closed-loop system is primarily determined by the controller design, resultingin varying performance for different control approaches or compensator parameters. Among the manycharacteristics of the converter’s dynamic response in the control of the DC-bus, voltage overshoot/un-dershoot deviation and recovery time are often considered among the most relevant ones for evaluatingthe dynamic performance [120, 121]. As previously discussed, a considerable research effort has beendevoted to improving the dynamic performance of three-phase converters. As a result, various PI tuningmethodologies and advanced control techniques have been developed, each achieving different dynamicperformance outcomes. However, assessing the effectiveness of these methods objectively is challengingdue to the absence of a standard reference for evaluating converter dynamic performance.One potential approach to addressing the challenge of evaluating the dynamic performance of three-phase converters is to define the converter’s achievable fastest responses as an ideal dynamic referencebaseline [122]. This would allow for an objective assessment of transient performance by comparing thesystem’s dynamics to the theoretical minimum time solutions. The minimum recovery time responses forthree-phase converters controlled in the SRF have yet to be defined, in comparison to DC-DC convert-ers [123, 124]. The lack of these ideal limits not only makes it harder to evaluate dynamic performancebut also leads to unnecessary control design efforts aimed at achieving responses that may be theoreticallyimpossible to accomplish.Efforts have been made to achieve minimum time current control for the three-phase converter byselecting optimal control actions while considering converter voltage limit constraints [125, 126]. Addi-tionally, control design methodologies have been introduced to design PI coefficients that result in nearlyminimum settling time and negligible overshoot values [127, 128]. However, these works only focus onthe dynamic performance of the inner current loop, neglecting the theoretical minimum time transients.Furthermore, the focus on current-loop design assuming a first-order model overlooks the influence ofthe DC-bus capacitor on the system’s dynamics, leaving the definition of the minimum time responses10for the three-phase converter unaddressed.The characterization of the theoretical minimum recovery time transients for three-phase controllersin the SRF is yet to be explored. Such characterization could provide a useful benchmark for evaluatingcontroller performance by comparing it to the theoretical limits of the dynamic performance of the system.Furthermore, it would help to identify limitations that may impact the control design, such as stability,bandwidth, or overshoot/undershoot constraints.1.3 Contributions of the WorkThe objective of this work is to conduct a comprehensive investigation into the dynamic behavior ofthree-phase PWM converters and to devise novel modeling and control techniques that have the potentialto significantly improve the dynamic performance, stability, and operating range of the system. Throughthe utilization of techniques such as normalization, state plane representation, and geometric analysis, thisresearch aims to derive simple yet detailed and comprehensive dynamic models of the three-phase PWMconverter. The derived models offer valuable insights into the behavior of these converters and providea solid basis for developing cutting-edge control solutions that surpass the shortcomings of conventionalapproaches. The areas in which this work contributes to advancing the state of the art are:• State plane large-signal modeling of the three-phase converter in the synchronous referenceframe: One of the main contributions of this work is the development of a novel framework for thestudy of the dynamic behavior of three-phase converters in the synchronous reference frame. The derivednormalized decoupled model of the converter allows for the representation of its large-signal dynamiccharacteristics on a state plane formed by the capacitor voltage and d-axis current component. In thisway, the evolution of the state variables can be represented by well-defined geometric surfaces in thestate plane, providing an insightful visual representation of the dynamic evolution of the operating point.Since the analysis is carried out in a normalized domain, it ensures the validity of the model for anycombination of resonant parameters, as well as input/output voltage and power levels.This framework forms the foundation for further analysis, characterization, and modeling of the con-verter behavior, as well as the development of advanced control methods aimed at improving the dynamicperformance of three-phase converters in various applications.• Characterization and derivation of the theoretical limits of the dynamic performance: Thiswork offers a comprehensive characterization of the three-phase converter’s theoretical dynamic perfor-11mance limits when controlled in the synchronous reference frame. The minimum recovery time transientsare determined by combining the developed large-signal model with the Time Optimal Control principle.By analyzing the system’s dynamic response using only saturation trajectories, given by the control sig-nal’s minimum and maximum limits, the theoretical optimal control law is derived. This enables a fulldefinition and characterization of the dynamic performance through geometric analysis, yielding closed-form mathematical expressions for various dynamic transients, such as DC-bus current step up/down,start-up, reference step, and others.The minimum recovery time transients offer valuable insights into the converter’s dynamic behaviorand provide a thorough, quantitative performance characterization by comparing the achieved dynamicperformance with the ideal transients. As the theoretical ideal performance cannot be exceeded, theproposed benchmarking method offers the feature of clearly indicating the extent to which the dynamicbehavior of the converters can be improved. The theoretical limit of performance serves as a referencepoint, and the benchmarking procedure in this thesis enables an objective quantification of performanceimprovements introduced by changes to the controllers.• Dynamic performance improvement with active loads: The implementation of a minimum re-covery time transient controller is proposed as an effective solution to enhance the dynamic stability ofthe DC-bus in three-phase converters that serve active loads. The fast response of the proposed controllermitigates the negative impact of constant power loads, thereby improving the system’s dynamic perfor-mance during sudden and large load transients. The controller uses a combination of natural saturationtrajectories to solve the transient, leading to near-time-optimal solutions that result in improved dynamicperformance and prevent system failures, such as DC-bus voltage collapse.This work presents a comprehensive analysis and implementation of the minimum time controller.The proposed solution is rigorously evaluated and compared to a conventional controller, demonstratingits superior performance and reliability in handling large active load transients. This study highlightsthe importance of the proposed control method as an efficient and effective solution for the control ofconstant power loads in three-phase converters.• Bidirectional dynamic performance improvement, the circular trajectory controller: The cir-cular trajectory controller is introduced in this work to enhance the dynamic performance of three-phaseconverters used in applications with widely varying conditions, such as bidirectional power flow opera-tion. The control technique aims to provide fast, reliable, and consistent transient responses throughout12the operating range, including rectifier and inverter modes. The control strategy is based on the geometricrepresentation of operating point dynamics in the state plane, allowing for control decisions based on thelocation of state variables in the geometric plane. By following the appropriate set of natural trajectories,the operating point follows sleek and well-defined circular paths, resulting in fast and smooth responsesthroughout the operating range while keeping implementation requirements simple. The circular tra-jectory control technique is fully explored in this work, including its derivation, analysis, and practicalimplementation. Additionally, the benefits of the proposed technique over traditional linear approachesand other advanced techniques are studied and experimentally validated.• Normalized models and universal controllers for PWM converters: The concept of normaliza-tion is used is explored throughout this work to obtain generalized models for three-phase converters thatare independent of specific parameters, such as values of reactive elements, voltage levels, and powerrating. By extending this concept to different types of PWM converters, normalized models and con-trol designs can be derived. This approach enables the description of PWM converters in a general andcomprehensive manner, simplifying the analysis of their dynamic behavior and facilitating the designof effective control solutions. The result of this is the development of universal controllers that exhibitconsistent normalized behavior when applied to converters with varying parameters.This research emphasizes the advantages of normalization in the modeling of PWM converters andthe creation of universal controllers. To demonstrate these benefits, the work presents the implementationof a universal normalized voltage-mode linear controller for a synchronous buck converter.1.4 Dissertation OutlineThe organization of this thesis is described below:• Chapter 2 introduces the normalized decoupled model of the three-phase PWM converter in thesynchronous reference frame. The complete derivation of the model is provided and the dynamic evolu-tion of the operating point in the state plane, formed by the capacitor voltage and the d-axis current, isstudied. This graphical representation of the converter’s dynamic behavior sets the basis for the analysis,characterization and development of different methods to study and improve the dynamic performance ofthe three-phase converter.• Chapter 3 explores the minimum recovery time solutions for the three-phase converter controlled inthe synchronous reference frame. The Time Optimal Control Principle is combined with the normalized13decoupled model to derive the theoretical minimum time solutions. The ideal optimal solutions for differ-ent dynamic transients, such as start-up and DC-bus current step-up/down, are thoroughly characterizedand closed-form expressions are derived for the diverse transient parameters. The minimum recovery timetransients are then used as an objective benchmark to evaluate the dynamic performance of the system,and the validity of the analysis is confirmed through simulation and experimental results.• In Chapter 4, the application of minimum recovery time transients as a means of improving thedynamic stability of three-phase converters with active loads is investigated. The constant power loadcharacteristics of tightly regulated load converters are analyzed, and their impact on system stability isevaluated. The limitations of conventional control methods that lead to the collapse of the DC-bus voltageare also explored. A state plane controller is introduced, which provides near-optimal time solutions andimproves transient responses, thus enhancing the stability of the system during large constant power loadtransients. The validity of the analysis and the implementation of the proposed controller are confirmedthrough simulation and experimental results, which demonstrate its improved dynamic performance com-pared to the conventional approaches.• In Chapter 5, the circular trajectory controller is presented as a solution to enhance the dynamicperformance of three-phase PWM converters in bidirectional applications. The working principle of thecontroller, which is based on the natural trajectories of the converter, is explained. The selection of theoptimal circular trajectories to achieve rapid, reliable, and consistent transient responses is analyzed andcharacterized. A thorough explanation of the practical implementation of the proposed control methodis provided, including the stability analysis of the controller. The performance of the proposed methodis evaluated through simulation and experimental testing under both inverter and rectifier modes andcompared with conventional approaches.• In Chapter 6, the application of normalization in control modeling and design of PWM converters isthoroughly examined. The normalization process and the definition of base quantities that produce modelsthat are independent of system parameters are presented. Additionally, the design of normalized universalcontrollers, which are independent of the system parameters, is discussed. To illustrate the normalizedmodel derivation and design of universal controllers, a voltage-mode linear controller for the synchronousbuck converter is designed and presented in the normalized domain. Finally, simulation and experimentalresults from two buck converter systems with different parameters, demonstrate that both systems exhibitsimilar normalized responses in terms of normalized settling time and overshoot/undershoot values.14• Chapter 7 contains the relevant conclusions, contributions, and planned areas of future work de-rived from this thesis. The findings of this work provide a significant contribution to the field of powerelectronics by improving the models and enhancing the dynamic performance of three-phase convertersin various applications. This new approach offers a universal and practical solution for modeling andcontrolling these systems, making it a valuable resource for researchers and engineers in the industry.The value of this work is demonstrated by the six relevant publications in international conferences andjournals from the Institute of Electrical and Electronics Engineers (IEEE).15Chapter 2Normalized Decoupled Model of theThree-Phase Converter in the SynchronousReference FrameIn this chapter, the normalized decoupled model of a three-phase converter in the synchronous referenceframe is derived. The model captures the dynamic evolution of state variables and provides a cleargraphical representation of the large-signal dynamics of the system. The operating point’s evolution isdescribed by circular trajectories with well-defined characteristics, which allows for the use of geometricanalysis to study the converter behavior.The understanding gained from the normalized decoupled model and its graphical representationserves as the foundation for analyzing the three-phase converter dynamics and developing control tech-niques to enhance system performance, as explored in later chapters of this dissertation.Portions of this chapter have been published in:• F. Degioanni, I. G. Zurbriggen and M. Ordonez, ”Enhanced DC-Link Voltage Dynamics for Grid-Connected Converters,”in IEEE Transactions on Industrial Electronics, vol. 69, no. 11, pp. 10787-10796, Nov. 2022.• F. Degioanni, I. G. Zurbriggen and M. Ordonez, ”Minimum Recovery Time Transients for Three-Phase Converters inthe Synchronous Reference Frame,” in IEEE Transactions on Power Electronics, vol. 37, no. 2, pp. 1579-1591, Feb.2022.• F. Degioanni, I. G. Zurbriggen and M. Ordonez, ”Fast and Reliable Geometric-Based Controller for Three-Phase PWMRectifiers,” 2020 IEEE Applied Power Electronics Conference and Exposition (APEC), 2020, pp. 1891-1896.• F. Degioanni, I. G. Zurbriggen and M. Ordonez, ”Characterization of the Minimum Recovery Time Transients for Three-Phase PWM Rectifiers,” 2021 IEEE Applied Power Electronics Conference and Exposition (APEC), 2021, pp. 2503-2509.• F. F. Degioanni, I. G. Zurbriggen and M. Ordonez, ”Improved Control for Three-Phase Rectifiers with Constant PowerLoads,” 2023 IEEE Applied Power Electronics Conference and Exposition (APEC), 2023, pp. 2775-2779.16LVsa ia Sa1Sa2 Sb2Sb1 Sc1Sc2ioCVsbVscLLNibicDCNetworkThree-Phase PWM ConverterCvFigure 2.1: Schematic of the three-phase PWM converter topology.2.1 Syncrhonous Reference Frame Model of the Three-Phase ConverterThe conventional schematic of the two-level three-phase PWM converter is shown in Fig. 2.1. The con-verter is connected to a three-phase voltage source Vsabc, such as the grid, on the AC side through aL-filter, whereas the capacitor C is connected to the DC-bus that may supply different types of loads. Thepolarity of the currents in the system is defined by the power flow direction and the converter operatingmode. During rectifier operation, the power is transferred from the grid to the DC-bus, and the DC currentio is defined as positive. On the other hand, io is defined as negative when the power is transferred fromthe DC-bus to the AC grid during inverter operation.The dynamic behavior of the three-phase converter of Fig. 2.1 can be described in a synchronousreference frame rotating at the same speed as the grid frequency ωg [54, 55]. In practice, a PLL is usedto estimate the phase angle θ of the voltage vectors for the grid synchronization and to transform all theAC variables, into corresponding DC variables. In this way, the converter currents and voltages can berepresented by two orthogonal dq components in the SRF.The set of differential equations that describes the behavior of the grid currents (iabc) and DC-buscapacitor voltage (vC) in the SRF can be derived as [61]:Ldiddt= Lωgiq+Vsd −ud vC2 (2.1)Ldiqdt=−Lωgid +Vsq−uq vC2 (2.2)CdvCdt=34(ud id +uqiq)− io (2.3)17The variables Vsd , Vsq and id and iq denote the three-phase grid voltages and currents represented inthe SRF, while ud and uq are the control switching variables in the d and q-axis respectively.2.1.1 Decoupled ModelThe converter model described by (2.1)-(2.3) shows the presence of two current controlled voltage sourcesLωgid and Lωgiq that creates a coupling between the d and q-axis. Commonly, the effect of these termsis canceled by appropriately adjusting the control inputs ud and uq as:ud = md +md,dec = md +2vCωgLiquq = mq+mq,dec = mq− 2vCωgLid(2.4)where the variables md,dec and mq,dec are the nonlinear feed-forward decoupling terms, and md and mqare the new control inputs of the decoupled d and q-axis respectively.The implementation of the decoupling network (2.4) enables the elimination of the cross-couplingeffect and both axes can be considered decoupled for the dynamic analysis. Moreover, the q-axis currentis usually controlled to be zero (iq = 0) in order to achieve near unity power factor conditions. Therefore,under balanced grid voltage operation (Vsq = 0), the differential equations (2.1)-(2.3) can be rewritten as:Ldiddt=Vsd −md vC2 (2.5)CdvCdt=34md id − io (2.6)Ldiqdt=−mq vC2 (2.7)The derived system of equations describes the decoupled dynamic model of the three-phase converterin the synchronous reference frame. According to (2.5)-(2.7), the converter dynamics can be divided intoa dual SISO system. The d-axis represents the dynamic behavior of the active power in the system, andconsequently, the regulation of the DC-bus voltage. It is modeled as a second-order nonlinear systemgiven by (2.5) and (2.6), with vC and id as state variables and md as the control input. On the other hand,the q-axis is modeled as a first-order system with iq as the state variable and mq as the control input andit is associated with the reactive power regulation.The conceptual block diagram of Fig. 2.2 illustrates the high-level modeling of the three-phase con-verter derived in the SRF. The decoupled model enables the analysis and design of the control loops forthe d and q-axis independently. In the following sections, the previous concepts are extended to developa comprehensive dynamic model of the three-phase converter in the synchronous reference frame.18dqabcdqabcDecoupling NetworkSRF ModelDecoupled SRF Control ModelThree-Phase ConverterPLLdidiqiqiduqugLωgLω2Cv2CvdmqmabcmCvabciabcvθFigure 2.2: Conceptual representation of the dynamic model of the three-phase converter in thesynchronous reference frame.2.1.2 Normalized ModelThe converter can be represented in a normalized domain that enables the elimination of the model de-pendence on the reactive components, voltage levels and power ratings. In this way, by performing theanalysis in the normalized domain, a generalized model and expressions can be derived, which are validfor any combination of parameters. The described normalization concept can be extended to differenttypes of PWM converters. Its application and benefits, such as the design of normalized controllers, arestudied in detail in Chapter 6.For the three-phase converter, the normalization parameters are given by the filter characteristicimpedance (Z0), the filter natural resonant period (T0), the desired closed-loop reference voltage of theDC-bus (vC,re f ) and the defined normalization current (inorm):Z0 =√L/C, T0 = 2π√LC, vnorm = vC,re f , inorm =vnormZ0(2.8)The normalized model is obtained by dividing the different variables by the corresponding base quan-tities, such as time, impedance, voltages and currents:txn =txT0, Zxn =ZxZ0, vxn =vxvnorm, ixn =ixinorm(2.9)19where the sub-indexes x and xn indicate the variables of the system before and after the normalization.By applying the normalization concept, the normalized decoupled model of the three-phase converterin the synchronous reference frame is obtained as:12πdidndtn=Vsdn−md vCn2 (2.10)12πdvCndtn=34md idn− ion (2.11)12πdiqndtn=−mq vCn2 (2.12)where all the variables are represented by their normalized states (vCn, idn, iqn, tn, etc.).In the normalized domain, the time variable is obtained by dividing by the filter resonant period T0.In this way, a normalized time duration tn = 1 p.u. represents one LC resonant period, resulting in anormalized natural frequency fn of 1 p.u.The obtained equations (2.10)-(2.12) describe the dynamic natural behavior of the three-phase con-verter, which enables the derivation of general expressions valid for any combination of parameters. Inthe following sections, the normalized decoupled model is studied under both steady-state and transientoperations to find a simple and comprehensive state plane representation of the converter’s dynamics.2.2 Steady-State Operating Points AnalysisThe converter is in steady-state operation if the state variables are unchanging in time; in other words,when the state variables reach an equilibrium position. The set of operating points that satisfy the equi-librium condition for the d-axis normalized SISO model can be obtained by making the differential equa-tions (2.10) and (2.11) equal to zero:dvCndtn= 0 =Vsdn−md vCn2 (2.13)didndt= 0 =34md idn− ion (2.14)The combination of all possible equilibrium points that the converter may take is defined by load lineλLL, which is given as the following parametric expression:λLL : 0 = idn− 23ionVsdnvCn (2.15)The derived expression can be represented as a straight line in the state plane formed by vCn and idn,20dn’iLLλmdλmdωmdcmdrCnv(0)Cnvdn’i (0)dmoni2’oni ’sdnV/dmsdnV2≈≈≈≈≈dn’iCnv(0)Cnvdn’i (0)dmoni2’dmsdnV2mdωpi2=mdTmdTntntFigure 2.3: Conceptual representation of the natural trajectories of the three-phase converter in thesynchronous reference frame. The sinusoidal waveforms of the state variables in the timedomain are represented as circles in the state plane with the center, radius and angular speeddetermined by md .in which the slope of λLL is defined by the normalized d-axis grid voltage component and the DC-buscurrent. Therefore, the slope of λLL is positive in rectifier mode (ion > 0) and it becomes negative forinverter operation (ion < 0). During steady-state operation, the converter reaches one of the equilibriumpoints contained in the load line and (2.15) becomes equal to zero.The desired closed-loop operating state is defined here as the target operating point vCn = vCnt andidn = idnt , which are obtained from (2.13) and (2.14) as:vCnt = 2Vsdnmd,ss(2.16)idnt =43ionmd,ss(2.17)where md,ss is the corresponding steady-state modulation index.Since the analysis is performed in the normalized domain, the normalized voltage is the absolutevalue of the desired output and vCnt results equal to 1 (vnorm = vC,re f ). On the other hand, the target d-axiscurrent component is contained in the load line and it is expressed as a function of ion and Vsdn.212.3 Dynamic Transient Analysis and State Plane RepresentationThe general expressions that describe the time domain evolution of the normalized state variables can bederived by solving the decoupled differential equations (2.10) and (2.11) as:vCn(tn) =[i′dn(0)−2i′onmd]sin(ωmdtn)+[vCn(0)−2Vsdnmd]cos(ωmdtn)+2Vsdnmd(2.18)i′dn(tn) =[i′dn(0)−2i′onmd]cos(ωmdtn)−[vCn(0)−2Vsdnmd]sin(ωmdtn)+2i′onmd(2.19)where the currents idn and ion are redefined as:i′dn =√3√2idn, i′on =√2√3ion (2.20)The solutions (2.18) and (2.19) represent the time domain evolution of the normalized DC-bus ca-pacitor voltage and d-axis current. The dynamic behavior of the system is determined by Vsdn, ion, mdand the initial conditions(vCn(0), i′dn(0)), which indicates that both state variables describe sinusoidalwaveforms with a normalized angular frequency given by:ωmd = π√3√2md (2.21)As shown, in Fig. 2.3, for a given set of operating and initial conditions, the sinusoidal time domainwaveforms can be represented in the state plane formed by vCn and i′dn. By combining (2.18) and (2.19),the state plane trajectories are found as:λmd :(vCn(tn)−cvmd︷ ︸︸ ︷2Vsdnmd)2+(i′dn(tn)−cimd︷︸︸︷2i′onmd)2=r2md︷ ︸︸ ︷(vCn(0)−2Vsdnmd)2+(i′dn(0)−2i′onmd)2(2.22)The derived parametric equation is independent of the normalized time tn and it describes the naturalevolution of the operating point in the state plane for different operating conditions. As illustrated inFig. 2.3, λmd is a circular surface, which center (cmd) it is located over the load line at(Vsdnmd; i′onmd)andradius (rmd) is a function of the initial conditions and md . According to (2.21), the operating point movesat the angular speed determined only by md . Therefore, for a fixed md , the operating point describescircular trajectories at constant angular speed and well-defined characteristics as shown in Fig. 2.3.22dmdn’idn’i (0)(0)Cnv=0mdλ1md,λ7md,λ6md,λ2md,λ3md,λ4md,λ5md,λ+satλ-satλ+satω-satω7md,c6md,c2md,c1md,c3md,c5md,c4md,c+satc-satc1md,ω7md,ω6md,ω2md,ω3md,ω4md,ω5md,ωLLλOut of Range-satcmdc <<-∞>>0 dm sat-mmd,ssc+satc mdc <<>> dmsat+m d,ssm∞md,ssc mdc <<>> dmd,ssm 0CnvFigure 2.4: State plane representation of several natural trajectories for different md values and sameinitial conditions.Several natural trajectories for the same set of initial conditions(vCn(0), i′dn(0))and different modu-lation indexes are conceptually illustrated in Fig. 2.4. The operating point describes circular paths withdifferent radii, centers and angular speeds defined by md . These trajectories are valid for the whole rangeof control inputs, including the saturation limits, [mdsat−, mdsat+]. Therefore, the saturation natural tra-jectories λsat− and λsat+ set the boundaries of the region at which the centers of the circles can be locatedover the load line. Since md can take positive or negative values, the direction of rotation of the circlesin the state plane is defined by the sign of md . For md > 0, the operating point rotates in the clockwisedirection, while for md < 0 it rotates counterclockwise. In the particular case of md = 0, the center of thecircle is located at the infinity and the trajectory becomes a straight line perpendicular to the load line asshown in Fig. 2.4.Conceptual natural trajectories for the same set of initial conditions and md , for different ion values,are illustrated in Fig. 2.5. According to (2.15), the slope of λLL changes according to the operatingconditions Vsdn and ion. Since the centers of the natural circular surfaces are located over the load line,the centers displace vertically for different DC-bus currents and horizontally for variations in normalizedgrid voltage, as the slope of λLL changes. For ion > 0, the center is located above the zero current axis,23dn’iCnvdn’i (0)(0)CnvicicvcLL3λLL1λLL2λmdλmdλmdλmdωmdωmdω=0oni>0oni<0oniFigure 2.5: State plane representation of several natural trajectories for different DC-bus currentvalues and same initial conditions with constant md .Cvdimdλdn’imdωmdcvcicCnvmdωpi2=mdTmdrFigure 2.6: Experimental natural circular trajectories of the three-phase converter in the syn-chronous reference frame.while it moves to the negative quadrant for ion < 0 as illustrated in Fig. 2.5.The experimental results in Fig. 2.6 show the dynamic evolution of the state variables in both timeand state plane domains for a real hardware-implemented three-phase converter. Further details of theexperimental setup are described in Appendix B. As predicted by the previous analysis, the capacitorvoltage and d-axis current show sinusoidal evolution in time that translates into a well-defined circulartrajectory in the state plane. The large excursions in the state variables, 80 V and 9 A for the voltage24and current respectively, illustrate the large-signal nature of the derived model. The natural resonanceperiod Tmd can be calculated from (2.21) and applying the proper denormalization according to (2.8)and (2.9). The value of the experimental parameters are 3 mH and 220 µF for the grid inductor andDC-bus capacitor respectively, which results in a theoretical Tmd of 3.32 ms. As shown, the experimentalTmd is obtained as 3.4 ms that closely matches the predicted value by the normalized decoupled model.2.4 SummaryThis chapter introduces the normalized decoupled model of the three-phase PWM converter in the syn-chronous reference frame. The developed large-signal model provides a powerful, intuitive and compre-hensive graphical tool to illustrate the natural dynamic behavior of the system.To derive the model, the differential equations describing the three-phase dynamic behavior in thesynchronous reference frame were analyzed, taking into account the implementation of a feed-forwardnonlinear term. By removing the cross-coupling between the d and q-axis, the converter dynamics weresimplified and represented as a dual SISO system. The decoupled model was represented in a normalizeddomain, which ensured generality and independence from the reactive component values, voltage levelsand power rating. The natural trajectories of the converter were then obtained through dynamic analysisin the state plane. These solutions represent the evolution of the operating point as circular trajectorieswith well-defined characteristics, accurately describing the large-signal behavior of the system.The methodology and analysis used to derive the normalized decoupled model and the concept ofnatural trajectories are applied in subsequent chapters to examine the dynamic behavior of the three-phase converter. This model serves as the foundation for characterizing the dynamic performance anddeveloping high-performance state plane controllers for the three-phase PWM converter.25Chapter 3Theoretical Minimum Recovery TimeSolutions of the Three-Phase Converter inthe Synchronous Reference FrameThis chapter presents a comprehensive analysis of the theoretical minimum recovery time transients forthree-phase converters controlled in the synchronous reference frame. The previously derived normalizeddecoupled model is utilized in combination with the Time Optimal Control principle to determine the the-oretical minimum time solutions. The results are represented by a combination of well-defined geometrictrajectories in the state plane, providing a simple and intuitive graphical representation for analyzing anddetermining closed-form expressions for various relevant transient parameters.These ideal minimum time responses not only provide the fastest possible dynamic responses forthe three-phase converter but also offer a useful benchmark for evaluating and comparing the dynamicperformance of the system. By comparing control designs and transients to the optimal time solutions,objective assessments of the dynamic performance can be made. Furthermore, as the minimum timetransients cannot be surpassed, this approach can also provide insight into the feasibility of improving thePortions of this chapter have been published in:• F. Degioanni, I. G. Zurbriggen and M. Ordonez, ”Minimum Recovery Time Transients for Three-Phase Converters inthe Synchronous Reference Frame,” in IEEE Transactions on Power Electronics, vol. 37, no. 2, pp. 1579-1591, Feb.2022.• F. Degioanni, I. G. Zurbriggen and M. Ordonez, ”Characterization of the Minimum Recovery Time Transients for Three-Phase PWM Rectifiers,” 2021 IEEE Applied Power Electronics Conference and Exposition (APEC), 2021, pp. 2503-2509.26-satrsatω+satr+satλ-satλsatω(0)CnvLLλCnvvsatcisatcisat-cvsat-cdn’idn’i (0)Cnvntnt(0)Cnvvsatcisatcisat-cvsat-cdn’i (0)dn’isatωpi2=satTFigure 3.1: Natural saturation trajectories λsat+ and λsat− in both time domain and state plane rep-resentation for the same set of initial conditions.dynamic transients and achieving the desired dynamic performance even before designing the controller.3.1 Time Optimal Control PrincipleThe Time Optimal Control Principle states that the operating point of a system with limited capabilitiesto absorb (or release) energy can be moved from one initial state to a final state in the shortest possibletime by taking (or releasing) the maximum available amount of power during the transient until the newequilibrium condition is reached [129]. In other words, the minimum time solutions are achieved bymaximizing the control effort through the implementation of a ’bang-bang’ type function that switchesthe control variable between its lower and upper bounds [130].According to the derived normalized decoupled model of the three-phase converter, the dynamic be-havior of the active power and DC-bus voltage is determined by the d-axis SISO system (2.10) and (2.11).Therefore, the evolution of the state variables vCn and idn is commanded by the control input md , whichcan take any value in the interval within [mdsat−,mdsat+]. By applying the concept of Time Optimal Con-trol, minimum time solutions are obtained by solving the transients using only a combination of saturationlimits of the variable md . Consequently, theoretical time optimal responses are achieved by analyzing thestate variables and system dynamics under the saturation states defined by mdsat− and mdsat+.273.2 Three-Phase Converter Saturation TrajectoriesAs discussed in Chapter 2, the dynamic evolution of the state variables can be represented in the stateplane, providing a graphical tool to study the system behavior. The operating point describes circulartrajectories moving at angular speed ωmd , with the center located at (cvmd , cimd) and radius rmd . There-fore, the saturation trajectories can be derived by replacing md by the saturation limits mdsat+ and mdsat−into (2.22):λmdsat+ : (vCn− cvsat)2+(i′on− cisat)2 = r2sat+ (3.1)λmdsat− : (vCn+ cvsat)2+(i′on+ cisat)2 = r2sat− (3.2)The evolution of the operating point for both saturation states for the same set of initial conditions isconceptually illustrated in Fig. 3.1. As shown, the state variables describe sinusoidal waveforms in thetime domain that can be represented by circles in the state plane. In most cases, mdsat− = −mdsat+, andconsequently, the operating point of λsat+ rotates in clockwise direction, while λsat− rotates in anticlock-wise direction at constant angular frequency:ωsat = π√3√2|mdsat±| (3.3)Moreover, the centers of the circles are located over the load line at the position given by:cvsat = 2Vsdn|mdsat±|cisat = 2i′on|mdsat±|(3.4)While the radii of the circles are defined by:r2sat+ = (vCn(0)− cvsat)2+(i′on(0)− cisat)2r2sat− = (vCn(0)+ cvsat)2+(i′on(0)+ cisat)2(3.5)The graphical representation of the saturation trajectories λsat− and λsat+ provides a simple interpre-tation of the system dynamics, which is used in the following sections to define, analyze and characterizedifferent minimum time solutions for the three-phase converter.28sat-dm=dm+satc-satc -satcsat+dm=dmTarget+tsatλ-tsatλ-0satλCnv Cnvdn’i dn’i, ]]Cn 1v (0) dnt’i (0) 1, ]]Cntv dnt’i, ]]Cn 2v (0) dnt’i (0) 2+satc+tsatλ-tsatλ-tsatλ+0satλ+0satλ+0satλdnt’i (0) 1Cn 1v (0)dnt’i (0) 2Cn 2v (0)Cnvdn’idmntntntCntvdnt’i+dsatm-dsatm-0satλ-0satλ(a) (b)(c)Figure 3.2: Conceptual representation of the theoretical minimum time solutions. (a) The stateplane is divided into two regions according to the saturation trajectories. (b) Two minimumtime responses in the state plane. (c) Tow minimum time responses in the time domain.3.3 Minimum Recovery Time Solutions for the Three-Phase ConverterIn order to achieve minimum time dynamic responses for the three-phase converter, the transients mustbe solved by employing only a combination of saturation states defined by mdsat− and mdsat+. The theo-retical time optimal solutions have an intuitive geometric interpretation in the state plane as conceptuallyillustrated in Fig. 3.2.For a given point in the plane, there is only one saturation trajectory (λsat0±) that guides the operatingpoint towards the corresponding circle (λsatt±) that intercepts the desired target. Once this surface is29reached, the operating point is guided toward the target by using the opposite saturation control limit.The proper saturation state to guide the operating point can be defined according to the position of thestate variables on the state plane as shown in Fig. 3.2 (a). By dividing the state plane into two majorregions delimited by the saturation trajectories λsatt+ and λsatt−, minimum time solutions are achieved byusing a combination of two complementary saturation states.Two conceptual minimum time responses for two different initial states are illustrated in both, geo-metric and time domains, in Fig. 3.2 (b) and (c) respectively. The saturation state is selected accordingto the position of the operating point in the state plane to follow the corresponding trajectory λsat0± thatmoves the operating point in direction of λsatt±. At the intersection of the two trajectories, md is selectedto be the opposite saturation state to guide the operating point directly toward the target.In this way, by using the graphical representation of the converter trajectories, the theoretical time-optimal solutions can be analyzed and completely characterized straightforwardly.3.4 Characterization of the Minimum Recovery Time SolutionsThe previous analysis and state plane representation are applied to define and characterize the ideal min-imum recovery time responses for different dynamic transients. The derived solutions are described inthe normalized domain providing general expressions valid for any combination of parameters. There-fore, the converter’s dynamics can be evaluated by either normalizing the converter’s response, or bydenormalizing the ideal solutions.3.4.1 Start-Up TransientThe theoretical time optimal solution for a start-up transient is illustrated in both time and geometricdomains in Fig. 3.3. In the beginning, the operating point is located at ➊ with a pre-charged DC-busvoltage vCn(0). In order to achieve a time optimal response, md is first set to mdsat− to guide the operatingpoint through the trajectory λsat0− until it reaches ➋. At this point, md is switched to mdsat+ to followthe trajectory λsatt+ that intercepts the target operating point located at ➌. Once the target is reached, theconverter operates in its steady-state condition.By analyzing Fig. 3.3 (a), the coordinates of ➋ are given by:vCn,min =δr4cvsat(3.6)30CnvCntvdmd,ssm-n,satt +n,satt-dsatm+dsatmntntntn,start-upt(0)Cnv231vsat-c Cntv Cnv+satαvsatcsatωsatω-satβ(0)Cnv123Start-UpMinimum Time Trajectory+satr-satrCn,minvtd0d-0satλ +tsatλdn’idn,max’idn,x’idn’idn,x’iFigure 3.3: Theoretical minimum recovery time start-up transient in state plane and time represen-tation.i′dn,x =√r2sat−−116c2vsat(δr +4c2vsat)2(3.7)with δr defined as:δr = r2sat−− r2sat+ (3.8)with the centers and radius given by (3.4) and (4.8) respectively.During the transient, the operating point covers the angles βsat− and αsat+:βsat− = acos(1− d2t2r2sat−)αsat+ = acos(1− d202r2sat+) (3.9)31where d0 and dt are the distances between the points ➊-➋ and ➋-➌ respectively:d20 =(vCn(0)− vCn,min)2+(i′dn,x)2d2t =(vCnt − vCn,min)2+(i′dn,x)2 (3.10)Since the operating point moves at constant angular speed, the time intervals can be read as anglesin the geometric domain. According to (3.3), the operating point moves at angular speed ωsat and thetheoretical normalized minimum time for a start-up transient is obtained as:tn,start−up =βsat−+αsat+ωsat(3.11)In this way, the theoretical minimum start-up time can be obtained from (3.6)-(3.11) by replacing withthe corresponding initial voltage vCn(0).The derived expressions for a start-up transient are also applicable for a change in the reference voltageby replacing vCn(0) by vCnre f 1 and vCnt by vCnre f 2 for a reference voltage increase, and by replacing vCn(0)by vCnre f 2 and vCnt by vCnre f 1 for the reference voltage decrease.3.4.2 DC-bus Current Step-Up TransientThe theoretical time optimal solution for a DC-bus current step-up is illustrated in Fig. 3.4. In thistransient, the final current in the DC-bus is larger than its initial value (ion(0) < iont); therefore, theoperating point is initially at the steady-state condition ➊ and it moves to ➌ after the step is applied.To reach the target in minimum time, the trajectory λsat0− is followed by setting md = mdsat−. At theintersection point ➋, the control input is switched to mdsat+ and the trajectory λsatt+ guides the operatingpoint towards the new steady-state position at ➌.By analyzing Fig. 3.4 (a), the coordinates of ➋ are:vCn,min =cvsat4δrcvi− cisat√cvi√√√√r2sat−−(δr +4cvi)216cvi(3.12)i′dnx =cisat4δrcvi+cvsat√cvi√√√√r2sat−−(δr +4cvi)216cvi(3.13)with cvi = c2vsat + c2isat .32Cnv231Cntvdmd,ssmCn,minv-dsatm+dsatmntntntn,supt-n,satt +n,sattCntv12-satβ+satαStep-UpMinimum Time Trajectory(0)dni’Cn,minvCnv3satωsatωvsatcisat-cvsat-cisatc+satr-satrtd0d-0satλ+tsatλdn’idn,max’idnt’i(0)dni’dn’idnt’iFigure 3.4: Theoretical minimum recovery time transient for a DC-bus current step-up in state planeand time representation.The minimum recovery time transient can be obtained by analyzing the angles αsat+ and βsat− coveredby the operating point during the transitory:αsat+ = acos(1− d2t2r2sat+)βsat− = acos(1− d202r2sat−) (3.14)where d0 and dt are the distances between the points ➊-➋ and ➋-➌ respectively.Since the operating point moves with constant speed given by (3.3), the minimum DC-bus currentstep-up transient time is:tn,sup =βsat−+αsat+ωsat(3.15)The theoretical minimum DC-bus current step-up transient times can be obtained by replacing the33d,ssm-dsatm+dsatmn,sdwtCnv23ntntnt1CntvCn,maxv-n,satt+n,sattsatωsatωtLLλ+satα123Step-DownMinimum TimeTrajectory+satr-satr0dtdvsatcvsat-cisatcCnvisat-cCntv-tsatλ+0satλ(0)dni’dn’idn,min’idnt’i(0)dni’dn’idn,min’idnt’iFigure 3.5: Theoretical minimum recovery time transient for a DC-bus current step-down in stateplane and time representation.corresponding initial ion(0) and final iont currents in (3.12)-(3.15) as illustrated in Fig. 3.6 (a).3.4.3 DC-bus Current Step-Down TransientThe time optimal solution for a DC-bus current step-down transient is illustrated in Fig. 3.5. In thistransient, the initial current in the DC-bus is larger than its final value (ion(0) > iont); therefore, theoperating point is initially located at ➊ and it is moved to ➌ after the current step. To achieve theshortest recovery time, the operating point follows the trajectory λsat0+ covering the angle αsat+. At ➋,md is switched to mdsat− and the trajectory λsatt− guides the operating point towards the new equilibriumcondition located at ➌.By analyzing Fig. 3.5, the following coordinates can be obtained:vCn,max = cvsat + rsat+ (3.16)34oniDC-Bus Current oniDC-Bus Current00 0.05 0.1 0.15 0.2 0.250.10.20.30.40.5(0) = 0oni(0) = 0.05oni(0) = 0.1oni(0) = 0.2oni(0) = 0.15oniCurrent Step-Upn,supt00.20.40.60.80 0.05 0.1 0.15 0.2 0.25(0) = 0.25oni(0) = 0.2oni(0) = 0.15oni(0) = 0.05oni(0) = 0.1oniCurrent Step-Downn,sdwt(a) (b)Figure 3.6: Theoretical minimum recovery times for a DC-bus current step transient for differentinitial values: (a) Step-up. (b) Step-down.vCnx =cvsat4δrcvi+cisat√cvi√√√√r2sat−−(δr +4cvi)216cvi(3.17)i′dn,min =cisat4δrcvi− cvsat√cvi√√√√r2sat−−(δr +4cvi)216cvi(3.18)The transient time is obtained from the angles covered by the operating point during the transient;therefore, the minimum time for a stet-down transient is given by:tn,sdw =βsat−+αsat+ωsat(3.19)The theoretical minimum DC-bus current step-down transient times can be obtained by replacing thecorresponding initial ion(0) and final iont currents in (3.16)-(3.19) as illustrated in Fig. 3.6 (b).The derived expressions (3.6)-(3.19) provide a complete characterization of different dynamic tran-sients for the three-phase converter. In this way, for a given set of initial and final conditions, the theoret-ical minimum recovery times to solve the transient can be calculated. These solutions not only providethe fastest dynamic responses that the converter is able to achieve but they also can be used as a simpleand useful tool for an absolute reference for assessing the dynamic performance of the system.3.5 Minimum Time Solutions Considering Different System ConstraintsIn this section, the examination of the influence of various control design constraints, including the mod-ulation technique and protective limits, is conducted.3500.20.40.60.800 0.05 0.1 0.15 0.2 0.250.10.20.30.40.50 0.05 0.1 0.15 0.2 0.25n,supt n,sdwtCurrent Step-Up Current Step-Down(a) (b)(0) = 0oni(0) = 0.125oni = 1dsatm = 1.15dsatm(0) = 0oni(0) = 0.125oni = 1dsatm = 1.15dsatmoniDC-Bus Current oniDC-Bus CurrentFigure 3.7: Theoretical minimum recovery times for a DC-bus current step with different saturationlimits for different initial values: (a) Step-up. (b) Step-down.3.5.1 Effect of the Modulation Technique in the Minimum Time SolutionsMinimum time solutions are obtained by solving the transient using only a combination of saturationlimits of md . These conditions mdsat+ and mdsat− represent the maximum and minimum possible valuesthat the modulation index can take in the SRF. The value of these limits will determine the maximumangular speed at which the operating point is going to move; and consequently, it defines the total transienttime.In practice, the maximum possible value of md is given by the modulation strategy and it needsto be considered when evaluating the theoretical minimum time solutions. In the conventional SinePulse Width Modulation (SPWM), md is limited between [−1;1]; therefore, the saturation conditionsare defined as mdsat+ = 1 and mdsat− = −1. However, advanced modulation techniques, such as ThirdHarmonic Injection (THPWM) and Space-Vector Modulation (SVPWM), among others, increase thelinear range of the converter by 15%, which extends the limits of the modulation index up to[− 2√3; 2√3],defining mdsat+ = 1.154 and mdsat− =−1.154.The theoretical minimum transient times for a DC-bus current step-up and down are provided inFig. 3.7 for both sets of saturation conditions |mdsat±| = 1 and |mdsat±| = 1.15. The obtained resultsreveal that modulation techniques with larger modulation index ranges, such as THPWM and SVPWM,enable to obtain of faster theoretical solutions due to the faster angular speed achieved by the operatingpoint during the saturation states.36αStep-UpMinimum Time Trajectory WithCurrent Limit1234Cntv Cnvvsatcisat-cvsat-cisatc-0satλ+tsatλβCnv(0)dn’i(0)dn’i231Cntvntnt-n,satt +n,satt4limn,suptn,limtdn’idnt’idn,lim’idn’idn,lim’iFigure 3.8: Theoretical minimum recovery time transient for a DC-bus current step-up with d-axiscurrent limit in state plane and time representation.3.5.2 Effect of the Protection Limits in the Minimum Time SolutionsIncluding different dynamic constraints, such as current and voltage protections, affect the dynamic re-sponse of the converter and; therefore, the derivation of the minimum recovery time solutions.The ideal minimum time transient for a DC-bus current step-up is modified as shown in Fig. 3.8 as-suming a maximum current limit idn,lim at which the converter is able to operate. Initially, the operatingpoint follows the saturation trajectory λsat0−; however, at ➋, the current hits its limit and the operat-ing point cannot continue moving along the saturation trajectory λsat0−. Therefore, the current remainsconstant at i′dn,lim until the operating point reaches the position ➌. At this point, the current state is be-low its limit and the operating point is guided through the saturation trajectory λsatt+ to achieve the newsteady-state condition ➍.By analyzing Fig. 3.8, the transient can be divided into three regions: from ➊ to ➋, from ➋ to ➌and from ➌ to ➍. For the initial and final intervals, the operating point follows well-defined circular37Table 3.1: Parameters of the three converters used in simulation and experimentsParameter Converter I Converter II Converter IIIvCre f 400 V 400 V 400 Vvsabc 110 Vrms 110 Vrms 110 VrmsL 3 mH 3 mH 5 mHC 220 µF 500 µF 500 µFT0 5.1 ms 7.7 ms 10 msZ0 3.69 Ω 2.45 Ω 3.16 Ωinorm 108 A 163 A 127 Atrajectories, describing the angles α and β respectively. Therefore, these angles can be obtained byapplying geometric analysis and the corresponding transient times (tn,sat− and tn,sat+) can be derived.However, during the time interval between➋ and➌, the current hits its protection limit and it remainsconstant at idn,lim. Therefore, the dynamic equations of the system under these conditions are given:12πidn,limdtn= 0 =Vsdn−md vCn2 (3.20)12πdvCndtn=34md idn,lim− ion (3.21)By replacing (3.20) into (3.21) and solving, the transient time for the converter operating at constantcurrent is derived as:tn,lim =(vCn,2− vCn,3)2πion+23idn,limi2onVsdn ln(1+2ion(vCn,2− vCn,3)2ionvCn,3−3idn,limVsdn)(3.22)where vCn,2 and vCn,3 are the capacitor voltage values at the points ➋ and ➌ respectively, which can beobtained by applying geometric analysis to Fig. 3.8.Finally, the theoretical minimum transient time for a DC-bus current step-up with the current limit is:tn,suplim =αωsat+ tn,lim+βωsat(3.23)3.6 Simulation and Experimental Validation of the Minimum RecoveryTime TransientsThe minimum recovery time responses are validated by Hardware-in-the-Loop (HIL) simulation and ex-perimental results. In both simulation and experimental tests, the SPWM technique (mdsat = 1) was em-38Table 3.2: Normalized minimum recovery parametersTransient ParameterTheoreticalSolutionsNormalized MeasurementsConverter I Converter II Converter IIIStart-Up tn,start−up 0.54 0.55 0.56 0.56vCn,min 0.73 0.7 0.72 0.73Current Step-Up idn,max 0.50 0.54 0.52 0.52tn,sup 0.47 0.49 0.48 0.5vCn,max 1.34 1.37 1.37 1.37Current Step-Down idn,min -0.44 -0.45 -0.45 -0.44tn,sdw 0.74 0.745 0.74 0.75ployed. Moreover, the experimental dynamic responses of two conventional PI controllers with differentparameter designs are compared with the theoretical minimum time solutions to illustrate the applicationof the theoretical limits as an objective reference for assessing the system’s performance.3.6.1 Hardware-in-the-Loop Simulation ResultsThe power converter is simulated in real-time using a HIL PLECS RT-Box, whereas the control and PWMsignal generation are also executed in real-time by an industry-standard 32-bit floating point TI-C2000microcontroller. Three different converters with parameters listed in Table 3.1 are implemented in theHIL platform to evaluate the theoretical minimum time solutions and derived expressions in Section 3.4.The power circuit implemented in HIL platforms is built using ideal physical models of the different com-ponents, including the switches. To compare the converters under the same conditions, equal normalizedtransients are applied to each converter. All the relevant transient parameters are indicated in the resultcaptures and the normalized values are highlighted in brackets.The obtained minimum time responses for a start-up transient and normalized DC-bus current stepof 0.25 p.u. for the three-phase converters are shown in Fig. 3.9. To achieve time optimal responses,combinations of saturation trajectories are implemented to solve the transient according to the conceptsof Time-Optimal Control. The three converters show similar normalized transient parameters such asovershoot/undershoot values and recovery times. The theoretical dynamic performance limits obtainedfrom (3.6)-(3.19) are summarized and compared with the results for each converter in Table 3.2. For allcases, the theoretical minimum time parameters closely match the obtained results.Although the obtained normalized minimum times shown in Table 3.2 are the same for the three con-verters, Converter I achieves faster minimum physical times than Converter II and III, whereas ConverterIII shows the slowest responses. Since the normalized times represent a fraction of the resonant period39(a)(b) (e)(d)+dsatm-dsatmCvd’iCvdidm5.6 ms (0.56)17.6 A(0.16)120 V(0.3)+dsatm-dsatmCvd’iCvdidm4.3 ms (0.56)17.6 A(0.16)120 V(0.3)+dsatm-dsatmCvd’iCvdidm2.8 ms (0.55)17.6 A(0.16)120 V(0.3)Cvd’iCvdidmoiai bi ci2.5 ms (0.49)120 V (0.3)3.8 ms (0.745)48.6 A (0.45)150 V (0.37)27 A (0.25)59 A (0.54)+dsatm-dsatmCvd’iCvdidmoiai bi ci3.7 ms (0.48)5.7 ms (0.74)73 A (0.45)150 V (0.37)40 A (0.25)85 A (0.52)+dsatm-dsatmCvd’iCvdidmoiai bi ci5 ms (0.5)7.5 ms (0.75)56 A (0.44)150 V (0.37)31 A (0.25)68 A (0.52)+dsatm-dsatmStart-Up Minimum Time Transients: Converter IStart-Up Minimum Time Transients: Converter IILoad Step Minimum Time Transients: Converter ILoad Step Minimum Time Transients: Converter II(c) (f)Start-Up Minimum Time Transients: Converter III Load Step Minimum Time Transients: Converter III115 V (0.28)110 V (0.27)Figure 3.9: HIL results of the minimum recovery time transients: (a) Start-up transient for ConverterI. (b) Start-up transient for Converter II. (c) Start-up transient for Converter III. (d) DC-buscurrent step transient for Converter I. (e) DC-bus current step transient for Converter II. (f)DC-bus current step transient for Converter III.T0, converters with smaller filter values achieve faster theoretical minimum transient responses. SinceT0I < T0II < T0III , Converter I achieves the fastest minimum recovery times under the same normalizedconditions.Since the switching frequency in three-phase converters is usually much larger than the natural LC res-onant period of the reactive elements, the control delay does not show a significant effect in the minimumtime transients and it can be neglected in the definition of the ideal solutions.40Cvdidm2 ms (0.39)2 ms (0.39)100 V (0.25)DC-Bus Voltage Reference Step Minimum Time TransientsCrefv= 500 V2Crefv2Crefv1Crefv= 400 V 1CrefvCvd’i+dsatm-dsatmFigure 3.10: HIL results of minimum recovery time transient for a DC-bus voltage reference step.DC-bus Voltage Reference Step Minimum Time TransientThe minimum recovery time transients for a DC-bus voltage reference for Converter I in Table 3.1 areshown in Fig. 3.10. The voltage reference vCre f is first changed from 400 V to 500 V and then it ischanged back to 400 V , which results in ∆vCre f = 100 V (0.25 p.u.).A combination of saturation control states is employed to solve the transient in minimum recoverytime, following similar trajectories to the start-up case. Therefore, the derived expressions for a start-uptransient can be used to estimate the transient time by replacing vCn(0) by vCnre f 1 and vCnt by vCnre f 2 for avoltage reference step-up, and vCn(0) by vCnre f 2 and vCnt by vCnre f 1 for a reference step-down.Both transients show the same recovery time of 2 ms (0.39 p.u.), which closely matches the theoreticalvalue of 1.94 ms (0.38 p.u.) calculated using (3.11). Moreover, the obtained responses are symmetric tothe horizontal axis, with almost zero DC-bus voltage overshoot and undershoot values.Minimum Recovery Time Transients Under Non-ideal Grid ConditionsThe minimum recovery time transients for a DC-bus current step with nonideal grid conditions for Con-verter I in Table 3.1 are presented in Fig. 3.11. The figure illustrates the time optimal responses underthree different grid conditions: a) grid voltages with high harmonic distortion (5% THD), b) a 15% gridvoltage imbalance, and c) grid voltages with both, harmonic distortion and imbalance. The results showthat the different grid condition scenarios do not have a significant impact on the dynamic response of thesystem. The recovery times are similar to the obtained values for the ideal case listed in Table. 3.2, evenunder the worst-case condition of unbalanced and distorted grid voltages. The main impact of nonideal41120 V (0.3)140 V (0.37)2.5 ms (0.49)3.7 ms (0.73)27 A (0.25)Cvdiai bi ciav cvbvCvd’i(a)Distorted Grid Voltages110 V (0.27)140 V (0.37)2.5 ms (0.49)3.7ms (0.73)28 A (0.26)Cvdiai bi ciav cvbvCvd’i(b)Unbalanced Grid Voltages(c)100 V (0.25)130 V (0.33)2.5 ms (0.49)4 ms (0.78)26 A (0.24)Cvdiai bi ciav cvbvCvd’iDistorted and Unbalanced Grid VoltagesFigure 3.11: Minimum recovery time transients for a DC-bus current step with nonideal grid con-ditions: (a) Grid voltages with harmonic distortion. (b) Grid voltages with 15% imbalance.(c) Grid voltages with harmonic distortion and imbalance.grid conditions is observed in the steady-state performance with oscillations in the DC-bus and currentdistortion.42Cvdidmai bi ci2.6 ms (0.5)70 V (0.35)29.7 A (0.55)22 A (0.4)13.5 A (0.25)4 ms (0.75)75 V (0.37)oiExperimental Minimum Recovery Time Current Step Transient+dsatm-dsatmd’iCvFigure 3.12: Experimental minimum recovery time transient for a DC-bus current step. The ob-tained results closely match the theoretical response.3.6.2 Experimental ResultsA scaled experimental version of Converter I in Table 3.1 is implemented and tested in the laboratoryprototype. The experimental prototype has the same parameter combination as the simulated Converter I;however, the power and voltage levels are scaled down to half. Therefore, the specifications of the exper-imental platform are: vCre f = 200 V, L = 3 mH, C = 220µF , T0 = 5.1 ms, Z0 = 3.69 Ω and inorm = 54A. The DC-bus voltage and converter’s power are 200 V and 2.7 kW respectively for the experimentalplatform. The converter is tested under the same normalized conditions as the HIL platform to evaluatethe practical implementation of the minimum time solutions. Further details of the experimental setup aredescribed in Appendix B. In addition, the transient responses for two PI controllers with different com-pensator parameters are compared with the theoretical minimum time responses to illustrate the dynamicperformance evaluation of the system.The experimental minimum recovery time transients in time and state plane representation are pre-sented in Fig. 3.13 The capture illustrates the employed experimental saturation trajectories to solve astart-up and a DC-bus current step-up and down transients. As shown, the obtained waveforms closelymatch the ideal responses previously discussed.The experimental results in Fig. 3.12 indicate all the relevant transient parameters for a DC-bus currentstep. The obtained results are similar to the theoretical values in Table 3.2. It is important to notice thatthe developed limits are theoretical and cannot be reached in practice; however, the experimental resultsare close enough to the ideal responses to validate the analysis.43didmCv 111333222132-satλ-satλ+satλ+satλCvd’isupt-satt +sattstart-upt-satt +satt+dsatm-dsatm(b)didmCv 3213-satλ+satλCvd’isdwt-satt+satt+dsatm-dsatm12(c)Experimental Current Step-UpExperimental Current Step-DownExeprimentalStart-Up(a)Cvdidm +dsatm-dsatmCvd’iFigure 3.13: Experimental minimum recovery time transients: (a) Start-up. (b) DC-bus currentstep-up. (c) DC-bus current step-down.44Experimental Dynamic Transients Benchmark PI Controller I13.5 A (0.25)sdwt x3.4x18 ms (1.5)13 ms (2.54)x3.2x1Cvd’idiCvoisuptExperimental Dynamic Transients Benchmark PI Controller IIdiCvoi13.5 A (0.25)supt sdwt x4.5x115 ms (2.94)17 ms (2.54)x6.2x1Cvd’iFigure 3.14: The experimental transients for two different dual-loop PI controller designs can becompared with the theoretical minimum time solutions to evaluate their performance.Finally, to illustrate the use of the theoretical dynamic limits as an objective performance reference, thetransient responses of the system controlled by the conventional dual-loop approach with two differentsets of PI parameters are compared with the minimum time solutions. The transient responses for anormalized DC-bus current step of 0.25 p.u. are shown in Fig. 3.14. For these control designs, the step-up transient is solved 3.2 and 6.2 times slower than the theoretical minimum time for PI controllers I andII respectively. On the other hand, the step-down transient is solved 3.4 and 6.2 times slower than thetheoretical solutions. Since the comparison is performed in a normalized domain, the evaluation can beextended for systems with different controllers and converter parameters.The provided simulation and experimental results validate the proposed analysis and derived expres-45sions. The definition of the minimum time solutions provides a useful tool to evaluate the dynamicperformance of the three-phase converter controlled in the synchronous reference frame.3.7 SummaryIn this chapter, the theoretical minimum recovery time for the three-phase converter controlled in thesynchronous reference frame is studied and characterized. The ideal time-optimal responses provide anin-depth insight into the dynamical limitations of three-phase converters, serving as both a theoreticalguide for the fastest possible responses and a standard reference for evaluating the dynamic performanceobjectively.To derive the theoretical minimum recovery time, the concepts of Time-Optimal Control and thepreviously developed normalized decoupled model of the converter were applied. The solution to thedynamic transients under saturation conditions was found by combining geometric trajectories in thestate plane. These trajectories, which are circular in nature, provide a comprehensive characterizationof different minimum time transient parameters, such as overshoot, undershoot, and recovery time. Byconducting geometric analysis on these responses, closed-form expressions for the transient parameterswere derived.The validity of the minimum time solutions was verified through Hardware-in-the-Loop and exper-imental platforms, which demonstrated that the near-time optimal responses obtained are similar to thetheoretical solutions for various converters with different parameter combinations.46Chapter 4State Plane Control of Three-Phase PWMConverters with Active LoadsThis chapter delves into the exploration of utilizing the concept of natural trajectories for enhancing thestability and dynamic performance of the DC-bus in three-phase converters feeding active loads. Thepresence of load-side converters that regulate their output voltage and/or current tightly may result inthe load behaving as a constant power load, which can negatively impact system stability and lead toDC-bus collapse and failure. To address this issue, the behavior of a three-phase converter system with aconstant power load is analyzed and a state plane control law is developed to mitigate the adverse effectsof constant power loads. The proposed control law operates on the saturation trajectories of the converterand aims to achieve near-time optimal responses, thus ensuring resiliency under large constant powerload steps and fast voltage regulation. The fast dynamic nature of the state plane controller enables thesystem to remain stable and responsive even under challenging operating conditions.The introduced approach provides a solution to mitigate the negative impact of constant power loadson three-phase converters. A robust and efficient control strategy for the DC-bus is crucial to enhancingthe performance and longevity of these systems, while also minimizing the risks and consequences ofpotential system failures.Portions of this chapter have been published in:• F. Degioanni, I. G. Zurbriggen and M. Ordonez, ”Improved Control for Three-Phase Rectifiers with Constant PowerLoads,” 2023 IEEE Applied Power Electronics Conference and Exposition (APEC), 2023, pp. 2775-2779.47∆v∆icplP=cplvcpli∆v∆i <0cplvcplicplV( ), cplIcpli˜ cplv˜cplPcplV2cplP2cplV=cplP- cpl2VcplPcplV2cplv˜cpli˜Linearized CPLcplPcplvcplicplvCPL(a)(b) (c)Figure 4.1: CPL model and characteristics: (a) CPL model. (b) Linearized equivalent circuit modelof a CPL around one equilibrium point. (c) Voltage-current characteristics of a CPL.4.1 Constant Power Load CharacteristicsThe presence of active loads in the DC-bus can be modeled as an equivalent constant power load at theirinput terminals. A CPL is a one-port device absorbing constant power Pcpl , which current at its terminalsis given by:icpl =Pcplvcpl(4.1)The nonlinear relationship (4.1) is illustrated in Fig. 4.1 and it represents the voltage-current charac-teristics of an ideal CPL.The dynamic characteristics of a CPL can be analyzed from the rate of change in the load current:∂ icpl∂vcpl=−Pcplv2cpl(4.2)The linear equivalent model for a CPL at a specific operating point (Vcpl, Icpl) can be derived fromequation (4.2) as follows:i˜cpl = 2PcplVcpl− PcplV 2cplv˜cpl (4.3)where i˜cpl and v˜cpl are the linearized current and voltage at the load terminals.According to (4.3) the voltage-current curve of a CPL can be approximated by a straight line thatis tangent to the curve at the operating point. The linearized model can be electrically representedby a negative resistor(rcpl =−V 2c /Pcpl)connected in parallel with an ideal controlled current load(Icpl = 2Pcpl/Vcpl)as shown in Fig. 4.1.48Imaginary AxisClosed-Loop Poles with Constant Power Load 20100-10-20Real Axis0-2.5-5-7.5-10-12.5-15cplnP IncreasescplnP = 0cplnP = 0.3Real AxisImaginary AxisImaginary Axis01-1-2-1-1.2-1.4-1.6-1.82Figure 4.2: Root-locus of the closed-loop linear control of the normalized three-phase converter fordifferent CPL conditions.Although the instantaneous value of the CPL impedance is always positive (Vcpl/Icpl > 0), the incre-mental impedance is always negative (∂vcpl/∂ icpl < 0). This has a destabilizing effect on the system,causing the current to increase when the voltage decreases and the current to decrease when the voltageincreases.By analyzing Fig. 4.1, an instantaneous power mismatch between the source and CPL would generatesudden voltage and current changes. Considering a disturbance around the equilibrium point (Vcpl, Icpl)that causes an increase of ∆i in the current and the load voltage is decreased by ∆v. At the new current, thesource voltage is larger than the load voltage; consequently, the current increases again and the operatingpoint continue drifting away. Finally, the voltage and current go to zero and infinity, respectively. Sim-ilarly, a decrease ∆i in current caused by a disturbance will make the load voltage lower than the sourcevoltage, resulting in more decrease in current, which will move the operating point away.According to the previous analysis, the CPL acts as positive feedback that introduces stability issuesthat may lead to the collapse of the DC-bus voltage and system failure if not compensated on time.490.50 1 1.5 2 2.5 30.210.500.30.400.050.10.150.20.250 0.12cpln,P∆3cpln,P∆4cpln,P∆1cpln,P∆DC-Bus Current Increases RapidilyDC-BusVoltageCollapsesCnv CnvonicplnPntFigure 4.3: Transient responses of the three-phase rectifier under different CPL steps. Although thesystem is stable around all loading conditions, the CPL characteristics may lead to the DC-buscollapse during transients.4.2 Analysis of the Negative Effects of a CPL connected to the DC-bus ofthe Three-Phase ConverterThe nonlinear characteristics of a CPL introduce dynamic challenges for the DC-bus voltage controllerof the three-phase converter. To analyze the negative impact of the CPL, the dynamic performance of thenormalized three-phase converter controlled with a conventional linear controller is studied. The DC-busvoltage is regulated using a dual-loop PI control architecture, which is designed following the proceduredescribed in Appendix A.The closed-loop pole map of the normalized three-phase converter loaded with an ideal CPL is pro-vided in Fig. 4.2. The root locus illustrates the eigenvalues of the system under different power loadconditions from 0 to 0.3 p.u. Each set of poles is obtained by linearizing the dynamic model of the con-verter with a CPL around the different operating points for each load. As shown, all the closed-loop polesare located on the left half side of the complex plane; consequently, the DC-bus si stable around eachpoint over the entire operating range.The dynamic response of the normalized three-phase converter loaded with an ideal CPL is provided50in Fig. 4.3. The simulation results show the DC-bus voltage, current and power transients for differentnormalized CPL steps ∆Pcpln. As can be observed, initially the converter operates in steady-state fordifferent loading conditions and the closed-cloop controller regulates the DC-bus voltage to its reference.This result matches the conclusion of the root locus and the system exhibits stable operation for all loadingconditions.Although the system is designed to be stable across all loading conditions, the nonlinear characteristicsof the CPL introduce dynamic stability issues that can be observed during transients. The dynamicperformance deteriorates for larger ∆Pcpl transients and the CPL destabilizing effect can be observed.Due to the negative incremental impedance behavior of the CPL, there is a maximum load step that theconverter can withstand before definitely failing. If the current injected by the converter to the DC-bus isnot fast enough to compensate for the rapid increase of the load current, the voltage continues decreasingleading to the final collapse of the DC-bus voltage as shown Fig. 4.3 for ∆Pcpl4.The previous analysis shows that the converter should be fast enough to recover the DC-bus voltagebefore the load current becomes too large. Therefore, faster transient responses of the source converterhelp to mitigate the negative incremental impedance dynamic behavior of the CPL.4.3 Control of the Three-Phase Converter with Active Loads Connectedto the DC-busThe negative dynamic effects of a CPL in the DC-bus of the three-phase converter are a consequenceof the load being faster than the source converter Since the response speed is defined by the controltechnique ruling the system, the implementation of fast dynamic controllers should help to mitigate thenegative CPL effects during transients.As discussed in Chapter 3, the use of the saturation states to solve the transient will bring the converterclose to its physical limits. Therefore, it is expected that using a control law that follows the theoreticalminimum recovery time trajectories would allow the converter to withstand larger CPL steps in the DC-bus and achieve improved dynamic performance over the entire operating range.4.3.1 State Plane Control LawThe converter saturation trajectories λsat+ and λsat− can be employed as switching surfaces that separatethe state plane region of maximum control effort from one of minimum control effort. Therefore, thecontrol law is defined based on the position of the operating point and the intersection of the switching51surfaces:Case I : (i′dn < i′dn,target)if : (σsat− < 0)then : md = mdsat+, else : md = mdsat−Case II : (i′dn > i′dn,target)if : (σsat+ > 0)then : md = mdsat−, else : md = mdsat+(4.4)The boundary conditions are given by the saturation trajectories that intercept the desired target point;therefore, their expressions are derived by replacing (2.16) and (2.17) into (3.1) and (3.2):σsat+ : λsatt+ : (vCn− cvsat)2+(i′on− cisat)2 = r2satt+ (4.5)σsat− : λsatt− : (vCn+ cvsat)2+(i′on+ cisat)2 = r2satt− (4.6)withcvsat = 2Vsdn|mdsat±| cisat = 2i′md|mdsat±| (4.7)andr2satt+ = (vCnt − cvsat)2+(i′dnt − cisat)2r2satt− = (vCnt + cvsat)2+(i′dnt + cisat)2(4.8)As can be observed, the control law employs two circular surfaces σsat+ and σsat− that define theboundary conditions to switch the state of md . Since only the saturation states are used, the converterachieves very fast recovery time transients, which helps to mitigate the CPL dynamic issues.4.3.2 Controller Operation with a Constant Power LoadWhen the three-phase converter is loaded with a CPL, the DC-bus current is a function of the capacitorvoltage according to the load characteristics (4.1). Therefore, the operating point trajectories deviate fromthe ideal circles described by (2.22). Despite these differences, it can be demonstrated that the controllaw (4.4) is always able to converge to the steady state condition by using the ideal saturation trajectories52dnt’idn’iCntv Cnvsat+λ’sat+,cplλ’sat+,cpl1λsat+,cpl2λsat+λ’’ sat+,cplλ’’+satσ-satσ-satλ+satλ ciciciLLλ-satσ-satλ+satλLLλdn’icicidnt’iCntv Cnvsat-λ’sat-,cplλ’sat-λ’’sat-,cplλ’’+satσsat-,cpl2λcisat-,cpl1λFigure 4.4: Conceptual representation of the proposed controller in the state plane. Although theCPL trajectories deviate from the ideal ones, the transient can be solved by using the controlsurfaces σsat− and σsat+.as control switching surfaces.To gain insight into the operating principle of the closed-loop controller when the converter is loadedwith a CPL, conceptual trajectories for different initial conditions ic are illustrated in Fig. 4.4. Eachsubfigure presents several state plane trajectories for the ideal case (λsat+ and λsat−) and the CPL case(λsat+,cpl and λsat−,cpl). Since the current in the CPL varies as a function of the voltage, the radii of theideal circles change along the trajectory and a difference can be observed between the ideal and the CPLtrajectories.The highlighted points ic1 and ic2 in Fig. 4.4 (a) and (b) correspond to different sets of initial conditionsfor which the CPL trajectories are sufficiently close to the ideal case. For these initial points, the controlleronly needs to perform one control action by setting the modulation index to guide the operating pointthrough an almost ideal trajectory during the transient time. However, the points ic3 represent the initialpoints for which the CPL trajectory deviates from the ideal case. In this situation, the circles that describethe target trajectories match different CPL trajectories. At the beginning of the transient, the operatingpoint follows the deviated trajectory until it reaches the boundary condition at ic2 and the structure isswitched according to the control law, reaching again the ideal target trajectory that allows the operatingpoint to arrive at the desired target.Although additional control actions may be required to solve the transient, compared with the idealcase, the control law is always able to select the correct structure to return to the converging area andreach the equilibrium point.53ntCnv0 0.5 1 1.5 2 2.5 30.6 0.7 0.8 0.9 1 1.1 1.2 1.300.50.2500.20.100.20.111.20.8dn’ iCnvdn’ i dn’ i0.40.60.50.20.3-0.100.1-satσ+satσ CCLCPLonPoniCPLCCL231Figure 4.5: Simulation results of the state plane controller for the normalized three-phase rectifier.Although the CPL trajectories deviate from the ideal case, the transients are solved by theproposed control law.4.4 State Plane Controller Performance Evaluation with a ConstantPower LoadIn order to validate the previous analysis and to evaluate the performance improvements of the proposedcontrol law when the three-phase converter is loaded with an active load, different simulation results areprovided in this section.The time domain and state plane representation of the normalized three-phase converter with an idealconstant power load is provided in Fig. 4.5. The simulation results compare the dynamic transient per-formance of the state plane controller for two types of loads connected to the DC-bus: constant powerload and constant current load (CCL). The use of a CCL represents the ideal case in which the converter’strajectories describe circular surfaces with a constant radius.For both types of loads, the converter initially operates in the steady state➊ under no load (ion = 0 p.u.and Pcpln = 0 p.u.). At the normalized time tn = 0.5 p.u., a corresponding power and current load step-upof 0.2 p.u. is applied to the converter. At the new loading condition, the target operating point moves to➋for both load types. According to (4.4), the controller sets md = mdsat− and the operating point followsthe corresponding trajectory λsat− until reaching the new boundary condition σsat+. At the intersection,md is changed to mdsat+ and the operating point is guided towards the new target point by following λsat+.Then, at tn = 2 p.u., a sudden normalized step-down of 0.1 p.u. CPL and CCL are applied respectively. In540.50 1 1.5 2 2.5 30.210.500.30.400.050.10.150.20.250 0.12cpln,P∆3cpln,P∆4cpln,P∆1cpln,P∆Cnv CnvonicplnPntDC-Bus Voltageis regulated forall the CPL steps0.50.310.90.80.7 0.9 1.1The negative effectsof the CPL are mitgated by the fast control dynamicsFigure 4.6: Transient responses of the state plane controller under different CPL steps. The DC-bus shows improved dynamic responses over the entire operating range, even when the largestpower step is applied.this case, the new target condition moves to the point➌ and the trajectory σsat− defines the new boundarycondition for this loading condition. After the step, the controller sets the modulation index to mdsat−and the trajectory λsat− is followed until reaching σsat−. At the intersection, the control law switches thecontrol input to guide the operating point towards the target by following the trajectory λsat−.As discussed in the previous section, the simulation results in Fig. 4.5 show some differences betweenthe trajectories of the CCL and CPL cases. However, the deviation on the CPL trajectories is not signif-icant and the controller solves the transient by using the ideal trajectories as boundary conditions. Thecorresponding control state is always selected in order to reach the trajectory σsat± that guides the oper-ating point towards the target. Therefore, the proposed control law provides very fast transient responsesand improved dynamic performance, even for the constant power load case.To evaluate the benefits of the state plane controller on the dynamic stability of the three-phase con-verter loaded with a CPL, the transient simulation results for the normalized three-phase converter areprovided in Fig. 4.6. The figure illustrates the response of the closed-loop system under different CPLsteps (∆Pcpln). The operation conditions are the same that have been applied to the linear controller in55Table 4.1: System parameters used in experimentsParameter Symbol ValueDC-bus voltage vC 200 VGrid inductor L 3 mHDC-bus capacitor C 220 µFSwitching frequency fsw 10 kHzNominal power P 2 kWGrid frequency fg 50 HzcplPCvd’idiCvoibiai ci-satσ+satσ1 cplP = 0 kW2 cplP = 1 kW3 cplP = 2 kW2 kW (0.2)10 A (0.2)1 kW (0.1)5 A (0.1)1 2 3Figure 4.7: Experimental results of state plane controller for the three-phase rectifier loaded with aCPL in both time and state plane representation.Fig. 4.3. By comparing both results, the proposed approach exhibits enhanced dynamic performance withrecovery times close to the theoretical limits and much lower voltage overshoot for all load steps. In par-ticular, for the largest applied CPL step ∆Pcpl,4 = 0.2 p.u., the linear controller is not able to compensatefor the negative effects of the CPL, which results in the collapse of the DC-bus voltage and system failure.On the other hand, by using the state plane controller and the natural trajectories of the converter to solvethe transient, the transient responses are much faster, which allows the converter to compensate for theCPL effects and tolerate the highest power step.4.5 Experimental ResultsIn this section, experimental results are provided to validate the previous analysis and the practical im-plementation of the proposed controller. The experimental setup with parameters listed in Table 4.1 wasimplemented in a hardware prototype. The AC side of the converter is connected to a three-phase ACpower supply, whereas the DC-bus capacitor is connected to an electronic load operating in constant560.5 kW=P∆ 0.5 kW=P∆0.5 kW=P∆ 0.5 kW=P∆cplPCvoibiai ci2 kW (0.2)Figure 4.8: Experimental results for the conventional dual-loop PI controller under small CPL steps.The system is stable under small transients around different operating points.power mode to emulate the CPL characteristics of tightly regulated converters. All the control algo-rithms, PWM signal generation, grid-synchronization, reference frame transformations, data samplingand variables normalization were executed in real-time by an industry-standard 32-bit floating point TI-C2000 TMS320F28335 microcontroller unit. Further details of the experimental setup are described inAppendix B. All relevant transient parameters in this section are indicated in the experimental capturesand their normalized values are highlighted between brackets.First, the experimental operation of the state plane controller for the three-phase converter with aCPL connected to the DC-bus is validated. Experimental transient results in the time domain and stateplan representation under different CPL steps are provided in Fig. 4.7. The converter initially operatesin the steady state at no load condition while the DC-bus voltage is regulated at its reference value (200V) when a sudden CPL step from zero to full-load (2 kW) is applied in the DC-bus. Consequently, theboundary condition σsat+ is first predicted by the controller and the corresponding control actions tofollow a combination of saturation trajectories to solve the transient is selected. Then, a CPL step-downof 1 kW is applied to the DC-bus and the boundary condition that intercepts the new target operatingpoint is defined by σsat− Therefore, the controller uses a combination of mdsat+ and mdsat− to reach theboundary condition and finally guide the operating point towards the target. As discussed in the previoussections, despite the differences in the converter’s trajectories due to the CPL characteristics, the proposedstate plane controller solves the transient by using the ideal trajectories as boundary conditions.To evaluate the improvements of the state plane controller on the system stability, its experimental57DC-Bus VoltageCollapses for largeCPL stepscplPCvoi(a)2 kW (0.2)DC-Bus Voltageis regulated forlarge CPL stepscplPCvoi(b)2 kW (0.2)Figure 4.9: Experimental results for a CPL step from no load (0 kW) to full load (2 kW). (a) Conven-tional dual-loop PI controller. (b) Proposed state plane controller. The conventional controllercannot compensate for the negative effects of the CPL which leads to the collapse of the DC-bus voltage. On the other hand, the proposed method can solve the large CPL transient.performance is compared with the conventional dual-loop PI controller. The conventional controller isdesigned to achieve stable closed-loop operation over the entire operating range from no load to full load(2 kW). The experimental results in Fig. 4.8 provide the operation of the conventional controller underseveral small CPL transients of 0.5 kW. As expected, for small perturbations, the converter exhibits stableresponses achieving steady state operation for all loading conditions.Although the closed-loop system in Fig. 4.8 seems to be stable, the dynamic characteristics of the CPLlead to unstable behavior when larger disturbances are present. The experimental results of applying a 2kW CPL step are provided in Fig. 4.9. The figure compares the transient response for the conventional58cplP12 ms (2.35)52 V (0.26)14 ms (2.74)(a)(b)biai cibiai cioicplPoiCvCv66 V (0.33)5 A (0.1)10 A (0.2)2.5 A (0.05)3 ms (0.58)36 V (0.18)5 ms (0.98)40 V (0.2)0.5 kW (0.1) 2 kW (0.2) =1.5 kWP∆10 A (0.2)2.5 A (0.05)0.5 kW (0.1) =1.5 kWP∆2 kW (0.2)Figure 4.10: Experimental CPL step-up/down transient from 0.5 to 2 kW (∆Pcpl = 1.5 kW). (a)Proposed state plane controller. (b) Conventional dual-loop PI controller.and the minimum recovery time controllers. As shown in Fig. 4.9 (a), the conventional approach is notfast enough to compensate for the CPL dynamics, leading to system failure. Due to the CPL, after the loadstep-up is applied, the capacitor voltage is decreased and the DC-bus current starts to increase rapidly.Since the converter is not able to inject current into the DC-bus fast enough to compensate the CPL, thevoltage continues decreasing generating the collapse of the DC-bus voltage. On the other hand, the stateplane controller reacts in a much faster way to the DC-bus voltage changes, which allows the converterto compensate for the CPL effects as shown in Fig. 4.9 (b).Finally, the dynamic performance of both controllers is compared over the entire operating range.The experimental results in Fig. 4.10 and 4.11 present the transient responses for different CPL steps.59(a)(b)biai cicplPoiCv3 ms (0.58)26 V (0.13)4 ms (0.78)33 V (0.16)10 A (0.2)=1 kWP∆2 kW (0.2)1 kW (0.1)5 A (0.1)biai cicplPoiCv12 ms (2.3)37 V (0.19)14 ms (2.74)45 V (0.22)10 A (0.2)=1 kWP∆2 kW (0.2)1 kW (0.1)5 A (0.1)Figure 4.11: Experimental CPL step-up/down transient from 1 to 2 kW (∆Pcpl = 1 kW). (a) Pro-posed state plane controller. (b) Conventional dual-loop PI controller.As shown, even for stable transients, the response of the conventional dual-loop PI controller showsdeteriorated dynamic performance with much larger settling times, oscillatory responses and larger over-shoot/undershoot values. Whereas the state plane controller achieves improved dynamic performanceachieving solutions close to the theoretical minimum times with lower overshoot/undershoot values.4.6 SummaryIn this chapter, a state plane controller that utilizes the natural trajectories of the three-phase converter isintroduced to enhance the stability of the DC-bus when active loads are present.Initially, the input characteristics of tightly regulated load converters are modeled as a constant powerload from the perspective of the DC-bus. The impact of the derived incremental dynamic resistance of60the constant power load on system stability is discussed. The conventional control approach is analyzed,revealing that if the converter response is not fast enough, large CPL steps may cause the collapse of theDC-bus voltage.To address this issue, a state plane controller is introduced to counteract the CPL dynamics duringtransients. The control law operates based on the ideal circular saturation trajectories of the three-phaseconverter in the synchronous reference frame. The principle of operation is analyzed in the state planeand compared to the CPL trajectories, demonstrating that the control law can consistently reach the targetoperating point.Finally, simulation and experimental results are used to validate the analysis and implementation ofthe state plane controller. The comparison with a conventional dual-loop PI controller highlights theimproved dynamic performance under different CPL conditions. The results show that the state planecontroller not only improves the transient responses but also stabilizes the system during large CPLtransients, avoiding the collapse of the DC-bus voltage and potential system failure.61Chapter 5Circular Trajectory Control forThree-Phase PWM ConvertersIn bidirectional applications, the three-phase converter must regulate the DC-bus voltage to its referencelevel, while minimizing its fluctuations during both rectifier and inverter modes of operation. The opera-tion of these systems across a wide range of operating conditions, with different dynamic characteristics,presents a challenge to conventional control designs, leading to poor responses and degraded dynamicperformance under large transients.To address these limitations, this chapter proposes a state plane control method for three-phase con-verters that provides reliable, consistent, and fast transient responses. This control strategy is based onthe natural trajectories discussed in Chapter 2 and involves guiding the operating point along well-definedpaths in the state plane using a combination of circular paths. By following the appropriate set of nat-ural trajectories, the operating points follow smooth and well-defined circular paths, leading to fast andconsistent transient responses throughout the operating range, including rectifier and inverter modes.The implementation of this circular trajectory control method can result in significant improvementsto the overall performance of the system, particularly in bidirectional power conversion applications thatrequire operation under a wide range of conditions.Portions of this chapter have been published in:• F. Degioanni, I. G. Zurbriggen and M. Ordonez, ”Enhanced DC-Link Voltage Dynamics for Grid-Connected Converters,”in IEEE Transactions on Industrial Electronics, vol. 69, no. 11, pp. 10787-10796, Nov. 2022.• F. Degioanni, I. G. Zurbriggen and M. Ordonez, ”Fast and Reliable Geometric-Based Controller for Three-Phase PWMRectifiers,” 2020 IEEE Applied Power Electronics Conference and Exposition (APEC), 2020, pp. 1891-1896.62+satλ-satλCntv(0)dni’(0)dni’(0)dni’(0)dni’(0)Cnv (0)Cnv (0)Cnv (0)Cnv Cnvdn’idn,t’i1md,c1md,r 2md,c2md,r3md,c3md,r4md,c 4md,r1md,ω2md,ω3md,ω4md,ωLLλCnvntnt(0)Cnv(0)Cnv(0)Cnv(0)Cnv(0)dni’(0)dni’(0)dni’(0)dni’dn’in,mdt n,mdtn,mdtn,mdtFigure 5.1: Conceptual representation of the circular control approach in state plane and time do-main for four different sets of initial conditions and one target point.5.1 Circular Trajectory Control ConceptThe normalized decoupled model in Chapter 2 enables the representation of the dynamic evolution of theoperating point as circular trajectories in the state plane, with a well-defined center, radius and angularspeed. Therefore, it is possible to find a circular surface that contains two given points in the plane asconceptually illustrated in Fig. 5.1. For a given set of initial conditions, the trajectory λmdx represents theunique circle that intercepts both the initial and target operating points.The trajectory λmdx can be obtained by evaluating (2.22) with the corresponding initial conditions(vCn(0), idn(0)) and the target operating point (vCnt ,idnt) given by (2.16) and (2.17):λmdx :(vCnt − cvmdx)2+(i′dnt − cimdx)2=(vCn(0)− cvmdx)2+(i′dn(0)− cimdx)(5.1)63where cvmdx and cimdx are the centers of the circles given by:cvmdx = 2Vsdnmdxcimdx = 2i′onmdx(5.2)The modulation index mdx in (5.2) is the required control state to guide the operating point from theinitial to the desired target point by following a unique circular path. By rearranging (5.1), mdx can beexpressed as the error between the current state of the operating and the target point:mdx = 4Vsdn(vCnt − vCn(0))+ i′on(i′dnt − i′dn(0))(v2Cnt − v2Cn(0))+(i′2dnt − i′2dn(0)) (5.3)The derived expression provides a straightforward and useful approach to guide the operating pointbetween two given points in the state plane by using a well-defined path. By setting the modulation indexto the constant value mdx, the target is reached by following a unique circular trajectory λmdx with angularspeed given by:ωmdx = π√3√2mdx (5.4)In this way, the operating point is guided between two states by using a defined circular path movingat constant angular speed. By selecting the appropriate combination of circular trajectories, consistentand reliable dynamic transient responses can be achieved.5.2 Complete Circular Trajectory Control LawThe described circular trajectory concept can be used for developing a closed-loop control law for reg-ulating the DC-bus voltage. In this way, the transients can be solved in a fast, reliable and consistentmanner, obtaining enhanced DC-bus voltage dynamic performance over the entire operating range.According to (5.3), the transient from an initial to final state can be solved by following a uniquecircular trajectory by setting the control input to mdx. However, the possible values of md are limited andsome points cannot be solved by using a unique trajectory. As discussed in Chapter 3, mmd is restrictedby the saturation limits mdsat+ and mdsat−, and consequently, the potential trajectories to solve the tran-sients are contained inside the saturation trajectories λsat+ and λsat− defined by (3.1) and (3.2). Sincethe saturation trajectories are the circles with the fastest angular speed, they provide improved dynamic641ic4ic2ic3icCnvdn’idxm=dmdxm=dm=dm +dsatm=dm -dsatm+satλ-satλLLλCnvdn’iTargetδ(a) (b)Figure 5.2: Conceptual graphical representation of the circular trajectory controller: (a) The controlstate is selected according to the position of the operating point in the state plane. (b) Idealcircular trajectory controller transient responses for four different initial states.regulation and recovery time transients for the DC-bus voltage. Therefore, to derive a complete controllaw, the consistent and reliable responses obtained by using circular trajectories are complemented withthe natural saturation trajectories.The control law that combines both circular and saturation trajectories is conceptually representedin the state plane in Fig. 5.2. The state plane is divided into five different regions according to thelocation of the operating point: one corresponding to a quiescent area of radius δ around the target, tworegions in which the transient is solved by (5.3) to achieve reliable circular trajectories, and two regionscorresponding to the saturation limits for fast-speed trajectories.The conceptual transient responses for four different initial states (ic1-ic4) following the trajectoriesdetermined by the proposed control law are illustrated in 5.2 (b). By using the capacitor voltage andd-axis current values, the position of the operating point is estimated and the control action to follow thecorresponding trajectory is selected. In this way, the operating point is guided through a combinationof circular λmdx and saturation λsat± trajectories to reach the target. As shown, well-defined paths arefollowed by the operating point, providing reliable and consistent transient responses. Once the targetregion is reached, the modulation index is set to its steady-state condition mdss (given by (2.16)) to operatethe system in the equilibrium condition.Theoretically, the control action calculation should be performed only one or two times per transient,since there is a unique combination of trajectories to reach the target. In practice, the control law is exe-65=dm -dsatmdxm=dmSteady-StateOperationNormalization and Synchronous Reference Frame TransformationCurrents and Voltages AcquisitionTarget Estimation2> δ2ε0>LLλ0>+satλ0<-satλ2)dnti−dni+ (2)Cntv−Cnv= (2εon, isdn, Vdn, iCnvdnt, iCntvYesNoNoNo NoYesYesYes=dm +dsatmdmFigure 5.3: Conceptual flow chart implementation for the circular trajectory controller.cuted each switching cycle to monitor the position of the operating point and to compute the correspond-ing control action for the next switching cycle. The periodic recalculation compensates any mismatchesin the component values, losses and any other non-ideal effect not being considered in the derivation ofthe proposed model. Moreover, a low-bandwidth linear controller can be implemented inside the quies-cent target operating region δ to eliminate potential steady-state error due to voltage and current drops inthe parasitic elements.The flow chart implementation of the complete control law for the circular trajectory controller isillustrated in Fig. 5.3. The proposed method requires the computation of different natural trajectoriesaccording to (2.15), (2.22) and (5.3). The involved mathematical operations are mostly additions andmultiplications, which are executed only if the operating point is located in the corresponding region ofthe state plane. Therefore, the computation burden of the proposed algorithm is similar to conventionalapproaches, maintaining low complexity and simple implementation.The complete block diagram of the proposed controller is shown in Fig. 5.4. The circular trajectory66i = 0qrefmdmqvCnidniqniongnωnLgnωnLGiq(s)dqabcduquThree-Phase ConverterabcmCvabcioidqabcPLLdiqiabcvθFigure 5.4: General block diagram of the complete circular trajectory controller.controller is implemented in the d-axis to achieve improved DC-bus voltage regulation, with fast, reliableand consistent transient responses. Whereas, the q-axis only controls the reactive power which can berealized by a simple PI controller Giq(s) to regulate iq to zero to obtain unitary power factor.5.3 Lyapunov Stability Analysis of the Circular Trajectory ControllerGiven the nonlinear nature of the circular trajectory controller, the stability of the closed-loop system isanalyzed by applying Lyapunov’s method.The proposed candidate Lyapunov function to evaluate the stability is defined as the sum of the squareerror of the capacitor voltage and d-axis current:V (evCn ,eidn) = e2vCn+ e2idn (5.5)with the error terms defined as:evCn = vCn(tn)− vCnteidn = i′dn(tn)− i′dnt(5.6)For a given equilibrium condition, which is defined by X0 = (vCnt , i′dnt), evCn and eidn are equal to zero,and consequently, V is a continuous and positive definite function with continuous partial first derivatives.67The derivative of V (V˙ ) can be expressed as:V˙(vCn(tn), idn(tn))=∂V∂ tn= 2(vCn(tn)− vCnt)dvCndtn+2(i′dn(tn)− i′dnt)di′dndtn(5.7)By replacing the derivative terms in (5.7) by the converter equations (2.10) and (2.11), as well as theequilibrium conditions (2.16) and (2.17):V˙(vCn(tn), idn(tn))= Kv(mdss−md(tn))(i′dn(tn)Vsdn− vCn(tn)i′on)(5.8)where Kv is a positive coeficient given by:4π√3√21mdss(5.9)In order to evaluate the stability of the system, V˙ must be a negative semi-definite function:V˙(vCn(tn), idn(tn))≤ 0 (5.10)According to (5.8), the sign of V˙ is defined by two multiplying factors, which must have oppositesigns to satisfy the Lyapunov stability condition. Since the sign of the last term is equivalent to load lineλLL expression in (2.15), the stability condition can be rearranged as:V˙(vCn(tn), idn(tn))=(mdss−mdx(tn))λLL ≤ 0 (5.11)The sign of the derived condition is determined by the value of mdx and λLL. Since mdx is the con-trol input, the stability condition must be evaluated for the circular trajectory control law illustrated inFig. 5.2. In this way, three stability regions can be defined according to the position of the operating point68(X(vCn, i′dn)) in the state plane:Region I : X(vCn, i′dn)< δ⇒ mdx = mDss ⇒ (mdss−mdx) λLL = 0⇒ V˙(vCn(tn), idn(tn))= 0 ∴ Bounded StabilityRegion II : X(vCn, i′dn)> λLL⇒ mdx ∈ (mdss,md,sat+] and λLL > 0 ⇒ (mdss−mdx) λLL < 0⇒ V˙(vCn(tn), idn(tn))< 0 ∴ Asymptotic StabilityRegion III : X(vCn, i′dn)< λLL⇒ mdx ∈ [md,sat−,Mdss) and λLL < 0 ⇒ (mdss−mdx) λLL < 0⇒ V˙(vCn(tn), idn(tn))< 0 ∴ Asymptotic StabilitySince V˙ ≤ 0 for any position of the operating point in the state plane, the Lyapunov stability conditionis satisfied in all regions, which is a sufficient condition for system stability.5.4 Transient Analysis of the Circular Trajectory ControllerDue to the trajectory-based nature of the developed control law, the different transient parameters such asrecovery times and overshoot/undershoot values can be estimated by performing a graphical analysis ofthe state plane theoretical responses.5.4.1 DC-Bus Current Step-Up TransientThe conceptual transient for a DC-bus current step-up solved by the circular trajectory controller is illus-trated in Fig. 5.5 in both time domain and state plane representation. In this type of transient, the finalDC current is larger than its initial value; therefore, the operating point is initially at the steady-state con-dition ➊ and it moves to ➌ after the step is applied. The circular trajectory controller solves the transientby first following the saturation trajectory λsat−. At the intersection point ➋, the control input is set atmdx according to (5.3) to follow the circular trajectory λmdx until the operating point reaches the newsteady-state position at ➌.69Cnv13satω-satr-satλγ δtLLλdn’i(0)dni’Cnxvmdxλisat-cvsat-cCntvdnt’ivmdxcimdxcdnx’imdxβmdxrdn,max’i-satα2 CircularControllerTrajectoryCnvCntvdmCn,minv-dsatm231n,mdxtdnt’i(0)dni’dn’intntntd,ssmdxm-n,sattn,suptFigure 5.5: Theoretical circular trajectory controller transient for a DC-bus current step-up in stateplane and time representation.By analyzing Fig. 5.5, the total recovery time for the circular trajectory controller is given by:tn,sup = tn,sat−+ tn,mdx (5.12)During each subinterval md is either msat or mdx; therefore, the operating point moves at a constantspeed and the transient time can be estimated as:tn,sup =αsat−ωsat+βmdxωmdx(5.13)where ωsat and ωmdx are given by (3.3) and (5.4) respectively and αsat− and βmdx represent the anglesdescribed by the operating point during the transient. By analyzing Fig. 5.5, βmdx is always equal to π70and αsat− is given by:αsat− = δ − γ (5.14)By replacing into (5.13) and rearranging terms, the current step-up recovery time for the circulartrajectory controller is obtained as:tn,sup =√23((δ − γ)π+1mdx)(5.15)By performing trigonometric analysis, the auxiliary angles δ and γ are given by:δ = tan−1(i′dnt + cisatvCnt + cvsat)(5.16)γ = tan−1(i′dn(0)+ cisatvCnt + cvsat)(5.17)with cvsat and cisat defined by (3.4).To solve (5.15), the value of mdx, which is the one computed by the circular trajectory controller, mustbe found. The coordinates of the point ➋ (vCnx,i′dnx) in Fig. 5.5 are given by:vCnx = rsat− cos(δ )− cvsat (5.18)i′dnx = rsat− sin(δ )− cisat (5.19)Then, mdx can be obtained by replacing (5.18) and (5.19) into (5.3), with rsati given by (4.8).Finally, the voltage and current deviation can be determined by:vCn,min = cvmdx− rmdx (5.20)i′dn,max = cidmx+ rmdx (5.21)In this way, the different transient parameters for a DC-bus current step-up solved by the circular tra-jectory controller can be estimated by replacing the corresponding initial and final DC currents in (5.12)-(5.21).71γδsatωtLLλ23vsatc Cnvisatc+satλdn’idn,min’idnt’idnx’i1Cntv(0)dni’vmdxcimdxcCnxv+satr+satαmdxβmdxλmdxrCn,maxvCircularControllerTrajectoryd,ssm+dsatmn,sdwtCnv 23ntntnt1CntvCn,maxvn,mdxtdnt’i(0)dni’dn’idxmdn,min’i+n,sattdmFigure 5.6: Theoretical circular trajectory controller transient for a DC-bus current step-down instate plane and time representation.5.4.2 DC-Bus Current Step-Down TransientThe conceptual transient for a DC-bus current step-down solved by the circular trajectory controller isillustrated in Fig. 5.6 in both time domain and state plane representation. In this type of transient, thefinal DC current is lower than its initial value; therefore, the operating point is initially at the steady-statecondition ➊ and it moves to ➌ after the step is applied. The circular trajectory controller solves thetransient by first following the saturation trajectory λsat+. At the intersection point ➋, the control inputis set at mdx according to (5.3) to follow the circular trajectory λmdx until the operating point reaches thenew steady-state position at ➌.By analyzing Fig. 5.6, the recovery time for a step-down transient is given by:tn,sdw = tn,sat++ tn,mdx (5.22)72Since the operating point moves at constant speed, tn,sdw is given by:tn,sdw =αsat+ωsat+βmdxωmdx(5.23)where αsat+ and βmdx are the angles described by the operating point during the transient. From Fig. 5.6,βmdx is always equal to π and αsat+ is given by:αsat+ = δ − γ (5.24)By replacing into (5.23) and rearranging terms, the current step-down recovery time for the circulartrajectory controller can be estimated by:tn,sdw =√23((δ − γ)π+1mdx)(5.25)where the angles δ and γ can be obtained as:δ = tan−1(i′dn(0)− cisatvCnt − cvsat)(5.26)γ = tan−1(i′dnt − cisatvCnt − cvsat)(5.27)with cvsat and cisat defined by (3.4).In order to find mdx, the point ➋ coordinates (vCnx,i′dnx) in Fig. 5.6 are obtained as:vCnx = rsat− cos(δ )− cvsat (5.28)i′dnx = rsat− sin(δ )− cisat (5.29)Then, mdx is obtained by replacing (5.28) and (5.29) into (5.3).Finally, the voltage and current deviation can be determined by:vCn,max = cvmdx+ rmdx (5.30)i′dn,min = cidmx− rmdx (5.31)73-100100.50.2511.250.7522 211CircularTrajectoriesSaturationTrajectories-0.40.4dmdn’ i dn’ idn’ i dn’ intCnvCnv334455oni0 2 4 6 80.5 0.75 1 1.250.50.250-0.25abcniFigure 5.7: Simulation results for start-up and DC-current step transients in rectifier operation forthe circular trajectory controller in state plane and time domain representation.In this way, the different transient parameters for a DC-bus current step-down solved by the circu-lar trajectory controller can be estimated by replacing the corresponding initial and final DC currentsin (5.22)-(5.31).5.5 Simulation ResultsIn this section, simulation results are provided and analyzed to validate the circular trajectory controllerdynamic performance. In order to obtain generalized results, the simulations are performed in the nor-malized domain. The normalized parameters are Cn = Ln = 1/(2π), fgn = 0.4 p.u., Vsn = 0.4 p.u. andfswn = 50 p.u.. The closed-loop system is tested under several dynamic operating conditions includingstart-up and sudden DC-bus current steps in both operating modes, rectifier and inverter.5.5.1 Dynamic Performance Analysis of the Circular Trajectory ControllerStart-up and step-up/down DC-bus current transients for the normalized converter operating in rectifiermode are shown in Fig. 5.7 in both time and state plane representation. During the start-up, the operatingpoint moves from the initial state➊ to the first target located at➋. The transient is solved according to thecontrol law (5.3), and consequently, the operating point follows a unique circular trajectory that interceptsboth points ➊ and ➋.74oniCircularTrajectoriesdn’ i dn’ iCnv0.75 1 1.2521SaturationTrajectories3-0.250-0.5nt0 2 4 6 821 13-1001-0.50-0.2511.250.75-0.40.4dmdn’ i dn’ iCnvabcniFigure 5.8: Simulation results for start-up and DC-current step transients in rectifier operation forthe circular trajectory controller in state plane and time domain representation.At the normalized time tn = 3 p.u., a DC-bus current step-up of ion = 0.25 p.u. is applied, and thecontroller employs a combination of saturation and circular trajectories to reach the new target pointat ➍. As shown in Fig. 5.7, the operating point follows the saturation trajectory λsat− until the point ➌.At the intersection with λLL, md is computed according to the control law (5.3) to find the correspondingcircular path that achieves the new steady-state conditon ➍. Then, at tn = 6 p.u., a step-down current stepis applied to the DC-bus. First, the operating point follows the trajectory λsat+ until ➎. Then, md is setto guide the operating point towards the new target condition at ➋ by following a well-defined circulartrajectory.Simulation results for inverter operation with the circular trajectory controller are shown in Fig. 5.8.At the time tn = 3 p.u., a normalized DC-bus current step ion = −0.25 p.u. is applied. To solve thetransient, the operating point follows the saturation trajectory λsat+ until intercepting the load line at ➋.Once the operating point reaches the new region, md is computed to follow the circular trajectory thatintercepts ➌. At tn = 6 p.u., an unloading current step is applied to the DC-bus. In this case, the initialoperating point ➋ is located in the same region that contains both the initial and final state. Therefore,the controller can solve the transient by following only one circular trajectory to reach the steady-statecondition ➌. Since the value of md is close to zero, the corresponding circular trajectory resembles astraight line with an infinity slope, which enables solving the transient with minimal voltage deviation750.050.10.150.20.25oni∆ =nt0 4 8 12 16nt0 4 8 12 160.5 0.75 1 1.25 1.5Cnv0.5 0.75 1 1.25 1.5Cnvdn’ i dn’ idn’ i dn’ i-0.2500.250.50.500.51011.50.50.500.51011.50.50.75-0.2500.250.50.75dmdn’ i dn’ iCnvdmdn’ i dn’ iCnv2LLλ1LLλ3LLλ4LLλ5LLλ2LLλ1LLλ3LLλ4LLλ5LLλ0.050.10.150.20.25oni∆ =Rectifier Operation - Circular Controller vs Dual-Loop-PIFigure 5.9: Simulation results for several DC-bus current steps in rectifier operation for the pro-posed controller (blue) and the conventional dual-loop PI controller (red).and almost zero recovery time.As illustrated in Fig. 5.7 and 5.8, the operating point follows well-defined circular trajectories accord-ing to its position in the state plane. In this way, dynamic transients can be solved in a fast, reliable andpredictable manner, achieving improved DC-bus dynamic performance in both operating modes, rectifierand inverter.76-0.25-0.2-0.15-0.1-0.05oni∆ =-0.25-0.2-0.15-0.1-0.05oni∆ =nt0 4 8 12nt0 4 8 120.5 0.75 1 1.25 1.50.5 0.75 1 1.25 1.5CnvCnv4LLλ5LLλ3LLλ2LLλ1LLλ4LLλ5LLλ3LLλ2LLλ1LLλdn’ i dn’ i-0.75-0.5-0.2500.25dn’ i dn’ i-0.75-0.5-0.2500.250.5-0.500.51011.5dmdn’ i dn’ iCnv0.5-0.500.51011.5dmdn’ i dn’ iCnvInverter Operation - Circular Controller vs Dual-Loop-PIFigure 5.10: Simulation results for several DC-bus current steps in inverter operation for the pro-posed controller (blue) and the conventional dual-loop PI controller (red).5.5.2 Dynamic Performance Comparison with the Dual-Loop PI ControllerThe circular trajectory controller is compared with the conventional dual-loop PI controller convention-ally implemented for the DC-bus control of three-phase converters. The conventional approach consists oftwo PI control stages, with an inner current and an outer-voltage loop. The design and tuning proceduresfor the PI parameters are discussed in Appendix A.Simulation results of the normalized three-phase converter are shown in Fig. 5.9 and 5.10 for thecircular trajectory and dual-loop PI controllers. The figure illustrates the dynamic performance of both77approaches under a large range of operating conditions including start-up and several current steps ap-plied to the DC-bus for both operating modes, rectifier and inverter. The proposed controller solves allthe transients by following well-defined circular trajectories, achieving faster recovery times and consis-tent transient responses. On the other hand, the dual-loop PI controller presents slower responses withdifferent dynamic characteristics that show longer and more oscillatory transients for larger current steps.For example, for a current step (ion = 0.15p.u.) the dual-loop controller has a voltage undershoot of23% and overshoot of 24% for step-up and down respectively, with a normalized recovery time of 3 forstep-up and 3.6 for step-down. While, the proposed method shows an improved dynamic response of 9%voltage undershoot for step-up and 17% overshoot for step-down, with a normalized recovery time of 0.6and 1.1 p.u. for step-up and down respectively. For the rest of the transients, the response of the dual-loop PI controller displays a more sluggish response with higher overshoots compared with the proposedcircular trajectory controller.In addition, it can be observed that the start-up transient response is improved by using the proposedmethod. As it is shown in Fig. 5.9, the circular trajectory controller achieves the steady-state operatingpoint by following a circular trajectory. In this way, the start-up is solved in a fast manner, without DC-bus voltage overshooting during the transient. On the other hand, the linear controller shows a slowerresponse with a large voltage overshoot during start-up.As discussed in detail in Appendix A, the DC-bus dynamic model of the three-phase converter presentsnonminimum phase characteristics during rectifier operation. The existence of a right-half-plane-zeroin the control-to-output transfer function imposes a limitation on the maximum achievable closed-loopbandwidth. On the other hand, the dynamic model shows a right-half-plane pole during inverter operation,which set a constraint for the minimum required bandwidth for stable operation. Since the operationmodel reflects the position of the right-half-plane-pole and zero, the voltage loop bandwidth has to bedesigned to fulfill the following condition:2ωRHPP ≤ ωvbw ≤ ωRHPZ2 (5.32)where ωvbw is the bandwidth of the voltage control loop, and ωRHPP and ωRHPZ the frequency of theRHPP and RHPZ respectively.For this reason, the conventional dual-loop controller shows deteriorated dynamic performance withpoor transient responses in bidirectional operation. On the other hand, due to the large-signal trajectory-78+20%+20%-20%-20%+20%-20%+20%-20%L∆ C∆nt0 2 4 6 8Cnv0.5 0.75 1 1.25 1.511.250.750.250.5010-1dmdn’ i dn’ iCnvdn’ i dn’ i0.50.250-0.25+20%+20%-20%-20%+20%-20%+20%-20%L∆ C∆nt0 2 4 6 8Cnv0.75 1 1.25 1.511.250.75-0.5-0.25010-1dmdn’ i dn’ iCnvdn’ i dn’ i0-0.25-0.5(a)(b)Figure 5.11: Circular trajectory controller simulation results for an LC parameter variation of±20%: (a) Start-up and rectifier operation. (b) Inverter operation.based nature of the introduced circular trajectory controller, these limitations are overcome and it providesimproved dynamic performance over the entire operating range.5.5.3 Parameter Variation Simulation AnalysisThe dynamic performance of the circular trajectory controller is evaluated under a large variation in thetolerance of the filter components. A parameter variation of ±20% on the L-C values is introduced forthe normalized converter as shown in Fig. 5.11. The results illustrate the transient response of the circulartrajectory controller for an absolute normalized DC current step of 0.25 p.u. in both operating modes,79Table 5.1: System parameters used in experimentsParameter Description ValuevC DC-bus voltage 200 VL Line inductor 3 mHC DC-bus capacitor 220 µFfsw Switching frequency 12 kHzTs Sampling period 83.33 µsP Nominal power 2.5 kWfg Grid frequency 50 Hzrectifier and inverter. As shown, the transients show fast and reliable responses with minor trajectorydeviation from the ideal case, even under large parameter variations.The variation in the reactive elements makes the trajectories differ from ideal circles, becoming el-lipses with a compressed or expanded ratio in direction of the current axis. However, the load line andsteady-state operating points remain unaltered under component variations. Since the control decisionsare based on the position of the operating point to the load line, it is always possible to find the corre-sponding trajectory to solve the transient.5.6 Experimental ResultsIn this section, experimental results are provided to validate the previous analysis and the practical im-plementation of the circular trajectory controller. The experimental setup with parameters listed in Ta-ble 5.1 was implemented in a hardware prototype. All the control algorithms, PWM signal generation,grid-synchronization, reference frame transformations, data sampling and variables normalization wereexecuted in real-time by an industry-standard 32-bit floating point TI-C2000 TMS320F28335 microcon-troller unit. Further details of the experimental setup are described in Appendix B. The conventionaldual-loop PI control coefficients are the same as the simulations and they are summarized in Table 5.1.All relevant transient parameters in this section are indicated in the experimental captures and their nor-malized values are highlighted between brackets.5.6.1 Experimental Dynamic Performance EvaluationStart-Up Transient ResponseThe start-up transient responses for the proposed circular trajectory and conventional dual-loop PI con-trollers are shown in Fig. 5.12. The conventional approach shows a voltage overshoot of 40 V (0.2 p.u.)80diCvbiai cidm(a)(b)Dual-Loop PI Controller (Start-Up Operation)Circular Trajectory Controller (Start-Up Operation)1122sav10 ms (1.96)12.5A (0.23)40 V (0.2)dsatmdsat-mdiCvbiai cidm dsatmdsat-mCvd’iCvd’i3 ms (0.59)6 A (0.11)Figure 5.12: Experimental start-up transient: (a) Circular trajectory controller. (b) Conventionaldual-loop PI controller.and a longer settling time of 10 ms (1.96 p.u.). Due to the large-signal nature of the start-up transient,the presence of voltage overshoot is a common drawback of linear controllers. To remove the overshoot,an additional soft-start procedure can be included in the control system by slowly ramping up the refer-ence voltage, which usually leads to much slower settling times. On the other hand, the proposed methodsolves the start-up by following a unique circular trajectory, achieving improved transient responses with-out implementing any soft-start procedure. In this way, the DC-bus voltage shows no voltage overshootand a settling time of 3 ms (0.59 p.u.).8112.5 A (0.23)22.5 A (0.41)6.5 ms (1.27)75 V (0.38)16 ms (3.13)80 V (0.4)(a)(b)Dual-Loop PI Controller (Rectifier Operation)Circular Trajectory Controller (Rectifier Operation)1 142 3dsatmdsat-mCvd’iCvd’idiCvbiai cidmoidiCvdmoisavbiai cisav12.5 A (0.23)22.5 A (0.41)4 ms (0.78)45 V (0.22) 8 ms (1.57)65 V (0.32)1 423dsatmdsat-mFigure 5.13: Experimental DC-bus current step up/down transients for rectifier operation: (a) Cir-cular trajectory controller. (b) Conventional dual-loop PI controller.Rectifier Operation - DC-Bus Current Loading and Unloading Transient ResponseThe response for a DC-bus current step for the converter operating in rectifier mode is illustrated inFig. 5.13. As shown in Fig. 5.13 (a), the proposed method solves the transient by following a combinationof saturation and circular trajectories achieving fast and reliable dynamic responses. The transients aresolved in 4 ms (0.78 p.u.) and 8 ms (1.57 p.u.), with voltage deviations of 45 V (0.22 p.u.) and 65 V(0.32 p.u.) for step-up and down respectively. Whereas the conventional control presents recovery timesof 6.5 ms (1.27 p.u.) and 16 ms (3.13 p.u.) with voltage overshoot/undershoot of 75 V (0.38 p.u.) and 80V (0.4 p.u.) for step-up and down respectively. The proposed method shows improved DC-bus dynamic8225 V (0.12)6 ms (1.17)Almost zero recovery time and overshoot-20 A (-0.37)-12.5 A (-0.23)diCvbiai cidmoi50 V (0.25)26 ms (5.3)(a)(b)Dual-Loop PI Controller (Inverter Operation)Circular Trajectory Controller (Inverter Operation)1 12 3savbiai cisav55 V (0.28)22 ms (4.3)dsatmdsat-mdsatmdsat-mCvd’iCvd’i123diCvdmoi-20 A (-0.37)-12.5 A (-0.23)Figure 5.14: Experimental DC-bus current step-up/down transients for inverter operation: (a) Cir-cular trajectory controller. (b) Conventional dual-loop PI controller.performance achieving faster recovery times (40% and 50%) and lower voltage deviation (40% and 20%)than the conventional approach.Inverter Operation - DC-Bus Current Loading and Unloading Transient ResponseThe dynamic responses of both controllers for the converter in inverter operation are compared in Fig. 5.14.In this case, the proposed controller solves the transient extremely fast compared with the conventionalapproach. The loading transient shows a recovery time of 6 ms (1.17 p.u.) and the voltage overshootis 25 V (0.12 p.u.), whereas the unloading transient shows almost zero current and voltage overshoot,83CvdisavCircular Trajectory Controller (Rectifier Operation)25 V (0.125) 5.1 ms (1.0)1 12 3 4Cvbiai ci10 ms (1.96)sbv scv40 V (0.2)d’i1 423Figure 5.15: Experimental results for a grid voltage step transient for the circular trajectory con-troller.and recovery time is under 1 ms. On the other hand, the linear controller shows recovery times of 22ms (4.3 p.u.) and 28 ms (5.5 p.u.) with larger voltage deviation for loading and unloading transient re-spectively. The circular trajectory controller shows an improvement around 3 times faster and 50% lessvoltage overshoot during step-up, and almost negligible response for step-down.5.6.2 Grid Voltage Step TransientThe performance of the circular trajectory controller under sudden grid voltage transients is presentedin Fig. 5.15. The results show the dynamic response of the system for a grid voltage decrease of 50%the nominal value, which is an extreme case of operation. As shown, the proposed method solves thetransients by following the natural trajectories of the converter, achieving fast and reliable circular pathseven during large grid voltage transients.5.6.3 Performance Analysis for Distorted Grid OperationExperimental dynamic transients for the circular trajectory control method for a DC-bus current step inboth modes of operation under nonideal grid conditions (grid voltage imbalance of 10% and THD of 5%)are provided in Fig. 5.16. The obtained transients under distorted grid conditions achieve similar perfor-mances to the ideal grid case and the controller solves the transients by following circular trajectories.Although the dynamic performance is not affected by the nonideal grid conditions, it affects thesteady-state operation. For a grid voltage imbalance of 10% and THD of 5%, the measured steady-84diCvbiai ci(a)(b)10% Voltage Imbalance5% Voltage THDsavCircular Trajectory Controller (Rectifier Operation)1 12 3 4Circular Trajectory Controller (Inverter Operation)1 12 3sbv scv4 ms (0.78)48 V (0.22) 8 ms(1.57)65 V (0.32)Cvd’idiCvbiai cisav sbv scvCvd’i25 V (0.12) 10 ms (1.96)10% Voltage Imbalance5% Voltage THD1142233Figure 5.16: Experimental capture for a current step transient for the circular trajectory controllerunder nonideal grid conditions: (a) Rectifier operation. (b) Inverter operation.state grid current THD for the circular trajectory controller is 1.61%, while the grid current THD for thedual-loop PI controller is 2.87%. This difference in the measured THD values is because the conventionalcontroller is designed with larger closed-loop bandwidth requirements to obtain satisfactory dynamic re-sponses. On the other hand, the proposed control law can be combined with a linear controller in thequiescent area around the steady-state operating point. Since the main objective of this controller is onlyto remove the potential steady-state error, the closed-loop bandwidth requirements are not critical, it canbe designed for low bandwidth. In this way, the steady-state performance can be improved by implement-ing a low-bandwidth linear controller, without impacting the large-signal dynamic performance.85Moreover, the steady-state performance under nonideal grid conditions can be further improved bycombining the proposed approach with specific steady-state solutions for voltage imbalance and harmonicmitigation. It can be implemented by modifying the proposed control law only around the equilibriumpoint and replacing the low-bandwidth compensator with one of these approaches.5.7 SummaryThis chapter presents a trajectory-based control approach for improving the DC-bus voltage regulation ofthree-phase converters in bidirectional operation. The proposed circular trajectory controller is derivedfrom the dynamic model and operating point graphical representation analyzed in Chapter 2. A unique setof circular trajectories, with well-defined characteristics, is utilized to guide the operating point from anyinitial state to the desired target, ensuring fast, consistent, and reliable transient responses. By selectingan appropriate modulation index, the proposed controller provides enhanced DC-bus voltage regulationthroughout the bidirectional operating range.The proposed control law and its implementation were thoroughly discussed and analyzed, includinga study of its stability using the Lyapunov method. The trajectory-based nature of the controller allowsfor the complete characterization of different dynamic transients, such as DC-bus current steps, and thederivation of closed-form expressions for their parameters. The effectiveness of the proposed controllerwas validated through extensive simulations and experimental results. A comparison was made betweenthe proposed circular trajectory controller and a conventional dual-loop PI control strategy, with resultsshowing that the proposed method outperforms conventional control by providing improved DC-bus volt-age regulation with fast, reliable, and consistent transient responses over the entire bidirectional operatingrange of the three-phase converter. Additionally, the impact of nonideal conditions, such as parametertolerances and grid voltage distortion, was analyzed through simulation and experimental results.86Chapter 6Normalization of PWM Converters andUniversal Control DesignIn Chapter 2, the normalization technique was used to derive a normalized dynamic model of a three-phase converter. However, this method can be applied to different PWM topologies, such as buck con-verters, boost converters, single-phase inverters, and others, to derive normalized models and controldesigns.The application of normalization to PWM converters is a powerful tool for dynamic analysis andcontrol design. This is achieved by deriving a normalized dynamic model that is independent of thesystem parameters such as reactive element values, voltage, and power ratings. As a result, universalcontrollers can be developed that exhibit the same normalized behavior when applied to converters withdifferent parameters.In this chapter, the concept of normalization for PWM converters and its benefits for system model-ing and control design are explored. To illustrate the advantages of normalization, a detailed design andanalysis of a voltage-mode linear controller for a synchronous buck converter is presented. This exam-ple highlights the importance of the normalization method as a tool for modeling and designing powerelectronics control systems.Portions of this chapter have been published in:• F. Degioanni, I. G. Zurbriggen and M. Ordonez, ”Universal Controllers for PWM Converters: a Normalized Approach,”2019 IEEE Applied Power Electronics Conference and Exposition (APEC), pp. 1770-1775.87AnalysisanddesignC LNormalizedCompensator UnifiedModels 12π12πPWM Converter IGcnCI LI fsI PI, , ,ixINorm.PWM Converter IIGcnCII LII fsII PII, , ,ixIIvsIIvsINorm.vxIvxIINormalizationAnalysis and Design Tool Implementation Toolfs fsnGcnsv xvxisnv xnvsi xnisnisIIisIiPWMConverterNormalized PWMConverterFigure 6.1: Conceptual representation of the normalization procedure. This method enables thederivation of unified models and controller designs that are not dependent on the particularconverter’s parameters.6.1 Normalization ProcedureThe conceptual representation of the normalization procedure is illustrated in Fig. 6.1. By selectingthe appropriate combination of base parameters, the normalization method enables the elimination ofmodel dependence on specific parameter values such as inductors, capacitors, input/output voltages andpower levels. In this way, by performing the analysis in the normalized domain, generalized models andexpressions can be derived, which are valid for any combination of parameters.The first step in the normalization procedure is to define the base quantities or normalization pa-rameters. For PWM converters, these parameters are given by the filter characteristic impedance (Z0),the natural period of the resonant elements (T0), the reference DC-bus voltage (vCre f ) and the definednormalization current:Z0 =√L/C, T0 = 2π√LC, vnorm = vC,re f , inorm =vnormZ0(6.1)The normalized converter’s variables such as impedances, time, voltages and currents are given by:Zxn =ZxZ0, txn =txT0, vxn =vxvnorm, ixn =ixinorm(6.2)where the sub-indexes x and xn represent different variables of the power converter before and after thenormalization respectively.By appliying (6.2) in a given PWM converter, the normalized converter model is derived which pro-vides a useful tool to analyze the characteristic behavior of the system. As illustrated in Fig. 6.1, the88practical implementation of the normalization procedure demands only extra gains on the sampled volt-ages and currents and a proper normalized switching frequency.Since the normalized time variable is obtained by dividing by the filter resonant period T0, a nor-malized time duration tn = 1 p.u. represents one LC resonant period in the system. It means that thenormalized natural frequency of the filter ( f0n) is equal to 1 p.u. and the normalized inductor (Ln) andcapacitor (Cn) are 12π . This value can be derived as:ZLn =ZLZ0⇒ (2π fn)Ln = (2π f )LZ0 (6.3)from (6.1), Z0 =√LC and fn =ff0with f0 = 1T0 ; therefore, by replacing and solving, Ln =12π is obtained.Similar steps can be followed to derive the value of the normalized capacitor Cn = 12π .6.2 Universal Linear Controllers in the Normalized DomainOne of the main advantages of the normalization concept is the design of universal controllers that offera general solution applicable to a different combination of parameters. In this way, it is possible toimplement the same controller to converters with different parameters and achieve the same normalizedbehavior, simplifying the control design methodology.In order to understand the benefits of using the normalized domain for the controller design, considerthe transfer function G(s) with h number of zeros and m number of poles:G(s) = Kh∏i=1(s+ωzi)m∏j=1(s+ωp j)(6.4)where K, ωzi and ωp j are the gain, zeros and poles of the system respectively.Assuming a power converter with switching frequency ωsw = 2π fsw, it is possible to express theposition of the zeros and poles of G(s) as a function of ωsw:G(s) = Kh∏i=1(s+αiωsw)m∏j=1(s+β jωsw)(6.5)the coefficients αi and β j represent the relative position of the zeros and poles and the switching frequency.89By applying (6.2), the transfer function G(s) can be normalized as:Gn(s) = Kh∏i=1(s+αiωswn)m∏j=1(s+β jωswn)(6.6)where ωswn = 2π fswn represents the normalized switching frequency of the system.In most PWM converter control applications, the sampling rate Tsn is constant; and consequently, itcan also be expressed as a function of the switching frequency:Tsn = ks1fswn= ks2πωswn(6.7)By using the Bilinear Transformation(s≈ 2Tsn 1−z−11+z−1), the discrete representation of the continuoustransfer function Gn(s) can be derived:Gn(z) = Kh∏i=1( 2Tsn1−z−11+z−1 +αiωswn)h∏j=1( 2Tsn1−z−11+z−1 +β jωswn)(6.8)Finally, by replacing (6.7) into (6.8) and rearranging terms, the normalized discrete transfer functionof G(s) is obtained as:Gn(z) = Kn∏i=1[(z−1)+αiksπ(z+1)]n∏j=1[(z−1)+β jksπ(z+1)] = Kn∏i=1[(z−1)+αn(z+1)]n∏j=1[(z−1)+βn(z+1)] (6.9)By inspecting (6.9), the terms containing ωswn of (6.8) are canceled and the coefficients K, αn andβn of Gn(z) are constant values for constant switching and sampling frequencies. Therefore, the previ-ous analysis reveals that different transfer functions can be mapped into the same normalized discreteexpression if they share the same normalized switching frequency. In other words, a controller designedin the normalized domain can be applied to systems with different parameter combinations and achievethe same normalized behavior.90iLioRfsLCBuck ConverterccV CviLnionRnNormalized Buck Converterfsn L 12π=nC 12π=nccnV CnvFigure 6.2: Synchronous PWM buck converter schematic diagram.6.3 Application Example: Normalized Voltage-Mode Synchronous BuckConverterThe normalized model derivation and the design of a universal voltage-mode linear controller for thesynchronous buck converter are studied as an example of the normalization approach.6.3.1 Normalized Dynamic Model of the Buck ConverterThe results of applying the normalization to the synchronous buck converter are shown in Fig. 6.2. Thedifferential equations that characterize the average behavior of the normalized model are:12πdiLndtn= dVccn− vCn (6.10)12πdvCndtn= iLn− vCnRn (6.11)where d is the duty cycle of the switches.The normalized average model can be linearized to derive the small-signal model for control design.By perturbing and linearizing (6.10) and (6.11), the normalized small-signal dynamic model of the buckconverter is obtained as:12πdi˜Lndtn=Vccnd˜+Dv˜ccn− v˜Cn (6.12)12πdv˜Cndtn= i˜Ln− v˜CnRn (6.13)where the perturbed variables are represeted as x = X + x˜.According to (6.12) and (6.13), the small-signal model of the normalized buck converter only dependson the linearized operating point defined by the steady state duty cycle D, rather than different circuitparameters.91Table 6.1: Universal voltage-mode compensator coefficientsa1 = 2N1 a2 = N1(2+N1) a3 = N21b1 = 2N2−1 b2 = N2(1−2N2) b3 =−N22kc = Dfsn∆B2√PP = 1−sin(φm)1+sin(φm) N1 =π√P−∆Bπ√P+∆B N2 =π−∆B√Pπ+∆B√PThe normalized expression for the control-to-output transfer function in the normalized buck converteris derived as:Gdvn(s) =1D1s2(2π)2 +1Rns2π +1(6.14)The characteristic double pole of the buck converter transfer function is located at the normalizednatural frequency ω0n = 2π , and it does not depend on any particular values of L and C. Moreover, thestatic gain Gdvn0 = 1D does not depend on the input voltage of the converter, and it is just defined by thesteady state duty cycle D. Since Gdvn(s) results the same for any buck converter operating at the sameduty cycle D, independently of their filter parameter and voltage levels, it provides a general expressionfor analyzing the converter’s dynamics.6.3.2 Universal Voltage-Mode Controller for Synchronous Buck ConvertersOnce the converter is modeled in the normalized domain, a normalized compensator can be designed.This controller can be directly implemented to any buck converters with different combinations of param-eters, ensuring the same normalized behavior. As an example, a voltage-mode linear controller followingconventional tuning approaches is designed; however, this approach can be extended to any control strat-egy.By analyzing the transfer function in (6.14), a lead compensator can be employed in order to achievethe desired phase margin (φm) at a given crossover frequency ( fc). A PI compensator is also included,in order to provide zero steady-state error at the output voltage. As the compensator is implemented ina microcontroller, the effects of the digital implementation must be considered in order to achieve stableoperation. The effects of the sampling and computational delay can be approximated as a first-ordersystem with a time constant equal to one and a half of the normalized sampling period Tsn. The effect ofthe computational delay is a significant phase reduction for frequencies close to the Nyquist frequency;therefore a lead compensator is added in order to account for the effects of the sampling. Finally, the92proposed compensator in the continuous domain is a third-order transfer function Gc(s):Gc(s) = Kc(1+ωLs)( 1+ sωz11+ sωp1)(1+ sωz21+ sωp2)(6.15)For a desired φm at a given fc, the pole and zero of the lead compensator can be obtained as follows:fz = fc√1− sinφm1+ sinφm; fp = fc√1+ sinφm1− sinφm (6.16)The relationship between the desired crossover frequency and the switching frequency is defined as∆B = fswnfc . Therefore, by replacing (6.16) into (6.15), the discrete transfer function of the compensatorusing Bilinear Transformation is given by:Gc(z) = kc1+a1z−1+a2z−2+a3z−31+b1z−1+b2z−2+b3z−3(6.17)where the coefficients ai, bi and kc are defined in Table 6.1. The derived parameters are constant valuesdefined only by the control design requirements φm and ∆B. The factor kc is the only term of Gc(z) thatdepends on the operating point D and the normalized switching frequency fsn.For example, for control design requirements φm = 52◦ and ∆B = 10, the normalized compensatorresults:Gc(z) = 0.027D f 2sn1−2.58z−1+2.21z−2−0.63z−31−1.14z−1+0.15z−2−0.0054z−3 (6.18)The voltage loop response for the normalized buck converter for different switching frequencies em-ploying the voltage compensator of (6.18) is illustrated in Fig. 6.3. The obtained loop responses havethe same design parameters φm and ∆B for all the switching frequency cases. Therefore, the normalizedcompensator Gc can be implemented into any buck converter with different combinations of parametersand achieve the same normalized response.6.4 Simulation and Experimental ResultsThe performance of two buck converters with a different set of parameters using the normalized controllerderived in the previous section is first compared with the normalized model. The parameters for each buckconverter are summarized in Table 6.2.The time domain responses for the normalized buck converter and the two converters of Table 6.2 are93Magnitude (dB)-180-90-2700Phase (deg)Frequency (p.u.)snf10snf10snf10snf10mφ = 52°0.1 1 10 1000.01-80-4004080snf 50=snf 100=snf 250=snf 500=Voltage Loop Response for Different snfFigure 6.3: Bode plot of the voltage loop for the normalized buck converter with the normalizedcompensator Gc(s) for different normalized switching frequencies.Table 6.2: Parameters of the two buck converters used in simulation and experimentsParameter Buck I Buck IIVcc 24 V 36 VvCre f 12 V 18 VL 240 µH 508 µHC 24 µF 33.3 µFZ0 3.16 Ω 3.90 ΩT0 476.86 µs 817 µsfsw 104 kHz 61 kHzprovided in Fig 6.4. The simulation results illustrate the transient response of the three systems under thesame normalized load transient using the normalized compensator of (6.18). Even though the convertershave different designs, they exhibit the same normalized closed-loop responses in terms of recovery timesand overshoot/undershoot values.Two experimental setups, with parameters listed in Table 6.2, were implemented to validate the practi-cal application of the normalized controller. The implementation of this approach only requires adjustingthe normalization gains on the sampled voltages and currents and a proper normalized switching fre-940 0.48 0.96 1.43 1.9 2.38 t [ms]0 0.81 1.62 2.43 3.24 4.05 t [ms]Norm.0.2500.510.9[p.u.][p.u.]0 1 2 3 4 5 tn [p.u.]Buck II1.1502.316.218[V][A]Buck I0.9501.910.812[V][A]Cv Cv Cnv0.0650.25 A0.30 A0.060.24 A0.27 A0.060.72 V1.08 V0.0540.65 V0.97 V0.730.35 ms0.59 ms0.730.35 ms0.59 msBuck Converter Normalized ResponseCvLiLniLiLiFigure 6.4: Step response for two different buck converters. The responses of the two buck converterare identical to the normalized responsequency. While the coefficients of the compensator given in (6.18) remains the same for any combinationof buck converter parameters. All the control algorithms, PWM signal generation, data sampling andvariable normalization were executed in real-time by an industry-standard 32-bit floating point TI-C2000TMS320F28335 microcontroller unit.The measured frequency response for both buck converters is illustrated in Fig. 6.5. The bode plotsillustrate that both converters have the same normalized voltage loop transfer function. Therefore, bothconverters show a closed-loop bandwidth of around ten times smaller than their corresponding switchingfrequency and phase margin of 50.91◦ and 51.24◦ for buck converter I and II respectively. These resultsclosely match the control design objectives of the normalized controller Gc previously derived.Moreover, the experimental time domain response for a load step for both buck converters is illus-trated in Fig. 6.6. All relevant transient parameters are indicated in the experimental captures and theirnormalized values are highlighted between brackets. The figure illustrates a normalized load transientstep from 0.25 p.u., which results in a current step of 0.95 A and 1.15 A for converter I and converter II95mφ = 50.91°Bandwidth = 9.41 kHz= 10.6ΔBSameNormalizedResponseBuck Converter Imφ = 51.24°Bandwidth = 6.81 kHz= 9.1ΔBSameNormalizedResponseBuck Converter IIFigure 6.5: Experimental voltage loop response for Buck I and II using the same normalized com-pensator.respectively. As shown, the absolute values for current and voltage overshoot/undershoot and recoverytime are different for each converter. However, both exhibit the same normalized response, achievingrecovery times of 0.73 and 0.74 p.u. for converter I and converter II respectively.6.5 SummaryIn this chapter, the concept of normalization and its benefits for control modeling and the design of PWMconverters are analyzed in detail. The normalization technique can be applied to various PWM topologies(such as buck, boost, three-phase converters, etc.) and control architectures (such as dual-loop control,current mode control, state plane controllers, etc.).The normalization procedure, which includes the definition of base quantities to obtain models inde-pendent to the different systems, was presented. The benefits of designing controllers in the normalizeddomain were also evaluated, showcasing the possibility of creating a universal normalized compensator,applicable to different power converter designs, ensuring consistent normalized behavior.To exemplify the process of normalization model derivation and universal controller design, a voltage-96Buck Converter I720 mV (0.06)280 mA (0.07)280 mA (0.07)600 mV (0.05)0.95 A (0.25)0.35 ms (0.73)0.35 ms (0.73)Cv Cnv( )Li Lni( )oi oni( )Buck Converter II1.15 A (0.25)1.08 V (0.06)340 mA (0.07)340 mA (0.07)990 mV (0.055)0.61 ms (0.74)0.61 ms (0.74)Cv Cnv( )Li Lni( )oi oni( )Figure 6.6: Experimental load step response for Buck I and II using the same normalized compen-sator.mode linear controller for a synchronous buck converter was presented. Simulation and experimentalresults of two buck converter systems with distinct parameters were then provided to demonstrate theeffectiveness of the normalization approach. The results revealed that both systems exhibit similar nor-malized responses with comparable normalized settling time and overshoot/undershoot values.97Chapter 7Conclusions7.1 Conclusions and ContributionsWith technology rapidly advancing, the stability, efficiency, and reliability of power electronics systemshave become crucial. The dynamic performance of three-phase converters, in particular, is a determiningfactor in the effectiveness of the control system in maintaining desired steady-state values in the face oftransient disturbances. Poor dynamic performance can result in significant voltage and current deviations,leading to system failures and decreased efficiency. Conventional control techniques may not effectivelyaddress the dynamic performance challenges posed by modern three-phase converter applications, such asactive loads in the DC-bus or the integration of bidirectional systems into the grid. Enhancing the transientperformance, operating range, and cost-effectiveness of three-phase converters is therefore essential formeeting the demands of modern systems and ensuring their efficient, stable, and reliable operation.This dissertation addresses these challenges by exploring novel modeling and control techniques forthe dynamic behavior of three-phase PWM converters. The research conducted includes the developmentof a large-signal model in the synchronous reference frame, the identification of theoretical limits of thedynamic performance for the converter, the implementation of high-performance controllers based onstate plane and geometric analysis for improved stability and performance in various applications, andthe introduction of a normalization concept and universal control design approach. The results of thisresearch contribute to the advancement of the field and have the potential to impact the integration ofrenewable energy and energy storage into modern power systems. The following subsections summarizethe contributions of this work.987.1.1 Large-Signal Modeling of Three-phase Converters in the Synchronous ReferenceFrameThe normalized decoupled model of the three-phase PWM converter was presented in Chapter 2. Thismodel provides a representation of the large-signal dynamics of the converter on the state plane, which isformed by the capacitor voltage and the d-axis current component. The evolution of the state variables canbe visualized through well-defined geometric surfaces in the state plane, providing a clear understandingof the dynamic behavior of the converter. The chapter includes a detailed mathematical procedure for de-riving the model and provides an in-depth analysis of the natural trajectories, highlighting their propertiesand characteristics under different operating parameters.The normalized decoupled model and the graphical representation of the natural trajectories serve asthe foundation for further research into the behavior and performance of three-phase converters. It sets thebasis for analyzing and characterizing the converter, as well as for developing advanced control methodsaimed at enhancing the dynamic performance of three-phase converters in various applications.7.1.2 Theoretical Limits of the Dynamic Performance for the Three-Phase PWMConverter in the Synchronous Reference FrameThe study and characterization of the theoretical minimum recovery time solutions for the three-phaseconverter controlled in the synchronous reference frame were presented in Chapter 3. The minimumrecovery time transients were determined by combining the developed large-signal model with the TimeOptimal Control Principle. By analyzing the dynamic response of the system using only saturation tra-jectories, defined by the minimum and maximum limits of the control signal, the optimal control law wasderived. Geometric analysis was performed on these responses, resulting in the derivation of closed-formexpressions for different parameters under different dynamic transients. The validity of the minimumtime solutions was confirmed through Hardware-in-the-Loop and experimental platforms, which showedthat the near-time optimal responses obtained were similar to the theoretical solutions for a variety ofconverters with different parameter combinations.The minimum recovery time transients provide a comprehensive, quantitative characterization of theconverter’s dynamic performance by comparing it to the ideal transients. As the theoretical ideal perfor-mance cannot be exceeded, this benchmarking method offers a clear indication of the extent to which thedynamic behavior of the converters can be improved.997.1.3 Performance Improvements of the Three-Phase PWM Converter with ActiveLoads ControlThe presence of nonlinear constant power loads in the DC-bus can have a detrimental effect on the sta-bility and reliability of the three-phase converter control. Traditional control approaches may not be ableto provide stable operation during large CPL transients, leading to a collapse of the DC-bus voltage.Chapter 4 presented a solution to this problem by introducing the implementation of a state plane con-troller based on the minimum recovery time solutions and their graphical representation. The proposedcontroller offers a fast response and effectively mitigates the negative impact of CPLs, improving thedynamic performance of the system during sudden and large load transients.The chapter discussed and analyzed the impact of incremental dynamic resistance on system stabilityand the limitations of conventional methods. The operating principle and implementation of the pro-posed control law were derived and analyzed, demonstrating its ability to consistently reach the targetoperating point. Simulation and experimental results showed that the proposed controller outperformsconventional methods, making it a reliable solution for handling large active load transients and avoidingsystem failures.7.1.4 Performance Improvements of the Three-Phase Converter in Wide OperatingRange ApplicationsChapter 5 introduced a novel geometric-based control approach for enhancing the dynamic performanceof three-phase converters with a wide range of operating conditions for bidirectional applications. Theproposed control method aims to achieve sleek and smooth transient responses across the entire operatingrange, including both rectifier and inverter modes.The control strategy is based on the geometric representation of the operating point dynamics in thestate plane, enabling control decisions based on the location of state variables. By following appropri-ate natural trajectories, the operating points follow smooth circular paths, leading to fast and consistentresponses throughout the operating range while keeping implementation requirements simple.The mathematical procedures for the control law derivation, implementation requirements, and stabil-ity analysis of the circular trajectory controller were thoroughly examined. The proposed controller wasvalidated through extensive simulations and experiments, which showed improved DC-bus voltage regu-lation and fast, reliable, and consistent transient responses over the entire bidirectional operating range ofthe three-phase converter, outperforming conventional control methods.1007.1.5 Normalized Modeling and Universal Controller Design for PWM ConvertersIn Chapter 6, the concept of normalization for PWM converters was thoroughly examined. This methodoffers a significant advantage as it enables the derivation of general and comprehensive converter modelsthat are independent of system parameters such as reactive element values, voltage levels, and powerratings. This simplification of the analysis of the converter’s dynamic behavior facilitates the design ofuniversal controllers.The chapter also provided a comprehensive examination of the mathematical procedures that supportthe normalization methodology. A clear, step-by-step design procedure was presented, along with anexample application, to demonstrate the practical use of the normalization approach. To validate theproposed theoretical framework and design procedure, a voltage-mode linear controller was designed andanalyzed for a synchronous buck converter. Simulation and experimental results of two buck converterswith different parameters were provided to show the effectiveness of the presented design. The resultsshowed that the normalized responses of both systems were similar, with comparable normalized settlingtimes and overshoot/undershoot values, emphasizing the significance of normalization in PWM converterdesign and analysis.7.1.6 Specific Academic ContributionsThe work presented in this dissertation has been reported in two journals and four published internationalconference papers.The following publications delve into the limits of the dynamic performance of three-phase converterscontrolled in the synchronous reference frame:• F. Degioanni, I. G. Zurbriggen and M. Ordonez, ”Minimum Recovery Time Transients for Three-Phase Converters in the Synchronous Reference Frame,” in IEEE Transactions on Power Electron-ics, vol. 37, no. 2, pp. 1579-1591, Feb. 2022.• F. Degioanni, I. G. Zurbriggen and M. Ordonez, ”Characterization of the Minimum Recovery TimeTransients for Three-Phase PWM Rectifiers,” 2021 IEEE Applied Power Electronics Conferenceand Exposition (APEC), 2021, pp. 2503-2509.The implementation and analysis of the circular trajectory controller for three-phase converters cul-minated in the following publications:101• F. Degioanni, I. G. Zurbriggen and M. Ordonez, ”Enhanced DC-Link Voltage Dynamics for Grid-Connected Converters,” in IEEE Transactions on Industrial Electronics, vol. 69, no. 11, pp. 10787-10796, Nov. 2022.• F. Degioanni, I. G. Zurbriggen and M. Ordonez, ”Fast and Reliable Geometric-Based Controller forThree-Phase PWM Rectifiers,” 2020 IEEE Applied Power Electronics Conference and Exposition(APEC), 2020, pp. 1891-1896.The work on the minimum recovery time transients to improve the dynamic performance of the three-phase converters feeding active loads was published in:• F. Degioanni, I. G. Zurbriggen and M. Ordonez, ”Improved Control for Three-Phase Rectifierswith Constant Power Loads,” 2023 IEEE Applied Power Electronics Conference and Exposition(APEC), 2023, pp. 2775-2779.The normalization approach and the design of universal controllers for PWM converters were pre-sented in:• F. Degioanni, I. G. Zurbriggen and M. Ordonez, ”Universal Controllers for PWM Converters: aNormalized Approach,” 2019 IEEE Applied Power Electronics Conference and Exposition (APEC),pp. 1770-1775.Some publications emerged from collaborative research focused on modeling and controlling powerelectronics converters for various applications:• M. Mohammadi, F. Degioanni, M. Mahdavi and M. Ordonez, ”Small-Signal Modeling of LLCConverters Using Homopolarity Cycle,” in IEEE Transactions on Power Electronics, vol. 35, no.4, pp. 4076-4093, April 2020.• I. Galiano Zurbriggen, F. Degioanni and M. Ordonez, ”Near-Time-Optimal Dynamics in PWMDC–DC Converters: Dual-Loop Geometric Control,” in IEEE Journal of Emerging and SelectedTopics in Power Electronics, vol. 9, no. 1, pp. 167-182, Feb. 2021.7.2 Future WorkThe dissertation explored novel techniques for modeling and developing high-performance controllersaimed at improving the DC-bus voltage dynamics of three-phase PWM converters. The concepts of nor-malization, state plane representation, and geometric analysis were central to the work presented in this102dissertation. The developed methodologies and results of this work not only demonstrate the effectivenessof these novel techniques but also open up new avenues for further exploration and study. The followingare some examples of the new research opportunities that may be based on the findings presented in thisdissertation:• Exploring the Impact of Cross-Coupling in the System’s Model: The inclusion of the cross-couplingterm in the system model can lead to a more comprehensive and accurate representation of the sys-tem dynamics. This opens up the possibility of developing innovative control strategies, resultingin superior control performance over both the d and q axes, ultimately leading to improvements inpower quality control applications.• Exploring Advanced Filter Designs in Modeling and Control Techniques: The current work fo-cused on three-phase converter systems connected to the grid using a L filter. However, in someapplications, a higher-order filter such as LC or LCL might be required. The higher-dimensionalequation systems posed by these designs present an exciting challenge for future research, requiringadvanced state plane analysis and visualization techniques to effectively analyze the solutions.• Expanding the Model and Control Techniques for Diverse Applications: This work lays the foun-dation for the continued development of cutting-edge geometric-based control strategies that arecustomized to address the unique requirements of various three-phase converter applications thatwere not covered in this work, such as grid-forming converters and motor drives. These appli-cations present unique challenges, such as varying frequency and/or AC voltage amplitude, thatrequire further exploration to fully incorporate these effects into the modeling and control tech-niques for optimal performance.• Expanding the Modeling and Control Techniques to Other AC/DC Topologies: The study presentedin this work provides avenues for the extension of modeling and control techniques to encompassa wider range of AC/DC converter topologies. In particular, it presents a promising outlook for thedevelopment of controllers for single-phase inverters, which are often characterized by a doublegrid frequency oscillation and require innovative design approaches for effective DC-bus voltagecontrol. Moreover, the methodology can potentially be applied to more complex topologies, suchas multilevel and matrix converters, which pose the additional challenge of balancing the DC-busvoltage. 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Proceedings of the National Academy of Sciences ofthe United States of America, 45(4):573, 1959. → page 27[130] H. Hermes and JP La Salle. Functional analysis and time optimal control. New York: Academic,1969. → page 27114Appendix ALinear Control of Three-Phase Convertersin the Synchronous Reference FrameThe conventional approach for controlling three-phase converters in the synchronous reference frame isillustrated in Fig. A.1. The figure illustrates a dual-loop control architecture with nested inner-currentand outer-voltage loops with the decoupling network to cancel the cross-axis dynamics. The outer loopregulates the DC-bus voltage and provides the reference for the d-axis current control loop, whereasthe q-axis regulates the reactive power. The compensators are usually PI regulators designed with well-established tunning conventions such as the technical Optimum (TO) and Symmetrical Optimum (SO)methods for the inner-current and outer-voltage loops respectively [63–65].A.1 Current-loop Controller DesignBy decoupling the dq-axes, the current control plan can be modeled as a first-order system with a timedelay as shown in Fig. A.2. Where L and rL are the inductance and parasitic resistance of the filter, idq isthe corresponding inductor current component in each axis, Td is the delay in the control loop (it includesthe PWM and the sampling delays) and udis represents the external disturbances to the plant. Finally, thecurrent loop is closed and controlled by using a PI regulator with proportional gain ki and integrator timeconstant Ti.Since the natural time constant of the current plant TL = LrL is usually much larger than the controldelay (TL >> Td), this dominant pole can be canceled by setting the integral time constant of the PIregulator equal to the that of the plant (Ti = TL). In this way, the closed-loop transfer function of Fig. A.2115vCvCrefvCve ie)s(vG )s(iGVoltageControllerCurrentControllerCurrentControllerdrefiqrefididiqiqidmd,decmq,decmie )s(iG qmLgωLgωduqudqabcdqabcThree-Phase ConverterPLLθabcmabciFigure A.1: General block diagram of the conventional dual-loop controller implemented in thesynchronous reference frame.becomes:Hci(s) =11+ srL Tiki + s2 TiTdki(A.1)The denominator of (A.1) is a second-order characteristic system equation which damping factor ζ isgiven by:ζ 2 =TiR4kiTd(A.2)Due to the very small product term TLTd , the closed-loop transfer function (A.1) can be further sim-plified by neglecting the s2 term:Hcis(s) =11+ s4Tdζ 2(A.3)For a desired closed-loop damping, the proportional gain ki of the PI regulator can be calculated as:ki =TiR4Tdζ 2(A.4)For the Technical Optimum approach, ζ is selected as√22 to obtain a critically damped system andthe ki term given by:ki =RTi2Td=L2Td(A.5)With the current loop adjusted to be critically damped and assuming a first-order system, the closed-116udisPIRegulatorControlDelayCurrentPlantdq,refidqi1dsT+1sL+Lr1isTikisT+1ie uFigure A.2: Simplified block diagram of the current-loop for Technical Optimum approach.loop bandwidth ( fbw) can be obtained from (A.3) as:fbw =12π4Tdζ 2≈ fsw20(A.6)where fsw is the switching frequency of the converter.Therefore, the bandwidth of the current regulator is approximately 20 times smaller than the samplingfrequency. These designs of Ti and ki provide good reference tracking response performance, but itresults in poor rejection capability for the disturbances udis. In order to improve the disturbance rejectioncapability, the time constant of the PI regulator can be selected to be ten times higher than the smallestdelay component in the system (Ti = 10Td).A.2 Voltage-loop Controller DesignThe model of the DC-bus voltage control plant can be obtained from the differential equations of theconverter in the synchronous reference frame. By linearizing (2.5)-(2.6) around one operating point, thedynamic equations for the decoupled model can be obtained. The small-signal transfer function from thed-axis current to DC-bus voltage is given by:Gvcid(s) =v˜C(s)i˜d(s)=−VCId(1−2 IdMdVC s)(1+ 43CVCMd Ids) (A.7)where the tilde terms represent the small-signal perturbation of the variables around the linearized oper-ating point. The terms Vc, Id and Md are the steady state DC-bus voltage, d-axis current and modulationindex components respectively.117RectifierOperationDifferent DC-Bus Current Operating PointsInverterOperationoni =0.25oni =0.05oni =|0.25|oni =|0.05|oni = 0.05-oni = 0.25-Magnitude (dB)Phase (deg)Frequency (p.u.)-90-45-505-151 100.10.01-135-180Voltage Loop Transfer FunctionFigure A.3: Bode plot of the voltage plant transfer function under a wide range of operating points.The transfer function Gvcid(s) is a first-order system with one pole and one zero given by:ωz =−MDVC2IdL ωp =3Id4VCC(A.8)The location of the RHP zero is dependent on the value of the grid inductor. The location of the RHPpole is dependent on the value of the DC-bus capacitor. Since the current d-axis current is defined aspositive for rectifier operation and negative for inverter operation, the sign of the pole and the zero aredefined by Id . For rectifier operation, the plant shows a nonminimum phase behavior with a right-half-plane zero and a stable pole located on the left side of the imaginary axis in the complex plane. Theposition of the zero moves to the left side when the power flow changes, while the pole moves to the rightside of the complex plane exhibiting nonstable operation with a right-half-plane pole.The effect of the power flow on the voltage loop transfer function is illustrated in Fig. A.3. Thefigure illustrates the bode plot of Gvcid for the normalized three-phase converter around different operatingconditions in both, rectifier and inverter modes. As shown, for the same absolute power level the gain plotis the same for both operating modes. However, the phase characteristic changes significantly depending118on the power flow showing an RHPZ for rectifier operation and an RHPP for inverter operation.Since the position of the pole and zero of Gvcid moves from the right and left sides of the imaginaryaxis for different operating modes, it imposes a limitation in the selection and design of the bandwidthfor the voltage control loop. In order to stabilize the system during inverter operation, the voltage controlloop bandwidth has to be at least twice the RHP pole frequency. Additionally, the RHP zero definesthe upper limit for rectifier operation. The voltage loop bandwidth ωvbw has to be designed to fulfill thefollowing condition:2ωRHPZ < ωvbw <ωRHPP2(A.9)In this way, the PI parameters for the voltage loop controller can be designed to obtain a loop crossoverfrequency that ensures the closed-loop operation in both modes. This limitation usually translates intopoor responses under large transients and deteriorated dynamic performance as different operating modesare reached.119Appendix BExperimental Hardware PlatformTo further validate the derived models and developed control techniques, an experimental platform forthe three-phase PWM converter was built. The overview of the experimental setup is despited in Fig. B.1.To emulate the behavior of an actual power AC grid, the experimental platform for the three-phasePWM converter is powered by a bidirectional AC power source, specifically the Ametek MX-30. Itoffers programmable three-phase voltage and current profiles that can simulate different grid conditions,including dynamic transients and non-ideal grid conditions.Connected to the DC-bus of the three-phase converter is a Chroma 63204 programmable DC load,which can be easily configured to operate in either constant current or constant power modes. Thisprovides a diverse range of experimental scenarios encompassing a wide variety of operating conditions.For scenarios involving bidirectional power operation, an additional programmable DC power source,Figure B.1: Experimental hardware platform of the three-phase PWM converter.120namely the Ametek Sorensen LXI Series, is integrated in parallel with the DC-bus. This supplementarysource functions as a constant current source, effectively injecting power into the DC-bus. This setupenables to explore bidirectional power operation dynamics.The execution of all control algorithms, including PWM signal generation, grid synchronization,reference frame transformations, data sampling, and variable normalization, takes place in real-timethrough the utilization of an industry-standard 32-bit floating point TI-C2000 TMS320F28335 micro-controller unit.121
Thesis/Dissertation
eng
Electrical and Computer Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Ordonez, Martin
Fast dynamic transient solutions for three-phase PWM converters
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