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Improved control of photovoltaic interfaces Xiao, Weidong 2008

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IMPROVED CONTROL OF PHOTOVOLTAIC INTERFACES by WEIDONG XIAO BESc., Shenyang University of Technologies, 1991 MASc., University of British Columbia. 2003  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Electrical and Computer Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA April 2007  CO Weidong Xiao, 2007  11  Ab stract Photovoltaic (solar electric) technology has shown significant potential as a source of practical and sustainable energy; this study focuses on increasing the performance of photovoltaic systems through the use of improved control and power interfaces. The main objective is to find an effective control algorithm and topology that are optimally suited to extracting the maximum power possible from photovoltaic modules. The thesis consists of the following primary subjects: photovoltaic modelling, the topological study of photovoltaic interfaces, the regulation of photovoltaic voltage, and maximum power tracking. In photovoltaic power systems both photovoltaic modules and switching mode converters present non-linear and time-variant characteristics, resulting in a difficult control problem. This study applies in-depth modelling and analysis to quantify these inherent characteristics, specifically using successive linearization to create a simplified linear problem. Additionally, Youla Parameterisation is employed to design a stable control system for regulating the photovoltaic voltage. Finally, the thesis focuses on two critical aspects to improve the performance of maximum power point tracking. One improvement is to accurately locate the position of the maximum power point by using centred differentiation. The second is to reduce the oscillation around the steady-state maximum power point by controlling active perturbations. Adopting the method of steepest descent for maximum power point tracking, which delivers faster dynamic response and a smoother steady-state than the hill climbing method, enables these improvements. Comprehensive experimental evaluations have successfully illustrated the effectiveness of the proposed algorithms. Experimental evaluations show that the proposed control algorithm harvests about 1% more energy than the traditional method under the same evaluation platform and weather conditions without increasing the complexity of the hardware.  111  Table of Contents Abstract ^ Table of Contents ^  ii iii  List of Tables ^  viii  List of Figures ^  x  List of Symbols ^  xv  List of Abbreviations ^  xvii  Acknowledgement ^  xix  Chapter 1^Introduction and Overview ^  1  1.1  Photovoltaic Power Status ^  1  1.2  Advantages and Disadvantages ^  2  1.3  Classifications of Photovoltaic Power Systems ^ 3  1.4  Techniques of Photovoltaic Power Systems ^  4  1.4.1  Protection ^  5  1.4.2  Control ^  6  1.5  Research Motivation and Objective ^  7  1.6  Thesis Overview ^  8  Chapter 2  Problem Statement ^  2.1  10  Fundamental Limitations of Maximum Power Point Tracking ^ 10 2.1.1  Variety of Photovoltaic Materials ^  10  2.1.2  Switching-Mode Power Converters ^  11  iv  2.1.3  Time-variant parameters ^  12  2.1.4  Non-Ideal Conditions ^  14  Techniques of Maximum power point Tracking ^  18  2.2.1  Heuristic Search ^  18  2.2.2  Extern= Value Theorem ^  19  2.2.3  Linear Approximation Methods ^  23  Summary ^  25  Photovoltaic Modeling ^  26  Equivalent Circuits ^  27  3.1.1  Simplified Models ^  27  3.1.2  Parameterization ^  28  3.2  Temperature Effect on Parameterization ^  30  3.3  Polynomial Curve Fitting ^  32  3.4  Modeling Accuracy ^  33  3.5  Summary ^  35  Topologies of Photovoltaic Interfaces ^  37  4.1  System Structures ^  37  4.2  Control Variables of Maximum Power Point Tracking ^  39  4.3  Converter Topologies ^  41  4.3.1  Component Comparison ^  42  4.3.2  Modeling Comparison ^  43  4.4  Test Bench ^  46  4.5  Summary ^  49  Regulation of Photovoltaic Voltage ^  51  5.1  Linear Approximation of Photovoltaic Characteristics ^  51  5.2  Linearization of System Model ^  55  5.3  Model Analysis and Experimental Verification ^  56  2.2  2.3 Chapter 3 3.1  Chapter 4  Chapter 5  V  5.4  Closed-Loop Design ^  60  5.4.1  Issues of Affine Parameterization used for Lightly-Damped Systems 60  5.4.2  Time delays ^  61  5.4.3  Controller Parameterization and Analysis ^  61  5.4.4  Controller Tuning ^  64  5.5  Digital Redesign ^  65  5.6  Anti-windup ^  67  5.7  Evaluation ^  68  5.8  Summary ^  69  Maximum Power Point Tracking ^  72  Improved Maximum Power Point Tracking ^  72  6.1.1  Euler Methods of Numerical Differentiation ^  72  6.1.2  Reduction of Local Truncation Error ^  73  6.1.3  Numerical Stability ^  74  6.1.4  Tracking Methods ^  75  6.1.5  Oscillation Reduction ^  75  6.1.6  Main Loop ^  80  6.1.7  Flowchart of Maximum Power Point Tracking ^  80  6.1.8  Evaluation of Improved Maximum Power Point Tracking ^  81  Real-Time Identification of Optimal Operating Points ^  89  6.2.1  Real-Time Estimation of the Maximum Power Point ^  90  6.2.2  Parameter Estimation Based on the Further Simplified Single-Diode  Chapter 6 6.1  6.2  Model ^  91  6.2.3  Recursive Parameterization of Polynomial Models ^  91  6.2.4  Determination the Voltage of the Optimal Operating Point ^  92  6.2.5  Convergence Analysis ^  94  6.2.6  Evaluations ^  95  6.2.7  Simulation ^  97  6.2.8  Experimental Evaluation ^  98  ^^  vi  6.3^Summary ^  100  Chapter 7^Summary, Conclusions and Future Research ^ 103 7.1^Summary ^  103  7.2^Conclusions ^  106  7.2.1 Topology Study of Photovoltaic Interfaces ^ 106 7.2.2 Regulation of Photovoltaic Voltage ^  106  7.2.3 Application of Centered Differentiation and Steepest Descent to Maximum Power Point Tracking ^  107  7.2.4 Real-time Identification of Optimal Operating Point in Photovoltaic  7.3  Power Systems ^  107  Future Research ^  107  Bibliograpy Appendix A  109 Hardware Data and Photos ^  115  A.1  Photovoltaic Modules ^  115  A.2  Power Interfaces ^  117  A.3  Laboratory Equipments ^  118  Comparative Design of Power Interfaces ^  119  B.1  Conceptual Design ^  119  B.2  Components ^  121  Appendix B  B.2.1 Inductor ^  121  B.2.2 Other components ^  121  B.2.3 Loss estimation ^  122  B.3  Signal Conditioners ^  122  B.4  Schematics ^  123  Appendix C C.1  Control of a DC/DC Converter Using Youla Parameterization ^ 125 Introducing Affine Parameterization ^  126  vii  C.2  Single Control Loop ^  C.3  Inductor Current Monitoring and Limiting ^ 130  C.4  Limitations in DC-DC Converter Control ^  127  131  C.4.1 Noise and disturbance ^  132  C.4.2 Model Error ^  132  C.4.3 Lightly Damped Systems ^  132  C.5  Anti-Windup ^  133  C.6  Evaluation ^  134  C.7 Appendix D  C.6.1 Modeling and Design ^  134  C.6.2 Simulation ^  136  Summary ^  140  Experimental Results ^  141  viii  List of Tables Table 2.1 Loss analysis in a particular module, BP350, caused by switching voltage ripples ^ 12 Table 2.2 Losses caused by series-connected modules under one-cell-shaded condition ^ 17 Table 2.3 Time variant characteristics of optimal ratios and maximum power levels ^24 Table 3.1 Comparison in modeling accuracy of SSDM, FSSDM and PCF ^ 34 Table 3.2 Modeling features of equivalent circuit and polynomial curve fitting ^ 36 Table 5.1 Frequency parameters of the open-loop model^  57  Table 5.2 Parameters of stability without considering the time delay caused by digital control ^  63  Table 5.3 Parameters of stability with considering the time delay ^  64  Table 5.4 Parameters of stability after further tuning ^  65  Table 6.1 Comparison of standard deviation in steady state ^  84  Table 6.2 Case definision ^  87  Table 6.3 Result oft-test ^  88  Table 6.4 Experimental test environment ^  99  Table A.1 Specification of photovoltaic module BP350 ^  115  Table A.2 Specification of photovoltaic module Shell ST10 ^  116  Table A.3 Specification of photovoltaic module MSX-83 ^  116  Table A.4 Specification of photovoltaic module Shell ST40 ^  116  Table A.5 Power converter specifications ^  117  Table B.1 Design specifications ^  119  Table B.2 Symbols ^  120  Table B.3 Inductance and capacitance ^  120  Table B.4 RMS values ^  120  Table B.5 Inductor parameters ^  121  Table B.6 Component list ^  121  ix Table B.7 System loss estimations ^  122  Table B.8 Measured variables and relative resolution ^  123  Table C.1 Controller configuration ^  137  Table C.2 Comparison of performance on step response ^  138  Table C.3 Comparison of performance on load disturbance ^  139  Table D.1 Results of long-term daily tests ^  142  x  List of Figures Fig. 1.1 Cumulatively installed photovoltaic power in IEA PVPS countries ^ 1 Fig. 1.2 Distribution of installed solar power reported by the IEA PVPS countries in 2004 ^2 Fig. 1.3 Assembly of the parabolic-trough system components for the 64MW Nevada plant (photo credit: Solargenix) ^  4  Fig. 1.4 (a) Photovoltaic modules integrated into the glass roof structure of Lehrter Station, Berlin, Germany (photo credit: PREDAC). (b) Photovoltaic modules integrated into the glass roof structure of the Fred Kaiser building of UBC, Vancouver, Canada^ Fig. 1.5 Block diagram of typical photovoltaic power system ^  5 5  Fig. 1.6 Normalized output characteristics of photovoltaic module at the standard test condition ^ Fig. 2.1 Plots of normalized I-V curves of ST10 and BP350. ^  7 11  Fig. 2.2 Simulated deviation from maximum power point caused by photovoltaic voltage ripple ^  12  Fig. 2.3 Simulated I-V curves of BP350 influenced by insolation when the cell temperature is constant 25°C ^  13  Fig. 2.4 Simulated I-V curves of BP350 influenced by cell temperature when the insolation is constant 1000W/m 2 ^  14  Fig. 2.5 Measured I-V curves of one-cell-shaded and non-shaded modules ^ 16 Fig. 2.6 Measured I-V curves when the one-cell-shaded module is in series connected with other non-shaded modules ^  16  Fig. 2.7 Measured P-V curves when the one-cell-shaded module is serially connected to other non-shaded modules ^  17  Fig. 2.8 Conceptual operation of hill climbing in P&O MPPT algorithm ^ 19 Fig. 2.9 The block diagram of P&O and IncCond MPPT topologies ^  20  xi  Fig. 2.10 Measured signals of photovoltaic panel with the operation of P&O ^21 Fig. 2.11 Measured P-V curves of two BP350 mounted on the same frame ^ 24 Fig. 2.12 Temperature effect on the relationship of power and current of MSX-83 solar panel with a constant insolation equal to 1000W/m 2 ^  25  Fig. 3.1 Model of photovoltaic cell ^  26  Fig. 3.2 Equivalent circuit of a single-diode model ^  27  Fig. 3.3 Equivalent circuit of a further simplified single-diode model ^ 28 Fig. 3.4 Temperature effects on ideality factor and series resistance of equivalent circuits ^ 30 Fig. 3.5 Relative simulation error in term of temperature change based on Shell ST40 ^ 31 Fig. 3.6 I-V curves of MSX-83 modeled by SSDM, FSSDM and 6 th -order polynomial ^ 34 Fig. 3.7 I-V curves of ST10 modeled by SSDM, FSSDM, and 4 th -order polynomial ^ 35 Fig. 4.1^Topology of grid-connected photovoltaic power systems with an AC bus configuration ^  38  Fig. 4.2 Topology of grid-connected photovoltaic power systems with a DC bus and a UPS function ^  38  Fig. 4.3 Quick start of IMPPT when the initial value is set to 80% of the open-circuit voltage ^  40  Fig. 4.4 Equivalent circuit of a buck DC/DC topology used as the photovoltaic power interface^  41  Fig. 4.5 Equivalent circuit of a boost DC/DC topology used as the photovoltaic power interface ^  42  Fig. 4.6 Equivalent circuit in the buck converter when the switch is on ^43 Fig. 4.7 Equivalent circuit in the buck converter when the switch is off ^44 Fig. 4.8 Bode diagram of nominal operating condition ^  47  Fig. 4.9 Measured photovoltaic power of BP350 acquired in Vancouver, from 10:00AM11:00AM in June 15, 2006 ^  48  Fig. 4.10 Block diagram of the bench system used to evaluate maximum power point tracking under the same condition ^  48  Fig. 5.1 Dynamic resistances versus photovoltaic voltage for four levels of cell temperature when the insolation is constant 1000W/m 2 ^  53  xii  Fig. 5.2 Dynamic resistances versus photovoltaic voltage for four levels of insolation when the cell temperature is constant 25°C ^  53  Fig. 5.3 Distribution of dynamic resistances at the maximum power point is influenced by cell temperature when the insolation is constant 1000W/m 2 ^  54  Fig. 5.4 Distribution of dynamic resistances at the maximum power point is influenced by insolation when the cell temperature is constant 25°C ^  54  Fig. 5.5 Linear approximation of photovoltaic output characteristics ^ 55 Fig. 5.6 Boost DC/DC converter^  56  Fig. 5.7 Bode diagram of the photovoltaic system G; (s) ^  57  Fig. 5.8 Open-loop step-response of photovoltaic voltage in power region I ^ 59 Fig. 5.9 Open-loop step-response of photovoltaic voltage in power region II ^ 59 Fig. 5.10 Nyquist plots of the loop transfer functions C(s)Gi (s) when the time delay is ignored ^  62  Fig. 5.11 Nyquist plots of the loop transfer functions C(s)Gi (s) when the time dealy is considered ^  63  Fig. 5.12 Nyquist plots of the loop transfer functions Cy (s)Gi (s) after further tuning ^ 65 Fig. 5.13 Bode plots of the controller frequency response in the continuous and discrete domain^ Fig. 5.14 Control loop for regulating the photovoltaic voltage ^  66 67  Fig. 5.15 Plots of photovoltaic voltage regulation with low solar radiation. The controller parameters are illustrated in (5.21). ^  69  Fig. 5.16 Simulated results of regulation performance against the disturbance caused by 100% step change of insolation level ^ Fig. 6.1 Normalized power-voltage curve of photovoltaic module ^  70 74  Fig. 6.2 Comparison of different algorithms for maximum power point tracking ^ 76 Fig. 6.3 Gauss-Newton method is theoretically efficient in maximum power point tracking ^ 76 Fig. 6.4 Flowchart to evaluate if the MPP was located ^  78  Fig. 6.5 Flowchart to determine if there is a new MPP ^  79  Fig. 6.6 Flowchart illustrating the main loop of improved maximum power point tracking ^ 80  Fig. 6.7 Flowchart of the improved maximum power point tracking ^  81  Fig. 6.8 Plots of startup procedures of IMPPT and P&O algorithms under low radiation. The available power is about 8W for 50W solar module ^  83  Fig. 6.9 Plots of startup procedures of IMPPT and P&O algorithms under weak radiation. The available power is about 3.53W for the 50W solar module. ^ 84 Fig. 6.10 Normalized power waveforms in steady state that the 50W photovoltaic module outputs about 22W power and the perturbation step Ad = 0.005 ^ 85 Fig. 6.11 Normalized voltage waveforms in steady state that the 50W photovoltaic module outputs about 22W power and the perturbation step Ad is 0.005. ^ 85 Fig. 6.12 Normalized power waveforms in steady state that the 50W photovoltaic module outputs about 22W power and the perturbation step Ad is 0.01 ^ 86 Fig. 6.13 Normalized voltage waveforms in steady state that the 50W photovoltaic module outputs about 22W power and the perturbation step Ad = 0.01 ^ 86 Fig. 6.14 Waveforms of power and voltage acquired by the eight-hour test in June 24, 2006, which is a sunny day. (a) Comparison of power waveforms controlled by PAMPPT and P&O, (b) voltage waveform controlled by PAMPPT, (c) voltage waveform controlled by P&O ^  89  Fig. 6.15 Waveforms of power and voltage acquired by the eight-hour test in July 09, 2006, which is a cloudy day. (a) Comparison of power waveforms controlled by PAMPPT and P&O, (b) voltage waveform controlled by PAMPPT, (c) voltage waveform controlled by P&O ^  90  Fig. 6.16 Flowchart of RLS estimation procedure to determine the system parameters ^ 93 Fig. 6.17 The flowchart of NRM operation in finding VOOP  94  ^  Fig. 6.18 The relationship off(v) and v ^  96  Fig. 6.19 The estimated I-V curves and maximum power points with FSSDM ^ 96 Fig. 6.20 The control structure of maximum power point tracking with real-time identification ^ Fig. 6.21 The simulated plots of the changes of temperature, PV voltage, PV current,  97 VoOP  and estimated parameters when the temperature increases from 25 ° C to 35 ° C with a constant insolation of 1000W/m 2  98  xiv  Fig. 6.22 The block diagram of experimental evaluation ^  99  Fig. 6.23 The plots of offline curve fitting and estimation of Voop by RLS and NRM ^ 100 Fig. 6.24 The estimation process of Voop and parameters using RLS and Newton method. ^ 101 Fig. A.1 Two photovoltaic modules installed on one frame at the same angle and direction ^ 117 Fig. A.2 DSP-controlled power interface for photovoltaic power conversion. ^ 118 Fig. B.1 Schematics and frequency response of a two-pole low pass filter ^ 123 Fig. B.2 Schematics of power interfaces ^  124  Fig. C.1 Block diagram of a negative feedback control system. ^  126  Fig. C.2 Representation of Youla parameterization in a closed-loop control system ^ 127 Fig. C.3 Schematic of DC-DC buck converter ^  127  Fig. C.4 Two-loop control of DC-DC converter ^  131  Fig. C.5 Relationship of the damping ratio versus the load condition in per-unit ^ 133 Fig. C.6 Anti-windup scheme with current limiter and saturation limit ^ 134 Fig. C.7 An example of DC-DC buck converter ^  135  Fig. C.8 Bode plots of DC-DC buck converter for different load condition ^ 135 Fig. C.9 Simulink model of ideal DC-DC buck converter ^  136  Fig. C.10 Simulation model of closed-loop control system with anti-windup and current limiter ^  137  Fig. C.11 Simulated step response with a set point changing from 0 to 18V ^ 138 Fig. C.12 Simulated transient response of step-changed load condition ^ 140  XV  List of Symbols A^Ideality factor empp^Relative modeling error at maximum power point eoop^Absolute voltage error of the optimal operating point ernipp^Relative modeling error of maximum power  Ga^Solar insolation I mpp^Current  at maximum power point  1ph^Photo current -  1 pv^Photovoltaic  current  Isar^Saturation current of diode in equivalent circuit ix^Short-circuit current  k^Boltzman constant KA^Temperature coefficient for ideality factor  K RS^Temperature coefficient for series resistance K,^Scale factor for steepest descent method P^Power (W) Pmax^  Maximum power  f^Arithmetic mean of a set of photovoltaic power values Rs^Series resistor  T^Temperature q^Magnitude of an electron  xvi  ^ 00P  V MPOP Vivipp  ^  ^  Voltage of optimal operating point  Voltage of maximum power operating point. Voltage at maximum power point  ^  Open-circuit voltage VOC Vpv^Photovoltaic voltage VI  ^  Thermal voltage of photovoltaic cell Ratio of the current of maximum power point over short-circuit current. Ratio of the voltage of maximum power point over open-circuit voltage  AP  ^  AV  ^  Incremental change of photovoltaic power Incremental step of photovoltaic voltage  xvii  List of Abbreviations AC^Alternating Current AM^Air Mass CIS^Copper Indium Diselenide DC^Direct Current DSP^Digital Signal Processor FSSDM^Further Simplified Single Diode Model IEA^International Energy Agency IEEE^Institute of Electrical and Electronics Engineers IncCond^Incremental Conductance method I-V^Photovoltaic Current versus Voltage LS^Least-Squares MOSFET^Metal Oxide Semiconductor Field Effect Transistor MPP^Maximum Power Point MPPT^Maximum Power Point Tracking or peak power point tracking NASA^National Aeronautics and Space Administration Norm.^Normalized NRM^Newton-Raphson method OOP^Optimal Operating Point PCF^Polynomial Curve Fitting PID^Proportional-Integral-Derivative PPP^Peak Power Point PREDAC^European Actions for Renewable Energies  xviii PV^Photovoltaic P-V^Photovoltaic Power versus Voltage PVPS^Photovoltaic Power Systems Programme PWM^Pulse Width Modulation RMS^Root Mean Square SSDM^Simplified Single-Diode Model STC^Standard Test Condition UBC^University of British Columbia ZOH^Zero Order Hold  xix  Acknowledgement I wish to express my deepest gratitude to my research supervisor Professor William G. Dunford, who supported me throughout the project. I appreciate the effort of former committee members, Professor Patrick Palmer and Professor Jun V. Jatskevich. I wish to acknowledge Dr. Guy A. Dumont's advice about control approaches during the qualifying exam. In particular, I would also like to express my thanks to Dr. Antoine Capel, who led me to this research direction. Numerous interactions with my colleagues, Magnus Lind and Kenneth Wicks, has as well inspired me throughout my graduate studies. Many thanks are due to both Alpha Technologies Ltd. and the Natural Science and Engineering Research Council for providing the necessary funding through the Industrial Post Graduate Scholarship program. Finally, I am expressing my sincerest gratitude to my parents, my wife, my brother, my sisters for their support during my study.  1  Chapter 1 Introduction and Overview  1.1 Photovoltaic Power Status The development of renewable energy technologies has become a necessity in our society. Photovoltaic power is experiencing a rapid growth because of its significant potential as a source of practical and sustainable energy. The use of solar energy displaces conventional energy; which usually results in a proportional decrease in green house gas emissions. Fig. 1.1 illustrates the growth rate of photovoltaic power installation in the countries registered with the Photovoltaic Power Systems programme of the International Energy Agency (IEA PVPS). In 2005, the installation capacity of photovoltaic power systems is 3,697MW, in contrast to 199MW in 1995 [1].  3500 3000 § 2500 43 2000 1500 2co  co 1000 500 ^  min o1992 1993  1997 1998 1999 2000 2001 2002 2003 2004 2005  Fig. 1.1 Cumulatively installed photovoltaic power in IEA PVPS countries  Chapter 1 Introduction and Overview^  2  Fig. 1.2 shows the global distribution of solar power installed in 2004. Germany, Japan, and the USA contributed about ninety-four percent of the growth in capacity. The dramatic growth in these countries was mainly inspired by a variety of government incentive programs. By the end of 2004, the established photovoltaic power in Canada reached a combined 13,884 kW, equal to 0.44% of the total installed capacity reported by IEA PVPS countries [1].  Fig. 1.2 Distribution of installed solar power reported by the IEA PVPS countries in 2004  1.2 Advantages and Disadvantages A typical photovoltaic cell consists of semiconductor material that converts sunlight into direct current (DC) electricity. In 1839, Alexandre Edmond Becquerel, a French experimental physicist, discovered the photovoltaic effect. During the 1950s, Bells labs produced photovoltaic cells for space activities in the United States. This was the beginning of the photovoltaic power industry. The high cost kept its major applications constrained to the space programs until the 1973 oil crisis. Recently, photovoltaic power generation has become more and more popular because of the following major advantages [2-7]: ■ Free energy source ■ Environmental Benefits •  Decrease of harmful green house gas emissions  ■ Reliability •  Established technology  Chapter 1 Introduction and Overview^  •  Static structure  •  Low maintenance  3  ■ Increasing efficiency and decreasing price ■ Economy of construction •  Scaleable design  •  Highly modular  ■ Regulatory and financial incentives •  Tax credits  •  Low interest loans  •  Grants  •  Special utility rates etc.  The disadvantages of solar power applications lie in: ■ The still relative high costs ■ The relative low conversion efficiency of the photovoltaic effect ■ The limited capacity in power generation, compared to wind power ■ The regional differences of solar radiation ■ The diurnal and seasonal variations.  1.3 Classifications of Photovoltaic Power Systems Photovoltaic power systems are typically classified as either stand-alone systems or gridconnected systems [2-6]. The operation of stand-alone photovoltaic systems is independent of the electric utility grid. The major applications of stand-alone systems include, but are not limited to: satellites, space stations, remote homes, villages, communication sites, and water pumps. In recent times, there have been an increasing number of grid-connected systems, which are in parallel with the electric utility grid and supply the solar power through the grid to the utility. In 2004, ninetythree percent of installed solar power systems were grid-connected structures [1]. Based on power capacities, photovoltaic power systems are classified as large, intermediate, and smallscale systems. According to the definition of photovoltaic power systems in [8], large systems  Chapter 1 Introduction and Overview ^  4  are greater than 500 kW in power capacities, intermediate applications range from over 10 kW up to 500 kW, and small systems are rated at 10 kW or less. An example of the large-scale photovoltaic power plant is illustrated in Fig. 1.3, which is a 64MW system located in Nevada State, USA. The latest development is building-integratedphotovoltaics, which is an integration of photovoltaics into the envelopes of residential and commercial buildings. These systems are generally intermediate or small-scale systems. One example of intermediate-scale architectural integration is the 325KW photovoltaic system built in the glass roof of Lehrter Station, Berlin, Germany, as shown in Fig. 1.4 (a). Small-scale power generation is usually near the sites of the end users, for example, the 10KW system of the Fred Kaiser Building at the University of British Columbia, as illustrated in Fig. 1.4 (b).  Fig. 1.3 Assembly of the parabolic-trough system components for the 64MW Nevada plant (photo credit: Solargenix)  1.4 Techniques of Photovoltaic Power Systems Unlike conventional power generation systems, the relative installation costs of photovoltaic power systems do not decrease significantly as power output grows. This allows customers to construct a flexible and affordable system based on individual needs. Therefore, more and more recent applications are the "small" photovoltaic power systems, rated at 10kW or less. In addition to the solar arrays, photovoltaic power systems require components that are designed, interconnected, and sized for a specific power output to properly conduct, control, convert, distribute, and store the energy produced. Static power converters, such as DC/DC and DC/AC,  Chapter 1 Introduction and Overview  ^  5  are widely used as the power interfaces. Fig. 1.5 illustrates a block diagram of a photovoltaic power system. The photovoltaic array produces electric power when exposed to sunlight. The power interface is the governing component for the proper control, protection, regulation, and management of the energy produced by the array. In stand-alone systems, the loads are DC and/or AC electrical loads. In grid-tied (or utility-connected) systems, solar power is transmitted to the electric utility grid.  ^A ILI"^ xlikr.1111011111 \‘^ALL( ^11114\_,L\LA ^ ^ A. _it_ IL MN AWN  „^IL_ MM.. 1111.‘^  It _JEW  151"‘^It  r:  L^il... ?t ..._lt  =,-.3*L•%••\••^  ws, as  • --IL^—1L-11C4  (a)  ^  (b)  Fig. 1.4 (a) Photovoltaic modules integrated into the glass roof structure of Lehrter Station, Berlin, Germany (photo credit: PREDAC). (b) Photovoltaic modules integrated into the glass roof structure of the Fred Kaiser building of UBC, Vancouver, Canada.  PV array  Power delivery  Power delivery  Fig. 1.5 Block diagram of typical photovoltaic power system  1.4.1 Protection In most applications, the photovoltaic array acts as a power source to energize devices capable of storing electricity and/or supplying power to the utility grid. However, the capacity of  Chapter 1 Introduction and Overview ^  6  solar generation depends heavily on the presence of light. At night, a current may flow back to the photovoltaic cells from devices that can supply electric power. The reverse current must be avoided because it can cause leakage loss, extensive damage, or even fire [9]. Blocking diodes are effective to prevent this reverse current flow. Unfortunately, the forward voltage drop of diodes results in considerable power loss when they are conducting current. The latest designs use some specific mechanisms to replace the use of blocking diodes, including relays or contacts controlled by the reverse current detectors [10]. Islanding is defined as a condition in which a portion of the utility system that contains both load and distributed resources remains energized while isolated from the remainder of the utility system [8, 11]. Grid-connected power generations should always avoid islanding because it can be hazardous and may interfere with the restoration of interrupted utility-grid service. Gridconnected inverters therefore require some anti-islanding algorithms to ensure that photovoltaic power generation is effectively disconnected from the grid during any interruptions that occur with the utility system.  1.4.2 Control Photovoltaic arrays produce DC electricity. However, the majority of electric loads and interconnected utility requires AC. The quality of the AC power that is produced by solar power systems must generally meet certain standards [8, 11], such as voltage level, frequency, power factor and waveform distortions. Sun trackers adjust the direction and angle of photovoltaic panels to face the sun all day long. They are widely used in satellites and large-scale power plants to maximize the power generation. However, the costly installation and maintenance limits their applications in small photovoltaic power systems. In a 24-hour daily period, sunshine is only available for a limited time and heavily depends on weather conditions. Under a given specific condition, a photovoltaic module has a maximum power point (MPP), which outputs the maximum power. This is occasionally called the peak power point (PPP) or optimal operating point (OOP) in the literature [8]. The electrical characteristics of photovoltaic modules are usually represented by the plot of current versus voltage, I-V curve, as illustrated in Fig. 1.6. The normalization is based on the maximum power  Chapter 1 Introduction and Overview^  7  point. The plot of power versus voltage (P-V curve) shown in Fig. 1.6 is also important in demonstrating the maximum power point.  o Maximum power point 0.2^0.4^0.6^0.8^1 1.2 Norm. voltage (VN) Fig. 1.6 Normalized output characteristics of photovoltaic module at the standard test condition The I-V curve and maximum power point of photovoltaic modules change with solar radiation and temperature, which will be discussed in Section 2.1.3. In most photovoltaic power systems, a particular control algorithm, namely, maximum power point tracking (MPPT) is utilized because it takes full advantage of the available solar energy. In some literature, such as [8], it is termed as the "peak power tracking". Maximum power point tracking is essential for most photovoltaic power systems because it results in a 25% more energy harvest [12]. The objective of maximum power point tracking is to adjust the power interfaces so that the operating characteristics of the load and the photovoltaic array match at the maximum power points. The techniques of maximum power point tracking will be briefly introduced in Section 2.2 and further developed in Chapter 6.  1.5 Research Motivation and Objective Photovoltaic power is an established technology with several advantageous characteristics as described in the previous sections. Although the photovoltaic cells are the key parts of most  Chapter 1 Introduction and Overview ^  8  systems, further work is required not only to improve the photovoltaic performance, but also to optimize the interaction between the photovoltaic cells and other components. The most important aspect of this study is to improve the control of the power interface and to optimize the operation of the overall power system. The primary concentration is on developing innovative topologies and methods of maximum power point tracking in order to improve the efficiency of solar energy harvest. To limit the scope of the analysis, a conventional photovoltaic test-bench system was designed for this study to provide a typical representation of a standard, small-scale, solar generation system. The contributions of this work comprise of the development of analysis and control methods for photovoltaic interfaces operating with maximum power point tracking. The strength of the proposed thesis is that the techniques are applicable to the majority of photovoltaic power systems. The details of the contributions are: ■ The development of comprehensive equivalent circuits and mathematical models of photovoltaic modules that are useful for analysis and simulation. ■ A unique experimental design and method for evaluating the performance of maximum power point tracking. ■ A proposed strategy in terms of photovoltaic voltage regulation and maximum power point tracking for photovoltaic power systems. ■ A complete simulation solution for a typical photovoltaic power system. ■ Practical implementation of a real-time digital-controlled photovoltaic power system with maximum power point tracking. ■ Application of advanced control algorithms (e.g., affine parameterization and anti-windup) in DC-DC converter control to achieve both robustness and performance.  1.6 Thesis Overview The current chapter first describes a literature review and an introduction to photovoltaic power systems. The background of solar power generation and control issues is also briefly discussed. This is followed by the goals and contribution of the thesis.  Chapter 1 Introduction and Overview ^  9  Chapter 2 introduces the problem statement, as well as the topology and operation of a specific photovoltaic power system with maximum power point tracking. It complements the literature review in Chapter 1 and gives numerous references and solution analysis, and reveals the general properties and characteristics of maximum power point tracking. Chapter 3 presents a detailed modeling process of photovoltaic cells used to configure a computer simulation model, which is able to demonstrate the photovoltaic characteristics in terms of environmental changes in irradiance and temperature. Additionally, methods to extract the circuit parameters are developed as well and experimental results prove the validity of the proposed methods. Chapter 4 describes a comprehensive analysis of various topologies suitable for photovoltaic power systems. It also proposes a specific bench system designed for evaluating maximum power point tracking. Chapter 5 presents the modeling, analysis, and control in regulating photovoltaic voltage. The controller synthesis is based on Youla parameterization. The background knowledge of Youla parameterization used for controlling DC/DC converters is also illustrated in Appendix-C. Chapter 6 proposes two approaches in searching for maximum power points. One algorithm is known as the Improved Maximum Power Point Tracking (LMPPT). Another method is about the real-time identification of the optimal operating point in photovoltaic power systems. The two methods are evaluated by computer simulations and experimental tests. Chapter 7 includes a summary of the experimental results, the conclusion, and future research directions.  10  Chapter 2 Problem Statement The following sections will define and state the issues and give the idea of the subject and direction of this study. The first part presents the fundamental limitations of maximum power point tracking. The second section reviews the existing techniques of maximum power point tracking.  2.1 Fundamental Limitations of Maximum Power Point Tracking Some fundamental limitations restrict achievable performance of maximum power point tracking in photovoltaic power systems.  2.1.1 Variety of Photovoltaic Materials Photovoltaic cells are mainly made of single-crystalline silicon, polycrystalline, amorphous silicon, and thin films [4], which vary from each other in terms of efficiency, cost, and technology. This also results in their different output characteristics. Fig. 2.1 illustrates the plots of measured I-V curves of two different photovoltaic modules, ST10 and BP350, of which specifications are given in Appendix A. For easy comparison, they are demonstrated in normalized formats and plotted together. The ST10 is made of Copper Indium Diselenide (CIS). The plots show that the I-V curve of ST10 is "softer" than that of BP350, which is made of polycrystalline silicon. By comparing the voltage of maximum power point to the open-circuit voltage, the percentages are 71.59% and 76.24% for ST10 and BP350 respectively. By  Chapter 2 Problem Statement^  11  comparing to the short-circuit current, the optimal percentages of current are 86.63% and 93.96% for ST10 and BP350 correspondingly.  0.9 0.8 Ze 0.7 — 0.6 a)  0.5  E 0.4 &:5 0.3 0.2 0.1 0  CIS thin film 0  multforystalline silicOn  0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Norm. voltage (VN)  Fig. 2.1 Plots of normalized I-V curves of ST I 0 and BP350.  2.1.2 Switching-Mode Power Converters Switching-mode power converters such as DC/DC and DC/AC are the major photovoltaic interfaces. However, they generally behave as nonlinear and time-varying systems, which require proper control techniques to achieve stability, robustness, and performance. The further analysis will be given in Section 4.3, Chapter 5, and Appendix C. Ripples in the photovoltaic voltage are unavoidable when a switching-mode converter is used as the power interface. Further, the continuous oscillation around the optimal operating point is an intrinsic problem of some algorithms of maximum power point tracking This generally results in conversion loss and degrades the control performance of maximum power tracking. Fig. 2.2 gives simulation results to demonstrate the impact on power generation caused by switching ripples on the photovoltaic voltage. This analysis is based on the simulation model that is developed in Chapter 3.  12  Chapter 2 Problem Statement^  1.021^ 1.01 Cr) 0 0- 0.99E 0.98  — Power ripples caused by 1% voltage ripple ^ Power ripples caused by 5% voltage ripple  10^15^20^25^30^35^40  — 1% voltage ripple ^ 5% voltage ripple  > 0.85 a)  0.8  > 0.75  E  0.7  5^10^15^20^25 Time step  30  35  40  Fig. 2.2 Simulated deviation from maximum power point caused by photovoltaic voltage ripple According to the product datasheet, the photovoltaic module, BP350, is able to output 50W electric power under the Standard Test Condition (STC). By comparing with this value, the losses resulting from the ripples are calculated and summarized in Table 2.1. It shows a ±5% ripple of photovoltaic voltage contributes about 0.70% power loss due to the deviation of the maximum power point, which is significant in contrast to the 96% conversion efficiency of the inverters [13, 14]. Table 2.1 Loss analysis in a particular module, BP350, caused by switching voltage ripples Voltage ripple 1%  Average power 49.9900W  Loss (%) 0.03%  Efficiency 99.97%  2%  49.9447W  0.11%  99.89%  5%  49.6494W  0.70%  99.30%  10%  48.5054W  2.99%  97.01%  2.1.3 Time-variant parameters The electrical characteristic of solar cell is normally specified under the standard test condition (STC), where an average solar spectrum at Air Mass (AM) 1.5 is used, the radiation is  13  Chapter 2 Problem Statement^  normalized to 1000W/m 2 , and the cell temperature is defined as 25°C. The yearly-averaged solar constant is 1368W/m 2 at the top of the earth atmosphere according to the statistics of NASA National Aeronautics and Space Administration [15]. Compared to space application, significant solar radiation is filtered and blocked by the atmosphere and cloud cover before it is received at the earth surface. Atmospheric variables dramatically affect the available insolation for photovoltaic generators. Consequently, I-V curves and maximum power points (MPPs) of photovoltaic modules change with the solar radiation, as illustrated in Fig. 2.3, where Ga symbolizes the solar insolation. 3.5 3 2.5 2 f2 5 1.5 o ' 1  * MPP at G a =1000W/m 2 o MPP at G a =800W/m 2 A  0.5  MPP at G a =600W/m 2 MPP at G a =400W/m 2  o MPP at G a =200W/m 2 00^  5  10^15 Voltage (V)  20  25  Fig. 2.3 Simulated I-V curves of BP350 influenced by insolation when the cell temperature is constant 25°C Besides insolation, another important factor influencing the characteristics of a photovoltaic module is the cell temperature, as shown in Fig. 2.4. The variation of cell temperature changes the MPP greatly along X-axis. However, a sudden increase or decrease in temperature seldom happens. The variation of cell temperature primarily depends on the insolation level, ambient temperature, and cell conduction loss. Fig. 2.3 and Fig. 2.4 illustrate the effects of insolation and temperature separately. Nevertheless, it is actually impossible to encounter one without the other because they are highly coupled under natural operating conditions. An increase in the insolation on photovoltaic cells normally accompanies an increase in the cell temperature as well. The  Chapter 2 Problem Statement ^  14  experimental test [16] has shown that the open-circuit voltage of the photovoltaic module is higher before the operation starts and lower after several cycle operations due to the change of cell temperature. 3.5 3 2.5 Q C  1 0.5 00^  * MPP when temperature=-25 ° C MPP when temperature=0 ° C M PP when temperature=25 ° C 0 MPP when temperature=50 ° C  10^15^20 Voltage (V)  ^  25  ^  30  Fig. 2.4 Simulated I-V curves of BP350 influenced by cell temperature when the insolation is constant 1000W/m2  2.1.4 Non-Ideal Conditions The non-ideal conditions refer to some specific situations that some solar cells cannot give the specific power. These phenomena are also referred to "non-optimal conditions" or "unbalanced generations" in some literature [17, 18]. The problem has drawn recent attention [17-22]. The common non-ideal conditions include partial shading, low solar radiation, dust collection, and photovoltaic ageing. The following paragraphs will address two major effects resulting from partial shading and low solar radiation.  2.1.4.1 Partial Shading Generally, it is preferable to build a solar array with the same panels and keep the photovoltaic array away from any shading. However, it is not easy to avoid shading for residential installations, since the sunlight direction changes from sunrise to sunset in a day  Chapter 2 Problem Statement^  15  period. The partial shading can be caused by obstacles, such as trees, other constructions, or birds etc. The study [21] have revealed that a significant reduction in solar power output is due to minor shading of the photovoltaic array. This study designed and conducted a specific experiment to quantify the significance of the shading effect on photovoltaic power systems. Shown in Fig. A.1 of Appendix A, two identical photovoltaic modules of BP350 were installed on the same frame with the same direction and angle on the roof. This allows testing them at the same time and under the same condition. Each BP350 module comprises of seventy-two photovoltaic cells as described in Table A.1 of Appendix A. The data acquisition system is configured and used to operate and record simultaneously the outputs of both photovoltaic modules. First, the experiment tested and calibrated the I-V curves of two photovoltaic modules under the same environmental condition. Second, the system is re-configured to measure the output characteristics when one cell is intentionally shaded. Again, the I-V curves of two photovoltaic modules are measured simultaneously while one of the two modules is partially shaded. The different output characteristics of one-cell-shaded module and non-shaded module are shown in Fig. 2.5 in terms of I-V curves. The peak power point degrades to 15.44W from 21.48W when one of seventy-two cells is shaded. When the shaded module is series-connected with other nonshaded modules, an additional loss will be produced and discussed in following paragraphs. The output characteristics are illustrated in Fig. 2.6 and Fig. 2.7 when the one-cell-shaded photovoltaic module is connected to other non-shaded modules in series. Fig. 2.7 demonstrates that there are two peak power points in P-V curves due to the shading condition. None of them can represent the true maximum power point that is equal to the sum of the individual maximum power of each photovoltaic module. Under this condition, the true maximum power point is hidden and the significant losses caused by series-connected structure are tabulated in Table 2.2. The study shows that the shaded cell contributes 14.06% power loss and additional 11.57% loss results from the hidden maximum power point in case that two modules are connected in series. When three or more modules are series connected, the situation is even worse as summarized in Table 2.2. So far, the discussion was based on one-cell-shaded situation. The shading condition could be more complicated in practical photovoltaic applications. This generally makes it difficult to perform the maximum power point tracking.  • • 16  Chapter 2 Problem Statement^  1.6 1.4 X: 16.16 ■ 1Y: 1.329  0  X: 17.19 Y: 0.8982  T., 0.8 0  E  0.6 0.4  —No cell shaded ^ One cell shaded  0.2  6  10^12^14^16^18^20^22 Photovoltaic voltage (V)  Fig. 2.5 Measured I-V curves of one-cell-shaded and non-shaded modules  . ..... . . * e n a, 3 6 c t . !II.V .41 ■1 ANA :AV . • + ■ • • • • • • • • ■ • ■ • -r .., , ,,,^  1. :^  \  ti f1  0 1 0.9 0 0 -0 0.8 0.7 0.6 0.50  —Two series-connected modules ^ Three series-connected modules ^ Four series-connected modules Five series-connected modules 20^40^60 Photovoltaic voltage (V)  80  100  Fig. 2.6 Measured I-V curves when the one-cell-shaded module is in series connected with other non-shaded modules To increase the conversion efficiency, some popular inverter designs [13, 14] use the series configurations of photovoltaic modules to obtain a moderate-voltage DC source, i.e. about 200V.  17  Chapter 2 Problem Statement^  This requires at least 300 solar cells connected in series. The non-ideal effect might be critical for these systems. 90  85.91^85.21  80 70  6761  64:43  g 60  \ 50.12  a 50 0  42'95  2 40  2 30  / 32.65  20 10 00^ 20  —Two series-connected modules ^ Three series-connected modules ^ Four series-connected modules Five series-connected modules 40^60 Photovoltaic voltage (V)  80  100  Fig. 2.7 Measured P-V curves when the one-cell-shaded module is serially connected to other non-shaded modules Table 2.2 Losses caused by series-connected modules under one-cell-shaded condition Connections Two series-connected modules  Hidden MPP Achievable MPP Losses (%) 36.92W 32.65W 11.57%  Three series-connected modules  58.40W  50.12W  14.18%  Four series-connected modules  79.88W  67.61W  15.36%  Five series-connected modules  101.36W  85.91W  15.24%  2.1.4.2 Low Radiation In a one-day period, the solar radiation varies in a large range. The performance of maximum power point tracking at low radiation is an issue for controller designs. It is pointed out that the Perturbation and Observation (P&O) algorithm does not work well at low radiation since the derivative of power over voltage (dp/dv) is small [23]. From these findings, it became clear that there was a need to develop a proper control algorithm, whose performance is not  18  Chapter 2 Problem Statement^  degraded in low radiation conditions. This condition also causes control issues, which will be addressed in Chapter 5.  2.2 Techniques of Maximum power point Tracking In recent years, many publications give various solutions to the problem of maximum power point tracking. Generally, the tracking algorithms can be classified in three categories, the heuristic search, extremum value theorem, and linear approximation methods.  2.2.1 Heuristic Search The heuristic search used for maximum power point tracking is called best-first search or hill climbing. The name "hill climbing" comes from the idea that you are trying to find the top of a hill, and you go in the direction that is up from wherever you are [24]. This technique often works well to find a local maximum since it is simple and only uses local information. Fig. 2.8 illustrates the application of hill climbing for maximum power point tracking. From the open-circuit position, ( v 0 ,0), the controller makes an adjustment of operating point to point ( v1 , p 1 ). The direction of movement is decided by the corresponding change of power level. The value of v1 is updated according to (2.1), where Av is a constant step. Since the new operating point ( v1 , p 1 ) make the photovoltaic modules output more power than that of former condition, the relocation to ( v 2 , p 2 ) is made. This continues until the movement from ( v 4 , p 4 ) to (v5 , p 5 ), the controller realizes the reduction of output power and moves the operating point back to ( v4 , p 4 ), then ( v, , p 3 ). This process continues until the operating point moves backward and forward around the maximum power point (v4 , p 4 ). The continuous perturbation and observation guarantees that the controller can always find the new maximum power point regarding the variation of solar insolation and cell temperature. Vk+1 - Vk ± 7-  AV  ^  (2.1)  The technique of heuristic search is more like a rule of thumb rather than a theory [24]. Therefore, the academic research [16, 25, 26] is not as extensive as the methods based on  Chapter 2 Problem Statement^  19  extreme value theorem. The benefit of this algorithm lies in its simplicity in digital implementation and requires no mathematical information about the shape of the hill curve. However, the perturbation of the photovoltaic voltage results in power loss due to the deviation from the maximum power point, which is illustrated in Fig. 2.2. The choice of the step size Av is critical for the tracking performance. The large step makes the system respond quickly to the sudden change in the optimal operating condition with the trade-off of poor steady-state performance. The small step size shows good steady-state performance. However, it slows down the tracking to the new optimal operating point caused by rapid changes of insolation. It also causes tracking instability which is explained in [16, 27]. The tuning of step size is awkward because the proper value relies on many factors, e.g. photovoltaics, power interfaces, and climate.  (  0.8 6 0.6 3 0  pi  1  a E 0.4  )  0  z  0.2  0.2^0.4^0.6^0.8^1 Norm. Voltage (VN)  ,0  )  Fig. 2.8 Conceptual operation of hill climbing in P&O MPPT algorithm  2.2.2 Extremum Value Theorem According to the extremum value theorem [28], the extremum, maximum or minimum, occurs at the critical point. If a function y = f (x) is continuous on a closed interval, it has the critical points at all point x o where f(x0 ) = 0 . In photovoltaic power systems, the local maximum power point can be continuously tracked and updated to satisfy a mathematical equation: dP/dx = 0, where P represents the photovoltaic power and x represents the control  Chapter 2 Problem Statement^  20  variable chosen from the photovoltaic voltage, photovoltaic current or switching duty cycle of power interfaces.  2.2.2.1 Perturbation and Observation Method Researchers have proposed many methods of maximum power point tracking based on the extreme value theorem. In photovoltaic power systems, one popular algorithm is the Perturbation and Observation method (P&O) [29]. The operation is regulating the voltage of photovoltaic array to follow an optimal set-point, which represents the voltage of the maximum power point (Vmpp ), as shown in Fig. 2.9. In some literature, The  Vmpp  is also symbolized by Voop, which  stands for the voltage of optimal operating point. To find this optimal operating point, their operation can be symbolized as (2.2). = Vk  dp dv  Ay • sign(—  MPP Tracker  ^  (2.2)  )  Measured PV voltage and current (Vpv and Ipv)  Limiter Control command  Controller VMPP  Power Conditioner  Photovoltaic Array  Power delivery  Vv  Load  Measured PV voltage  Fig. 2.9 The block diagram of P&O and IncCond MPPT topologies By investigating the power-voltage relationship (P-V curve) of a typical photovoltaic module shown in Fig. 2.8, the maximum power points can always be tracked if we keep dP/dV equal to zero for any solar insolation or temperature, since all local maximum power points have the same mathematical attribute. In recent applications, dc-dc converters and dc-ac inverters are widely used as the power interfaces between photovoltaic arrays and loads. Most converters or inverters are realizable with semiconductor switches operated by Pulse-Width Modulation (PWM). Since the modulation duty cycle is the control variable of such kind of systems, another implementation of P&O  ^  21  Chapter 2 Problem Statement^  method is presented in [30-32], which is based on a hill-shape curve representing the relationship of array output power and switching duty cycle. Therefore, a variation of these methods is to directly use the duty cycle of switching mode converter or inverter as a MPPT control parameter and force dP/dD to zero, where P is the photovoltaic array output power and D is the switching duty cycle. This topology ignores the inner loop of voltage regulation, as shown in Fig. 2.9, and adjusts the switching duty cycle directly by the maximum power point tracker. Continuous oscillation around the optimal operating point is an intrinsic problem of the P&O algorithms as shown in Fig. 2.10. The plots illustrate the measured signals of the photovoltaic voltage, Vpv, the photovoltaic current, Ipv, and the output power Ppv. In the steady state, the continuous oscillation of the operating point around the Vinpp make the averaged power level biased from the maximum power point. It is also pointed out that the P&O algorithms sometimes deviates from the optimal operating point in the case of rapidly changing atmospheric conditions, such as broken clouds [27].  - P&O tracking ^ True V  1.2  mPP  > 1.1 ••••• 1 6 2^6.3 ".•••••••••  z  .01n 1 es  6.4  1^  6.5  6.6  6.7  6.8  wl•••-.% • 1....s-uo.  ^0.8^ _0 0.6• 0 .4 ^ .6 0.2  z  g  6 2^6.3  6.4  6.9  6.5  ^  6.6  ^  P&O tracking True I mpp 6.7  6.8  6.9^7  s.....r.our■v•-•• rnomm....wegbao•  0.8 8. 0.6 0- 0.4 0.2 ^ 0 z^6.2^6.3^6.4  P&O tracking ^ True P mPP  6.5^6.6^6.7^6.8 Time (s)  6.9^7  Fig. 2.10 Measured signals of photovoltaic panel with the operation of P&O  Chapter 2 Problem Statement^  22  2.2.2.2 Incremental Conductance Method The Incremental Conductance method (IncCond) [27] is developed to eliminate the oscillations around the maximum power point (MPP) and avoid the deviation problem. It is also based on the same topology as shown in Fig. 2.9. However, the experiments showed that there were still oscillations under stable environmental conditions because the digitalized approximation of maximum power condition of dI/dV = -I/V, which is equivalent to dP/dV = 0, only rarely occurred. The problem is caused by the local truncation error of numerical differentiation, which will be discussed in Chapter 6. Another drawback for both algorithms is the difficulty in choosing the proper perturbation step of photovoltaic voltage [31]. Although both P&O and IncCond were developed based on extreme value theory, its operation is similar to the hill climbing method. The reason is that both rely on the numerical approximation of differentiation, of which the stability and accuracy is difficult to be guaranteed in practical applications considering noise and quantization error etc.  2.2.2.3 Method of Steepest Descent One principle is originally from the optimization method in applied mathematics. The method of steepest descent, also called gradient descent method [33], can be used to find the nearest local minimum of a function. The same principle can be used to find the maximum by reversing the gradient direction. This can be applied to find the nearest local maximum power point when the gradient of the function can be computed. Different from the typical P&O method, an algorithm of maximum power point tracking [16, 25, 34] is demonstrated by (2.3), AP where AM is the step-size corrector, and ^ isi the digitalized approximation of the derivation, V dP/dV. AP and A V are the incremental change of photovoltaic power and the incremental step  of photovoltaic voltage, respectively. Vk+1 = Vk ±  AM  AP AV  (2.3)  Theoretically, this method is better than the ordinary P&O method in terms of fast tracking dynamics and smooth steady state. However, this tracking method uses numerical differentiation  Chapter 2 Problem Statement ^  23  to represent the local uphill gradient since the mathematical function of power versus voltage is time-variant due to the environmental change. Numerical differentiation is a process of finding a numerical value of a derivative of a given function at a given point. Generally, numerical differentiations are more tricky than numerical integration and requires more complex properties such as Lipschitz classes [35]. Further, the Euler methods of numerical differentiation have severe drawbacks, which will be discussed in Chapter 6.  2.2.3 Linear Approximation Methods The linear approximation methods try to derive a fixed percentage between the maximum power point and other measurable signals, such as the open-circuit voltage and short-circuit current etc. These methods generally lead to a simple and inexpensive implementation. They are also designed to avoid the problems caused by "trial and error". Some studies [36-40] indicated that the optimal operating voltage of photovoltaic module is always very close to a fixed percentage of the open-circuit voltage. This implies that the maximum power point tracking can simply use the open-circuit voltage to predict the optimal operating condition, which is called voltage-based MPPT (VMPPT). Similarly, studies [41] illustrate the linear approximation between the maximum power point and the short-circuit current. This is the so called current-based MPPT method (CMPPT). However, this condition varies with different material of photovoltaic cells. The experimental study in Fig. 2.1 shows that the optimal percentage of voltage is 71.59% and 76.24% for ST10 and BP350 respectively. According to the product datasheet [23, 42], there is a gradual decrease of module power, which is up to 3% during the first few months of deployment. Besides this, the same models mounted under the same condition can show different performance as illustrated in Fig. 2.11. The installation of two BP350 modules is demonstrated in Appendix A as Fig.A.1 These optimal ratios are also time-variant due to the variation of insolation and cell temperature in a daily period. Table 2.3 provides the measured data acquired at different times. The  max  represents the available maximum power level, the 2 v symbolizes the ratio of the  voltage of maximum power point over open-circuit voltage, and the^stands for the ratio of the  24  Chapter 2 Problem Statement^  current of maximum power point over short-circuit current. This study shows that these ideal ratios and maximum power level vary with the environmental change in terms of insolation and temperature. 18 16  Module #1 Module #2  14 12 , 10 0  8 6 4 4^ -6-  8  10 12 14 16 18 20 22 V pv  Fig. 2.11 Measured P-V curves of two BP350 mounted on the same frame Table 2.3 Time variant characteristics of optimal ratios and maximum power levels Module #1  Module #2  Time 2„ (VN) 27 (A/A) ''max (W) (V N) 2 (A/A) ''max (W) 10:09 AM 12.85 0.7840 0.9221 12.70 0.8038 0.9545 11:18 AM 17.38  0.7673  0.9018  17.26  0.7608  0.9438  11:58 AM 19.82  0.7574  0.9053  19.62  0.7514  0.9434  21.94  0.7339  0.8965  21.77  0.7275  0.9343  2:40 PM  Other approaches [43] were presented in past years. They are based on a linear approximation between the maximum output power and the optimal operating current. It supposes that the change of operating condition results only from the variation of insolation and assumes the temperature is constant. However, the linear relationship does not exist when the temperature effect illustrated in Fig. 2.12 is considered. Although a sudden increase or decrease in temperature seldom happens, the continuous solar irradiance on the surface of photovoltaic cell and conduction though the photovoltaic cells will cause a gradual increase in cell temperature, which results in gradual reduction of the photovoltaic output power and shift the optimal operating point to a different value. Further, the dynamics of photovoltaic cell  Chapter 2 Problem Statement^  25  temperature is difficult to be measured and modeled, since it is also affected by other factors, such as ambient temperature, insolation level and cell conduction loss etc. 100  0°C --- 25° C ^ 50 ° C 75 ° C —  80  g 60 a_ 40 20  3^4^5^6 Current (A)  Fig. 2.12 Temperature effect on the relationship of power and current of MSX-83 solar panel with a constant insolation equal to 1000W/m 2  2.3 Summary This thesis carries out extensive research to obtain knowledge regarding the issues of maximum power point tracking. This chapter complemented the literature review in Chapter 1 in that numerous references were given to similar systems and phenomena as those discussed in the sections above. The simulated and experimental results presented in this chapter were a motivation factor that initiated the solutions to existing problems. First, the study shows some important limitations that affect the performance of maximum power point tracking. It also emphasizes the partial shading effect, which significantly degrades the solar power generation. Then, the study summarizes the existing approaches to the problem of maximum power point tracking. The methods include the Heuristic search, the extremum value theorem, and the linear approximation. To avoid the drawbacks addressed in this chapter, further development is desirable to find an improved control algorithm to operate solar power systems efficiently.  26  Chapter 3 Photovoltaic Modeling 'Mathematical Modeling is an important subject to improve the design of products and simulate the behavior of a certain system. This work presents several comparative modeling approaches of photovoltaic cells to configure a computer simulation model, which takes into account of environment impacts of irradiance and temperature. Two major types of cells, namely crystalline silicon and thin film, are used as examples to demonstrate that there is a good agreement between the solution of the model and the practical data. The study shows that the proposed model will be valuable for design and analysis of the photovoltaic power system. Fig. 3.1 illustrates a typical model to represent the characteristics of photovoltaic cell. The insolation G a and cell temperature T are input variables. The output is the direct current. The photovoltaic voltage is the feedback signal forcing the output current to follow the I-V curves according the present condition of insolation level and cell temperature.  Model of Photovoltaic cell  Fig. 3.1 Model of photovoltaic cell  A version of this chapter has been published. W. Xiao, W.G. Dunford, and A. Capel, "A Novel Modeling Method for Photovoltaic Cells", IEEE PESC 2004, June 20-25, 2004, Aachen, Germany.  Chapter 3 Photovoltaic Modeling^  27  3.1 Equivalent Circuits The traditional equivalent circuit of a solar cell is represented by a current source in parallel with one or two diodes. A single-diode model from [44] is illustrated in Fig. 3.2, including four components: a photo current source, /ph, a diode parallel to the source, a series resistor, R 3 , and a shunt resistor, Rp . There are five unknown parameters in a single diode model. In the doublediode model [45], an additional diode is added to the single-diode model for better curve fitting. Due to the exponential equation of a p-n diode junction, the mathematical description of the current-voltage characteristics for the equivalent circuit is represented by a coupled nonlinear equation, which is generally complex for analytical methods. The proposed modeling process is divided into two steps. First, the modeling methods are presented. Finally, the accuracy of modeling method is evaluated and analyzed through the comparison of simulation to the practical data.  V  • Fig. 3.2 Equivalent circuit of a single-diode model  3.1.1 Simplified Models The study [46] proves that the Simplified Single-Diode Model (SSDM) is sufficient to represent three different types of photovoltaic cells when the temperature effect on parameterization is taken into account. The shunt resistor Rp is removed in the model, as shown in Fig. 3.2. The current through a photovoltaic cell is given by (3.1) and (3.2).  ^1 e (  i = I ph  —  v+iR,  I sat [  vi =  )  vt  AkT q  —1  (3.1) (3.2)  Chapter 3 Photovoltaic Modeling^  28  where /ph represents the photo current, v is the voltage across the cell, v t is the thermal voltage, Rs is the series resistor, and /sat is the saturation current of the diode. The thermal voltage 1/, is a function of the temperature T, where A represents the ideality factor of diode, k is the Boltzman constant, and q stands for the charge of an electron. The series resistor of SSDM is neglected to form a Further Simplified Single-Diode Model (FSSDM) developed in [47], as shown in Fig. 3.3. The significance of this model is that it decouples the current-voltage relationship as v  i = Iph  —  ID = 'Ph  —  'sat  em —1  (3.3)  Fig. 3.3 Equivalent circuit of a further simplified single-diode model  3.1.2 Parameterization There are four and three parameters in SSDM and FSSDM respectively that need to be derived from the data obtained from experimental tests of photovoltaic cells under three conditions: short-circuit, open-circuit, and maximum power point. According to the short circuit situation in equivalent circuit, the photo current iph is approximated as 'ph '"'"I  1  sc  (3.4)  where iph represents the estimated photo current /ph, and i„ is the measured short-circuit current at a certain test condition with constant irradiance and temperature. The photo current is a function of insolation and temperature. Based on the open-circuit condition in the equivalent circuit, the saturation current /sat of the diode is expressed as:  29  Chapter 3 Photovoltaic Modeling^ t =^ph  ^  (3.5)  e v` —1  The variable is., represents the estimated saturation current Isat , and J70c is the measured open-circuit voltage at the same test condition as that of short-circuits measurement. The thermal voltage v t is a function of temperature T, which needs to be determined. The maximum power point (vmpp , impp) is measured through experimental tests under the identical testing environment. From (3.1), the maximum power point of SSDM is described as impp = -I ph —^sat  e^vi^  Substitution of (3.4) and (3.5) into (3.6) gives eI  -1  (3.6)  _1  v„,pp+vii'mppRs)  iMpp^'Sc  (3.7)  Isc e (:) —1  Re-organizing (3.7), we obtain the relationship of Rs and vt in a SSDM v In  i  riTPP  e (17:1 d-i;PP  sc  '  V  mPP  Sc  (3.8)  Rs = mPP  According to the power-voltage characteristics of photovoltaic cell, the maximum power points occur when dP/dV = 0, where P represents the photovoltaic module output power and V represents the photovoltaic voltage. The equation can be represented by di dv  imPP =  0  vmpp  (3.9)  From (3.1), we have  = 1 sat{e  v+vitR,^  di — dv^vt  which gives,  )dii} + ( Rv dv s  t  (3.10)  30  Chapter 3^Photovoltaic Modeling  '  di dv ^R  sat  +: v e(" -  —  R  vt V= Vmpp  "1  (3.11)  ^( v"'PP+4 PPE  sat^s  Vt  The parameters, A and R„ in (3.1) that best represent the output characteristics of the solar cell can be numerically determined by solving (3.9) and (3.11). Examples of available numerical analysis methods are Bisection and Newton-Raphson (NRM). Although the parameterization described above is only based on the SSDM, it is easy to follow the same procedure and make Rs = 0 to determine the parameters in FSSDM.  3.2 Temperature Effect on Parameterization This study also demonstrates that the parameters of ideality factor and series resistance are influenced by temperature. Fig. 3.4 shows temperature characteristics of ideality factor A and series resistance Rs for a specific photovoltaic module, Shell ST40. 2 0  0  42 1.5  10^20^30 Temperature ( ° C)  40  50  10^20^30 Temperature ( ° C)  40  50  :CI 0.045 U)  ad 0.04 •a5 a)  0.035  ce 0.03 a) 1?) 0.0250^  Fig. 3.4 Temperature effects on ideality factor and series resistance of equivalent circuits  • Chapter 3 Photovoltaic Modeling^  31  The temperature characteristics of ideality factor and series resistance can be represented by (3.12) and (3.13), correspondingly. The K A and K 1,s are the temperature coefficients for ideality factor and series resistance, respectively. These can be easily implemented into the model shown in Fig. 3.1 to improve the modeling accuracy. The relative modeling error of the maximum power, er„, is defined as (3.14), where P nwp is the maximum power derived by mathematical models and p, represents the measured value of maximum power. Fig. 3.5 illustrates the plots of relative modeling error of maximum power. Compared to the result (dash line) when the temperature effect on model parameters is ignored, the relative modeling error of the maximum power is dramatically reduced when the temperature influence is integrated with the simulation model. The specification of Shell ST40 is shown in Appendix A. A(T) = A o + A T  (3.12)  R S (T) = R, o + K izs T  (3.13)  e nnpp  P  mpp P mpp  (3.14)  P mpp  ".".Z.' 3.5 3 3 0 a_ E 2.5 2 E 1 5 a) _c 0 1 .  0.5 rn  •  0  E  -0.5  •  -I  a)  ____ ---0—ConsIderation of temperature effect ^  •••*. ^ ^ No consideration of temperature effect **-. ^ ^ ^ -20^-10^0^10^20 30 40 50 Temperature (°C)  Fig. 3.5 Relative simulation error in term of temperature change based on Shell ST40  32  Chapter 3 Photovoltaic Modeling^  3.3 Polynomial Curve Fitting Besides the equivalent circuit modeling methods, this study proposes the Polynomial Curve Fitting (PCF) to represent the electric characteristics of photovoltaic modules. A n th degree polynomial can be represented by (3.15). The residual is the sum of deviations from a best-fit curve of arbitrary form [48]. The residual is given by (3.16), where m represents the number of samples. y = a o +^+ a 2 x 2  ± • • • ±  (3.15)  X fi  R2 =E[y —(a 0 +^+ a 2 .4 + • • • + b„x;')]^ (3.16)  It was determined that a fourth-order polynomial can demonstrate the output feature, the I-V or P-V curve, of photovoltaic cells made of CIS thin film. For the cells made of crystalline silicon, a higher order, up to sixth-order, equation is needed. The number of orders is decided according to the measure of deviation in term of the norm of residual. The unknown parameters of a polynomial model are determined by the Least-Squares method (LS) when the measured data of photovoltaic voltage and current are available. To illustrate the parameterization process, a sixth-order I-V relationship of silicon cell is defined as (3.17), where k is the sample index. The physical meaning of the parameter bo is the short-circuit current, which is proportional to the insolation level. i(k) = bo + bi v(k) + b2 v 2 (k)+19/ 3 (k)+ b4 v 4 (k)+ b5 v 5 (k)+ b6 v 6 (k)  (3.17)  The variables i(k) and v(k) are the photovoltaic output current and voltage respectively, in the discrete time domain. The b, are the parameters of polynomial. For the convenience of representation, the following notations of vectors and matrix are introduced as (3.18), where m equals the number of samples. 1 1 1 1 •• • bo i(2) v(1) v(2) v(3) •• • v(m) Y = i(3) X = v 2 (1) v 2 (2) v 2 (3) •• • v 2 (m) ,B = b2 i(1)  ,  i( m )  v6(1) v 6 (2) v 6 (3) •• • v 6 ( m )_  b6  Then, a simple regression model can represent the measured output vector Y  (3.18)  Chapter 3 Photovoltaic Modeling^ Y = XTO  33 (3.19)  where X is the vector of regression variables observed from measurements. According to the theorem of least-squares curve fitting, the vector of estimated parameters (e) is given by [30, 31] .  (xTx) xTy  (3.20)  3.4 Modeling Accuracy To compare different modeling methods correctly, the performance indices are defined. The relative modeling error at the maximum power point is defined as emPP=  ^(17 nIPP ^2  —  V mPP )2^mPP imPP )2 ^•^2  V nipp ± 1 nipp  (3.21)  where the point (cmpp , impp ) is the estimated maximum power point derived by mathematical models and the point ( v mPP ,  mpp  ) represents the actual maximum power point. The absolute  voltage error of the optimal operating point is described as eoop  mpp — V  mppl  (3.22)  where the V' mpp stands for the estimated voltage of maximum operating point derived by mathematical models, and the -v mpp is the actual voltage of maximum operating point. Shown in (3.16), the residual, R 2 , demonstrates the curve fitting performance To analyze the modeling accuracy of the proposed models, two popular types of photovoltaic modules made of polycrystalline silicon and CIS thin film are implemented and evaluated by the presented modeling methods. The specifications of photovoltaic modules used in evaluation are summarized in Table A.2 and Table A.3 of Appendix A. The modeling accuracy is also analyzed in quantity through comparison between the practical data and simulation results. The data representing the modeling accuracy are summarized in Table 3.1. For the solar cells made of polycrystalline silicon, all models of SSDM, FSSDM and sixthorder polynomial shows good matching-performance, as shown in Fig. 3.6. The I- V curves of ST10 (CIS thin film) generated by three different models are illustrated in Fig. 3.7. The data of  34  Chapter 3 Photovoltaic Modeling^  modeling accuracy are summarized in Table 3.1. The evaluations show that the SSDMs shows the most accurate estimation of maximum power point and the PCF models demonstrates the best curve fitting performance. It also illustrates that the FSSDM shows good accuracy in modeling the cells made of crystalline silicon, but it has a problem to correctly represent the output characteristics of cells made of CIS thin film as shown in Fig. 3.7. Table 3.1 Comparison in modeling accuracy of SSDM, FSSDM and PCF Photovoltaic panels and Measured MPP Model: MSX83 Material: Crystalline silicon Measured MPP: (17.23V, 4.85A) Model: Shell ST10 Material: CIS thin film Measured MPP: (14.46V, 366mA)  Modeling Method SSDM  empp  eoop  (17.22V, 4.85A)  0.01%  0.007V  1.82x10-2—  FSSDM  (17.22V, 4.81A)  0.21%  0.007V  2.61x10-2  PCF  (16.96V, 4.94A)  1.59%  0.270V  6.02x10-4  SSDM  (14.47V, 366mA)  0.00%  OV  5.89x10-5  FSSDM  (14.47V, 312mA)  0.39%  OV  9.35x10-4  PCF  (14.65V, 356mA)  1.28%  0.18V  6.59 x 10 -6  (;)^ )  mPP mPP  R2  6 5 Q4 C' 3  ca. 0  2  Output of SSDM Output of FSSDM _s_ True MPP True I-V curve ---- 6 th -order polynomial fit  00^5^10^15  20  PV voltage (V)  Fig. 3.6 I V curves of MSX 83 modeled by SSDM, FSSDM and 6 th order polynomial -  -  The accuracy of estimating the voltage of optimal operating point,  -  V Glop,  is important since it  represents the output of maximum power point tracker and the reference for the voltage regulator in the presented MPPT scheme, as shown in Fig. 2.9. In terms of modeling accuracy of  V 0013 ,  both SSDM and FSSDM modeling methods show good estimations because both modeling processes are based on the maximum power point and contain the algorithm to minimize the  35  Chapter 3 Photovoltaic Modeling ^  modeling error at this point. The entries in Table 3.1 also show that the estimation of maximum power point in PCF model is unfavorable because the least-squares operation is designed to minimize the loss function of curve fitting and the matching of MPP is not considered into the modeling process. This situation can be improved by directly identifying the P-V curve instead of I-V curve as discussed in Chapter 6.  0.4 0.35 :5- 0.3 t 12 0.25 5 2, 0.2 -g. 0.15 0 0.1 0.05  Output of SSDM ^ Output of FSSDM _._ Measured MPP — True I-V curve ---- 4th-order Polynomial fit  ---.  -  00^5^10^15 PV voltage (V)  20  Fig. 3.7 I-V curves of ST10 modeled by SSDM, FSSDM, and 4 th -order polynomial  3.5 Summary The mathematical description of current-voltage characteristics for photovoltaic cells are traditionally represented by coupled nonlinear equations. This chapter introduces a simplified equivalent electrical circuit model that relates to the physical structure of a photovoltaic cell and proposes two modeling approaches that is suitable for computer simulation and controller design. This study also presents the use of polynomials to model the current versus voltage relationship of photovoltaic modules. This representation can be used for the real-time estimation of the maximum power point, which will be introduced in Chapter 6. The effectiveness of the proposed methods is successfully demonstrated by computer simulations and experimental evaluations of two major types of photovoltaic panels, namely crystalline silicon and CIS thin film. To compare different modeling methods correctly, the performance indices are also defined. The modeling accuracy is also analyzed in quantity through comparison between the practical data and  Chapter 3 Photovoltaic Modeling  ^  36  simulation results. The major modeling features of equivalent circuit and polynomial curve fitting are analyzed and summarized in Table 3.2. Table 3.2 Modeling features of equivalent circuit and polynomial curve fitting Models  ^  Pros  ^  Equivalent circuit  ^  It has physical meaning  0 It demonstrates the features of cell output in terms of environment changes in irradiance and temperature.  Cons  ^  It requires only three or four points for modeling process  ^  It demonstrates good performance in offline modeling  ^  It is difficult to solve by analytical methods  Polynomial curve fitting  ^  It is an ideal mathematical model useful for parameter identifications and numerical solutions in real time  ^  The overall modeling error is minimized when the least-square method is used  ^  It is a purely mathematical representation, no physical meaning  ^  The modeling accuracy is related to the number of available measurements  37  Chapter 4 Topologies of Photovoltaic Interfaces  1  This chapter will discuss topologies used for photovoltaic power systems to optimize the  operation of maximum power point tracking. The first section presents two system structures suitable for the photovoltaic features and maximum power point tracking. Second, the study compares two converter topologies that are widely used as photovoltaic interfaces. Finally, the test bench system will be proposed and designed to evaluate the performance of maximum power point tracking in the future.  4.1 System Structures According to the study in Section 2.1.4, the non-ideal conditions considerably downgrade the performance of maximum power point tracking, especially when photovoltaic modules are connected in series. To alleviate the complexity caused by interconnected photovoltaic modules, this study recommends an individual power interface for each photovoltaic module. Two gridconnected topologies are proposed and illustrated in Fig. 4.1 and Fig. 4.2, respectively. Both configurations ensure that other defective modules will not influence the overall array so that each individual photovoltaic module is able to work at its optimal operating point. Fig. 4.1 shows a photovoltaic power configuration, which consists of DC/AC MPPT modules, an AC voltage bus, and an anti-islanding device. The bus voltage is equal to the grid voltage. The anti-islanding device constantly senses the grid status and the power quality of solar  I A version of this chapter has been accepted for publication. W. Xiao, N. Ozog, and W.G. Dunford, "Topology Study of Photovoltaic Interface for Maximum Power Point Tracking", accepted for publication in IEEE transaction on Industrial Electronics, December, 2006.  Chapter 4 Topologies of Photovoltaic Interfaces^  38  generation. It connects the photovoltaic power system to the grid when the system is in a normal operation. It also is able to disconnect the system from the grid to avoid any islanding problem, which has been introduced in Section 1.4.1. The DC/AC MPPT module is the power interface between the photovoltaic module and the AC voltage bus. This is controlled by the algorithm of maximum power point tracking to generate the maximum possible solar power. The energy is eventually transferred to the grid via AC bus and the anti-islanding device. Photovoltaic module #1  DC/AC MPPT module #1  Photovoltaic module #2  DC/AC MPPT module #2  •  •  •  •  •  •  Photovoltaic module #n  DC/AC MPPT module #n  Anti- Islanding device  Grid  AC BUS  Fig. 4.1 Topology of grid-connected photovoltaic power systems with an AC bus configuration  d Photovoltaic module #1  DC/DC MPPT module #1  Photovoltaic module #2  DC/DC MPPT module #2  •  •  •  •  Photovoltaic module #n  DC/DC MPPT module #n  Critical DC loads Critical AC toads  DC/AC inverter with anti-I slanding  Grid  Battery or supercapacitor storage DC BUS  Fig. 4.2 Topology of grid-connected photovoltaic power systems with a DC bus and a UPS function  Chapter 4 Topologies of Photovoltaic Interfaces^  39  Fig. 4.2 demonstrates another topology that uses a DC bus instead of AC bus. The DC/DC MPPT modules track the maximum power points of photovoltaic modules and deliver power to DC bus. The solar energy is eventually transferred to the grid via a centralized DC/AC inverter with the anti-islanding function. This configuration is flexible to be added an uninterrupted power supply (UPS) function when some storage devices such as batteries and/or supercapacitors are installed as shown in Fig. 4.2. During blackouts, the electricity can be continuously supplied to the critical loads by both the storage units and the photovoltaic modules. To simplify the analysis, the block diagram shown in Fig. 4.2 ignores a charge and discharge interface for the storage devices. This is necessary to keep the normal charge or discharge cycle between the storage unit and DC voltage bus. To limit the scope of the analysis, only the interface of the DC/DC MPPT module is designed and investigated in the following studies.  4.2 Control Variables of Maximum Power Point Tracking Both the photovoltaic voltage and the photovoltaic current at the maximum power point can represent the maximum power point, as shown in Fig. 1.6 and Fig. 2.12. For a particular operating condition, the control of maximum power point tracking normally regulates either the voltage or current to a certain value that represents the local maximum power point. However, these conditions are time-variant with the change of insolation and temperature. The ideal control variable that characterizes the maximum power point should be constant, or change slowly within a certain range. As a result, the photovoltaic voltage is preferable because of the advantages described below. Fig. 2.3 illustrates that the changing radiation varies the photovoltaic current dramatically. The fast dynamics of insolation are usually caused by a cover of mixed, rapid moving cloud. If the photovoltaic current is used as the set point, the maximum power point tracking requires fast dynamics to follow a wide operating range from zero amperes to the short-circuit current depending heavily on weather conditions. Nevertheless, the changing insolation slightly affects the voltage of maximum power point  (VMpp),  temperature is the major factor to shift the  as shown in Fig. 2.3. Fig. 2.4 shows that the cell  VMPP varies  significantly to follow the temperature  change. However, the cell temperature has slow dynamics and is always within a certain range.  Chapter 4 Topologies of Photovoltaic Interfaces^  40  Unlike the current of the maximum power point, the photovoltaic voltage of the maximum power point is usually bounded to 70%-82% of the open circuit voltage. This gives a lower bound and upper limit of tracking range. When regulation of photovoltaic voltage is implemented, the maximum power point tracker can quickly decide the initial point according to the percentage of the open-circuit voltage. Fig. 4.3 shows a measured start of maximum power point tracking from the open circuit condition. The IMPPT shown in Fig. 4.3 represents the improved maximum power point tracking, in which the initial setpoint of photovoltaic voltage is set to 80% of the open-circuit voltage and is close to the true  VMPP.  By comparison, the voltage  regulation loop makes the starting time of maximum power point tracking much shorter than the operating of P&O method, which is introduced by [40]. The moment of 4.43s is the start point of the maximum power point tracking, as shown in. Before the moment, the photovoltaic modules are at the open-circuit condition.  •  IMPPT^P&O  0.5 0  0 rt 4 2^4.4  4.6^4.8 Time (s)  5^5.2  Fig. 4.3 Quick start of IMPPT when the initial value is set to 80% of the open-circuit voltage The IMPPT shown in Fig. 4.3 represents the improved maximum power point tracking, which is developed in Chapter 6. The study [49] shows that the photovoltaic current value at MPP is close to about 86% that of the short circuit current. Because the photovoltaic current  Chapter 4 Topologies of Photovoltaic Interfaces^  41  dramatically varies with insolation, the transient response of maximum power point tracking can occasionally send the photovoltaic current to saturate at the short-circuit current. This should be prevented because its nonlinear feature causes a sudden voltage drop and results in power losses. However, for the regulation of photovoltaic voltage, voltage saturation can be easily avoided because a controller knows its operating range is bounded about to 70%-82% of the open-circuit voltage. Furthermore, a good-quality measurement of voltage signal is cheaper and easier than that of current measurement. As a result, the recommended control structure has been shown in Fig. 2.9. The controller regulates the photovoltaic voltage to follow a time-variant set-point, which represents the voltage of maximum power point  (VMpp).  The controller design will be  introduced in Chapter 5. The value of Voop is continuously tracked and updated by the MPP tracker, which will be presented in Chapter 6.  4.3 Converter Topologies This section provides a comparative study to choose a suitable converter topology for the applications of DC/DC MPPT modules shown in Fig. 4.2. Non-isolated buck and boost DC-DC converters are widely used in photovoltaic power systems due to their advantages of simplicity and efficiency. A buck DC-DC converter shown in Fig. 4.4 has a discontinuous input current and a continuous output current devoid of considering the input filter. On the contrary, the boost converter illustrated in Fig. 4.5 has a continuous input current and a discontinuous output current. These characteristics make their applications different when they are used as photovoltaic interfaces. The battery symbols in Fig. 4.4 and Fig. 4.5 represent a constant voltage of DC bus.  1pv>  vpv  SW  /bat  AD C  2  //bat  -  T_T  Fig. 4.4 Equivalent circuit of a buck DC/DC topology used as the photovoltaic power interface  Chapter 4 Topologies of Photovoltaic Interfaces^  42  Fig. 4.5 Equivalent circuit of a boost DC/DC topology used as the photovoltaic power interface  4.3.1 Component Comparison For comparison, one buck and one boost converter are physically designed and described in Appendix B. They operate as the power interface between a photovoltaic module and a constantvoltage load. Different from the output voltages, both converter designs follow the same specifications tabulated in Table B.1. The boost topology shows some advantages over the buck converter for this application. The comparison is discussed in following paragraphs according to the conceptual design illustrated in Appendix B. In the design of inductors, no topology shows significant advantage over another. To achieve the same ripple of inductor current, the boost topology needs more inductance than the buck converter, as shown in Table B.3. However, the RMS current through the inductor is much less than that of buck converter as shown in Table B.4. The parameters of inductor are shown in Table B.5 according to the design requirement. In the selection of input capacitors, the buck topology requires a large and expensive capacitor to smooth the discontinuous input current from the photovoltaic module, and to handle significant current ripple. On the other hand, the photovoltaic current is as smooth as the inductor current without any input capacitor when the boost converter is used. A small and cheap capacitor can make the photovoltaic current and voltage even smoother. The comparison is illustrated in Table B.6. In the selection of power MOSFETs and driver, the current rating is lower in the boost topology than in the buck configuration, as shown in Table B.4. The buck converter requires a high-side MOSFET driver, which is more complex and expensive than the low-side driver used in the boost converter. The comparison is tabulated in Table B.6.  Chapter 4 Topologies of Photovoltaic Interfaces^  43  According to Section 1.41, photovoltaic power systems should prevent reverse current that flow back to photovoltaic cells from devices that can supply electric power. Blocking diodes are effective to prevent the reverse current flows. In the selection of blocking diodes, the boost topology shows significant advantage over the buck converter. In boost topology, the freewheel diode can serve as the blocking diode to avoid reverse current discussed in Section 1.4.1. However, in the buck interface, the blocking diode is an additional component that need conduct the full photovoltaic current. This results in an increased cost and an additional power loss due to the forward voltage drop as demonstrated in Table B.7.  4.3.2 Modeling Comparison Appendix B describes two converter designs that follow the same requirements except of the different output voltages. This section will analyze and compare their frequency characteristics according to the parameters derived in Appendix B. All symbols used in following paragraphs refer to Table B.2. These are also illustrated in Fig. 4.4 and Fig. 4.5, respectively. In this system, the voltage of battery Vbat is a constant DC bus due to the slow dynamics of the batteries. Furthermore, the voltage change across the capacitor  C2  can be ignored. The analysis only covers  continuous inductor current mode, which is the nominal operation of the proposed DC/DC converters. When the switch in Fig. 4.4 is on, the equivalent circuit is shown in Fig. 4.6. The inductor voltage and capacitor current are expressed as (4.1) and (4.2), respectively. The RL symbolizes the equivalent series resistance of inductor.  ipv>  iL  That  >  +^•  ici  Olipv  C1  1c2  Vbat ^ —  T  Fig. 4.6 Equivalent circuit in the buck converter when the switch is on  Chapter 4 Topologies of Photovoltaic Interfaces di L---L.---R i +v —V L L^pv^bat dt  dv Cl ^' = dt  i pv — iL  With the switch "off' position, the converter circuit is shown as Fig. 4.7. The inductor voltage and capacitor current are represented by (4.3) and (4.4), correspondingly. The VFW stands for a forward voltage drop of the freewheel diode. L  diL dt  =  —RL i L —Vbat +VFW  Cl  ipv  dvP, =i  dt '  (4.3) (4.4)  > icl  V  C1  Fig. 4.7 Equivalent circuit in the buck converter when the switch is off When the buck converter is on the continuous inductor current mode, the averaged equilibrium equations are derived as (4.5) and (4.6) according to the state-space averaging method. (4.5) is derived by averaging (4.1) and (4.3). Similarly, (4.6) is obtained by combining (4.2) and (4.4). The same results can be developed by the circuit averaging method described in [50]. m denotes the duty ratio of switching and mi =1- m . Without the loss of generality, the nonlinear relationship of photovoltaic voltage and current is represented by (4.7), which refers to (3.1) and (3.2) in Section 3.1.  ^L  ^C  ^I  di dt  L L -— - -R i +mv - Vbat^FW L^pv^ bat m'V dv^ ^ P^=i pv^L — Mi dt i pv = f (v pv)  (4.5) (4.6) (4.7)  Chapter 4 Topologies of Photovoltaic Interfaces^  45  From (4.5), (4.6), and (4.7), the averaged small-signal state space model can be expressed as (4.8) and (4.9) when the buck converter is the photovoltaic interface. rpv is the dynamic resistance that is defined as (4.10), where v pv represents the small increment of photovoltaic voltage and ipv stands for the small increment of photovoltaic current. rii and m' symbolize the small increments of m and m', respectively. The linearization process is based on the fact that a model is linear in the incremental components of inputs and outputs around a chosen operating point. RL^m  d aL dt vpv  vpv  aL  L^L _ m^1  vpv  +  C,^0 C'1 Y=  L _ lL  rim^(4.8)  Cl  [o^1] rpv  —VFW  (4.9)  iL  vpv  V -  (4.10)  pv  ipv  For easy comparison, both the boost converter and the buck converter adopt the same symbols as the inductors, the input capacitors, and the output capacitors. However, their values may differ. Similarly, the averaged model can be formulated as (4.11) and (4.12) when the boost converter is used as the interface. Combining (4.10), (4.11), and (4.12), the averaged small-signal state space models can be derived as (4.13) and (4.14). The dynamic resistance, rpv is in fact a time-variant parameter that depend on operating conditions. This issue will be further discussed in Chapter 5. L f = RL iL +v,,, C,  datV  RL^1 ^' L L  =ipv - iL  r1+ aL  1^1 ^ vpv  Ci  rpv  Cl  (4.11)  ^—m'VFW  —  (4.12)  Vbat — VFW  L 0  in'  ^  (4.13)  Chapter 4 Topologies of Photovoltaic Interfaces^  46  SAO 1^ vpv  (4.14)  By analyzing the mathematical model of the buck interface (4.8), there are four time-variant parameters including the switching duty cycle in , the photovoltaic voltage v p , , the inductor current it, , and the dynamic resistance rpv . However, when the boost topology is used for photovoltaic interface, there is only one time-variant parameter, rpi, , shown in (4.13). By comparison, there are less time-variant factors in the mathematical model when the boost converter is used as the photovoltaic interface. In (4.13), rpv can be modeled as piecewise linear parameter that will be introduced in Chapter 5. Based on the parameters derived in Appendix B and the nominal operating condition, the system frequency response can be illustrated by a Bode diagram in Fig. 4.8. The boost interface shows better dynamic characteristics than the buck interface. We can see the advantages in terms of wide bandwidth and small resonance due to the small input capacitance of the boost converter.  4.4 Test Bench Outdoor evaluations of maximum power point tracking have advantages that the actual behavior will be examined with real photovoltaic arrays and natural sunlight to avoid potentially unrealistic effects of artificial sunlight, simulators, or emulators. The efficiency of maximum power point tracking is defined as a ratio of the practical power output divided by the true maximum power value. However, a fair comparison of maximum power point is never easy because a true maximum power point is unknown at a specific moment when the system is operated by maximum power point tracking One recommendation [31] is to periodically interrupt the operation of maximum power point and switch the photovoltaic output to an I-V curve tracer, which is able to acquire the I-V curve and show the maximum power point quickly. Nevertheless, this method is only suitable for evaluating the steady state performance since it assumes that environmental conditions do not change significantly in a short-time period. Furthermore, it is important to assess the dynamic performance of maximum power point tracking caused by rapid and unpredictable change of environmental conditions.  47  Chapter 4 Topologies of Photovoltaic Interfaces^  50  Co -0  -  a)  -o o 0  0.) Buck co 2^Boost -50 ^ 1 03 10 2  1 04  1 06  0  rn a) w -100 _c -200 10 2^10^10^10^106 Frequency (rad/s)  Fig. 4.8 Bode diagram of nominal operating condition Fig. 4.9 shows the weather effect on the photovoltaic power of a specific solar module, BP350. In one-hour period, a specific photovoltaic power system is operated by the P&O MPPT algorithm to deliver the maximum power of the photovoltaic module. The photovoltaic power variation is sampled by a data acquisition system recording the photovoltaic voltage and current. This measurement demonstrates a dramatic change of maximum power point in one-hour period. Under this condition, periodical interruptions will eventually affect the dynamic evaluations of maximum power point tracking in response of the fast variation of solar insolation. To evaluate the maximum power point tracking properly, the study demonstrates a unique bench system as illustrated in Fig. 4.10. A DSP-controlled photovoltaic power system is specially designed and constructed for evaluation purposes. It consists of two identical photovoltaic modules, two equal DC-DC boost converters as the power interfaces, and a 24V battery bank. The eZDSPTM LF2407 acts as a digital controller, in which the control algorithms are implemented. The capacity of the battery bank is sufficient to handle the maximum power outputs from both photovoltaic modules. To avoid any overcharge of batteries, the system also includes a charge protection unit and a discharge circuit, which are not directly related to this study and not illustrated in Fig. 4.10.  ^  48  Chapter 4 Topologies of Photovoltaic Interfaces^  ^45,^ 40 a 35  ^Q  30 ^  g 25 .  201 4  E^V -  (13 15 ^ ^)7  10 ^ 50^  10^20^30^40^50^60 Time (min)  Fig. 4.9 Measured photovoltaic power of BP350 acquired in Vancouver, from 10:00AM11:00AM in June 15, 2006  Uniform solar radiation  Photovoltaic module #1  Boost DC-DC #1  Drivers  Photovoltaic module #2  Boost DC-DC #2  Fig. 4.10 Block diagram of the bench system used to evaluate maximum power point tracking under the same condition The system design and implementation of power interface are described in Appendix B. Without loss of generality, this bench system can represent most of photovoltaic applications with maximum power point tracking. In a stand-alone system, the battery bank is commonly  Chapter 4 Topologies of Photovoltaic Interfaces^  49  used to prevent power interruptions from insolation variations. The load is AC voltage source in a grid-connected photovoltaic power system; however, the output voltage is constant. The structure allows testing two photovoltaic modules independently and simultaneously under the same environment. The experimental results will be used to illustrate the effectiveness of the proposed control method and maximum power point tracking. With this structure, one of two identical modules is used as the benchmark sharing with the same load and control bandwidth, while another is implemented with a developed algorithm. This bench system has been proven effective for the evaluation purpose, further introduced in Chapter 6. The benchmark algorithm of maximum power point tracking is the well-established method of perturbation and observation (P&O). The parameterization and implementation refer to the study [31] published in 2005, which gives detailed analysis and optimization for the P&O method. According to the analysis [51], the perturbation time interval for this bench system should be longer than 0.0072s. The system parameters are illustrated in Appendix B. Therefore, we choose the tracking frequency as 100Hz, which satisfies the requirement. The perturbation level of switching duty cycle is alternatively chosen either 0.01 or 0.005.  4.5 Summary Chapter 2 illustrates that the non-ideal conditions considerably downgrade the performance of maximum power point tracking, especially when photovoltaic modules are connected in series. To solve this problem, this study recommends an individual power interface for each photovoltaic module. As a result, two structures are proposed in Section 4.1. The control variable that represents the maximum power point can be either the photovoltaic voltage or photovoltaic current. Analysis in Section 4.2 showed that the photovoltaic voltage has advantages to be the regulated variables so that the maximum power point can be tracked. Section 4.3 provides a comparative study to choose a right converter topology for the applications of DC/DC MPPT modules. The boost topology shows some advantages over the buck converter for this application. The features include cheaper implementations and better dynamic response comparing to the buck converter.  Chapter 4 Topologies of Photovoltaic Interfaces^  50  A fair comparison of maximum power point is difficult because a true maximum power point is unknown when a system is operated by maximum power point tracking. Section 4.4 proposed a specific bench system designed for the evaluation of maximum power point tracking. For a fair comparison, two identical systems can be operated at the same time and the same condition.  51  Chapter 5 Regulation of Photovoltaic Voltage ' There are several advantages to regulate photovoltaic voltages as discussed in Section 4.2 However, both photovoltaic modules and switching mode converters demonstrate nonlinear and time-variant characteristics, which make a regulator design difficult. This chapter presents a mathematical analysis of a specific photovoltaic system that includes a photovoltaic module and a boost DC/DC converter. The nonlinear system is characterized as several linear models within a certain range and time period. The linearization process is based on the fact that a model is linear in the incremental components of inputs and outputs around a chosen operating point (i.e., small signal model). These models are suitable for controller design by using the Youla parameterization, which is advantageous and introduced in Appendix C. The system design is based on the DSP-controlled bench system, which has been introduced in Section 4.4 and Appendix B. Experimental results and simulation will confirm the effectiveness of the presented analysis, design, and implementation.  5.1 Linear Approximation of Photovoltaic Characteristics The characteristics of photovoltaic module are nonlinear and time-variant as shown in Fig. 2.3 and Fig. 2.4 of Chapter 2. In electrical circuits, the resistance is defined as the ratio of the voltage across a circuit element to the current through it. However, the ratio of the voltage across a photovoltaic cell to the current through it varies with either voltage or current. The ratio of the change in voltage to the change in current is known as dynamic resistance, which represents the IA version of this chapter have been accepted for publication. W. Xiao, W.G. Dunford, P. Palmer, and A. Capel, "Regulation of Photovoltaic Voltage", accepted for publication in IEEE transaction on Industrial Electronics, November, 2006.  Chapter 5 Regulation of Photovoltaic Voltage^  52  slope of resistance curve. The dynamic resistances of photovoltaic cells are negative although the static resistance of the circuit element is defined as positive. Derived from the equivalent circuit shown in Fig. 3.2, the current through a photovoltaic cell can be represented by (5.1) and (5.2). The derivation of (5.1) leads to a dynamic resistance, rp , , represented by (5.3) and (5.4), which are functions of the photovoltaic voltage. The cell temperature can vary the values of v, gpv , R„ and isa „ as discussed in Section 3.1 and Section 3.2. The following analysis is based on a specific photovoltaic module, BP350. Fig. 5.1 shows the plots of the dynamic resistance versus photovoltaic voltage for various cell temperatures. Although the relationship of rpv and insolation G a is not shown in (5.4), there is some minor interaction, which can be shown as Fig. 5.2. Therefore, rpv has the time-variant characteristics affected by both insolation and temperature. The dynamic resistance can be numerically approximated as (5.5), where Ppv symbolizes a small increment of voltage and stands for a -  small increment of electric current. ph^sat (g pv  —1 )  (5.1)  v+ip, g pv  (5.2)  e dv  r — Pv di sat RSg pv  rPv =  (5.3) (5.4)  sag pv  r  pv  (5.5)  pv  To improve the maximum power point tracking, it is important to understand the distribution of dynamic resistances at the maximum power points regarding different temperatures and insolation. Fig. 5.3 and Fig. 5.4 show the distribution variation influenced by different temperature and insolation, respectively. The plots show that their amplitudes increase with the decreasing cell temperature or with the decreasing insolation. The largest amplitude happens when the insolation is at its lowest level and the cell temperature is at its lowest level. The upper  • Chapter 5 Regulation of Photovoltaic Voltage ^  53  bound of rp „ is useful for a controller to estimate the present operating condition and to detect if the operating point deviates from the maximum power point.  or  .0. •^•  .. • ^"^  ...... . .  ..  ..  111.  ......  .....  -50 -100 G -150 a) -200 .• -250 'ff -300 C  0  - -T = -25 ° C ^ T = 0°C — T = 25 ° C T = 50 ° C  ♦  -400 -450 ^ -500 10  12^14^16 Voltage (V)  18  20  Fig. 5.1 Dynamic resistances versus photovoltaic voltage for four levels of cell temperature when the insolation is constant 1000W/m 2  -50  a -100 ^  40,  a)^, ♦  cti -150  . •••  0 -200 ^ co  c>2 -250  i  —G a = 1000W/m 2 G a = 800W/m 2  •  -300  —•— G a = 600W/m 2  i  cti  -350 10  G a = 400W/m 2 12^14^16 Voltage (V)  18  20  Fig. 5.2 Dynamic resistances versus photovoltaic voltage for four levels of insolation when the cell temperature is constant 25°C  Chapter 5 Regulation of Photovoltaic Voltage ^  54  0  a  .  -2 ih  c 8 -4 ^ co  g -6 ^0  E -8 ^ T = -25 ° C, co T = 0°C >, ° -10 ^ R T = 25° C T = 50 ° C  -1  0 11 12 13 14 15 16 17 18 19 20 21 Voltage (V)  Fig. 5.3 Distribution of dynamic resistances at the maximum power point is influenced by cell temperature when the insolation is constant 1000W/m 2  0 * G a 1000W/6 2  G -5  o'^ Ga*800W/m2 v G a600W/62 L + G a 4001/V/m  •`c_; -15  41-  ^37  G a =2001.Nim  E  >, -20 -25  10  11 12 13 14 15 16 17 18 19  Voltage (V) Fig. 5.4 Distribution of dynamic resistances at the maximum power point is influenced by insolation when the cell temperature is constant 25°C To simplify the analysis, a piecewise linear approximation of photovoltaic output characteristics is illustrated in Fig. 5.5. The I-V curve of photovoltaic output is divided as four regions, current-source region, power region I, power region II, and voltage-source region according to the slope of the curve that is proportional to the dynamic resistance. The absolute value of dynamic resistance is small in voltage source region and big in current source region. The operating condition will eventually affect the system dynamics that will be discussed in  55  Chapter 5 Regulation of Photovoltaic Voltage ^  following sections. A normal operation generally starts from the voltage-source region at the beginning and stays inside power regions in steady state so that the photovoltaic module can deliver maximum available power.  — I-V curve 1 0 mpp  Current source region "C 0.8  Power region  C  ,- 0.4  Voltage source reg ion  0  0.2 00^  Power region II  0.2^0.4^0.6^0.8 Norm. Voltage ( VN)  1.2  Fig. 5.5 Linear approximation of photovoltaic output characteristics  5.2 Linearization of System Model Based on the test bench system introduced in Section 4.4 and Appendix B, the averaged small-signal state space models are illustrated in (5.6) and (5.7). All symbols in (5.6) - (5.11) refer to Fig. 5.6. The derivation has been described in Section 4.3.2. m denotes the duty ratio of switching and m' =1– m . rp, is the dynamic resistance that is defined as (4.10). 1), represents -  the small increment of photovoltaic voltage and apv stands for the small increment of photovoltaic current. III and th' symbolize the small increments of m and m' , respectively. The dynamic resistance, rp „ is a time-variant parameter, which depends on the operating condition and is represented by rpvi in (5.6). The plant can be represented by an input-output transfer function as (5.8)-(5.11), where co; is the undamped natural frequency, K0 is the gain, and^is its damping factor. Due to the time-varying characteristics, we are going to have four transfer functions according to the four regions defined in Fig. 5.5.  Chapter 5 Regulation of Photovoltaic Voltage ^  56  Fig. 5.6 Boost DC/DC converter  1  _ RL d [L 1 dt pv]  L —  1 Cl  -  L ^11, [ 1 ^''V pv]+ rpvi Ci  [ tl  j) = o l  V  u r bat  VFW  (5.6) 0  L  (5.7)  Pv  (s) K G ,(s) = ^ = ^ + i coi s + co i2 (s) s2^  (5.8)  Where, V  + VFW  LCi co, =  —rpvi + RL —  rpvI LC1  —rpvi RL C1 +L  2 rpvi LC1 coi  5.3 Model Analysis and Experimental Verification Based on the system parameters described in Section B.2, the magnitude and phase plots in  G  Fig. 5.7 depict the frequency response of pv (s) for four different operating conditions, which are defined as the voltage region, current region, power region I and power region II. The  ^  • 57  Chapter 5 Regulation of Photovoltaic Voltage ^  parameters of undamped natural frequency coi and damping factor changes with the variation of operating points, as demonstrated in Table 5.1.  loon m a 50!L  —  a)  0 ^ Power region I - ---Power region II Cu -50 - -Voltage source region — Current source region -100 ^ 3 10 1 04 10 C  10  5  10  6  0  rn -csa)  -50  a) -100 m a,  0_ -150 -200  ^Power region I - ---Power region II - -Voltage source region — Current source region  10  103  4  10 Frequency (rad/s)  10  5  10  6  Fig. 5.7 Bode diagram of the photovoltaic system G; (s)  Table 5.1 Frequency parameters of the open-loop model Operating point at Current source zone  r„ (0) DC gain -350.00 26.52  10.60 x10 3  0.0303  Power zone I  -15.71  26.28  10.64x10 3  0.1167  Power zone II  -3.20  25.34  10.84 x 10 3  0.4624  Voltage source zone -1.67  24.34  11.06x10 3  0.8470  coi (rad/s)  From Table 5.1, the different operating condition slightly changes the DC gain and undamped natural frequency, coi . However, it significantly affects the damping factor, . The system shows well-damped characteristics when the operating point is in the voltage source region. When the operating point is close to current source zone, the damping factor reduces while the dynamic resistance increases. When the operating point enters the current source  Chapter 5 Regulation of Photovoltaic Voltage^  58  region, the plant becomes a lightly damped system, which is a difficult control problem as described in Appendix C.4. The system is open-loop-stable because there is no unstable pole. The normal operation of maximum power point tracking starts at the voltage source region, which demonstrates a medium -damped feature. It is desirable that the operating point stays constantly in power zones at steady state to achieve the maximum power delivery from the photovoltaic cells. It turns out to be a lightly damped system when the operating point enters the current source zone. This condition should be avoided since it results in reduced power output and some control issues of lightly damped system explained in Appendix C.4. A lower bound of photovoltaic voltage cannot prevent the operating point deviating of power zones because the cell temperature shifts the optimal operating voltage over a large range, as illustrated in Fig. 2.4. An alternative way is to use the lower bound of dynamic resistance of the photovoltaic module. For example, the lower limits of the operating condition of insolation and cell temperature are 200W/m 2 and -25°C, respectively, the value of the dynamic resistance is from -400 to 00. When the detected dynamic resistance is lower than -400, the controller should increase the setting of photovoltaic voltage to prevent the system from going into the oscillatory state because the value of dynamic resistance has changed the damping factor. Based on the derivations of the linear models, computer simulations were performed to compare the behaviors of the mathematical model to that of the physical device. The experimental tests illustrate the transient response in open-loop control with the 5% step change of the duty cycle. The plots in Fig. 5.8 and Fig. 5.9 show the traces of the photovoltaic voltages regarding the measured values and those from the simulations of the mathematical model. The study focuses on the power regions, which are the normal operating area. The dynamic response in Fig. 5.8 is different from that in Fig. 5.9 because they are based on two different operating conditions. As expected, in power zone I, the step response shows lightly damped characteristics, as shown in Fig. 5.8. In power zone II, it shows medium-damped feature, as shown in Fig. 5.9. This generally proves the effectiveness of the linearization of system model since the plots of the computer simulation are compatible with those obtained experimentally.  Chapter 5 Regulation of Photovoltaic Voltage ^ 13.4 13.2 13  E 12.8  a- 12.4  Measured step-response Simulated step-response 820^821^822^823 824 Time (ms)  Fig. 5.8 Open-loop step-response of photovoltaic voltage in power region I  17.9 17.8 17.7 17.6 S 17.5  17.1 17  Measured Simulated. step-reSponse V85.5 1586 1586.5 1587 1587.5 1588 1588.5 1589 1589.5 1590 Time (ms)  Fig. 5.9 Open-loop step-response of photovoltaic voltage in power region II  59  Chapter 5 Regulation of Photovoltaic Voltage ^  60  5.4 Closed-Loop Design A successful closed-loop design needs to consider major control limitations. Three major issues will influence the control performance. These consist of the time-variant characteristics, the lightly damped system, and the time delay caused by digital control. First, the plant model changes with the variation of operating points and environmental condition. The control design should consider four regions and the worst case to ensure the system stability. Second, the system shows lightly damped characteristics when the operating point accesses the power region I or current source region. All control algorithms will be implemented in a digital controller that inevitably introduces the time delay caused by computation time and sampling rate.  5.4.1 Issues of Affine Parameterization used for Lightly-Damped Systems According to the investigation in Appendix C, a design limit (C.18) needs to be applied to the affine parameterization to ensure the proportional gain Kp is positive. A negative proportional gain can cause problems when an anti-windup scheme takes effect. In the bench system, the damping factor is 0.12 when the dynamic resistance is -400. According to the rule of thumb [52], we choose the closed-loop damping factor to be equal to 0.7. To ensure the positive proportional gain, the design should satisfy the condition (5.12), where represents the closed-loop undamped natural frequency and coi stands for the plant undamped natural frequency. According to the analysis in [52], this design is very sensitive to the modeling errors and input disturbance because the closed loop is beyond the location of the open-loop resonance. Therefore, this study demonstrates that it is difficult for the affine parameterization to synthesize a controller for a lightly damped plant. To solve this problem, a further tuning after the affine parameterization is necessary to guarantee the system robustness, which will be introduced in following sections by a practical example. cod > 2.98w1^(5.12)  Chapter 5 Regulation of Photovoltaic Voltage^  61  5.4.2 Time delays The time lag introduced by the digital controller must be taken into account for the stability analysis. Time delays limit the achievable control bandwidth. To control the switching-mode converters, the new control variable can be updated only at the beginning of the next switching period. The time delay can be considered equal to one switching period, which is 25pts in this system because the switching frequency is 40kHz. In the worst case, the delay is two switching cycles. There are two ways to analyze the effect of time delay quantitively. According to the first-order Pad& approximation [53], the time delay caused by the digital implementation can be expressed as (5.13), where Ts represent the time dealy. —s Ts e^ 2 sTs 2 + sTy  (5.13)  The influence of time delay on a closed-loop system can also be explained as the reduced phase margin [51], which can be illustrated as (5.14). yonnip represents the non-minimal phase margin caused by the time delay. cocp is the crossover frequency. The final phase margin at the crossover frequency, com , can be calculated as (5.15), where pmp is the phase margin of minimal phase part. Pnmp^W CP  Ts  Pm = 7r +  (5.14) (5.15)  5.4.3 Controller Parameterization and Analysis According to the rule of thumb [54, 55], the damping ratio is selected as 0.7 and the undamped natural frequency is chosen to 5.08kHz to reshape the closed-loop performance. Based on the nominal converter model in power region and following the procedure of affine parameterization shown from (C.7) to (C.12), the transfer functions of Q(s) and controller C(s) are determined as (5.16) and (5.17) correspondingly. For stability analysis, the system is internally stable since Q(s) has no pole on the right hand side.  Chapter 5 Regulation of Photovoltaic Voltage ^  62  s' + 2484s +113300000 Q(s)=^ 2.92s 2 + 149200s + 2978000000 C(s) =  S2  (5.16)  + 2484s +113300000 2.92s 2 +149200s  (5.17)  According to the study of Section 5.3, there are large variations in the plant model since the the damping ratio changes with operating conditions from 0.03 to 0.85. The worst case is from the current source region that shows lightly damped characteristics. Generally, the Nyquist diagram gives a graphical illustration of the system in terms of stability margins and maximum sensitivity. When the time lag is ignored, the plots in Fig. 5.10 demonstrate the Nyquist curves to represent the degree of stability. The definition of four regions refers to Fig. 5.5. The plots also show the shortest distances between the Nyquist curve and the critical point (-1+j0), which can be interpreted as the stability margin, s m . The parameters of stability and robustness are summarized in Table 5.2.  0.2 0 ^4  o Current source region 13 Power region I Power region II t* Voltage sour e region  c  -0.2 >7  <  5) cts  E  -0.6-  ................^.  -0.8-  .... ........... . .  -0.8^-0.6^-0.4^-0.2 Real Axis  ..  ........  .......  ...  ^ ^ 0 0.2  Fig. 5.10 Nyquist plots of the loop transfer functions C(s)Gi (s) when the time delay is ignored  Section 5.4.2 states that the time lag introduced by the digital controller is the major component of non-minimum phase. According to the approximation of the time delay as (5.13),  63  Chapter 5 Regulation of Photovoltaic Voltage^  the time delay can be added to the system transfer function and the Nyquist curve can be replotted as Fig. 5.11. The parameters of stability and robustness are summarized in Table 5.3. By comparing Table 5.3 and Table 5.2, the phase margin and stability margin are noticeably reduced by the time delay introduced by the digital control. According to the rule of thumb, the phase margins in Table 5.3 are not significant enough to guarantee the system robustness. As a result, a further tuning is required for this specific system after the affine parameterization. Table 5.2 Parameters of stability without considering the time delay caused by digital control Performance criteria Phase margin  Current source region 62.6°  Power region I  Power region II  69.9°  45.9°  Voltage source region 34.8°  3.01  2.97  1.39  1.20  oo  oo  oo  oo  >0.57  >0.57  >0.57  0.57  Cross freq (kHz) Gain margin Stability margin s.  0.2  -0.2 -OA al -0.6 )  E -0.8 o Curr btit.source region O Power reglarrl. .................. * Power region II t* Voltage source region ..  -0.8^-0.6^-0.4^-0.2 Real Axis  ..............  0  02  Fig. 5.11 Nyquist plots of the loop transfer functions C(s)Gi (s) when the time dealy is considered  Chapter 5 Regulation of Photovoltaic Voltage ^  64  Table 5.3 Parameters of stability with considering the time delay Performance criteria  Current source region 35.9°  Power region I  Power region II  43.5°  33.4°  Voltage source region 24.1°  Gain cross freq (kHz)  6.04  6.26  7.15  7.97  Gain margin (dB)  2.49  3.07  3.76  11.5  Phase cross freq (kHz)  3.01  2.98  3.01  1.20  Stability margin s„,  >0.40  >0.40  >0.40  0.40  Phase margin  5.4.4 Controller Tuning According to the analysis in Section 5.4.3, the affine parameterization delivers a controller that cannot illustrate enough stability margins for four operating regions of the specific system. When the time lag of digital controller is considered, the phase margin reduces to 24.1° in the worst case, which is not big enough for the issues caused by modeling error and disturbance. Therefore, a re-tuning is necessary to increase the system stability and robustness. The further tuning introduces another tuning parameter ii c , which can adjust the proportional gain. A reduced proportional gain can increase both the stability margin and phase margin. According to (5.17), the re-tuned controller is expressed as (5.18). The range of y c is shown in (5.19). The stability margin can be adjusted by changing the value of 7 c, . For example, the Nyquist plots are shown in Fig. 5.12 when yc = 0.2 . Thus, the parameters of stability and robustness are summarized in Table 5.4. By comparing Table 5.3 and Table 5.4, it clear that the system stability is improved but the bandwidth is reduced by the additional tuning after the affine parameterization. After re-tuning, the controller transfer function becomes (5.20).  Cy (s) =  Cy (S) = 7 c C (S)^  (5.18)  0<;,c,_1^  (5.19)  0.2s 2 + 497s + 22660000 2.92s 2 +149200s  (5.20)  Chapter 5 Regulation of Photovoltaic Voltage^  0.2  o Current source region ‘\'■ Ob -.mat o Power region I v „a A ^ ... i Power region II i t* Voltage source/region, NIP  -0.2 .u) < -0 4  ..... . .. .  .  2) -0 6 8  65  .  ....  ..... .. .  .....  -0.8  Ili  -1  ............  -0.8^-0.6^-0.4^-0.2^0^0 2 Real Axis  Fig. 5.12 Nyquist plots of the loop transfer functions Cr (s)Gi (s) after further tuning  Table 5.4 Parameters of stability after further tuning Performance criteria  Current source region 49.0°  Power region I  Power region II  79.9°  66.7°  Voltage source region 57.8°  Gain cross freq (kHz)  6.04  6.26  7.15  7.97  Gain margin (dB)  11.81  12.44  15.34  18.78  Phase cross freq (kHz)  1.71  0.63  0.58  0.51  Stability margin s n,  0.56  >0.56  >0.56  >0.56  Phase margin  5.5 Digital Redesign The analog controller has been developed in Section 5.4. By extending it to a digital control system, we need a digital redesign to obtain a digital controller by discretizing the analog controller. The sampling rate is chosen to 40kHz. In this section, two approximation methods, Bilinear (Tustin) and Matched pole-zero, are used and compared. According to the analog controller transfer function in (5.20), we have two digital controllers as (5.21) and (5.22) based  Chapter 5 Regulation of Photovoltaic Voltage ^  66  on Bilinear transformation and Matched pole-zero, respectively. The comparison results of frequency responses between the discrete approximations and continuous-time controller are illustrated by Bode plots in Fig. 5.13. Both demonstrate satisfactory approximation inside the border of the Nyquist frequency. \ 0.04384 —0.08212z" + 0.04124z -2 1-1.2205z-1 + 0.2205z -2  (5.21)  C y _Tustin k Z =  Cy  \  MatchedlZ =  (5.22)  0.04052 —0.07583z -1 + 0.03808z -2 1-1.2788z" + 0.2788z'  -10  V  -0.4.4-- Tustin - Matched pole-zero  -20  .11=  ■=...0.■ • ..1•11..  at  1' -30 "E  rn  3 -40 -50 90  rn  45  a)  Cu  -  as _c  0  :MN  e- -45 -90  1 03  ^  10  4^  10  5^  10  5  Frequency (rad/sec)  Fig. 5.13 Bode plots of the controller frequency response in the continuous and discrete domain The controller is a 16-bit-fixed-point DSP. The constraints of digital implementation should be carefully handled to avoid the degradation of control performance. The major issues of 16-bitfixed-point controller are from the quantization errors of controller parameters and control variables because all data have to be truncated and stored as finite-length words. The clock of DSP dramatically limits the resolution of duty cycle of high-frequency switching converter. In this system, the achievable resolution of duty cycle is 0.1%. The details have been described in Appendix B.  Chapter 5 Regulation of Photovoltaic Voltage ^  67  5.6 Anti-windup The discrete controller can be represented by a difference equation as shown in (5.23), which is transformed from (5.21) where represents the small-signal control variable and e(k) is the error between the setpoint and the measured signal Other parameters including a i ,a 2 ,b0 ,bi and b2 , are derived from (5.21). The regulation flowchart is shown in Fig. 5.14, where Vp , represents the photovoltaic voltage,  Vref  is the target voltage, th' x symbolizes  the upper limit of the control variable, and /k % m stands for the lower limit of the control variable. ini(k) =^(k —1)— a 2 r17 (k — 2) + bo e(k)+bl e(k —1) + b2 e(k — 2)^(5.23) 1  (Regulation Um) 4 Read Vpv(k) ,  e(k) Vref - Vpv(k)  (k)= —a/n"(k —1)— a 2 in"(k — 2) + bo e(k)+b,e(k —1)+b 2 e(k — 2)  yes no yes (k)= rh o,' ax no  "'AO= Rm..  Update historic data in"(k — 2) = in"(k —1) tiqk —1) = 111'(k) e(k— 2) = e(k --1) e(k —1) = e(k)  ( Return )  Fig. 5.14 Control loop for regulating the photovoltaic voltage  Chapter 5 Regulation of Photovoltaic Voltage ^  68  When the control variable reach the limit, we can see rir 1 (k —1) = (k — 2) according to the flowchart shown in Fig. 5.14. Therefore, the controller difference equation, (5.23), becomes (5.24). From the parameters in either (5.21) or (5.22), we know a l + a, = —1 . Hence, the controller difference equation can be written as (5.25) when the saturation happens. The controller shown in (5.25) is a proportional controller in an incremental format. With the proposed controller structure, the integrator windup can be avoided because the recursive terms are automatically eliminated when the control variable reaches a limit. in" (k) + —1) + a 2 rni (k —1) = b o e(k) + bi e(k —1) + ke(k — 2) (5.24)  ric' (k) 712' (k —1) = bo e(k) + bi e(k —1) + ke(k — 2) (5.25)  5.7 Evaluation The effectiveness of the proposed design is demonstrated by computer simulations and experimental evaluations with natural sunlight. Fig. 5.15 illustrates the regulation performance based on a low solar radiation, which is one of the worst operating conditions according to the analysis in Section 5.1, 5.2, and 5.3. The experiments intentionally applied periodical step-up and step-down changes in the voltage setpoints. It shows stabilized regulations of the photovoltaic voltage at three operating regions, namely, voltage source region, power region, and current source region. To demonstrate the output characteristics during the test condition, the I-V curve was measured and plotted before each regulation evaluation, as shown in Fig. 5.15. The major disturbance that has fast dynamics is the insolation variation, which is usually caused by the cover of mixed, rapid moving cloud. To illustrate the effect of insolation disturbance, the simulated plots in Fig. 5.16 show the regulation performance against the disturbances caused by the 100% step-up and step-down insolation changes, which is difficult to be repeated in experimental test with natural sunlight. Vp „ is the photovoltaic voltage, Ipv symbolizes the photovoltaic current, and m is the control variable. The simulation shows the dynamic response against the disturbances caused by step changes of insolation.  ^  Chapter 5 Regulation of Photovoltaic Voltage ^  (a) Measured I-V curve  69  (b) Voltage source region 20 ^ 19.5 19 18.5 18 17.5  0.05  17  5^10^15^20 V Pv (V) (c) Power region I  150  (d) Current source region  16.5 ^  15  16 wM.►wr  14.5  15.5  14  Za 15  Z 13.5  14.5  13  14  12.5  13.5 ^ 0  50^100 Time (ms)  a  50^100^150 Time (ms)  12 0^  50^100^150 Time (ms)  Fig. 5.15 Plots of photovoltaic voltage regulation with low solar radiation. The controller parameters are illustrated in (5.21).  5.8 Summary This study presents an in-depth analysis and modeling to discover the intrinsic characteristics of photovoltaic power generation systems. It also presented the use of Youla Parameterization to design a control system for regulating the photovoltaic voltage. In the bench system, both photovoltaic modules and switching mode converters present nonlinear and time-variant characteristics. It is important to approximate the non-linear feature by a linear model since the techniques of linear control are more well-established and simpler than the case of nonlinear control. To ease the analysis, the operating range was divided as four regions, current-source region, power region I, power region II, and voltage-source region  Chapter 5 Regulation of Photovoltaic Voltage ^  70  according to relative output characteristics. The method of successive linearization simplifies the nonlinear problem back to the linear case. The system also shows the time-variant characteristics because both the mathematical models and experimental results show that the frequency response is different from one region to another. The major variation lies in the damping factor caused by the variation of dynamic resistance. The change of insolation and temperature also affect the values of dynamic resistance. When the operating point enters the current source region, the plant becomes a lightly damped system, which is a difficult control problem as described in Appendix C.4.  19 18  >2 17 > 16 15 0.008  0.01  0.008  0.01^0.012  42 40 38 — 36 34 32 30  0.012  0.014  0.016  0.018  0.014^0.016  0.018  F  0.008  ^  0.01  ^  ^ ^ ^ 0.012 0.014 0.016 0.018 Time (ms)  Fig. 5.16 Simulated results of regulation performance against the disturbance caused by 100% step change of insolation level Based on these characteristics, a controller is designed through Youla parameterization. However, the computational lag that is introduced by the digital control reduces the system phase  Chapter 5 Regulation of Photovoltaic Voltage ^  71  margin significantly. Therefore, a further tuning is also presented in Section 5.4.4 to guarantee the system stability and robustness. Finally, Experimental results and simulation confirm the effectiveness of the presented analysis, design, and implementation.  72  Chapter 6 Maximum Power Point Tracking 1 This  chapter introduces two different approaches in searching for maximum power points.  One algorithm is the Improved Maximum Power Point Tracking (IMPPT). Another method is the real-time identification of optimal operating point in photovoltaic power systems. Experimental evaluations have successfully proved the effectiveness of both methods.  6.1 Improved Maximum Power Point Tracking According to the analysis in Chapter 2, deviations from maximum power points result in power losses. There are two key aspects to improve the performance of maximum power point tracking. One requirement is to locate accurately the position of maximum power point. Another effort is to reduce the oscillation around the maximum power point in steady state. The following paragraphs will discuss these issues and provide solutions.  6.1.1 Euler Methods of Numerical Differentiation As discussed in in Section 2.2, most efforts for maximum power point tracking are based on a mathematical equation of dp/dv = 0, where p represents the output power and v represents the photovoltaic voltage. They generally rely on numerical differentiations, which are processes of finding a numerical value of a derivative of a given function at a given point. Both P&O and IncCond methods trace maximum power points by way of Euler methods, which are simple in 1 A versions of this chapter have been published and accepted for publication. W. Xiao, M.G. J. Lind, W.G. Dunford, and A. Capel, "Real-time identification of optimal operating points in photovoltaic power systems", IEEE Trans. on Industrial Electronics, Vol. 53, No. 4, AUGUST 2006. W. Xiao, W.G. Dunford, P. Palmer, and A. Capel, "Application of Centered Differentiation and Steepest Descent to Maximum Power Point Tracking", aceepted for publication in IEEE transaction on Industrial Electronics, Febuary, 2007.  Chapter 6 Maximum Power Point Tracking^  73  use, but show local truncation errors [56]. The definition of the local truncation error can be explained as "how well the exact solution satisfies the numerical scheme"[57, 58]. It is usually defined as the consistent of order. To demonstrate this subject clearly, we define the derivative in a general form as (6.1). Consequently, the numerical differentiations of forward Euler and backward Euler are demonstrated in (6.2) and (6.3), correspondingly. A V is the incremental step of photovoltaic voltage. Both backward Euler and forward Euler are called first order accurate, or consistent of order 1, because their local truncation error is equal to O(A V 2 ) . They also demonstrate both magnitude errors and phase errors in the frequency domain [57]. The local truncation error of Euler methods is the reason that both P&O and IncCond algorithms can never locate the optimal operating point accurately by finding the root off (v, p) = 0 . dp — = f (v, 13 ) dv  +oov)  Pk - Pk-1^ 2 f( vk,,pk_i)=  AV  f  ,,, ) Pk Pk-, ± 00V (vk, 1-'k 1 -^ AV -  2)  6.1.2 Reduction of Local Truncation Error The centered differentiation is symmetric as expressed in (6.4) and (6.5). The local truncation error is consistent of order 2, which is represented by 0(A V') . This means this method is more accurate than the Euler methods for numerical differentiation. Further, it was proved that the centered differentiation method does not have a phase error in the frequency domain [59]. For maximum power point tracking, a controller needs to find the point where f (v, p) = 0 . The symmetric characteristics allow the extremum value to be located with a better precision, as demonstrated by the normalized power-voltage curve of a photovoltaic module shown in Fig. 6.1. When the calculation shows  P k+1 %;"j" Pk-1  or  P k+1 - Pk-1 ^".". 0 5  the  dp dv  differentiation approximation of — is very close to zero and the maximum power point is located at (v k ,p k ) instead of either (vk+, , P k+i) or  (v k _ i  D k-1 ) •  5 A.  However, Euler methods can  Chapter 6 Maximum Power Point Tracking ^  74  never make the operating point remain stably at the actual maximum power point (vk ,p k ) because their approximations are always biased due to the phase error. dp — = f (v, p)  (6.4)  dv  f (v k , p k ) = Pk+1— Pk-1 ± 0(A )^ 2A V  (6.5)  11 1.05  1 g 0.95 0.9 3 0.85 a_ g 0.8 0 Z  0.75 0.7 0.65 0.6 u 4^0.5^0.6^0.7^0.8^0.9 Norm. Voltage (VN)  Fig. 6.1 Normalized power-voltage curve of photovoltaic module  6.1.3 Numerical Stability Numerical stability is an essential property for any numerical algorithm. In general, numerical differentiation is more difficult than numerical integration because numerical differentiation requires Lipschitz classes [60]. According to the definition, the numerical differentiation in (6.4) shall satisfy the Lipschitz condition shown in (6.6), where C LIP is a constant independent of AV . In maximum power point tracking, the value of  CLIP  can be  dp selected according to the maxima of — , because this value is known through offline analysis of dv  power-voltage curves. To promise numerical stability, the controller will evaluate this condition (6.6) after each numerical differentiation. Disturbances and measurement noises are two major reasons that cause a malfunction of numerical differentiation.  Chapter 6 Maximum Power Point Tracking ^ ^ 1Pk+1  -  Pk-11-< CLIPI2AVI  75 (6.6)  6.1.4 Tracking Methods Two mathematical methods are relevant to applications of maximum power point tracking. One optimization method is the steepest descent, which is originally from the optimization method in applied mathematics. Another is the Newton's method, also called Newton-Raphson method, which is a root-finding algorithm [60]. The method of steepest descent was introduced in Section 2.2.2.3. The Newton-Raphson method uses a first and second derivative of the change with parameter value to estimate the direction and distance the program should to go to reach a better point. When it is used to track maximum power points, the computation of operating point can be illustrated in (6.7). This algorithm needs to perform numerically both single and double differentiations. dp Vk+1 = Vk  dv  V= Vk  d2  p dv 2  (6.7)  V=Vk  In theory, the Gauss-Newton method is the fastest algorithm by comparing to the steepest descent and hill climbing. The plots of simulation results shown in Fig. 6.2 and Fig. 6.3 prove its efficiency in seeking the maximum point. Shown in Fig. 6.3, the algorithm can access to the maximum point after four steps of movement. Nevertheless, this procedure can be unstable regarding the initial condition [61]. To avoid this problem, the improved maximum power point tracking proposed in this study chooses the method of steepest descent, which shows faster dynamic response and smoother steady state than the method of hill climbing, as illustrated in Fig. 6.2.  6.1.5 Oscillation Reduction Continuous tracking operations cause unnecessary oscillations around the maximum power point. This can be reduced by stopping continuous perturbations when a local maximum power point (MPP) is accurately located. The controller shall be able achieve these operations: (1) to  Chapter 6 Maximum Power Point Tracking^  76  evaluate if the true MPP is found, (2) to stop the perturbation operation to make the operating point stay at MPP, (3) to estimate if the MPP has drifted to a new location. Gauss-Newton — Hill climbing I ^ Steepest descent  0.95 0 >  0.9  .1P., 0.85 0 > 0 0.8  0.70  20  40^60 Time step  80  100  Fig. 6.2 Comparison of different algorithms for maximum power point tracking  ^ Gauss-Newton MPPT -^ Measured P-V curve - Simulated P-V curve 5^10^15 Photovoltaic voltage (V)  20  Fig. 6.3 Gauss-Newton method is theoretically efficient in maximum power point tracking  6.1.5.1 Estimating the Existing Maximum Power Point According to extremum value theorem, any maximum power point shall satisfy a condition, either (6.8) or (6.9). Section 6.1.2 has shown an increasing accuracy of numerical differentiation by introduction of centered differentiation. However, centered differentiation cannot eliminate the local truncation error and the tracking iteration continues until the condition illustrated in  Chapter 6 Maximum Power Point Tracking ^  77  (6.10) is continuously satisfied for several cycles. Then, the local extremum has been determined within a chosen accuracy, E mpp . The choice of e mpp depends on a required sensitivity and a signal-noise-ratio of measurements. dp dv  =0 v„,pp  dp di dp dv  =0 tmpp  6 ,npp  A flowchart in Fig. 6.4 demonstrates the operation. When the location of maximum power point is located, the controller records the value of  I mpp  and V mpp for further estimation  described in following section. As shown in Fig. 6.4, the variable, MPP_index, records how many times the condition (6.10) is continuously satisfied. When the number of MPP_index is accumulated to a certain threshold, MPP_th, the controller presumes a local MPP is temporarily found under current condition. The controller needs to clear the index variable, MPP_index, and records the current location of MPP. Otherwise, the maximum power point tracking will continue until the MPP is successfully located.  6.1.5.2 Shift of Maximum Power Point Both insolation and temperature are time-variant parameters of a photovoltaic power system in a daily period. A changing environment can make the MPP drift to a new location. From (6.8) and (6.9), we can derive (6.11) and (6.12) respectively. Each maximum power point corresponds to a specific value of resistance, R mpp or conductance, G ,npp . Consequently, the controller can estimate a shift of maximum power point by monitoring the variation of either resistance, or conductance, G mpp  .  Rmpp  The absolute resistance error, e R , is characterized as (6.13), which  illustrates the difference of present measurement and recorded conductance error, e G , of present measurement and recorded  R mpp .  G mpp  Likewise, the absolute  is characterized as (6.14).  The averaged values of these absolute errors are symbolized as (6.15) and (6.16) for resistance and conductance, correspondingly. Nth is the number of samples for each averaging window.  Chapter 6 Maximum Power Point Tracking ^  yes  78  no  I MPP_index--  yes  no  MPP_index =  • (Record (Vm„,/,„)  (Found MPP)  Fig. 6.4^Flowchart to evaluate if the MPP was located di dv  G mpp =  0  (6.11)  R mpp =  0  (6.12)  v,,„p  dv di imPP vpv  e R =-- RA,ipp pv # 0 pv  eG =  pv v  5 V pv #  pv  O  (6.13) (6.14)  N/h  E e (i) R  ER  = i=1  (6.15)  N th N, h  Ee  G (i)  EG = 1=1  N th  (6.16)  Chapter 6 Maximum Power Point Tracking^  79  The averaged errors, E Ror E G , are updated in each control cycle. A maximum power point that has drifted can be detected by monitoring the change of either E R or E G . If there is no variation of maximum power point, the values of E R or E G are generally within a certain range. When the cumulative error is larger than a specified threshold, the controller supposes that the maximum power point has drifted to a new level and initializes a procedure of maximum power point tracking. The controller stops the calculation of the averaged errors when it is tracking the new maximum power point. ER or EG will be reset to zero after the existing maximum power point is successfully located in steady state. Then, the detection restarts for a new shift of maximum power point. With a regulation of photovoltaic voltage as recommended in Chapter 6, the controller needs to monitor the error of the photovoltaic current change since the voltage is principally constant due to the regulation process. The absolute error of photovoltaic current and the averaged error are symbolized as (6.17) and (6.18), respectively. The threshold of averaged error in photovoltaic current, E1 TIRED , can be chosen by analyzing the nominal current of the photovoltaic array and the dynamics of weather condition. The flowchart in Fig. 6.5 illustrates the determination if a maximum power point has drifted to a new level. This is applicable for systems with a photovoltaic regulation function. For the general case, the ER or E G shall be used as the criterion to determine a shifted MPP.  Fig. 6.5 Flowchart to determine if there is a new MPP e 1 = i pv -1MPP  (6.17)  Nth  E, (i) E1 =  ^ Nth  (6.18)  Chapter 6 Maximum Power Point Tracking ^  80  6.1.6 Main Loop Fig. 6.6 illustrates a main loop of the proposed maximum power point tracking. The tracking can be activated or deactivated according to the estimation if a maximum power point has been changed or located, correspondingly. The evaluation of drifting or locked MPP refers to the section 6.1.5.  Call MPPT yes  I  Activate MPPT  no  Deactivate MPPT no  (Return )  Fig. 6.6 Flowchart illustrating the main loop of improved maximum power point tracking  6.1.7 Flowchart of Maximum Power Point Tracking The flowchart in Fig. 6.7 demonstrates an implementation of the improved maximum power point tracking. The numerical differentiation is based on the centered differentiation according to the analysis in section 6.1.2. To avoid numerical instabilities, the Lipschitz function is evaluated in each cycle. The tracking goes back to P&O method when the stability condition is not satisfied. The AV is a constant, which represents the step change of photovoltaic voltage for the purpose of numerical differentiation. The value of K, determines how big the step takes in the gradient direction. In this study, the system starts with a chosen value for K. Then, a further tuning is proceeded to make sure that the tracking converges to the local extremum value. When a fixed-point DSP or microcontroller is used, it is important to choose binary numbers for  Chapter 6 Maximum Power Point Tracking ^  81  both AV and K, for efficient computation. The variable of "Sched" is used for scheduling the work load of maximum power point tracking.  • K  =Vim,.  PK-1 - PPV  'Iv Sched = 0  i F  •  = VK - AV I  dp dv  REF  VK +  dp dv  ,''''  = PPV  K+1 v=vk-  =  ^  -  F  +AV  PK-,  2AV  K  K,  = V K ± AV„,„„sign(  dp dv v=v„  Return )4  Fig. 6.7 Flowchart of the improved maximum power point tracking  6.1.8 Evaluation of Improved Maximum Power Point Tracking The evaluations will show the dynamic response, steady-state performance and daily 8-hour tests. The detailed description of test bench used for evaluation is in Section 4.4.  Chapter 6 Maximum Power Point Tracking^  82  6.1.8.1 Startup Performance As discussed in Chapter 2, the voltage of optimal operating point (Voop) is about 70-84% of the open-circuit voltage. For the improved algorithm of maximum power point tracking (IMPPT), the startup time can be shortened noticeably because the controller knows the range of maximum power point by measuring the open-circuit voltage in the initial condition. However, the optimal duty cycle for maximum power point is generally unknown at the beginning because it depends on the load condition. A routine operation of photovoltaic power systems is that the controller starts the maximum power point tracking in the morning whenever sunlight is available. However, the radiation is generally low at this moment. Fig. 6.8 demonstrates the IMPPT can locate the maximum power point much faster than the P&O method at low power condition, where the available power is about 8W for a 50W photovoltaic module. Comparing to 0.26 second for P&O method, the IMMP used 0.1 second to stabilize the operating point at the maximum power point. The moment of 4.34s is the start point of maximum power point tracking, as shown in Fig. 6.8. Before the moment, the photovoltaic modules are at the open-circuit condition. Ideally, systems shall start even earlier to harvest more solar energy in the early morning. This requires the maximum power point tracking can start and work well under weak radiation. Fig. 6.9 shows the IMPPT can initialize the tracking operation and stabilize the operation at maximum power point when the achievable photovoltaic power is only 3.53W for a 50W module due to the weak insolation. However, when the perturbation step is 0.005, the P&O method was not able to find the maximum power point because it was trapped due to the low Signal Noise Ratio and large local truncation errors. Experiments show that the P&O method can operate normally when the perturbation step is increased to 0.03 under the weak insolation. However, the large ripples of photovoltaic voltage and power result in significant loss in steady state.  Chapter 6 Maximum Power Point Tracking ^  a  8_ 0.5  IMPPT and module #21 P&O and module #1  0  z  4.2^4.4^4.6  4.8^5^5.2^5.4  -----------------^-----------------^----------------- ----------  z^  4  83  4.2  4.4  4.6  4.8  IMPPT and module #2  5.2  5.4  - P&O and module #1  0  z  4.2  4.4  4.6  4.8  4.2  4.4  4.6  4.8  5.2  5.4  -IMPPT and module #2 5.2  5.4  ;7t 0.5 0 z  - P&O and module #1 4.2  4.4  4.6  4.8 Time (s)  5.2  5.4  Fig. 6.8 Plots of startup procedures of IMPPT and P&O algorithms under low radiation. The available power is about 8W for 50W solar module.  6.1.8.2 Steady-state Performance The benefit of IMPPT is that the controller stops the tracking process and runs only the voltage regulation when the maximum power point is located. In steady state, this dramatically reduces ripples in photovoltaic power and voltage comparing to continuous perturbations caused by P&O-type algorithms. As discussed in Chapter 2, the value of perturbation step affects the steady-state performance when P&O-type algorithms are used. Fig. 6.10 and Fig. 6.11 illustrate the photovoltaic power and voltage waveforms, respectively, when the step size is 0.005 for the P&O method. Fig. 6.12 and Fig. 6.13 show normalized power waveforms and normalized  ^  Chapter 6 Maximum Power Point Tracking ^  84  voltage waveforms, correspondingly, when the step size is 0.01 for the P&O method. Comparing to the P&O methods with two difference step sizes, the improvement of IMPPT in steady state can be demonstrated by the values of standard deviation summarized in Table 6.1, where the M stands for the perturbation step of duty cycle. The standard deviation of photovoltaic voltage controlled by IMPPT is much smaller than that commanded by P&O algorithm.  0.4.,,,A.004411 spar.ro....., .s.r.0.47101.4.06iinrimr 4PommapebhbilatisitiblibmixtairliiiAritissibillikidireilifigiiir 1 1^  - -- • I MPPT and module #2 - - - P&O and module #1  35 ;  >7  0  4.5  ^  5.5  1.15  IMPPT and module #2  1.1  1.05  ^3.5^4^4.5  k.  5.5  Z1.25 ^ - P&O and module #1  1.2 ^  •^1.15 ^  2  1.1  5^4^4.5^5^5.5^6  -78- 0.5 ^  E  z 0  -IMPPT and module #2 -  35  4.5  5.5  4.5  5.5  0.3 0.2 0.1 P&O and module #1  8 0 35  Time (s)  Fig. 6.9 Plots of startup procedures of IMPPT and P&O algorithms under weak radiation. The available power is about 3.53W for the 50W solar module. Table 6.1 Comparison of standard deviation in steady state Algorithms IMPPT P&O (M=0.005) P&O (Ad=0.01) Voltage  0.0312  0.1513  0.2214  Power  0.0531  0.0929  0.1174  Chapter 6 Maximum Power Point Tracking ^  1 g 0.995 0.99  0_ 8- 0 . 985 E  0 z  0.98 0.975 0.97  IMPPT and module #2 1 0.1^0.2^0.3^0.4^0.5^0.6^0.7^0.8^0.9  1 g. 0.995 0.99 o_8. 0.985  E  0.98  z 0.975 0.97  0.1^0.2^0.3^0.4^0.5^0.6^0.7^0.8^0.9 Time (s)  Fig. 6.10 Normalized power waveforms in steady state that the 50W photovoltaic module outputs about 22W power and the perturbation step Ad = 0.005  1.03 1.02  - IMPPT and module #2  1.01 8_^ > 1  g  0.99  0  Z 0.98 0.97  0.1^0.2^0.3^0.4^0.5^0.6^0.7^0.8^0.9  1.03 1.02 ?„ 1.01 >°- 1 E 0.99 0 Z 0.98 0.97  0.1^0.2^0.3^0.4^0.5^0.6^0.7^0.8^0.9 Time (s)  Fig. 6.11 Normalized voltage waveforms in steady state that the 50W photovoltaic module outputs about 22W power and the perturbation step Ad is 0.005.  85  •^  ▪  Chapter 6 Maximum Power Point Tracking ^  86  ------ -------- --------------------------- ------------------  1 S 0.995 0.99  RI  da 0.985, 0.98 z 0.975  -IMPPT and module #2  0.97 ^  0.1^0.2^0.3^0.4^0.5^0.6^0.7^0.8^0.9  1 g 0.995  - P&O and module #1  ---  0.99 n_8- 0.985  g  0.98  a 1 11-r  4(1 ;  z 0.975 0.97  0.1^0.2  ^  0.3  0.4^0.5^0.6^0.7 Time (s)  ^  ., 0.8  11 \  ^  0.9  Fig. 6.12 Normalized power waveforms in steady state that the 50W photovoltaic module outputs about 22W power and the perturbation step M is 0.01  1.03 - IMPPT and module #2  - 1.02 1.01  ---  >°-^1  g  0.99 0 z 0.98 0.97  0.1^0.2^0.3^0.4^0.5^0.6^0.7^0.8^0.9 ----------------P&O and module #1  - 1.02 1.01  ii >^1  •  g 0.99  It  z 0.98 -^ 0.97  0.1^0.2^0.3^0.4^0.5^0.6^0.7^0.8^0. 9 Time (s)  Fig. 6.13 Normalized voltage waveforms in steady state that the 50W photovoltaic module outputs about 22W power and the perturbation step Ad = 0.01.  87  Chapter 6 Maximum Power Point Tracking^  6.1.8.3 Results of long-term Test To evaluate the efficiency of photovoltaic power systems sufficiently, daily long-term tests with natural sunlight have been performed for fourteen days in June and July of 2006. Most of them are conducted for eight hours a day. Both IMPPT and P&O were being tested under the same weather conditions, since two power interfaces are available for operations at the same time. A multi-channel data acquisition system is available to record the photovoltaic voltages and currents every second for eight-hour periods. The arithmetic mean of a set of power values, 13 , is expressed as (6.19). Appendix D summarize the important data of the fourteen-day tests.  1 iV (k)I(k)^ N k=i  (6.19)  Calibrations have shown that two photovoltaic modules used for tests give slightly different output characteristics. One of the two photovoltaic modules consistently produced more power than the other under the same conditions. For better comparison, two modules were alternatively connected to the power interfaces in a daily base. Two cases were defined and illustrated in Table 6.2. Table 6.2 Case definision Case #^Test Condition 1^The module 2 was operated by IMPPT and module 1 was controlled by P&O. 2^The module 1 was operated by IMPPT and the module 2 was controlled by P&O. The t-test is a commonly used method to evaluate the differences in means between two groups. This statistic method assesses whether the means of two groups are statistically different from each other. In this study, the t-test was used to test for a difference in power output between a group of data in case 1 and that of case 2 to show the performance improvement of IMPPT. The independent t-test is shown in (6.20) because the two sample sizes of the study cases are equal [46].  Chapter 6 Maximum Power Point Tracking^ t  Where, In (6.20), X, and  ••  2  2^2 SA7i _x,2 = \155,71 + Sy X2  (6.20)  2  S  88  ,  (6.21)  the averaged difference between case 1 and case 2, respectively. s xi _ x2  is the grand standard deviation and expressed in (6.21), where s xi and sA-,2 are the standard deviation for case 1 and case, correspondingly. By using MATLAB function, the test result is demonstrated in Table 6.3, where hx = 1 means that there is a difference between two cases, Px stands for the significance, and CI represent the confidence interval on the averaged difference. MATLAB is a simulation product of the MathWorks Inc.The significance, Px, is 0.0014, which means that by chance you would have observed values oft-test more extreme than the one in this example in only 14 of 10,000 similar experiments. The 95% confidence interval on the mean is [0.3617 0.9212], which indicate there is a difference between two cases. This t-test evaluation statistically demonstrates that the IMPPT algorithm is better than the P&O method regardless of the output difference of the two photovoltaic modules. Table 6.3 Result oft-test 71'2^  S-^sx-1 .4 2  x2  ^hx^Px^CI  1.7686^1.1271^0.2639^0.4010^0.4801^1^0.0014^[0.3617 0.9212] The module # 2 demonstrates more power output than the module #1 When the module #2 is operated by IMPPT and the module #1 is controlled by P&O, the module #2 outputs average 5.30% more energy than the module #1 in an eight-hour test period. When the module #1 is operated by IMPPT and module #2 is controlled by P&O, the module #2 harvests average 3.28% more energy than the module #1. The eleven daily tests prove that the IMPPT is an efficient algorithm that always harvests about 1.01% more energy than the P&O method under different weather conditions. Fig. 6.14 demonstrates waveforms of power, voltage in one eight-hour test performed in June 24, 2006, which is a perfect sunny day. Evaluations show that the IMPPT performs even better in rapidly changing atmospheric conditions. One example is illustrated in Fig. 6.15, where the IMPPT made the module #2 harvest 5.81% more solar energy than P&O  Chapter 6 Maximum Power Point Tracking ^  89  algorithm that controlled the module #1. Broken clouds happened very often in the eight-hour testing period of July 09, 2006. (a) 45 J5; 40 35 1 30 125 0  2  -  PAMPPT and module #2 ^ P&O and module #1 2  3  4  5  6  7  (b) 18  PAMPPT and module #2  >2> 16 14  2  ^ ^ ^ 6 4 7 (c)  18  P&O and module #1  > 16 a 14  2^3^4 Time (h)  5  6  7  Fig. 6.14 Waveforms of power and voltage acquired by the eight-hour test in June 24, 2006, which is a sunny day. (a) Comparison of power waveforms controlled by PAMPPT and P&O, (b) voltage waveform controlled by PAMPPT, (c) voltage waveform controlled by P&O.  6.2 Real-Time Identification of Optimal Operating Points In the operation of maximum power point tracking, the ideal operation is to determine the maximum power point of the photovoltaic array directly rather than track it by the active operation of trial and error that causes the undesirable oscillation around the maximum power point. Since the features of a photovoltaic cell vary with the environment changes in irradiance and temperature from time to time, real-time operation is required to trace the variations of local maximum power points in photovoltaic power systems. The method of real-time estimation proposed in this paper uses polynomials to demonstrate the power versus voltage relationship of photovoltaic panels and implements the recursive least squares method and Newton-Raphson method to identify the voltage of the optimal operating point. The effectiveness of the proposed  Chapter 6 Maximum Power Point Tracking ^  90  methods is successfully demonstrated by computer simulations and experimental evaluations of two major types of photovoltaic panels, namely crystalline silicon and CIS thin film. The study presents the principle and procedure of real-time estimation in details based on the recommended PV model. Finally, the effectiveness of the proposed method is demonstrated by both computer simulation and experimental evaluation.  60  (a) — — PAMPPT anc mocule #2  40  0)k  °- 20 2  14 18  >  P&O and module #1  3^4 (b)  3  4 (c)  6  PAMPPT and module #2 5^6^7  P&O and module #1  0. 16 140^  3^4 Time (h)  7  Fig. 6.15 Waveforms of power and voltage acquired by the eight-hour test in July 09, 2006, which is a cloudy day. (a) Comparison of power waveforms controlled by PAMPPT and P&O, (b) voltage waveform controlled by PAMPPT, (c) voltage waveform controlled by P&O.  6.2.1 Real-Time Estimation of the Maximum Power Point The major modeling features of equivalent circuit and polynomial curve fitting are analyzed and summarized in Chapter 3. Although SSDM shows good modeling performance in both types of photovoltaic cells according to the analysis in Section 3.4, the interactive representation of current-voltage relationship, the I- V curve, makes its real-time application difficult. Based on the evaluation of modeling accuracy in the last section, the conclusions were made that the FSSDM is applicable in modeling of solar cells made of polycrystalline silicon, and the PCF is a general approach to describe the characteristics of photovoltaic cells made of either thin films or crystalline silicon.  Chapter 6 Maximum Power Point Tracking^  91  6.2.2 Parameter Estimation Based on the Further Simplified Single-Diode Model In (3.3), the thermal voltage Vt , the saturation current 'sat, and the photo current Iph are constant when there is no variation in insolation and cell temperature. These parameters of FSSDM can be determined with the measurements of photovoltaic voltage and output current. Then, the voltage of the optimal operating point, Voop, can be derived thereby. The detailed process of parameters has been published in [48]. This method requires four measurements of photovoltaic voltage and current.  6.2.3 Recursive Parameterization of Polynomial Models In solar power system, the parameters change continuously with the variation of insolation and cell temperature, and the parameter observations are obtained sequentially in real time. Therefore, it is desirable to make the computation recursively to save computation time. By using the Recursive Least Square estimation (RLS) [48], the parameters of system model are obtained by minimizing the error between prediction and measurement. The vector of estimated parameters B is updated as (6.22). d(k) = O(k —1) + M (k)[y(k)— y(k)] ^  (6.22)  where (j(k —1) is the previous estimation, the components of vector M(k) are the weighting factors that determine the correction, y(k) is the measurement, and y(k) is the prediction based on the previous parameter estimate. The predictions and corrections are operated recursively in real time to make the model estimation accurate. The RLS estimate can also be interpreted as a Kalman filter [62]. In maximum power point tracking, the P-V curve of the PV array is more useful than I-V curve since the former clearly shows the situation of MPP. Therefore, the proposed modeling process is based on the relationship of power and voltage outputted by photovoltaic panels. Considering a panel made of thin film, the output power is represented by (6.23). p(k) = bl (k)v(k)+ b2 (k)v 2 (k)+ b3 (k)v 3 (k)+ b4 (k)v 4 (k)^(6.23)  where b i are the model parameters that change with environmental condition, and v(k) is the voltage output. For convenience, they are defined as (6.24).  Chapter 6 Maximum Power Point Tracking^ bi (k) v(k) b2 (k) v2 (k) 0(k) = ,p(k) = b3 (k) v3 (k) b 4 (k) _ _v4 (k)  92  (6.24)  Therefore, (6.12) can be written as (6.25). p(k) = p T (k)0 (k)^  (6.25)  According to the theorem of RLS, the estimation of parameter vector d can be obtained by satisfying the following recursive equations. 0(k) = 8 (k –1) + M (k)[p(k)– (D T (101) (IC - 1)]^(6.26) W (k –1)p(k) M (k)=W (k)co (k) = ^T^ ^ 2 + p (k)W (k –1)p(k) [I – M (k)p  T  (6.27)  (k)iW (k – 1)  ^(6.28) 2 where p(k) represents the measured power condition, M(k) is the weighting factor that W (k ) =  determines the direction of correction, W(k) is a nonsingular matrix defined for RLS operation, and X, is the forgetting factor that ranges from zero to unity. The parameters of the proposed model are time-varying due to the variation of insolation and temperature, so the forgetting factor is needed to base the identification on recent data rather than old data. The recursive equations need to start with the initial conditions, the matrix W(0) and vector t'Si(0) . The identification process is illustrated by a flowchart as shown in Fig. 6.16.  6.2.4 Determination the Voltage of the Optimal Operating Point As analyzed in Section 2.2.2, Voop satisfies the condition (6.29). dp ^ =0 dv v .,,, 0,  (6.29)  where p is the output power and v represents the PV voltage. For convenience, it is defined by (6.30). dp dv  f (v) = —  (6.30)  ^  Chapter 6 Maximum Power Point Tracking ^ (  93  start ) *  Set initial values of W(0), 0(0), A., and k  ►1 k  +  -  k+1 I  *  Measure the voltage v(k) and current i(k)  *  Calculate the power p(k) = v(k) * i(k) lir Compute M(k)  ^W (k -1)(o(k)  M(k)2 + (k)W (k -1)(o(k) Update  0(k)  0(k) = 0(k -1) + M (k)[p(k)- (o r (k)0 (k -1)]  +  Calculate W(k) [/ M(k)CO T (k)1W(k - 1) W (k)- ^ 2 -  +  Identify the maximum operating point Voop  Output the new estimation of Voop  Fig. 6.16 Flowchart of RLS estimation procedure to determine the system parameters Then, Voop can be found by setting f(v) = 0. This can be solved by using the NewtonRaphson method (NRM) [63]. This numerical solution requires the derivative f'(v) to be computed. In FSSDM (3.1), dp  ^p  _Iv +i I e (:,:) +i 1 dv^) vt  f(v)=— =i  h ^sat  d2p^. N1 ( v ,..,) f'(v)= = is e ' — —+ z. ^dv 2^Vt \ Vt  (6.31)  (6.32)  Chapter 6 Maximum Power Point Tracking ^  94  With the correctly identified parameters of the fourth-order PCF model (6.23), Voop can be found to make f (v) = dP — = + 2b 2 v + 3b,v 2 + 4b 3= 0 dv A  ^  (6.33)  A  where the parameters^are estimated. In the mathematical model, the f' (v) can be calculated as (6.34). f (v) = ^= d P 2i•2 + 6b3 v + lAv 2^(6.34) dv 2  The flowchart that illustrates NRM's iteration operation in finding the roots of f(v) = 0 is demonstrated in Fig. 6.17.  (start  Initialize v(0)  t = t +1  no  Calculate f [v(t —1)] v(t) = v(t — 1) f [v(t -1)  ]  yes  Fig. 6.17 The flowchart of NRM operation in finding  VoOP  6.2.5 Convergence Analysis Convergence is the most fundamental design requirement of system identification and numerical solvers. The exponential forgetting of data was the technique chosen to identify the time-varying parameters of PV panels changing with insolation and temperature. When a constant forgetting factor is used in the application of RLS, the smaller forgetting factor X is, the  Chapter 6 Maximum Power Point Tracking ^  95  faster the algorithm can track the parameters. However, a small X may lead to an unstable estimation by causing the matrix W to blowup [63-66]. It is desirable to make the X adaptive to the change of system dynamics. Many effective solutions to this problem in real-time system identification are found in [31]. In this study, the forgetting factor is adjusted according to the prediction error as (6.35) and (6.36).  2(k) =1— kc 2 (k)  (6.35)  s(k) = y(k) — S;(k)  (6.36)  where k is a chosen parameter according to system dynamics, E represents the error between prediction and measurement of the output power. In the system of maximum power point tracking, the dominant factor shifting the Voop is the cell temperature. Generally, the cell temperature in photovoltaic power system shows slow dynamics since a sudden increase or decrease of temperature seldom occurs. Therefore, the forgetting factor needs to be tuned properly according to the dynamics caused by the change in temperature. The NRM is a fast approach in finding the roots off(v), but it does not always converge. It is known that the choice of the initial value v(0) is critical to avoid the divergence from the root and to prevent the problem of division by zero when  df dv  (6.37)  To demonstrate this problem, the relationship of f(v) and v is illustrated in Fig. 6.18. The point A represents the short circuit situation, B is the point that causes division by zero, point C is the maximum power point need to be identified, and point D symbolizes the open circuit condition. Through the analysis of this plot, the convergence is guaranteed if the initial value of PV voltage is chosen in the interval (E, D). Generally, it is recommended to choose the initial value for NRM ranging from the possible VOOP, point F, to the open-circuit voltage, point D, to avoid this divergence problem.  6.2.6 Evaluations It is realized that the distribution of the selected points used in FSSDM is sensitive to the estimation accuracy. This problem is demonstrated by the FSSDM estimation process of a MSX83 solar module made of crystalline silicon. The experimental I-V curve is shown in Fig. 6.19 in a solid line, and the estimated I-V curves are illustrated in dashed lines.  96  Chapter 6 Maximum Power Point Tracking^  df/dv = 0 f=0 H  1  -1.5  Pt^a  E-  10^15^20 PV voltage (V) Fig. 6.18 The relationship off(v) and v  l B A^ 01...... ____  rAAPP  ^•  \ A2 4  MPP  a`l) 3- -----------------------  0^  1  -  B  2  c  C2  1  EL 2 ^ 10P2  -  0^5^10^15^20  PV Voltage (V) Fig. 6.19 The estimated I-V curves and maximum power points with FSSDM When the four points, A1, B 1 , C1, and D1, are chosen, the prediction gives a reasonable result, MPP1, indicates that the Voop = 16.95V, compared to the actual Voop of 17.22V. When another four points, A2, B2, C2, and D2, are selected in the evaluation, the estimation shows incorrect maximum power point, MPP2, illustrating Voop = 15.71V, which is far from the true value. This issue makes it impossible to use the FSSDM in the real-time estimation, since the system controller cannot change the operating points in a large range to satisfy the matching requirement  Chapter 6 Maximum Power Point Tracking ^  97  during the practical operation. Therefore, the evaluation results described in following paragraphs are based on the real-time estimation of polynomial model as discussed in Section 3.3. The effectiveness of the proposal with PCF is verified by both computer simulation and experimental tests. During the system operation, the controller senses both the voltage and current of a PV array, while the identification process runs simultaneously to reveal the  estimated Voop, as shown in Fig. 6.20. The value of Voop is identified by RLS and NRM. It is used as the reference in the regulation loop of photovoltaic voltage. The digital filter facilitates the attenuation of the high-frequency noise, which may otherwise lead to incorrect estimates.  Measured signals  A/D Converters  Vpv & Ipv  Scaling  Digital Filter  Vpv  Control Controller commands  Voop  PCF model RLS parameters  Fig. 6.20 The control structure of maximum power point tracking with real-time identification  6.2.7 Simulation It is known that temperature is the dominant factor that affects the value of Voop. The variation of cell temperature primarily depends on the insolation level, ambient temperature, and cell conduction loss. The simulation is designed to show the effect of the constant irradiance on cell surface and the continuous conduction through the photovoltaic cells will cause a gradual increase in cell temperature. As shown in Fig. 6.21, these phenomena will reduce the PV output power and shift the voltage of optimal operating point to lower levels until thermal balance is reached. During the system operation, the PV voltage is supposed to be regulated at a constant level, which is a typical operation mode in photovoltaic power system. The performance is evaluated  by comparing the estimated Voop to the true Voop measured in real time. It is clearly noticed in Fig. 6.21 that the value of estimated Voop smoothly decreases following the change of actual Voop caused by the continuous increasing of cell temperature and converge with the actual Voop  when thermal balance is reached. The plot also shows the unavoidable time delay in estimation  Chapter 6 Maximum Power Point Tracking ^  98  during the real-time operation. The plots also show the variation of parameters, which is caused by the temperature change and estimated by RLS operation in real time. The simulation software is the MATLAB & Simulink, the products of the MathWorks Inc.  U 35 -qs c 30  ,I  E 25  0^1^2 20  2  >8. 10 00^  2  3  4  6  5  7  0.5 0  6^7^  1  16 ...  . ............................................  — Estimated VMPOP ^ Actual V MPOP ^ .... .... ^  ...... .  1^2^3^4^5^6^7  4 Time (s)  Fig. 6.21 The simulated plots of the changes of temperature, PV voltage, PV current, V oop and estimated parameters when the temperature increases from 25 ° C to 35 ° C with a constant insolation of 1000W/m 2 .  6.2.8 Experimental Evaluation The validity of the proposed identification method was examined with a photovoltaic panel and natural solar irradiance. As illustrated in Fig. 6.22, the operating point of the PV panel can be varied randomly by adjusting the load condition. The transducers sense the voltage and  Chapter 6 Maximum Power Point Tracking ^  99  current of panel output, and the data acquisition system (DAQ) provides the interface between transducers and Personal Computer (PC). With the measurements, the real-time identification is performed by computer software (Matlab ®). The testing specification is listed in Table 6.4.  Transducers  Adjustable Load  Fig. 6.22 The block diagram of experimental evaluation Table 6.4 Experimental test environment Photovoltaic module^Shell ST10 (CIS/thin film) Testing conditions^Sunny day, no cloud Ambient temperature: 25.3 ° C Load condition  ^  Resistive load: Resistance ranges from 051 to infinity  Data ac• uisition system LabviewTM DAQ board and software In the beginning, the polynomial curve fitting of P-V characteristics and identification of Voop are analyzed offline with the measured voltage and current of photovoltaic outputs. The  experimental results show the P-V curve is successfully recovered by the operation of RLS and the Voop is estimated accurately by NRM, as shown in Fig. 6.23. Further, the operating points of photovoltaic panel are adjusted in a large range by varying the load conditions. The PV voltage and current are measured in real-time, illustrated in Fig. 6.24 (a) and Fig. 6.24 (b), respectively, and used to estimate the system parameters and to identify Voop in real-time. During the test period, the measured maximum power point is constant at the  point (14.51V, 0.366A). The plot shows the model parameters are successfully identified after the initial condition. The Voop converges to a steady value of 14.53V and does not change with  Chapter 6 Maximum Power Point Tracking ^  100  the variations of operating points as illustrated in Fig. 6.24 (c). The experimental results show some minor variations in the model parameters, which are illustrated in Fig. 6.24 (d). These have insignificant impact on the estimation of Voop. The estimation error in steady state is less than 0.14%, which is more accurate than the identification based on I- V curve demonstrated in Section 3.2 and Section 3.3. 6  5 4  hi 3 2  — Actual P-V curve ^ Estimated P-V curve e Predicted MPP True MPP —  00^  —  5^10^15 PV voltage (V)  20  Fig. 6.23 The plots of offline curve fitting and estimation of Voop by RLS and NRM.  6.3 Summary There are two sections in the chapter to improve the maximum power point tracking. The first part presents an improved algorithm of maximum power point tracking, which can locate accurately the position of maximum power point and reduces the oscillation around the maximum power point in steady state. Instead of the Euler method of numerical differentiation, the study proposes the centered differentiation, which improves the approximation to a secondorder accuracy. The algorithm also occasionally stops tracking operations to avoid unnecessary oscillations around the maximum power point. Long-term evaluations show that it is an efficient algorithm that always harvests about 1% more energy than the P&O method under different weather conditions. However, the proposed algorithm requires more computation power than the simple P&O algorithm proposed in [67, 68], which needs 0.01% of total computational power in  Chapter 6 Maximum Power Point Tracking  ^  101  this bench system. However, the IMPPT requires about 40% computational power due to the fast regulation loop of photovoltaic voltage.  •••■  0.11••■  10  20  •••■  •  IIM  ama  •  I  30^40^50^60^70 (a)  1•••  0.4  'VP  •  -0 ;12 0.2  Lo co a) 2 0  0  10  20  — Estimated V IP ^ Actual V oop  5  0  10^20  a) 0.6 'c-,5 0.4 .0 °- 0.2 t 0 . E -0.2  50^60^70  I  15  >8  30^ 40 (b)  3040^50^60^70 (c)  E  ^ b2 b3  k  im••• •”1.1  b4  10^20  30^(d) 40^50^60^70 Time (s)  Fig. 6.24 The estimation process of Voop and parameters using RLS and Newton method. A general real-time identification method is proposed to estimate the optimal operating point in photovoltaic power systems. It uses the polynomial equations to demonstrate the electric characteristics of photovoltaic panels. This makes the application of the RLS method and NRM method effectively identify the voltage of optimal operating point ( Vo op), which is the key parameter in the operation of maximum power point tracking. The effectiveness of the proposed methods is demonstrated by computer simulations and experimental evaluation. The experimental test shows that the estimation error of Voop in steady state is less than 0.14%. It is also discovered that the polynomial model based on the power-voltage relationship, P-V curve, of photovoltaic panels shows more accurate estimation on Voop than using the model of I-V curve.  Chapter 6 Maximum Power Point Tracking^  102  Although the FSSDM is applicable in the real-time prediction for cells made of silicon, the estimation accuracy is sensitive to the distribution of the selected measurement points. This makes the identification unavailable to use the FSSDM in real time, because the system only operates the voltage of photovoltaic array in a certain range, which may not cover the range of measurements required by the modeling process. Despite the focus of real-time identification on the photovoltaic module made of CIS thin film, the same concept can be applied to the panels made of polycrystalline silicon when the sixth-order polynomial models are adopted. It is proved that the polynomial equation is a general approach in modeling photovoltaic modules made of crystalline silicon and CIS thin film.  103  Chapter 7 Summary, Conclusions and Future Research  7.1 Summary To significantly increase the potential application of photovoltaic (solar electric) systems as a practical, sustainable, energy option, this study has focused on the control and power interfaces better suited for employment in photovoltaic power systems. The main objective is to find effective control algorithms and topologies better suited for extracting the maximum possible power from the photovoltaic modules. The thesis consists of the following major subjects: photovoltaic modeling, topology study of photovoltaic interfaces, regulation of photovoltaic voltage, and maximum power tracking. The thesis contribution has been summarized in Section 1.2. The first chapter first describes a literature review and provides an introduction to photovoltaic power systems. The essential background of solar power generation and control was briefly discussed. Chapter 2 introduced the problem statement and the topology and operation of a specific photovoltaic power system with maximum power point tracking. It complemented the literature review in Chapter 1 and gave plentiful references and analysis in solutions. It revealed the general properties and characteristics of maximum power point tracking. The simulated and experimental results presented in this chapter were a motivation factor that initiated the solutions to existing problems. Section 2.1.4.1 addressed the major effect of the partial shading on the photovoltaic power generation.  Chapter 7 Summary, Conclusions and Future ^  104  Chapter 3 presented a modeling process of photovoltaic cells to configure a computer simulation model, which is able to demonstrate the photovoltaic characteristics in terms of environment changes in irradiance and temperature. This chapter introduced a simplified equivalent electrical circuit model that relates to the physical structure of a photovoltaic cell and proposes two modeling approaches that is suitable for computer simulation and controller design. The study also presented the use of polynomials to model the current versus voltage relationship of photovoltaic modules. The effectiveness of the proposed methods was successfully demonstrated by computer simulations and experimental evaluations of two major types of photovoltaic panels, namely crystalline silicon and CIS thin film. The non-ideal conditions considerably downgrade the performance of maximum power point tracking, especially when photovoltaic modules are connected in series. Chapter 4 describes a comprehensive analysis in various topologies suitable for photovoltaic power systems. This study recommended an individual power interface to be used for each photovoltaic module. Furthermore, it presented that the photovoltaic voltage has advantages to be the regulated variables so that the maximum power point can be efficiently tracked. It also illustrated that the boost topology shows some advantages over the buck converter for this application. Finally, the study proposed a specific bench system designed for evaluation of maximum power point tracking. With this bench system and natural sunlight, two identical control algorithms can be operated at the same time and the same condition. Chapter 5 presented the comprehensive modeling, analysis, and design of a control system to regulate the photovoltaic voltage. In the bench system, both photovoltaic modules and switching mode converters present nonlinear and time-variant characteristics. The method of successive linearization simplifies the nonlinear problem back to the linear case. The system also shows the time-variant characteristics because both the mathematical models and experimental results show that the frequency response is different from one region to another. The major variation lies in the damping factor caused by the variation of dynamic resistance. The change of insolation and temperature also affect the values of dynamic resistance. Furthermore, when the operating point enters the current source region, the plant becomes a lightly damped system, which is a difficult control problem.  Chapter 7 Summary, Conclusions and Future^  105  To ease the analysis, the operating range was divided as four regions, current-source region, power region I, power region II, and voltage-source region according to relative output characteristics. Based on these characteristics and the worst case, a controller is designed through Youla parameterization. However, the computational lag that is introduced by the digital control reduces the system phase margin significantly. Therefore, a further tuning is also presented in this chapter to guarantee the system stability and robustness. Finally, Experimental results and simulation confirm the effectiveness of the presented analysis, design, and implementation. Chapter 6 proposes two approaches in searching for maximum power points. One algorithm is named as the Improved Maximum Power Point Tracking (IMPPT). Another method is about the real-time identification of optimal operating point in photovoltaic power systems. The improved algorithm of maximum power point tracking can locate accurately the position of maximum power point and reduce the oscillation around the maximum power point in steady state. Instead of Euler method of numerical differentiation, the study proposes the centered differentiation, which improves the approximation to second-order accuracy. The algorithm also occasionally stops tracking operations to avoid unnecessary oscillations around the maximum power point. Long-term evaluations show that it is an efficient algorithm that always harvests about one percent more energy than the P&O method under different weather conditions. A general real-time identification method is proposed to estimate the optimal operating point in photovoltaic power systems. It uses the polynomial equations to demonstrate the electric characteristics of photovoltaic panels. This makes the application of the RLS method and NRM method effectively identify the voltage of optimal operating point (Voop), which is the key parameter in the operation of maximum power point tracking. The effectiveness of the proposed methods is demonstrated by computer simulations and experimental evaluation. Additionally, Appendix C investigates how Youla Parameterization can be used in control configurations for DC/DC converters as an alternative to widely used lead-lag-compensator methods. It complemented the controller design in Chapter 5 and gave plentiful analysis in solutions. The main attraction is that the Youla Parameterization simplifies the control design by allowing its parameters to be chosen directly from the desired closed loop characteristics. The basic method is to describe the closed loop control system as an open loop system, using affine  Chapter 7 Summary, Conclusions and Future ^  106  parameterization, replacing the current controller with a simpler current limiter, and integrating an anti-windup mechanism. This method is explained and demonstrated generally, while specific parameters and an anti-windup scheme are shown. Simulations illustrate that this parameterization is an effective method for constant output voltage control in handling transient disturbances as well as startup.  7.2 Conclusions The conclusions from this study of the improved control of photovoltaic interfaces are summarized as follows.  7.2.1 Topology Study of Photovoltaic Interfaces This study proposes an individual power interface for each photovoltaic module. The interface is shown to be suitable for the photovoltaic system. A maximum power point tracking algorithm is also given to minimize the performance reduction caused by the non-ideal conditions. The study shows that regulating the photovoltaic voltage has advantages to improve the performance of maximum power point tracking. This study also demonstrates the boost topology shows some advantages over the buck converter for this specific photovoltaic power system.  7.2.2 Regulation of Photovoltaic Voltage This study presents in-depth analysis and modeling that uncover the intrinsic characteristics of photovoltaic power generation systems. It also presents the use of Youla parameterization to design a control system for regulating the photovoltaic voltage. Experimental evaluation shows the stable regulation of the photovoltaic voltage for three operating regions, namely, the voltage source region, the power region, and the current source region.  Chapter 7 Summary, Conclusions and Future ^  107  7.2.3 Application of Centered Differentiation and Steepest Descent to Maximum Power Point Tracking This study proposed an algorithm of maximum power point tracking, which can locate accurately the position of the maximum power point and can reduce the oscillation around the maximum power point in steady state. Instead of the Euler method of numerical differentiation, the study proposes the centered differentiation, which improves the approximation accuracy. The algorithm also occasionally stops tracking operations to avoid unnecessary oscillations around the maximum power point. Experimental evaluations show that it is an efficient algorithm that harvests about 1% more energy than the traditional method under equal operating conditions.  7.2.4 Real-time Identification of Optimal Operating Point in Photovoltaic Power Systems This work shows that the location of the maximum power point can be estimated in real time rather than tracked by the active perturbation that causes undesirable oscillation around the maximum power point. The effectiveness of the proposed methods is demonstrated by the use of computer simulation. The method was also experimentally validated. The experimental test shows that the estimation error of the maximum power point in steady state is less than 0.14%.  7.3 Future Research One interesting subject for future work is the parallel operation of photovoltaic power modules. The existing MPPT algorithms have not been tested when the modules are connected in parallel. Further evaluation is important. The advantages of parallel operation can be summarized as: ^ Increase the flexibility in power capacity, ^  Optimization of load sharing can increase the overall system efficiency since the efficiency of each module is related to load condition,  ^ Reduce the power loss caused by centralized PV-MPPT systems by separating a big PV array into several small groups.  Chapter 7 Summary, Conclusions and Future^  108  A novel MPPT algorithm will be further developed according to the real-time identification of maximum power point illustrated in section 6.2. This could be the ideal operation of maximum power point tracking because it can determine the maximum power point of the photovoltaic array directly rather than track it by the active operation of perturbation. When the dynamics of insolation and temperature is considered, the identification will become more complicated. A further development based on real-time identification is necessary in the future. The performance of maximum power point tracking is an important factor for photovoltaic power systems. The accuracy and dynamic response of maximum power point tracking affects the system overall conversion efficiency. The existing methods are generally good for testing steady-state performance. However, there is no standard testing method to evaluate the dynamic performance with natural sunlight. 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They include four photovoltaic modules, one digital signal processor, and some laboratory equipments.  A.1 Photovoltaic Modules Four photovoltaic models used for the simulation and experiments include a BP350, Shell ST10, Solarex MSX-83, and Shell ST40, of which parameters are respectively summarized in Table A.1, Table A.2, Table A.3, and Table A.4. STC stands for the standard test condition, which refers to a set of reference photovoltaic device measurement conditions consisting of irradiance of 1000W/m2 , AM 1.5, and 25°C cell temperature. Fig. A.1 demonstrates two BP350 modules installed on one frame with the same angle and direction. Table A.1 Specification of photovoltaic module BP350 Model^  BP350  Photovoltaic Cells^Multicrystalline silicon Cell number^  72  Manufacturer^  BP Solar  Voltage at MPP @ ST*^17.3V Current at MPP @ STC^2.89A Short-circuit current (Isc) @ STC^3.20A Open-circuit voltage (Voc) @ STC ^21.8V  Appendix A Hardware Data and Photos^ Table A.2 Specification of photovoltaic module Shell ST10 Model  Shell ST10  Photovoltaic Cells  CIS thin film  Cell number  42  Manufacturer  BP Solar  Voltage at MPP @ STC  15.6V  Current at MPP @ STC  0.64A  Short-circuit current (Isc) @ STC  0.77A  Open-circuit voltage (Voc) @ STC  22.9V  Table A.3 Specification of photovoltaic module MSX-83 Model^  Solarex MSX-83  Photovoltaic Cells^Polycrystalline silicon Cell number^  36  Manufacturer^  Solarex  Voltage at MPP @ STC^17.1V Current at MPP @ STC^4.85A Short-circuit current (Isc) @ STC^5.27A Open-circuit voltage (Voc) @ STC^21.2V Table A.4 Specification of photovoltaic module Shell ST40 ^ Model Shell ST40 ^ Photovoltaic Cells CIS thin film Cell number^  42  Manufacturer^  BP Solar  Voltage at MPP @ STC^16.6V Current at MPP @ STC^2.41A Short-circuit current (Isc) @ STC^2.68A Open-circuit voltage (Voc) @ STC^23.3V  116  117  Appendix A Hardware Data and Photos 1rI^Radiation sensor  PlioN °Sidle  module  Pliovoltaic module  Fig. A.1 Two photovoltaic modules installed on one frame at the same angle and direction.  A.2 Power Interfaces In photovoltaic power systems, power interfaces are necessary to adapt module characteristics to load requirements. Shown in Fig. A.2, this study presents a DSP-controlled photovoltaic power system that is specially designed and constructed for evaluation purposes. The power conversion modules are based on boost topologies. The system specifications is summarized in Table A.5 Table A.5 Power converter specifications Topology^ Boost Nominal input voltage^17.3V Maximum input voltage^24V Nominal output voltage^24V Maximum output voltage^30V Nominal solar power^50W Maximum solar power^60W Switching frequency^40kHz Max inductor current ripple^10% Maximum input voltage ripple^0.5% Maximum output voltage ripple (%)^0.5%  Appendix A Hardware Data and Photos^  118  onneetions to PhotovolMie modules andDSP module loads Power conversion modules  Fig. A.2 DSP-controlled power interface for photovoltaic power conversion. The digital controller used in the control system is an eZDSPTM LF2407. The core is a TMS320LF2407, 16-bit-fix-point digital signal processor (DSP), manufactured by Texas Instruments Incorporated. The detailed information refers to the product website: of www.ti.com or www.spectrumdigital.com .  A.3 Laboratory Equipments Following equipments were used to obtain the experimental results and calibrate the onboard transducers and signal conditioners. Oscilloscope: Lecroy WaveRunner 6050A. Multimeter: Fluke 45 Dual Display Data acquisition card: National Instruments PCI-6025E. Data acquisition software: National Instruments Labview 7.0  119  Appendix B Comparative Design of Power Interfaces This study presents two conceptual designs used for photovoltaic power interfaces to compare buck and boost topologies, as shown in Fig. 4.3 and Fig. 4.4 respectively. The design specifications of both buck and boost converters are summarized in Table B.1. Table B.1 Design specifications Buck^Boost Topology^ Nominal input voltage^17.3V^17.3V Maximum input voltage^24V^24V Nominal output voltage^12V^24V Maximum output voltage^15V^30V Nominal solar power^50W^50W Maximum solar power^60W^60W Switching frequency^40kHz^40kHz Max inductor current ripple^10%^10% Maximum input voltage ripple^0.5%^0.5% Maximum output voltage ripple (%) 0.5% ^0.5% Convection^ Natural^Natural Efficiency^ >90%^>90%  B.1 Conceptual Design Table B.2 lists symbols used in following paragraphs. For easy comparison, both boost and buck share the same definition and symbols. However, they can be different values. Table B.3 and Table B.4 list important parameters for the conceptual design according to the system requirements tabulated in Table B.1. RMS stands for Root Mean Square.  Appendix B Comparative Design of Power Interfaces ^ Table B.2 Symbols Symbol Description Input capacitor or capacitance CI Output capacitor or capacitance C2 L Inductor or inductance Steady-state photovoltaic current or input I pv current Value of photovoltaic voltage ripple Avpv Switching frequency _I:. D Switching duty cycle Steady-state output voltage Vbat Value of inductor current ripple Ai L Steady-state photovoltaic voltage Vp v Value of output voltage ripple Av Steady-state output current I0  Table B.3 Inductance and capacitance ^ Description Buck^Boost Input capacitance Ci  I' (1— D) — 127.94g 2Av p,f,,,  Inductance L^V(1— D) _ 2AiL fs ,  Output capacitance C2  AlL — 10.44g 8Avpv f5 „, Vpv D — 208.9[tH 2AiL fs „,  110.3µH  I oD  Ail'^— 21.7011F  8Avf,,,,  8Avf,„  — 60.5811F  Table B.4 RMS values Description  ^  RMS current through input capacitor C1  Buck  ^  Boost  1.9311A 0.1669A  RMS current through output capacitor C2 0.2405A 1.3041A RMS current through MOSFET  3.4758A 1.5295A  RMS current through diode  2.3099A 2.4578A  RMS current through inductor  4.1734A 2.8948A  120  Appendix B Comparative Design of Power Interfaces^  121  B.2 Components The important components in the converter design include inductors, aluminum electrode capacitors, MOSFET, diode and MOSFET drivers, as shown in Fig. 4.3 and Fig. 4.4.  B.2.1 Inductor MAGNETICS Kool Mu® powder cores are used to construct the inductor. The inductor parameters are summarized in Table B.5. Table B.5 Inductor parameters ^ ^ Topology Buck Boost ^ Core Two 77071 core in stack ^ Adjusted inductance 1401.1E^260 pH ^ Winding turns 25^34 ^ Wire resistance ^ 0.0120/m 0.0120/m Wire length 1.4m^2.0m ^ Estimated Series Resistance (ESR) 0.0170 0.02452  B.2.2 Other components Other components are listed in Table B.6. The price information was quoted from www.digikey.com in October 2005.  Topology  ^  Input capacitors C1  Output capacitors  Table B.6 Component list ^ Buck Boost One EEUFC2A471 Capacitance: 4700/100V Cost: $2.47/EA 33 µF/50V  22 µF/50V  MOSFET  IRFZ48V24V, US$1.47/EA  Two EEUFC2A101L in parallel Cost $1.86/EA IRFZ48V24V, US$1.47/EA  Power diode  40CTQ045, US$2.15  40CTQ045, US$2.15  MOSFET driver IC  Model: IR2125 Cost: US$3.73/EA  MIC4420ZN Cost: US$1.85/EA  C2  Appendix B Comparative Design of Power Interfaces^  122  B.2.3 Loss estimation An approximation of losses and efficiency is summarized in Table B.7 according to the system specifications and product datasheets. Table B.7 System loss estimations Losses^  Buck Boost  Loss on inductor due to series resistance: ^0.30W 0.20W Loss caused by forward voltage drop of freewheel diode 1.22W 1.30W Loss caused by forward voltage drop of block diode^1.53W n/a Loss on capacitor due to ESR^  0.18W 0.13W  Loss on high-side MOSFET^  0.25W 0.16W  Loss on low-side MOSFET^  n/a^n/a  Scaling factor^  120% 120%  Total estimated loss^  4.18W 2.15W  Expected peak efficiency^  91.6% 95.7%  B.3 Signal Conditioners The DSP controller measures three signals to perform operations of maximum power point tracking. Signal-conditioners scale and offset the measured signals to desired levels for the Analog-to-Digital Converters (ADC) of DSP. The DSP has 16 channels of 10-Bit ADC. The measurement and relative resolutions are listed in Table B.B. A two-pole-low-pass filter is designed to attenuate high-frequency noise, as illustrated in Fig. B.1(a). The cut-off frequency is 10 kHz. A bode plot demonstrates the low pass feature as shown in Fig. B.1 (b). The transfer function can be derived as (B.1). All symbols in (B.1) refer to Fig. B.1. v , , (s)^1  v n (s) R2 R,C2 C1 s 2 +(R1 C1 +R2 C,)s+1  ^  (B.1)  Appendix B Comparative Design of Power Interfaces^  123  B.4 Schematics The complete schematics of power interfaces are illustrated in Fig. B.2. 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T^s 0.0 RE7.48V  1--.  1-33oF  1110 CON_  -,C139 I IOu  158 10Ki 103V 15  1403  ;X-. VDD VDU  1"  4X -  OtT  NC^our x— cAD cm)  f1 1  Z 14 S V  1C109^MIC44 NI2N .15  In  C142 C137 ^11-1.5a0 D104  0135  8155 0  k 74 In  MC33074  310uFf  R143 1880  14 R152  1k  tOkt  L103  RI41  --x  33 34 35 36 37 38  4  IC I 0 3IL  RI ,  31 2014 R147  OUT  GND  .1.15V  C140  11210613  1 C151-1.  _Env T i0(151t12 .....-- •  125  Appendix C Control of a DC/DC Converter Using Youla Parameterization 1 Proportional-Integral-Derivative  (PID) controllers are dominant in power electronics  although more and more new control methods are being proposed for industrial applications. One way to design a PID controller is to tune the control parameters based on experimental tests and practical measurements, using, for example, the Ziegler-Nichols method. For better performance, however, it is preferable to apply model-based techniques. In power electronics, model-based designs are commonly presented with lead-lag-compensators [49] to improve phase margins and closed-loop gains. This generally leads to a controller structure, which is a traditional PI or PID format. The combinations of proportional, integral, and derivative actions usually provide enough flexibility to design a linear controller with a stable loop control and the required performance. Rapid developments in modern control theories and digital control techniques have provided more and more design methods applicable for power electronics. One useful methods is affine parameterization [52], which in some literature is also called Youla parameterization. When applying affine parameterizations, controller designs do not have to follow a standard PID format because the affine parameterization guarantees closed-loop stability. Furthermore, the proposed design procedure is simpler and more straightforward than lead-lag compensator methods.  1 A version of this chapter has been submitted for publication. W. Xiao and W.G. Dunford, "Control of a DC/DC Converter using Youla Parameterization", IEEE Trans. Industrial Electronics.  Appendix C Control of a DC/DC Converter Using Youla Parameterization ^126  This chapter will discuss how affine parameterization is used to design a controller for DC/DC converters. The first section introduces the principle of affine parameterization. Then, the study uses a DC/DC converter example to provide design details. Further, it recommends an anti-windup scheme and a current limiting method suitable for controlling DC/DC converters. Simulation results confirm the effectiveness of the presented solution.  C.1 Introducing Affine Parameterization In the S-domain, a single-input-single-output (SISO) control loop is generally demonstrated as Fig. C.1. The closed-loop transfer function To (s) is characterized in (C.1), where Go (s) represents the nominal plant model, C(s) stands for the feedback controller, R(s) is the control reference (set-point), U(s) symbolizes the control action or plant input, and Y(s) is the plant output or control variable. 0  (s)C(s)  To (s) = G 1 + Go (s)C(s) R(s) + ^  C(s)  U(s) ►  (C.1) G o (s)  Y(s)  Fig. C.1 Block diagram of a negative feedback control system. The relationship between To (s) and C(s) is not proportional as shown in (C.1), causing the correlation between the controller parameters and the closed-loop characteristics to be unclear and making it difficult to tune the parameters of C(s) with respect to the required closed-loop performance. In industry, the model-based designs are sometimes discarded and the control parameters are determined exclusively through experimental tests. Affine parameterization simplifies model-based controller design by introducing a transfer function in terms of the variable Q(s) as shown below in (C.2). This allows that a synthesis of a stable controller is performed by choosing various stable Q(s) according to the system requirement. Consequently, its closed-loop representation is illustrated in Fig. C.2.  Appendix C Control of a DC/DC Converter Using Youla Parameterization^127  U(s) = Q(s)R(s)^  R(s)  Q(s)  1  U(s)p, Go(s)  (C.2)  Y(s) ►  Fig. C.2 Representation of Youla parameterization in a closed-loop control system. This simple transformation allows the transfer function of the closed-loop system to become (C.3) and the transfer function of the controller to be expressed as (C.4). To (s) = Q(s)Go (s)^  (C.3)  Q(s) C(s)= ^ ^(C.4) 1— Q(s)G,,(s)  For a stable plant, it has been proven that stability of Q(s) is sufficient to ensure internal stability of the closed loop system [52]. A control example presented below will illustrate the application of affine parameterization. It is a design of a controller for the DC-DC buck converter system shown in Fig. C.3.  Fig. C.3 Schematic of DC-DC buck converter  C.2^Single Control Loop In many cases, one control loop is adequate in regulating DC-DC buck converters for output of a constant voltage. According to the averaging method [49], the transfer function of the DC/DC converter can be shown in the S-domain as (C.5). The input variable is the switching duty cycle symbolized by d(s), and the controlled variable is the output voltage represented by v(s). The other parameters refer to the values of physical components, including resistance,  inductance, and capacitance, as shown in Fig. C.3.  Appendix C Control of a DC/DC Converter Using Youla Parameterization ^128 ii, v(s)^LC  Go (s)= ^ = 1^1 d(s)^S2^ S+ + RC LC  (C.5)  For analysis, it is convenient to convert (C.5) into the normalized form of a second-order transfer function as (C.6), where K = —V5_, wo = 1____ 1 , and =  1 L LC^LC^2R C  Go (s) =  .  K0 2  S ± 2COn S + (On  (C.6)  Since G0 (s) represents a minimum-phase transfer function, it can be inverted as (C.7). -1  {Gc, (s)] =  s 2 + gcon s + co n2  K0  (C.7)  For minimum order design and unit steady-state output, the nominal closed-loop transfer function is defined as (C.8), which represents a second-order system with unit DC gain. FQ^ (s) =  2  1  a 2 S +a1 s+1  (C.8)  The parameters ai and a2 are specified according to the closed-loop specification in terms of a desired closed-loop natural frequency w a and damping factor c, as expressed in (C.9) and (C.10). 2 1 al = ^ cod  a2,. —  1  2^ Mc/  (C.9) (C.10)  A reasonable value of ci chosen for the closed-loop transfer function in (C.8) is generally between 0.4 to 0.9 corresponding to 25.38% to 0.15% overshoot in a step response. Choosing the correct value of cod is somewhat complicated because it is closely related to control limitations, dynamic responses, and robustness. This will be discussed further in following sections. With the specified closed-loop parameters, the transfer function Q(s) is derived as (C.11). Subsequently, the equivalent transfer function of the controller becomes (C.12).  Appendix C Control of a DC/DC Converter Using Youla Parameterization^129  Q(s) =  FQ (s)G,'D (s)  2 ± 4co + can =s Ko (a2 s 2 + al s +1)  ^s 2 +gw n s + co n2 C(s) = Q(s) 1— FQ (s) K o s(a 2 s + al )  It can be seen by analyzing (C.12) that the transfer function of controller can also be expressed as an equivalent PID controller, in a parallel form (C.13) [52]. The proportional, integral, and derivative gains are expressed in (C.14), (C.15), and (C.16), respectively. The derivative time constant is given as (C.17). K s  C(s) =Kp + K1 + D S r D s +1 KP —  2c.o n al — a 2 co ,27 K o ce  (C.13)  (C.14)  2  KD=  con Ko ai  (C.15)  a 2 2 co n a 2 a + a22 con2 K 0 a12  (C.16)  —  TD  =  a2  (C.17)  1  To ensure the proportional gain Kp is positive, a limit is applied to select the closed-loop parameters of undamped natural frequency cod and damping factor cr. This condition is expressed as (C.18). c/cocr >  con  (C.18)  With Q(s), sensitivity functions can be easily established as (C.19), (C.20), (C.21), and (C.22). To (s) is the function of nominal complementary sensitivity, So (s) represents nominal sensitivity, S io (s) is the nominal input-disturbance sensitivity, and Suo (s) symbolizes the nominal control sensitivity. In the light of lemma of affine parameterization, the closed loop system is internally stable if the plant model Go (s) and Q(s) are stable. The stability of functions  Appendix C Control of a DC/DC Converter Using Youla Parameterization ^130 (C.19)-(C.22) guarantee that the outputs of controller and plant are always bounded, regardless of input disturbances, output disturbances, measurement noise, etc... 7'0 (s) = Q(s)G0 (s)  (C.19)  So (s) = 1 — Q(s)G0 (s)  (C.20)  .5,0 (s) = [1— Q(s)G0 (s)] Go (s)  (C.21)  Suo (s) = Q(s)  (C.22)  Neglecting the inductor current, it is straightforward to design a single control loop for the DC/DC converter. An averaged transfer function showing the relationship of inductor current and duty cycle can be derived as (C.23). Comparing to the transfer function of (C.5), it is seen that the DC gain is different and a zero has been added. The zero causes significant inductor current overshoots when a step change of duty cycle is applied. Large inductor current overshoots should be prevented because they result in harmful effects, such as inductor core saturations or component damage. To ease this problem, an effective method is to apply a setpoint filter, of which the slew rate can be determined according to derivative part of the plant model. In this study, a function of l st-order low-pass filter is employed as (C.24), where 13 is a tuning parameter ranging from 0-1. V^1 (s + iL(s)^L RC ) d(s) s2+ 1 s+ 1 RC LC La =  1 fiRCs +1  (C.23)  (C.24)  C.3 Inductor Current Monitoring and Limiting A high-performance DC-DC converter should be able to maintain a stable and accurate output voltage and keep the inductor current within a certain range. To do this, two-loop control topologies have been defined and developed in past studies [67], as shown in Fig. C.4. These include an outer voltage loop and an inner current loop. Two controllers, Gv and Gi, are needed and designed with consideration of mutual interaction. To simplify this complex two-loop  Appendix C Control of a DC/DC Converter Using Youla Parameterization ^131 control scheme, the inner current loop can be replaced by a current limiter with anti-windup scheme, which is done in this study. Voltage setpoint  Current^Duty ^ command + ^Cycle DC-DC • u, Converter - A —  Output Voltage  Inductor Current  Fig. C.4 Two-loop control of DC-DC converter The quality of output voltage is important for DC/DC converters. It should be accurate during steady state and stable against load disturbance. The inductor current is an internal variable, whose value should be limited to a certain range to avoid any negative effects. Moreover, there is actually no need to regulate the inductor current if the power is smoothly converted and is always within a safe range. Therefore, this current control function can be reasonably thought of as a current limiter instead of an inner current control loop. To avoid additional dynamics appending to the closed-loop system, a proportional-type controller is sufficient because a zero-steady-state error is not necessary. The proportional gain of current limiter, Ki, needs to be designed as (C.25), where R represents the equivalent load resistance, Vg is the input voltage, and lc is the tuning parameter. The lc can be selected from 1-10 to account for different dynamic responses. K =  Vg  (C.25)  C.4 Limitations in DC-DC Converter Control In affine parameterizations, the closed-loop parameters need to be specified properly because control parameters are derived from these. Feedback designs need to consider the fundamental trade-off between performance and robustness. The issues that generally limit the achievable performance of steady state and dynamic response are presented below. Good analysis helps to answer the question: How high can closed-loop bandwidth be assigned for a specific system?  Appendix C Control of a DC/DC Converter Using Youla Parameterization^132  C.4.1 Noise and disturbance Measurement noise is unavoidable in most control systems. In switching-mode converters or inverters, switching noise always appears in the ground plane and power supply. In analog control, the duty cycle signal can be contaminated with some of this high frequency noise. In digital control systems, there is always noise caused by quantization errors. The frequency of these various contributions of noise should be considered as an upper limit on the closed-loop bandwidth; this affects the closed-loop setting of cod . Variation of input voltage and load is a typical disturbance, which causes unwanted transients. The closed-loop bandwidth must be high enough to bring the system back to steady state quickly after these disturbances.  C.4.2 Model Error According to a nominal Linear-Time-Invariant (LTI) model, the controller is generally synthesized to accomplish a required closed-loop performance. The mathematic models of DC/DC converters are usually represented by parameters of voltage, current, inductance, capacitance, and resistance. However, a true LTI model does not exist because the characteristics of inductors, capacitors, and resistors are generally nonlinear. Temperature effect is another factor that introduces time variances and nonlinear factors. In DC/DC converters, even variation of input voltage and load can dramatically change the system model. Despite the uncertainties discussed above, the difference between the nominal model and the actual model can be estimated and used to design a control system robust to the model variation. This usually adds another upper bound on closed-loop bandwidth, especially for lightly damped systems, which are explained in the following sections.  C.4.3 Lightly Damped Systems Some special consideration shall be given to the lightly damped systems, which are also called high Q-factor systems in power electronic literature. It has been proven that these systems are sensitive to modeling error, which is unavoidable in a DC/DC converter model. The damping  Appendix C Control of a DC/DC Converter Using Youla Parameterization ^133  ratio is proportional to per unit load. An example is illustrated in Fig. C.5, which shows that in an ideal converter, the damping ratio goes to zero without any load in the ideal converter and the system is well damped with heavy load. For a lightly damped plant, it is recommended that the closed loop bandwidth is chosen lower than the open-loop resonance to reduce the sensitivity to disturbance [52]. However, the condition in (C.18) must be satisfied to avoid the effect of negative proportional gain. 0.8 0.7 0.6 0 lu: 0.5 a) •g,.. 0.4 , Lit  E  43 0.3  a) _c  I 0.2 —  0. 1  ^  0.5^1^1.5^2 Load condition in per-unit  25  Fig. C.5 Relationship of the damping ratio versus the load condition in per unit -  C.5 Anti-Windup For PWM-type converters, the control variable is the duty cycle, which ranges from zero to one-hundred percent. Saturation of the control variable is commonly caused by a step change of set-point or load, which in a PD controller implemention results in integral windup. A current limiter is useful keeping the inductor current in its constraints as discussed in section C.3. However, current limiters can cause windups because they directly change the control variable without notifying the voltage loop controller. Therefore, an anti-windup control is essential for good closed-loop performance under these conditions. Combined with the voltage regulation controller (C.7) - (C.12) and anti-windup scheme, the overall control configuration proposed is illustrated in Fig. C.6. The block "Sat" restricts the limiter from activating unless the measured inductor current is higher than the upper-level limit,  Appendix C Control of a DC/DC Converter Using Youla Parameterization ^134 Imax• C of is  the direct feedthrough term, which can be equal to the DC gain of the plant model  (C.5). The [C(s)t' represents the inverse form of controller transfer function, which must be stable. 'max is the upper limit of inductor current. Ki is the proportional gain, which reduces the current when it is higher than  Imax .  d(t) and d(t) represent the desired intput and actual input,  respectively. The block "Lsat" represents the saturation of duty cycle and G(s) denotes the transfer function of the plant. The effectiveness of this approach is evaluated in next section.  Fig. C.6 Anti-windup scheme with current limiter and saturation limit  C.6 Evaluation To prove the effectiveness of the proposed control method, a computer simulation has been developed through a DC/DC buck converter as shown in Fig. C.7. All control functions are integrated in the controller block, C(s). The input voltage is 24V DC. The output voltage is expected to be constant 18V DC. The nominal value of load resistance is 252, but the load resistance is variable from 2 - 60. The practical implementations and experimental evaluations refer to Chapter 6.  C.6.1 Modeling and Design The nominal averaged-model is derived according to (C.5). It is realized that the damping ratio changes from 0.51 to 0.17 due to load variation. The effect of load condition on system's dynamic nature is shown in Fig. C.8 by way of bode diagram. It is noted that temperature effects and other non-ideal characteristics of resistors, capacitors, and inductors also affect the dynamic  Appendix C Control of a DC/DC Converter Using Youla Parameterization ^135 characteristics. These effects are ignored in this study because their influence to the linear model is minor and bounded  Fig. C.7 An example of DC-DC buck converter  Bode Diagram  ^40 ^  a 20 17  a)^0 -a a .E  -20cm co 2 -40-  — R = 2 ^ R–6  n n  -68 iFfi) -45  a) -a  a) -90 ci) cu if -135 -180 10 2  10 3^104  ^  10  5^  10°  Frequency (rad/sec)  Fig. C.8 Bode plots of DC-DC buck converter for different load condition The ideal closed-loop performance is decided upon having a damping ratio of 0.60 and an undamped natural frequency of 4.68 kHz. Based on the nominal converter model and following the procedure of affine parameterization shown from (C.7) to (C.12), the transfer functions of Q(s) and the controller C(s) are determined as (C.26) and (C.27), respectively. The system is internally stable because Q(s) is has no poles on the right hand side of the S-domain.  Appendix C Control of a DC/DC Converter Using Youla Parameterization^136  Q(s) =  s2 +15150s + 216500000 ^ (C.26) 6s 2 + 211900s + 5195000000 s 2 +15150s + 216500000 6s 2 +211900s  (C.27)  C.6.2 Simulation With Simulink®, the simulation model of an ideal DC/DC buck converter can be easily constructed and shown in Fig. C.9. The advantage of Simulink lies in its flexibility to configure the controller and converter. The inputs include the input voltage, Vg, the PWM signal and load conditions. The system outputs inductor current i_L and voltage Vo, which can be monitored or used as feedback variables. Referring to [68], this model can be easily extended by implementing some non-ideal factors, such as inductor winding resistance, diode voltage drop, and switch series resistance etc.  Vg PWM ^ 0 G nd CaD Load  Switch  -  Fig. C.9 Simulink model of ideal DC-DC buck converter The controller is constructed in an inverse format with a set-point filter, an inductor current limiter, and an anti-windup scheme, as shown in Fig. C.10. The upper-limit of inductor current is set to 10A. To make the simulation closely mimic ao practical environment, disturbances and noise are intentionally added, including variation in load condition, measurement noise, and input noise. Consequently, low-pass filters are necessary and have been employed to attenuate the high-frequency measurement noise coupled into feedback signals. To illustrate the effectiveness of the functions presented above, several controller models are configured with Simulink® models. All are based on the same parameters illustrated in (C.27).  Appendix C Control of a DC/DC Converter Using Youla Parameterization ^137 The first simulation demonstrates the effect of a step change of set-point, which steps from OV to 18V at the moment of O.lms. Fig. C.11 shows plots of transient responses. The different configurations of controllers are described in Table C.1. The detailed comparison of performance among them is summarized in Table C.3. The results confirm that the sudden change of set-point can result in windup, exhibited significant overshoot and undershoot. It also demonstrates that the overshoot on both output voltage and inductor current can be reduced without sacrificing the response speed when the setpoint filtering works with the anti-windup arrangement. Further, the combination of anti-windup, set-point filter, and inductor current limiter exhibits the best overall performance in terms of overshoot and quick setting time. lmax  den(s) Sat^Ki  ^•  dente) Setpoint^  L^ass 1  Cinf  Setpoint filter  Via Duly LL  Lsat  DC/DC C onverted Don_CO(s)  Scope  Num CO(s) Cinv Noise  1/Cinf  FB noise  Rf*Cf /01 Low-pass 2 N cAse  Input noise  Fig. C.10 Simulation model of closed-loop control system with anti-windup and current limiter Table C.1 Controller configuration Configuration # Anti-Windup Set-point Filtering Inductor current limiter 1 Yes Yes No 2 Yes No No 3 No No No 4 Yes Yes Yes  Appendix C Control of a DC/DC Converter Using Youla Parameterization^138 The second simulation is about the step change of the load condition, which changes from 54W to 162W at the moment of 10ms. This is a common consideration in a converter.design, which is seen as a significant load disturbance by control systems. Fig. C.12 shows the plots of transient responses. The performances among the different controllers are compared in terms of recovery speed and transient overshoot on voltage and current, and are summarized in Table C.3. Table C.2 Comparison of performance on step response Performance indices  Configurations 1 2 3 4 % of overshoot on output voltage 5.00% 13.33% 36.67% 1.11% % of overshoot on inductor current 32.22% 45.56% 62.22% 26.67% 0.32ms 0.33ms 0.52ms 0.20ms ±2% Setting time 90% rising time 0.13ms 0.11ms 0.11ms 0.15ms .......^.............. ......  20  .......  ........^  .......... ....................^..............^.......  E 15  O ^ o *  j 10 5 0.1 14 12 10 •t-' 8 6 4 2 00  0 . 2^0 . 3  0 .4^0.5  Configuration 1 Configuration 2 Configuration 3 Configuration 4 0.6  O Configuration 1 V Configuration 2 o Configuration 3 * Configuration 4 0.1  0.6  1 0.8  a) 7 5; 0.6  0 Z" 0.4 0  0.2 0  0  Configuration 1 Configuration 2 a Configuration 3 * Configuration 4 * ^  *  I^t  0.1^0.2^0.3^0.4 Time (ms)  0.5  0.6  Fig. C.11 Simulated step response with a set point changing from 0 to 18V  Appendix C Control of a DC/DC Converter Using Youla Parameterization ^139 Table C.3 Comparison of performance on load disturbance Performance indices  Configurations 1 2 3 4 % of overshoot on output voltage 17.22% 17.22% 25.00% 7.22% % of overshoot on inductor current 32.22% 32.22% 41.11% 20.00% ±2% recovery time 0.45ms 0.45ms 0.56ms 0.45ms It shows that the sudden change of load condition can result in integral windup, which exhibits large overshoot. It also proved that the overshoot on both output voltage and inductor current could be diminished without sacrificing the speed of response, when the anti-windup is applied. In the case of load disturbance, the set-point filtering is invalid so that the two controllers demonstrate the same performance. Further, the combination of anti-windup, setpoint filtering, and inductor current limiter demonstrates the best overall performance in terms of low overshoot and quick setting time.  22 20 S18  O Configuration 1 V Configuration 2 o Configuration 3 * Configuration 4  • ..  16 14  10.7^10.8  10^10.1^10.2^10.3^10.4^10.5^10.6  O ^ o * 10^10.1^10.2^10.3^10.4^10.5  10.6  Configuration 1 Configuration 2 Configuration 3 Configuration 4 10.7^10.8  O Configuration 1 ^ Configuration 2 o Configuration 3 * Configuration 4 10.2  10.3^10.4 Time (ms)  10.5^10.6  10.7^10.8  Appendix C Control of a DC/DC Converter Using Youla Parameterization ^140  Fig. C.12 Simulated transient response of step-changed load condition  C.7 Summary This study proposes a comprehensive approach to design the controllers for DC/DC converters by ways of affine parameterization and anti-windup. Besides the traditional lead-lagcompensator design based on Bode diagram, this method can be treated as an alternative approach for designing a linear controller for a stable control loop. The advantage of affine parameterization is that the controller parameters are directly derived from the desired closedloop model. This means a designer can easily make the trade-offs decision between system performance and robustness by selecting Q(s). Two cases are investigated separately, one includes only a voltage feedback controller, and the other has a supplementary current limiter implemented using an inductor current measurement. This study also proposes an anti-windup scheme incorporating a current limiter to reduce the current overshoot in controlling DC/DC converters. The simulation shows that the set-point filter is effective in reducing an overshoot caused by step response and the anti-windup scheme combined with an inductor current limiter is useful in reducing an overshoot or undershoot resulting from a sudden load change. It is also important to mention that the anti-windup scheme is unnecessary for some conservative controller designs because the duty cycle changes smoothly and never reaches its upper-level limit.  141  Appendix D Experimental Results Table D.1 summarizes the important data of the 14-day tests to verify the improved algorithm of maximum power point tracking.  142 Table D.1 Results of long-term daily tests Date and weather Configurations  ^  Results  ^  Comparisons  July 09, 2006  IMPPT and module #2  Pave2 =  25.61W  Module #2 output 5.81% more  Most cloudy  P&O and module #1 (M=0.005)  Pave_i =  24.12W  energy than module #1 in 8 hours  July 08, 2006  IMPPT and module #2  Most sunny  P&O and module #1 (Ad=0.005)  Pave 2 = 35.72W P„, i = 33 .85W  July 07, 2006  IMPPT and module #1  Pave_2  -  38.14W  Module #2 output 3.55% more  Most sunny  P&O and module #2 (M=0.005)  Pave_i =  36.79W  energy than module #1 in 6 hours  June 24, 2006  IMPPT and module #1  Pave_2 = 38.76W  Module #2 output 3.63% more  Most sunny  P&O and module #2 (&1 0.005)  P,,, i = 37.35W  energy than module #1 in 8 hours  June 23, 2006  IMPPT and module #2  Pave2 =  37.90W  Module #2 output 5.20% more  Most sunny  P&O and module #1 (4d=0.005)  Pave l =  35.93W  energy than module #1 in 8 hours  June 22, 2006  IMPPT and module #1  Pave_2 =  38.43 W  Module #2 output 3.56% more  Most sunny  P&O and module #2 (Ac1=0.005)  June 21, 2006  IMPPT and module #1  Pave_2 = 38.5 7 W  Most sunny  P&O and module #2 (M=0.01)  Pave_i =  June 20, 2006  IMPPT and module #2  Pinez 3 5.9 5 W  Module #2 output 5.57% more  Most sunny  P&O and module #1 (M=0.01)  Pay. = 33.95W  energy than module #1 in 8 hours  June 19, 2006  IMPPT and module #2  Pave2 =  Most sunny  P&O and module #1 (M=0.01)  Pavei = 37.39W  energy than module #1 in 8 hours  June 18, 2006  IMPPT and module #1  Pave_2 = 36.78W  Module #2 output 3.51% more  Most sunny  P&O and module #2 (M=0.01)  Pave = 35.48W  energy than module #1 in 8 hours  June 17, 2006  IMPPT and module #1  Pave_2 =  26.97W  Module #2 output 2.71% more  Most cloudy  P&O and module #2 (Ad=0.01)  Pave 1 =  26.24W  energy than module #1 in 8 hours  June 16, 2006  IMPPT and module #2  Pave2 ="" 28.49W  Module #2 output 5.18% more  Most cloudy  P&O and module #1 (Ad=0.01)  Pavo = 27.02W  energy than module #1 in 3 hours  June 15, 2006  IMPPT and module #2  Pave_2 =  30.91W  Module #2 output 4.92% more  Most sunny  P&O and module #1 (Ad=0.01)  Pavel =  29.39W  energy than module #1 in 8 hours  June 14, 2006  IMPPT and module #1  Paw 2 = 15.38W  Module #2 output 2.52% more  Most cloudy  P&O and module #2 (Ad=0.01)  P„vei = 14.99W  energy than module #1 in 8 hours  = 37.06W  37.23W  39.45W  Module #2 output 5.24% more energy than module #1 in 8 hours .  energy than module #1 in 8 hours Module #2 output 3.46% more energy than module #1 in 8 hours  Module #2 output 5.21% more  

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