ABSTRACT AN IMPROVED MAXIMUM POWER POINT TRACKING ALGORITHM USING FUZZY LOGIC CONTROLLER FOR PHOTOVOLTAIC APPLICATIONS

ABSTRACT AN IMPROVED MAXIMUM POWER POINT TRACKING ALGORITHM USING FUZZY LOGIC CONTROLLER FOR PHOTOVOLTAIC APPLICATIONS This thesis proposes an advanced maximum power point tracking (MPPT) algorithm using Fuzzy Logic Controller (FLC) in order to extract potential maximum power from photovoltaic cells. The objectives of the FLC are to increase tracking velocity and to simultaneously solve inherent drawbacks in conventional MPPT algorithms. The performances of the conventional Perturb & Observe (P&O) algorithm and the proposed algorithm are compared by using MATLAB/Simulink, and the theoretical advantages of FLC were demonstrated. To further validate the practical performance of the proposed algorithm, the two algorithms were experimentally applied to a DSP-Controlled boost DC-DC converter. The experimental results indicated that the proposed algorithm performed with faster tracking time, smaller output power oscillation, and higher efficiency, compared to that of the conventional P&O algorithm. Pengyuan Chen August 2015

AN IMPROVED MAXIMUM POWER POINT TRACKING ALGORITHM USING FUZZY LOGIC CONTROLLER FOR PHOTOVOLTAIC APPLICATIONS by Pengyuan Chen A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Engineering in the Lyles of College of Engineering California State University, Fresno August 2015

2015 Pengyuan Chen

APPROVED For the Department of Electrical and Computer Engineering: We, the undersigned, certify that the thesis of the following student meets the required standards of scholarship, format, and style of the university and the student's graduate degree program for the awarding of the master's degree. Pengyuan Chen Thesis Author Woonki Na (Chair) Electrical and Computer Engineering Nagy Bengiamin Electrical and Computer Engineering Ajith Weerasinghe Mechanical Engineering For the University Graduate Committee: Dean, Division of Graduate Studies

AUTHORIZATION FOR REPRODUCTION OF MASTER S THESIS I grant permission for the reproduction of this thesis in part or in its entirety without further authorization from me, on the condition that the person or agency requesting reproduction absorbs the cost and provides proper acknowledgment of authorship. X Permission to reproduce this thesis in part or in its entirety must be obtained from me. Signature of thesis author:

ACKNOWLEDGMENTS I wish to thank my major professor, Dr. Woonki Na most deeply for his support, guidance, and encouragement through my graduate study. I would like to thank to Dr. Nagy Bengiamin who helped me to establish my background of the power electronics and control theory solidly. I want to thank Dr. Ajith A. Weerasinghe who provided me valuable suggestions of photovoltaic applications. Also, I would like to thank Dr. Daniel Bukofzer who helped me to enhance my background of the mathematics and system modelling. Finally, I would like to extend my heartfelt gratitude to my parents, Lin Chen and Xiaomeng Chen, and my friends for their love, support, and encouragement while pursuing my course of study.

TABLE OF CONTENTS Page LIST OF TABLES... viii LIST OF FIGURES... ix 1 INTRODUCTION... 1 1.1 Characteristics of Photovoltaics... 1 1.2 Topology of Stand-Alone Photovoltaic Systems... 4 1.3 Topology of Grid-Connected Photovoltaics Systems... 8 1.4 Scope of This Thesis... 11 2 PHOTOVOLTAICS MODELLING... 13 2.1 Structure of Photovoltaics... 13 2.2 PV Modelling and Simulation... 15 2.3 The Internal Impedance of Photovoltaics... 25 3 MAXIMUM POWER POINT TRACKING ALGORITHM... 30 3.1 Conventional MPPT Algorithms... 30 3.2 Performance of the Conventional P&O Algorithm... 37 3.3 Fuzzy Logic Controller (FLC)... 42 3.4 Simulation and Comparison... 50 4 BOOST DC-DC CONVERTER... 56 4.1 Topology of the Typical Boost DC-DC Converter... 56 4.2 Small Signal Model... 61 5 PROTOTYPE IMPLEMENTATION... 70 5.1 Parameters of the Boost DC-DC Circuit... 71 5.2 Peripheral Circuits... 75 5.3 Signal Process System... 88

vii Page 5.4 Implementations of the MPPT Algorithm... 96 6 CONCLUSION... 99 REFERENCES... 101 APPENDICES... 105 APPENDIX A: MATLAB CODE... 106 APPENDIX B: SYSTEM SCHEMATICS... 111

LIST OF TABLES Page Table 2-1 The Specification of SW-260-mono [31]... 22 Table 2-2 Simulated Parameters of the SW-260-mono... 22 Table 2-3 The Rmpp of the SW-260-mono under Different Irradiation Conditions... 28 Table 2-4 The Rmpp of the SW-260-mono under Various Temperature Conditions... 28 Table 3-1 Parameters of Photovoltaics... 31 Table 3-2 The Numerical Unions Corresponding to the Fuzzy Sets... 45 Table 3-3 Rules for the Proposed FLC... 47 Table 3-4 The Configuration of Simulations... 50 Table 4-1 Parameters of the Designed PV System... 66 Table 4-2 Linear Approximations with Different values of Rpv... 67 Table 4-3 Effects of Independently Increasing a Parameter in a PI Controller [22]... 68 Table 4-4 Step Response of the Closed-Loop Compensated System... 69 Table 5-1. Parameters of Components of the Boost Circuit... 71 Table 5-2 Parameters of the Solar Panel under Testing Conditions... 72 Table 5-3 Parameters of the Voltage Divider... 83 Table 5-4 Parameters of the Analog Low Pass Filter for Voltage Measurement.. 85 Table 5-5 Parameters of the Current Sensing Circuit... 88

LIST OF FIGURES Page Figure 1-1 Definitions of the solar cell, solar panel, and solar array... 3 Figure 1-2 I-V curve (left) and P-V curve (right)... 4 Figure 1-3 The topology of the stand-alone photovoltaic system.... 5 Figure 1-4 The topology of the voltage regulation of photovoltaics [10]... 6 Figure 1-5 A power system with its PWM signal... 7 Figure 1-6 The general topology of a grid-connect photovoltaic system... 9 Figure 1-7 A grid-connect photovoltaic system with micro inverters... 10 Figure 1-8 A grid-connected photovoltaic system with power optimizers... 10 Figure 2-1 P-N junction of a solar cell [16]... 14 Figure 2-2 The simplest single diode model... 15 Figure 2-3 The improved signal diode model... 16 Figure 2-4 The double diode model... 16 Figure 2-5 Short circuit current... 17 Figure 2-6 Open-circuit voltage... 18 Figure 2-7 The equivalent circuit model of a photovoltaic matrix... 20 Figure 2-8 The diagram of the algorithm for finding the parameter pair (Rs,Rp)... 21 Figure 2-9 The simulated I-V curves of the SW-260-mono operating under different irradiation conditions.... 23 Figure 2-10 The simulated P-V curves of the SW-260-mono operating under different irradiation conditions.... 23 Figure 2-11 The simulated I-V curves of the SW-260-mono operating at different temperature conditions.... 24 Figure 2-12 The simulated I-V curves of the SW-260-mono operating at different temperature conditions.... 25

x Page Figure 2-13 The I-V curve for different resistive load... 26 Figure 2-14 A wrong topology for changing the internal resistance of a solar panel... 27 Figure 2-15 The proper system diagram of a photovoltaic system.... 27 Figure 3-1 The P-V curve of the photovoltaics under STC... 31 Figure 3-2 The general mechanism of the P&O algorithm... 32 Figure 3-3 The flow chart of the conventional P&O algorithm [19]... 33 Figure 3-4 Derivative photovoltaic power with respect to photovoltaic voltage.. 35 Figure 3-5 The flow chart of the InC algorithm [19]... 35 Figure 3-6 Power with P&O p-i 0.1 vs p-i 2.0... 38 Figure 3-7 Average power conducted by P&O with p-i 0.1 vs p-i 2.0... 38 Figure 3-8 Tracking time of P&O with p-i 2.0 volts... 39 Figure 3-9 Tracking time of P&O with perturbation intensity 0.1 volts... 39 Figure 3-10 Energy with P&O: p-i 0.1 vs p-i 2.0 (the 1000 th second)... 40 Figure 3-11 Energy with P&O: p-i 0.1 vs p-i 2.0 (the 7000 th second)... 41 Figure 3-12 An illustration of the membership function μax... 43 Figure 3-13 The sectionalized P-V curve with different operating zones.... 44 Figure 3-14 The membership function E... 46 Figure 3-15 The membership function CE... 46 Figure 3-16 The membership function PT... 46 Figure 3-17 The output surface of the proposed FLC... 48 Figure 3-18 Unexpected problem... 49 Figure 3-20 The Simulink block diagram of the Fuzzy Logic Controller... 51 Figure 3-21 The variable short circuit current of the simulated solar panel.... 51 Figure 3-22 The MPPT traces of the two MPPT strategies... 52 Figure 3-23 MPPT traces of two MPPT strategies in the time interval (0s,3s)... 53

xi Page Figure 3-24 MPPT traces of the two MPPT strategies during the decrease of the irradiation... 53 Figure 3-25 The decisions of FLC in the transition period (3.0s, 3.16s)... 54 Figure 3-26 The responses of the two strategies during the change in the irradiation condition... 55 Figure 4-1 The typical topology of a boost DC-DC converter.... 57 Figure 4-2 The equivalent circuit during switching on periods... 58 Figure 4-3 Inductor current and voltage during switching on periods... 58 Figure 4-4 The equivalent circuit during switching off periods... 59 Figure 4-5 Inductor current and voltage during switching off periods... 59 Figure 4-6 The average dynamic model of a boost DC-DC converter... 61 Figure 4-7 The equivalent circuit of small signal model (a) [21]... 62 Figure 4-8 The equivalent circuit of small signal model (b) [21]... 63 Figure 4-9 Small signal model: output terminal of a boost DC-DC converter [21]... 63 Figure 4-10 Equivalent circuit of a boost converter with irregular input source.. 64 Figure 4-11 Bode plot of the variable-parameters system... 67 Figure 4-12 Step response of the closed-loop compensated system.... 69 Figure 5-1 The proposed topology of the PV boost DC-DC circuit... 71 Figure 5-2 Amplitude of the ripple current versus photovoltaic voltage... 74 Figure 5-3 Current waveforms of the PV model, inductor and input capacitor... 75 Figure 5-4 The gate drive circuit... 76 Figure 5-5 The peak-peak voltage of the noise on the 5 volts DC bus (without filtering capacitor)... 77 Figure 5-6 The fundamental frequency of the noise on the 5 volts DC bus (without filtering capacitor)... 78 Figure 5-7 The suppressed switching noise... 78

xii Page Figure 5-8 The drain-source voltage of the IRFP460A (without gate resistor and RC snubber circuit)... 81 Figure 5-9 The drain-source voltage of the IRFP460A (with gate resistor and RC snubber circuit)... 81 Figure 5-10 The topology of the voltage divider... 82 Figure 5-11 The bode plot of the proposed low pass filter... 84 Figure 5-12 The low pass filter for voltage sensing... 85 Figure 5-13 The topology of High-Side Current Sensing... 86 Figure 5-14 The topology of Low-Side Current Sensing... 87 Figure 5-15 The Low-Side Current Sensing circuit.... 87 Figure 5-16 The layout of the MPPT system.... 89 Figure 5-17 The topology of the MPPT system.... 89 Figure 5-18 The designed MPPT system... 90 Figure 5-19 Simulink block of the digital PI controller... 91 Figure 5-20 The voltage regulation of photovoltaics... 92 Figure 5-21 The inductor voltage waveform (CCM)... 93 Figure 5-22 The inductor voltage waveform (DCM)... 93 Figure 5-23 The illustration of DCM detection mechanism... 94 Figure 5-24 The flow chart of the DCM detection mechanism.... 95 Figure 5-25 The performance of the conventional P&O algorithm... 96 Figure 5-26 The performance of the improved MPPT algorithm using FLC... 98

1 INTRODUCTION As the demand for solar energy is dramatically increasing, solar energy applications have been massively studied for the last decade. Solar panels can conveniently convert the received light energy to electricity without any pollution. However, the potential maximum power generated by a solar panel heavily depends on irradiation and temperature conditions. Additionally, due to the nonlinear current-voltage (I-V) characteristics of photovoltaic cells, the output voltage of photovoltaics is determined by the photovoltaic current so that the output power cannot be forthrightly predicted by the load impedance. To achieve the maximum power point (MPP) of photovoltaics, a photovoltaic MPPT control system is normally needed. A tracking control system can continuously changes its operation status, and keeps perturbing the voltage or current level of its input power in order to find the potential maximum power point. Photovoltaic systems can be generally categorized into stand-alone and grid-connected photovoltaic systems. In this thesis, the proposed MPPT control strategy for stand-alone photovoltaic systems has been discussed and validated throughout simulation and experimental results. The characteristics of photovoltaics are briefly addressed in the section 1.1. The topologies of the two types of photovoltaic systems and their components are introduced in sections 1.2 and 1.3. The scope of this thesis is described in section 1.4. 1.1 Characteristics of Photovoltaics In photovoltaic systems, the core elements for converting solar energy into electricity are the photovoltaic (PV) cells. Irradiated PV cells can generate DC power and supply their direct-connected load. However, the photovoltaic power and its voltage level may not always meet the desired requirements. This is

2 because the photovoltaics are well-known by their nonlinear voltage-current and voltage-power characteristics [1]. Given the load impedance and environmental conditions, photovoltaics can perform as irregular current source or voltage source, which may be unacceptable for most power electronic applications. To compensate for these disadvantages, photovoltaic systems are designed to regulate the performance of photovoltaics in terms of their output voltage and power. The two primary objectives of a photovoltaic system are to extract maximum power from photovoltaics and to regulate the voltage level of the photovoltaic power. A photovoltaic system generally contains variable system structures in order to shift the operation point of photovoltaics. Hence the stability and efficiency of a photovoltaic system is commonly challenged by the variable power load, irradiation, temperature, and shading condition [1-2]. To enhance stability, robustness and efficiency of photovoltaic systems, sufficient statistical efforts and uncommon control strategies are normally involved in the system design. A solar panel normally consists of numbers of inter-connected solar cells. The pattern of the connection can be cascaded, paralleled or both. The size and rated power of a solar panel is determined by the number of its solar cells, by the area of each solar cell, and by the efficiency of each solar cell. If a solar panel can be defined as a matrix of inter-connected solar cells, a solar array can be defined as a matrix of inter-connected solar panels. A straightforward illustration is shown in Figure 1-1. Throughout the photovoltaic effect, irradiated photovoltaics can generate DC power. The electronic characteristics, such as the output current, output voltage and internal resistance of a solar cell are generally determined by the intensity of the received irradiation, by the temperature of the cells surface, by the efficiency of the photovoltaic conversion, and by the load impedance. For each

3 Figure 1-1 Definitions of the solar cell, solar panel, and solar array solar cell, the model expression related to its output voltage and output current is nonlinear such that the calculation of the cell s power is not straightforward. In this thesis, the photovoltaic voltage and current will be abbreviated by PV voltage and PV current, respectively. To illustrate the nonlinear characteristics, power-voltage (P-V) curve and current-voltage (I-V) curves are seen in Figure1-2. Note that any photovoltaic application will show a unique I-V curve and a unique P-V curve under an arbitrary environmental condition. On an ideal P-V curve, there will be only one point that contains two parameters, the photovoltaic voltage and photovoltaic power, where the value for the photovoltaic power is maximized. This point is named as the maximum power point (MPP).

4 Figure 1-2 I-V curve (left) and P-V curve (right) 1.2 Topology of Stand-Alone Photovoltaic Systems According to objectives of photovoltaic systems, photovoltaic systems can be generally classified into stand-alone and grid-connected photovoltaic systems [3]. Stand-alone photovoltaic systems are designed to supply local electric load, and generally consist of energy storage devices for meeting excessive electricity demands. Grid-connected photovoltaic systems are designed to deliver photovoltaic power to electric grids [4]. In this section, a brief introduction of stand-alone photovoltaic systems will be presented. The fundamental topology of a stand-alone photovoltaic system is shown in Figure 1-3. A stand-alone system consists of the following components: - Solar Cells/Solar Panels/Solar Arrays - Maximum Power Point Tracking Controller - Voltage regulator of photovoltaics - PWM Generator - DC-DC Converter - DC Electric Load - DC-AC Inverter (Optional)

5 Figure 1-3 The topology of the stand-alone photovoltaic system. Maximum Power Point Tracking (MPPT) controllers are popular in both stand-alone and grid-connected photovoltaic systems. A MPPT controller can be designed as a physical analog circuit or an embedded system. The main objective of a MPPT controller is to extract potential maximum power from photovoltaic cells by continuously perturbing the operation point of the photovoltaic cells. The operation point of photovoltaics consists of two parameters, the photovoltaic voltage and photovoltaic power. It can be treated as a point on a P-V curve. The operation point will reach the maximum power point if the MPPT controller rationally perturbs the photovoltaic voltage. At the end of every control interval, a new photovoltaic voltage reference is calculated by the MPPT algorithm and sent to the photovoltaic voltage regulator. Recently, even though numerous MPPT algorithms have been researched [5-9], the adopted MPPT algorithms of commercialized solar energy applications are still based on the conventional

6 Perturb & Observe (P&O) algorithm due to its easy implementation and robust performance. Related discussions will be presented in chapter 3. A Voltage Regulator of photovoltaic cells is essential for a MPPT controller. The voltage regulator is to make the photovoltaic voltage trace its reference value, which is provided by the MPPT algorithm. There are few MPPT research papers that mention photovoltaic voltage regulators by showing a PI/PID controller in their control loop. Additionally, how to design a voltage regulator for a photovoltaic power source has rarely been explained thoroughly. In this thesis, a theoretical discussion related to photovoltaic voltage regulation will be presented in chapter 4. The fulfillment of the photovoltaic voltage regulation requires a proper compensator which can improve the transient response of a photovoltaic system. The design of such a compensator must consider the photovoltaic model and its associated power electronic system. The proposed feedback control loop for the voltage regulation is shown in Figure 1-4 [10]. Figure 1-4 The topology of the voltage regulation of photovoltaics [10] The fundamental control signal of a photovoltaic system is a Pulse-Width- Modulation (PWM) signal, which can be generated by an analog circuit or by a microcontroller. In a photovoltaic system, the PWM signal causes the system to perform two structures in every switching interval. The widths of switch-on and

7 switch-off intervals determine system dynamics. In other words, by changing the duty-ratio of the PWM signal, the DC-DC converter (which is shown in Figure 1-3 and 1-5) can change the proportion of its input terminal voltage to its output terminal voltage. The equivalent internal impedance of PV cells is able to be perturbed. In consequence, the photovoltaic power can be changeable [11]. Figure 1-5 A power system with its PWM signal A DC-DC converter can step-up/step-down the voltage level of its input DC power. In a photovoltaic system, the input photovoltaic voltage level may not exactly meet the requirement. Therefore, the first objective of a photovoltaic DC- DC Converter is to change the voltage level of input photovoltaic power. The second objective is to fulfill the voltage regulation of photovoltaics, as associated with a voltage or current control. Several MPPT algorithm research assumed that the electric load of photovoltaic MPPT systems can be only resistive. Such assumption may be impractical. The transient response of a power converter may be undesirable and unpredictable if electric load is only resistive. In a boost or buck-boost converter, a resistive load introduces a variable Right-Hand-Plane (RHP) zero into the system s transfer function, as shown in equation (4.11). The RHP zero may result in difficulties to the design of a compensator which regulates the system s output. The parameters of the transfer function, which is a linear approximation of the system, and the RHP zero both depend on the duty ratio of the PWM signal. Thus

8 the output voltage regulation of the converter will be further laborious. To avoid the above issue related to the converter s output voltage regulation, the appropriate electric load for a stand-alone photovoltaic system should consist of depthrecycled batteries and ultra-capacitors. These can absorb the increasing photovoltaic power, and stabilize the voltage of the output terminal at a relative fixed level if the load s capacitance is sufficiently large. Many photovoltaic systems are designed to supply to AC loads, like motors or pumps. In such case, a DC-AC Inverter is added into the system topology. A DC-AC Inverter can be directly cascaded to a DC-DC converter, or can be connected to the medium energy storage devices, such as ultra-capacitors and batteries. 1.3 Topology of Grid-Connected Photovoltaics Systems The fundamental components of a grid-connected photovoltaic system involve photovoltaic arrays and a DC-AC inverter. The basic topology is shown in Figure 1-6. To convert the standard AC power (120V/60Hz), the required voltage level of the input DC power should be greater than 240 volts. However, to meet this voltage requirement, the size of the input photovoltaics has to be enlarged. Given the fact that the size of a general 240W solar panel, which has nominal 30V/8A output, is normally 1.35 m 2. Throughout calculation, the size of a solar array with a 240V rated voltage is about 108m 2. Such size may cause multiple issues when the partial shading happens. The partial shading on a photovoltaic array will cause two typical problems, the reduction in power output and thermal stress on the photovoltaic array [15]. The photovoltaic current of solar cells normally diminishes whenever the received irradiation reduces. With the shaded cascaded connection pattern, the photovoltaic current of the PV cells will reduce

9 due to those shaded solar cells. The residual power, which cannot be utilized by the electric load, because of the shading condition, will be partially transformed to thermal energy, which may affect the photovoltaics efficiency. Recently, with the help of micro-inverters, photovoltaic engineers are glad to divide a large-size photovoltaic array to several small-size arrays, for solving the shading effects. As shown in Figure 1-7, every micro inverter processes power for one panel, and consists of a DC-DC converter and a DC-AC inverter. The DC-DC stage is used to boost the voltage level of the photovoltaic power to about 240 volts for the DC- AC conversion. The MPPT function for the PV panels is performed centrally at the inverter stage [29]. Hence, each panel can be isolated from other panels in the process of the power transmission. Figure 1-6 The general topology of a grid-connect photovoltaic system Similar to the micro inverter, alternative applications for optimizing power of photovoltaic cells are power optimizers. As shown in Figure 1-8, the output of each DC-DC converter is connected in series prior to the DC-AC inverter. At each DC-DC state, the MPPT function is fulfilled. Different from the micro inverter, the objective of this topology is to deliver the maximized photovoltaic power to a universal DC bus [29].

10 Figure 1-7 A grid-connect photovoltaic system with micro inverters Figure 1-8 A grid-connected photovoltaic system with power optimizers

11 1.4 Scope of This Thesis The advanced MPPT algorithms for photovoltaic systems have been significantly researched in the past decade. Y. Gaili and H. Hongwei from Xi an University of Science & Technology directly shifted the operating point of photovoltaics by perturbing the duty ratio of the switching signal of their photovoltaic boost DC-DC converter with an invariant scale [26]. By referring to the previous changes in the photovoltaic power, their algorithm varies the duty ratio for the next control period. Even though, their method is doable, the possible values for the input photovoltaic voltage can be predicted, given that the input voltage is proportional to the voltage of the output terminal with respect to the duty ratio. A fixed perturbation intensity in the duty ratio may also cause the inherent issue within the conventional P&O algorithm. Note that the inherent drawback of the P&O algorithm is that increasing the tracking velocity will definitely affect the MPPT efficiency and vice versa. To solve this drawback, S. Tao et al from North China Electric Power University designed a gradient method to perturb the photovoltaic voltage with the gradient intensity, which is proportional to the derivative value of the change in photovoltaic power with respect to the change in voltage [27]. Their algorithm has only been simulated in the MATLAB/Simulink environment. Therefore, the practical performance of their algorithm may be needed to experimentally validate. Given that switching noise is hard to eliminate in switching circuits, the voltage and current measurement signals generally contain the switching noise. Consequently, a derivative operation will boost the noise level, and may break their MPPT control. Y. EI Basri et al, introduced a discrete-time PI controller to create a variable perturbation offset in the photovoltaic voltage [28]. The error signal for the PI controller is the change in photovoltaic power. Hence, by a simple conjecture, the

12 operation point may stay on a point, which can result in a zero (0) watts change in photovoltaic power, and the MPPT control may stop. In practice, the P-V and I-V curves of a photovoltaic application keep changing with environmental conditions, and the position of the potential MPP may continuously shift. Therefore, the MPPT controller should not stop tracking the shifting MPP. At present, commercialized MPPT controllers for photovoltaic systems are still based on the conventional Perturb & Observe algorithm due to its easy implementation and control robustness, though it is not efficient [5]. Therefore, there are two objectives for this research: 1) to design an advanced MPPT algorithm with a Fuzzy Logic Controller (FLC) for generating flexible perturbation intensities, and 2) to validate the proposed algorithm throughout a designed PV system. In chapter 2, the characteristics of photovoltaics will be generally reviewed, and the photovoltaic modeling will be introduced in detail via a single diode model. In chapter 3, the concepts of several fundamental MPPT algorithms will be discussed. Based on the mechanism of the conventional P&O algorithm, the derivation of the proposed MPPT algorithm will be addressed. Chapter 4 will emphasize the modeling of a DC-DC boost converter and the voltage regulation of photovoltaics. In chapter 5, the implementation of the proposed MPPT algorithm, and the design of the photovoltaic boost DC-DC converter will be discussed, along with the related problems.

2 PHOTOVOLTAICS MODELLING Solar cells are the basic elements of solar panels/solar arrays which provide renewable electricity without any pollution. Solar cells can convert received light energy into electricity and generate DC power. In the current solar energy market, the price per watt of solar panels varies from 0.36 dollars to 1.44 dollars. Customers only need to pay the cost of solar panels, with no additional charges for using permanent renewable solar energy. Nevertheless, the cost of solar panels is still high. For example, the cost of a six kilowatts-per-hour photovoltaic (PV) system may be about six thousand dollars. Fortunately, photovoltaics generating their maximum power can reconcile for their high cost. To extract the maximum power from PVs, their mathematical model, which can predict their nominal voltage and nominal current, should be investigated. In this chapter, the structure of photovoltaics is briefly reviewed in section 2.1. Section 2.2 introduces an approach to model solar panels with a signal diode model. The internal impedance of photovoltaics is discussed in section 2.3. 2.1 Structure of Photovoltaics The process of PVs converting received light energy into electricity is known as the photovoltaic effect. When the light irradiates the surface of a solar cell, part of the photons of the light may get reflected or consumed immediately when they impact the surface of the solar cell. This is because the energy that they carry is too weak to be converted into electricity. Only the photons, which are absorbed near the P-N junction of the solar cell, can work for the photovoltaic effect. By absorbing the energy of the photons, the atoms in the P-N junction generates plentiful hole-electron pairs. Under the force of the electrical field of the P-N junction, the holes carry the positive charge and shift from the N-type layer to

14 the P-type layer. The electrons carrying the negative charge, escape from the P- type layer, and eventually migrate to the N-type layer [16]. By connecting an electric load to the P-N junction, such as resistor, the electrons in the N-type layer flow through the load, and finally enter the P-type layer. The holes in P-type layer combine with the coming electrons. The P-N junction of a PV cell is shown on Figure 2-1[16]. Figure 2-1 P-N junction of a solar cell [16] The size of the surface of a solar cell normally varies from 4cm 2 to 225cm 2. The nominal power of a solar cell, under standard test condition (STC), is less than 4 watts. The STC means an irradiation of 1000W/m 2 at 25 temperature. The nominal voltage of a silicon solar cell is about 0.5 volts, while the nominal current is about 8 amperes. Multiple solar cells are generally interconnected for enhancing the rated power. Connecting solar cells in parallel can increase the rated current, while connecting them in series can increase the rated voltage.

2.2 PV Modelling and Simulation 15 2.2.1 Fundamentals of Photovoltaics Figure 2-2 illustrates the simplest model of a solar cell, which is presented by an equivalent current source and a diode. Based on the simplest solar model, the computation for obtaining the I-V and P-V curves requires three parameters: the short-circuit current (I sc ), the open-circuit voltage (V oc ), and the diode ideality factor. This solar model may exhibits serious deficiencies when the irradiation and temperature vary [18]. Figure 2-3 illustrates an improved solar cell model, which has an additional shunt resistance R sh and a series resistance R s. Figure 2-4 illustrates a two-diode model. However, the main challenge in using the two-diode model is in the complexity of computing multiple parameters and the associated long simulation time [17]. Given the practical requirements, the model shown in Figure 2-3 is adopted for the system design. Figure 2-2 The simplest single diode model

16 Figure 2-3 The improved signal diode model Figure 2-4 The double diode model Equation (2.1) illustrates the mathematical expression related to the photovoltaic voltage and photovoltaic current of a solar cell model. It involves the short circuit current, reverse saturation current, temperature, irradiation, diode ideality factor, electron charge, Boltzmann s constant, series resistance, and shunt resistance [1]. Where I I sc I o V I= I sc I o (e (qv+ir s akt ) 1) ( V+IR s R p ) (2.1) is the photovoltaic current (A) is the short circuit current (A) is the reverse saturation current (A) is the photovoltaic voltage (V)

17 q k T a R s R p is the electron charge (1.6 10 19 C) is the Boltzmann s constant (1.381 10 23 J/K) is the junction temperature (K) is the diode ideality factor is the series resistance (Ω) is the parallel resistance (Ω) Figure 2-5 illustrates the condition, which leads a solar cell to generate its short circuit current. The short circuit current (I sc ) is the output current of a solar cell, when its load impedance is extremely small such as 0.01 ohms and 0.001 ohms. Figure 2-5 Short circuit current The short circuit current of a solar cell could be reasonably predicted by using equations (2.2) and (2.3) [17]. G I sc = (I sc_stc + C i T) (2.2) G STC T= T - T STC (K) (2.3) Where, I sc_std T C i is the short-circuit current (STC) is the temperature error (K) is the short circuit coefficient(a/ )

18 G STC is the STC irradiation (1000W/m 2 ) T STC is the STC temperature(25 ) T denotes the solar cell s actual temperature and G denotes the actual received irradiation on the solar cell. Figure 2-6 illustrates the condition, which leads a solar cell to generate its open-circuit voltage. The open circuit voltage (V oc ) is the voltage between the positive lead and negative lead of a solar cell when the current that flows through the connected load is almost zero. Figure 2-6 Open-circuit voltage 2.2.2 Parameters Calculations The value of the reverse saturation current, I o is rarely provided by the manufactures. However, it may be approximated by using equation (2.4) [17]. I o = (I scstc +C i T) (2.4) e (qv SC_STC +C v T ) akt 1 Where C v and a are the open-circuit voltage coefficient and the diode ideality factor, respectively. The value of diode ideality factor generally varies from 1 to 2. The unknown parameters in equation (2.1) are R s and R p. The accuracy of these two parameters determines the similarity between simulated I-V and

experimentally measured I-V curves provided by manufactures. Fortunately, [18] provides a reasonable approach to compute R s and R p. The core concept of [18] is to keep increasing the value for R s, while simultaneously calculate the value for R p to best match the calculated maximum power to the experimental maximum power provided by manufactures. Equation (2.5) can be used for fulfilling the above procedures [18]. Where P mpp I mpp V mpp 19 R p = V mpp(v mpp +I mpp R s ) [V mpp (I sc I d )] P mpp (2.5) is the maximum power point power is the maximum power point current is the maximum power point voltage The photovoltaic current of the maximum power point is called Maximum power point current and the corresponding photovoltaic voltage is called Maximum power point voltage. In the following discussions the abbreviations, I mpp and V mpp will denote the maximum power point current and maximum power point voltage, respectively. By using equations (2.1) through (2.5), the parameters of a single solar cell can be reasonably computed. However, those equations may not be sufficient for solving the parameters of a solar panel or a solar array, which is a matrix of interconnected solar cells. Figure 2-7 illustrates the equivalent circuit of a photovoltaic matrix. The photovoltaic current of a solar panel/array can be calculated by using equation (2.6). In equation (2.6), N parallel is the number of columns shown in Figure 2-7, while N series is the number of rows.

20 Figure 2-7 The equivalent circuit model of a photovoltaic matrix I pv = I sc N parallel I o N parallel [e V+IRs( N series ) N parallel (q ) aktn series 1] ( N V+IR series s N parallel R p ( N series N parallel ) ) (2.6) By reviewing the equation (2.5), which is provided by [18], the relationship between the R s and R p in a matrix of solar cells may not be sufficiently accurate. Therefore, according to the equations (2.5), (2.6), and the structure of the photovoltaic array, the R p can be computed by using equation (2.7). R p = V mpp (V mpp +I mpp R s ( N series N parallel )) ( N series N parallel )[V mpp (I sc I d )] P mpp (2.7) Throughout equation (2.7), for any given pair of {V mpp, I mpp }, according to an arbitrary value for R s, there must be an unique solution for R p. By substituting any solution pair of {R p, R s } and a given V mpp into equation (2.6), a corresponding value for I pv can be calculated. However, the product of the calculated I pv and V mpp may not optimally match to the experimental maximum

21 power provided by manufactures. Therefore, the significant step for modelling PVs is to duplicate the above calculations for finding a solution pair of {R p, R s }, which can best match the calculated maximum power to the provided P mpp. After R p and R s are found, the continuous work is to iteratively use the Newton Raphson Method for solving the numerical equation (2.6) and for drawing the I-V and P-V curves of the modeled photovoltaics. Given the above principles, an algorithm is designed and programed as a MATLAB *.m file. The flow chart of such algorithm is shown in Figure 2-8. Figure 2-8 The diagram of the algorithm for finding the parameter pair (Rs,Rp) 2.2.3 Validations of the Single Diode Model The proposed single-diode model was validated by the simulation results. The specification of the SW-260-mono is summarized in Table 2-1 [31]. Table 2-1

demonstrates two series of parameters that present the characteristics of the SW- 260-mono operating under two different testing conditions. R s and R p are found by implementing the algorithm shown in Figure 2-8 via MATLAB. The simulated parameters of the SW-260-mono are listed in Table 2-2. Table 2-1 The Specification of SW-260-mono [31] Parameter Mono-Crystalline SW-260-mono-silver Test condition 1000W/m 2, 25 Mono-Crystalline SW-260-mono-silver Test condition 800W/m 2, 25 P max 260W 194.2W V oc 38.9V 35.6V V mpp 30.7V 28.1V I sc 9.18A 7.42A I mpp 8.47A 6.92A C i 0.004%/K 0.004%K C v -0.3%/K -0.3%K Table 2-2 Simulated Parameters of the SW-260-mono Parameter Mono-Crystalline SW-260-mono-silver Test condition 1000W/m 2, 25 Mono-Crystalline SW-260-mono-silver Test condition 800W/m 2, 25 P max 260.2W 194W V oc 38.9V 35.6V V mpp 30.7V 28.1V I sc 9.18A 7.42A I mpp 8.47A 6.91A Rs 0.0038 Ω 0.0038 Ω Rp 5.8737 Ω 5.8737 Ω Io 4.6618 10 7 A 2.8392 10 7 A Figures 2-9 and 2-10 illustrate the simulated I-V curves and P-V curves of the SW-260-mono operating under the same temperature condition and different irradiation conditions, respectively. The MATLAB code for the PV modelling can be found in Appendix A. 22

23 Figure 2-9 The simulated I-V curves of the SW-260-mono operating under different irradiation conditions. Figure 2-10 The simulated P-V curves of the SW-260-mono operating under different irradiation conditions.

24 By observing Figures 2-9 and 2-10, decreasing irradiation obviously leads V mpp and I mpp to nonlinearly reduce when the temperature condition is invariant. Besides the irradiation condition, the cell s temperature can also affect the characteristics of photovoltaics. Figures 2-11 and 2-12 illustrate I-V and P-V curves of the SW-260-mono, which operates under different temperature conditions and a fixed irradiation condition of 1000 W/m 2.The various temperature conditions do not heavily change the I mpp of the SW-260-mono, while they obviously affect the V mpp and P mpp. Figure 2-11 The simulated I-V curves of the SW-260-mono operating at different temperature conditions.

25 Figure 2-12 The simulated I-V curves of the SW-260-mono operating at different temperature conditions. 2.3 The Internal Impedance of Photovoltaics The internal resistance of PV cells is V pv /I pv. To extract the potential maximum power from PV cells, the load impedance is supposed to be V mpp I mpp. Throughout a simple experiment, this concept can be easily validated. As the load impedance linearly increases from 0 ohms to V mpp I mpp, the photovoltaic voltage will nonlinearly increase from 0 volts to V mpp, and the photovoltaic current will nonlinearly decrease from the short circuit current to I mpp while the photovoltaic power will nonlinearly increase from 0 watts to P mpp. As the load impedance increases from V mpp I mpp to a sufficient large value, such as hundreds ohms or thousands ohms, the photovoltaic current will nonlinearly decrease from I mpp to 0 amps, and the photovoltaic voltage will nonlinearly increase from V mpp to the open circuit voltage while the photovoltaic power will nonlinearly decrease from P mpp to 0 watts. Given the changeable photovoltaic voltage level being sensitive to the load impedance, photovoltaic cells may not be used to directly supply to the

26 electric load which require different input voltage level. Consequently, in a photovoltaic system, a power converter is needed to successfully deliver the irregular power with a variable voltage level to a DC-Link or electric grids. Further details related to the internal impedance of the SW-260-mono are demonstrated by Figure 2-13. This figure illustrates the output power of the SW- 260-mono and its corresponding internal resistance. The curve is based on simulated data. The proper load resistance that can extract maximum power from the solar panel is about 3.62 ohms. Increasing or decreasing this critical impedance will result in a reduction in the available photovoltaic power. Figure 2-13 The I-V curve for different resistive load How can a power converter change the internal resistance of photovoltaic cells, and how can the voltage level of the output terminal of the power converter be simultaneously stabilized? One may want to use the topology shown in Figure 2-14 to explain how to use a power converter to perturb the internal resistance of photovoltaic cells. If a boost DC-DC converter with a linear power source operates

27 in a steady-state of the continuous conduction mode, its input dynamic resistance will equal (1 d) 2 R load where d is the duty ratio of the PWM signal [21]. Hence, the dynamic internal resistance of the linear power source can be linearly changed by gradually increasing the duty ratio, d. Figure 2-14 A wrong topology for changing the internal resistance of a solar panel However, the above principles cannot be applied for photovoltaic power converters. As the duty ratio, d, increases, given the nonlinearity of the I-V characteristics of photovoltaic cells, the variable photovoltaic current may not ensure that the converter always operates in continuous conduction mode. If a power converter operates in discontinuous conduction mode, the relationship between its input dynamic resistance and output load impedance may be unpredictable. Hence, the topology shown in Figure 2-14 may not be an ideal way to change the internal resistance of PV cells. In fact, the proper topology to change the internal resistance of PV cells is shown in Figure 2-15. The related detail for the topology shown in Figure 2-15 is presented in chapter 4. Figure 2-15 The proper system diagram of a photovoltaic system.

Table 2-3 illustrates the maximum power point internal resistance (R mpp ) of the SW-260-mono, which operates under variable irradiation conditions and the invariant 25 temperature condition. Table 2-4 illustrates the R mpp of the SW- 260-mono, which operates under variable temperature conditions and the invariant 1000 W/m 2 irradiation. Table 2-3 The R mpp of the SW-260-mono under Different Irradiation Conditions Irradiation condition The R mpp of the SW-260-mono 1000 W/m 2 3.62 Ω 800 W/m 2 4.07 Ω 600 W/m 2 6.00 Ω 400 W/m 2 9.41 Ω 200 W/m 2 16.91 Ω Table 2-4 The R mpp of the SW-260-mono under Various Temperature Conditions Temperature condition Output resistance of the SW-260-mono at its maximum power point 75 3.08 Ω 50 3.38 Ω 25 3.58 Ω 0 3.98 Ω According to patterns shown in Tables 2-3 and 2-4, the R mpp is sensitive to the environmental conditions. This is because photovoltaic cells present different I-V and P-V curves under different environmental conditions. Therefore, to extract the potential maximum power from PV cells, a photovoltaic MPPT control system should exhibit the following three significant abilities: - The ability to change the internal resistance of photovoltaic cells - The ability to detect migration and transformation of P-V curves - The ability to predict the location of the potential MPP. 28

29 The next chapter mainly discusses: how to efficiently predict the MPP; how to increase the PV system s tracking velocity; how to improve the system s MPPT efficiency.

3 MAXIMUM POWER POINT TRACKING ALGORITHM The MPPT algorithm of a photovoltaic system is used to continuously set new photovoltaic voltage references in order to sense P-V curves and to perturb the PV operation point towards the potential maximum power point (MPP). In section 3.1, several conventional MPPT algorithms are briefly introduced by analyzing their advantages and disadvantages. In section 3.2, the performance of the Perturb & Observer (P&O) algorithm is discussed in detail because it provides fundamental concepts for other MPPT algorithms. Section 3.3 explains concepts of the proposed tracking algorithm using Fuzzy Logic Controller (FLC) for a PV system. The objectives of the FLC are to accelerate the MPPT velocity and to suppress the power oscillation around the maximum power point (MPP). In section 3.4, MATLAB/Simulink based results are presented and validate the advantages of the proposed controller in terms of the tracking speed and tracking accuracy. 3.1 Conventional MPPT Algorithms 3.1.1 The Conventional Perturb & Observe Algorithm The concepts of the MPPT algorithms are derived from the characteristics of P-V curves of photovoltaic cells. Therefore, the illustrations of MPPT algorithms can be rationally conveyed through graphs. In this section, the simulation results and related discussions are all based on the specified parameters shown in Table 3-1. Figure 3-1 reminds the shape of a P-V curve. In Figure 3-1, the region covered by the P-V curve is divided into two areas. In area A, when the operation point of the photovoltaic cells moves towards the MPP, the photovoltaic power continuously increases until it reaches the MPP.

Table 3-1 Parameters of Photovoltaics Short circuit current (I sc ) Open circuit voltage (V oc ) Maximum power point (P mpp ) Maximum power point voltage (V mpp ) Maximum power point current (I mpp ) 4.75 (A) 27.03 (V) 98.23 (W) 22.37 (V) 4.39 (A) 31 Figure 3-1 The P-V curve of the photovoltaics under STC In other words, in area A, iteratively increasing photovoltaic voltage leads the photovoltaic power to increase. On the contrary, in area B, increasing photovoltaic voltage results in a reduction in photovoltaic power. By concluding the above phenomenon, several logical cases can be constructed: Case 1: if the operation point is located within area A, then a positive perturbation in photovoltaic voltage results in an increase in photovoltaic power. Case 2: if the operation point is located within area A, then a negative perturbations in photovoltaic voltage results in a decrease in photovoltaic power. Case 3: if the operation point is located within area B, then a positive perturbations in photovoltaic voltage results in a decrease in photovoltaic power. Case 4: if the operation point is located within area B, then a negative perturbations in photovoltaic voltage results in an increase in photovoltaic power.

32 As the above patterns indicate, the conventional Perturb & Observer (P&O) algorithm is designed to continuously perturb the photovoltaic voltage with an invariant intensity, in order to gather the information of the present location of the operation point and to shift the operation point towards the real MPP. It is expected that the operation point will keep oscillating around the real MPP with a fixed scale due to the nature of the conventional P&O algorithm. Thus, some MPPT algorithms research attempt to prevent the perturbation after the operation point reaches its MPP, while this may be irrational due to the instability of the MPP. Note that PV cells will vary their I-V and P-V characteristics after temperature and irradiation conditions changes so that the position of the real MPP is variable in practical environments. So to speak, the practical MPPT control is not a single-time trace. In this thesis, such oscillations around the MPP are reserved for the sakes of detecting changes in environmental conditions. The fundamental mechanism of the conventional P&O algorithm can be summarized as what is shown in Figure 3-2. Figure 3-2 The general mechanism of the P&O algorithm In Figure 3-2, the detection function involves the photovoltaic current and voltage sensing. By reviewing the change in the photovoltaic power and the previous perturbation in the photovoltaic voltage, the detection function roughly

33 concludes the present location of the operation point, while the prediction function predicts the direction and intensity for the next perturbation. For example, if an increase in the photovoltaic power is detected and the previous perturbation is positive, given the case 1, the present operation point should be located within area A. Hence, the next perturbation is presumed to be positive. On the contrary, if a decrease in the photovoltaic power is detected and the previous perturbation is positive, then the location of the operation point should be located within area B. Therefore, the next perturbation of photovoltaic voltage is presumed to be negative. By summarizing all the possibilities, the Perturb & Observe algorithm is derived. The flow chart of P&O is shown in Figure 3-3 [19]. Figure 3-3 The flow chart of the conventional P&O algorithm [19] It is obvious that the P&O algorithm is easy to implement. Moreover, the related implementation is not heavily affected by the measurement noise, because the P&O algorithm does not involve any derivative operation. Currently, the

34 conventional P&O algorithm are widely adopted by electronic companies, such as Texas Instruments and Linear Technology for manufacturing MPPT controllers. However, due to the nature of the algorithm, the efficiency of MPPT is sacrificed in order to accelerate the MPPT velocity. In section 3.3, the proposed MPPT algorithm that can solve the drawback of the conventional P&O algorithm will be described. 3.1.2 Incremental Conductance Algorithm The incremental conductance algorithm (InC) is an updated version of the conventional P&O algorithm. Different from the conventional P&O algorithm, the InC algorithm uses the slope at the operation point as the position indicator. P V I = (V I) V = I sc I o (e = I + V I (q V V (3.0) akt ) 1) (3.1) Equation (3.0) is the derivative of the photovoltaic power with respect to the photovoltaic voltage. Equation (3.1) is the simplified expression of equation (2.3), and ignores the series and parallel resistance of the photovoltaic model. I = I V o q P V akt e(q V akt ) (3.2) I = I + V = I V sc I o (e (q akt ) 1) I o qv V akt e(q V akt ) (3.3) By substituting the equation (3.1) and (3.2) into (3.0), the equation (3.0) is reformed to equation (3.3), which involves the photovoltaic current, voltage and short-circuit current. 2 P 2 V = I o q V akt (e(q akt ) 1) I o I o ( q akt )2 V e (q V akt ) q V akt e(q akt ) (3.4)

35 Taking the derivative of equation (3.3) with respect to V yields equation (3.4) which demonstrates the monotonically decreasing characteristic of (3.3). In Figure 3-4, the slope of the P-V curve crosses zero at the MPP. Therefore, on a P-V curve, if the operation point moves along the left hand side curve of the MPP, it will be greater than zero. Otherwise, the slope is less than zero. The InC algorithm is derived from the above characteristic. The flow chart of the InC algorithm is shown in Figure 3-5. Figure 3-4 Derivative photovoltaic power with respect to photovoltaic voltage Figure 3-5 The flow chart of the InC algorithm [19]

Given the flow chart of the InC algorithm, to lock the MPP, the condition shown in equation (3.5) must be satisfied. In fact, the value of dp/dv is difficult to converge to the exact zero (0), even in a computer-based simulation environment. To generate a result, 0, by using equation (3.5), the InC algorithm may keep perturbing the operating point, while such perturbations may not contribute to extract more power from PV cells. Additionally, the operation point will abruptly jump to another P-V curve when environmental conditions rapidly change. In this case, the calculated slope may not be meaningful to predict the location of the MPP. Therefore, an error tolerance, e t should replace the 0 in equation (3.5) for helping the InC algorithm to lock the potential MPP. The new condition for helping the InC algorithm to lock the potential MPP is given by equation (3.6). The further computation based on equation (3.6) is shown in equation (3.7). dp = 0 dv (3.5) dp dv t (3.6) I + V I V t I V e t I V (3.7) To implement InC algorithm, the requirement of noise filtering related to the voltage and current measurement may be more enforced than that of the P&O algorithm because derivative operations will boost the magnitude of the measurement noise and further make the slope-detection mechanism meaningless. Moreover, the InC algorithm does not solve the inherent issue of the conventional P&O algorithm, but makes the MPPT control more complex. Hence, the InC algorithm is not considered in this thesis. 36

37 3.1.3 Constant Voltage Method Instead of perturbing the photovoltaic voltage, a reasonable PV power can be obtained by clamping the photovoltaic voltage at a certain level. By occasionally measuring the open circuit voltage of photovoltaics, an updated clamped voltage level (V cp ) can be obtained by using equation (3.8). V cp = βv oc (3.8) The value of β is normally selected in range from 70% to 80%.The constant voltage method is derived from experimental experiences: V mpp generally is located within the range from 70% to 80% of V oc [16]. This characteristic of photovoltaics can be validated by Figures 2-9 and 2-10 (p. 23). If the requirement for the MPPT efficiency is not extremely strict, the constant voltage method may be a good choice due to its relatively low cost and easy implementation, which may merely require a simple analog circuit. Under the various test conditions, the constant voltage method may collect 70% of the potential maximum power from photovoltaics. 3.2 Performance of the Conventional P&O Algorithm This section discusses the performance of the conventional P&O algorithm. The conventional P&O algorithm generally exhibits a trade-off between the tracking velocity and MPPT efficiency. This nature can be seen by simulating behaviors of the conventional P&O algorithm with two different perturbation intensities, 0.1 volts and 2.0 volts. In this simulation, the perturbation frequency is set to 1-Hz. In the following analysis, the term perturbation intensity is denoted by p-i. Figure 3-6 illustrates that the P&O algorithm with a larger perturbation intensity shows a faster tracking velocity, while Figure 3-7 presents that the algorithm with a weaken perturbation intensity presents a higher MPPT efficiency.

38 Figure 3-6 Power with P&O p-i 0.1 vs p-i 2.0 Figure 3-7 Average power conducted by P&O with p-i 0.1 vs p-i 2.0

39 Given Figures 3-8 and 3-9, a larger perturbation intensity 2.0 results in a shorter tracking time, 9 seconds, while the weaker one results in 223-seconds tracking time. Therefore, the tracking process may waste time and energy if the initial operation point is distant from the potential MPP, or if the fixed perturbation intensity for the MPPT algorithm is relatively weak. For instance, to perturb the photovoltaic voltage from 0 to 23 volts with an invariant perturbation intensity 0.1 volts, 230 computational loops are needed. Figure 3-8 Tracking time of P&O with p-i 2.0 volts Figure 3-9 Tracking time of P&O with perturbation intensity 0.1 volts

40 The MPPT efficiency can be calculated by using equation (3.9) [30]. η MPPT = average power conducted by algorithm the potential MPP (3.9) Given Figure 3-7 (p.39), a larger perturbation intensity attenuates the MPPT efficiency. The MPPT efficiency of the P&O algorithm with perturbation intensity 2.0 volts is 97.25%, while the MPPT efficiency of the P&O algorithm with perturbation intensity 0.1 volts is 99.99%. Figures 3-10 and 3-11 present the performance of the P&O algorithm with regard to energy. Although a larger perturbation intensity makes photovoltaic cells generate more energy in a short period of time, more energy losses are expected in a longer period of time due to its low MPPT efficiency. Figure 3-10 Energy with P&O: p-i 0.1 vs p-i 2.0 (the 1000 th second)

41 Figure 3-11 Energy with P&O: p-i 0.1 vs p-i 2.0 (the 7000 th second) To accelerate the tracking velocity, a larger perturbation intensity is required when the operation point is distant to the potential MPP, while to improve the MPPT efficiency, a weaker one is needed when the operation point is nearby the MPP. Therefore, the perturbation intensity of an advanced MPPT algorithm should be adaptive with respect to practical conditions. In fact, the precisely mathematical expression related to the proposed perturbation intensity and practical electronic characteristics of photovoltaics may not be easily obtained. Additionally, given the nonlinear and environment-dependent I-V and P-V curves, a traditional controller which fulfills a fixed differential equation or single logic control rule may not be suitable for generating adaptive perturbation intensities. An ideal controller for the MPPT control should contain multiple control rules. It is worthy to note that an interesting fact: although the exact mathematical models of a whole PV system are not available, we may still manually shift the operations point to the potential maximum power point with very few trials, by trying different perturbation intensities and by checking the corresponding consequences. This is because human may approximately predict proper actions with a given

42 observation, without knowing the exact model. For instance, note that the MPP showing in Figure 3-1 (p. 31) is 98.23 watts, we may decrease the perturbation intensity when the solar panel s output power exceeds 90 watts, because the operation point could be close to the MPP. On the contrary, we may increase the perturbation intensity when the operation point is considered as distant to the MPP. Given the change in power, the perturbation intensity may be increased when it is considered as small, or be diminished and vice versa. In the above processing, the numerical elements such as the perturbation intensity and photovoltaic power are converted into linguistic variables so that we may easily make decision by using their logic principles. For example, if the present operation point is close to the MPP, and if the perturbation intensity is large, a reasonable next perturbation is supposed to be small, then we make a decision which is to reduce the perturbation intensity. This is a simple type of Fuzzy Logic Control which is proceeded in our mind. To improve the performance of the fuzzy logic control, the advanced Fuzzy Logic Controller is designed in this research. To make efficient decisions, the proposed Fuzzy Logic Controller not only depends on the rough considerations such as large and small, but also considers the degrees of truth, for example, 50% large, 90% small, 40% distant, 70% close, etc. 3.3 Fuzzy Logic Controller (FLC) Fuzzification, logic judgment and defuzzification are successive three stages of a FLC [5]. At the stage of fuzzification, the numerical ratio, E (a change in solar power to a change in solar voltage, P/ V ) is translated into a linguistic variable via membership functions, as well as the numerical error, CE, which is the perturbation intensity, V. E and CE are two input linguistic variables of the

43 FLC. The next perturbation intensity, the output variable of the FLC is referring to the control rules seen in Table 3-2 (p.46). The output membership functions are used for translating the linguistic output variable, PT to a numerical variable. The notations of two input variables, E and CE are expressed by equation (3.10) and (3.11) E = P[k] P[k 1] V[k] V[k 1] (3.10) CE = V[k] V[k 1] (3.11) 3.3.1 Fuzzification Each linguistic variable consists several fuzzy sets [20]. The general expression of a fuzzy set is given by equation (3.12). A = {(x, μ A (x)) x U} (3.12) In equation (3.12), μ A (x) is the membership function which represents the certainty of x the fuzzy set A. U is the comprehensive union that contains all the possible values of x. For example, x varies from -10 to 10, and the set A denotes the union, (3, 8). Then the point where x equals 6 is belong to the set A. Generally, membership functions are presented graphically. Figure 3-12 illustrates a sample of membership function, and the mathematical statements are given by equation (3.13). According to (3.13), the certainty of x the fuzzy set A is 33%. Figure 3-12 An illustration of the membership function μ A (x)

μ A (x) = { 0 if x < 3 1 if 3 x < 5 x 5 3 if 5 x 8 0 if x > 8 44 (3.13) Based on the case shown in Figure 3-1 (p.31), to properly select the fuzzy sets for CE and E, the P-V curve should be truncated into several zones for the purposes of timely changing the perturbation intensity, and for preventing the two incidences: 1) the operation point moves slowly when it is far from the real MPP; 2) the operation point moves quickly when it is nearby the real MPP. In Figure 3-4, the curve shows relative linearity in the range of V pv (0,18). The slope of the PV curve nonlinearly decreases towards zero in the range of V pv (18,23) and deviates from zero towards -78 in the range of V pv (23,27.03). Based on the above features, the P-V curve can be deliberately sectionalized as what is shown in Figure 3-13. Figure 3-13 The sectionalized P-V curve with different operating zones. Positive Big, Positive Small, Positive Zero, Negative Zero, Negative Small and Negative Big are correspondingly denoted by PB, PS, PZ, NZ, NS and NB. In the PB zone, the slope is relatively constant because the points in this zone are distant from the MPP. Therefore, the

perturbation intensity is supposed to be enlarged for quickly pushing the operation point out of this zone. In the PS zone, it is obvious that the value of slope gradually decrease towards zero, but still there is a short distance to the MPP. So, the perturbation intensity is definitely needed to be diminished but not to be thoroughly eliminated. If the operation point shifts in the PZ and NZ zones, where points within in these zones are extremely close to the MPP, the ideal perturbation intensity is supposed to be very weak for keeping the consequent oscillation as small as possible. Based on the above considerations, the membership functions of each fuzzified variable are determined. The Table 3-2 explains the proposed fuzzy sets in greater detail. Table 3-2 The Numerical Unions Corresponding to the Fuzzy Sets Fuzzy set CE E PT NB (-2.0, -0.1) (, -1) (-1.5, -0.5) NS (-0.1,-0.01) (-1.5,-0.1) (-0.5,-0.01) NZ (-0.01, 0) (-0.2, 0) (-0.01, 0) PZ (0, 0.03) (0, 0.5) (0, 0.05) PS (0.03, 0.3) (0.3, 2.5) (0.05, 1) PB (0.3, 5.0) (2.5, + ) (1, 3) In [5-9], the oscillation around MPP is commonly treated as an undesirable byproduct of a MPPT algorithm, because the oscillation obviously degrades the MPPT efficiency. However, without such an oscillation the algorithm cannot detect the changes of P-V curves due to the changes in environmental conditions. In this thesis, the objective is to keep operation point oscillating around the MPP with an extremely small deviation once the operation point goes into the PZ and NZ zones. Figures 3-14, 3-15, and 3-16 illustrate the graphical membership functions, E,CE and PT, respectively. 45

46 Figure 3-14 The membership function E Figure 3-15 The membership function CE Figure 3-16 The membership function PT

47 3.3.2 Fuzzy Rule Base A fuzzy logic controller determines its fuzzified output variable by looking up its fuzzy rule base, which consists of a set of fuzzy IF-THEN rules. The general format of a fuzzy logic rule is that [20]: Rule#: IF x 1 is A 1 and x 2 is A 2 and and x n is A n, THEN y is B j Where A i and B j are fuzzy sets in U i and V, respectively, and U i consists of any possible value for the input variable, x i. And the V is the union consisting of any possible value for the numerical output. The proposed FLC has two input linguistic variables, E and CE, and one output linguistic variable PT. Therefore, for example, the rules can be written as: Rule#: IF CE is PB and E is PB then PT is PZ When the FLC detects the incoming input pair {CE,E}, it will apply all rules for recording any possible logic outputs. This is the outstanding characteristic of the FLC, compared to other binary logic controller. In the sense of statistics, multiple logic consequences will improve the accuracy of final weighted results which are expectation-type solutions. Table 3-3 is the rule base of the proposed FLC. The output surface of the FLC is illustrated by Figure 3-17, which is derived from Table 3-3. Table 3-3 Rules for the Proposed FLC CE E NB NS NZ PZ PS PB NB NZ NZ NZ PZ PZ PZ NS NZ NZ NZ PZ PZ PZ NZ NB NS NZ PZ PZ PZ PZ NZ NZ NZ PZ PS PB PS NZ NZ NZ PZ PZ PZ PB NZ NZ NZ PZ PZ PZ

48 Figure 3-17 The output surface of the proposed FLC Throughout trials and calibrations, some difficulty has been observed in Figure 3-18. Occasionally, the operation point may go beyond the maximum power point, and the value of E is still classified into the fuzzy set PS. As the slope shown in Figure 3-18, the undesired logic judgment will happen if the logic operation follows such a rule as IF E is PB and CE is PS THEN the PT is PS. In this case, such rule can perfectly accelerate the tracking velocity when the operation point is distant from the MPP. Although the previous operation point has already been close to the MPP, according to the rule, the operation point will be forced to keep moving towards a wrong direction, and deviating from the MPP. To avoid this undesirable situation, as seen in Table 3-3, a self-correction mechanism is added into our FLC. The strategy is to check the present slope with a smallest scale perturbation if a large scale shifting of the operation point happened. If the sign change of E is detected by FLC, the direction of perturbation will be inversed immediately. Moreover, the wrong perturbation direction will be detected timely.

49 Figure 3-18 Unexpected problem 3.3.3 Defuzzification Defuzzification is an inversed procedure with respect to the fuzzification. In the processing of defuzzification, a linguistic output will be translated into a numerical value by adopting a weighting operation. The general expression of such operations is that: Next perturbation = μ A i (x) B i μ Ai (x) (3.14) Where μ Ai (x) is the membership function of the output variable fuzzy set and B i is the fuzzy set s numerical solution. The operation for using equation (3.14) can be explained by the following example: If three output possibilities, μ A1 (x), μ A2 (x) and μ A3 (x) are generated by the FLC, if the certainties of the three possibilities are 20%, 30%, and 40%, respectively, and if the numerical solution corresponding to the three possibilities are 1, 2, and 3, respectively, the equation (3.14) will be written as: Weighted result = 20% 1+30% 2+40% 3 20%+30%+40% = 2.22 (3.15)

3.4 Simulation and Comparison To validate the proposed Fuzzy Logic Controller, the following simulations show that the performance of the conventional P&O algorithm with a fixed perturbation intensity 0.1 volts, and the proposed Fuzzy logic controller with adaptive perturbation intensities. Simulation settings are summarized in Table 3-4. Table 3-4 The Configuration of Simulations The parameters of the solar panel under STC Short circuit current, I sc 4.75 (A) Open circuit voltage, V oc 27.03(V) Maximum power, P mpp 98.23 (W) The parameters related to the algorithms and the simulation configurations Initial operation point, (V,P) (0 V, 0 W) 50 Perturbation intensity of P&O Perturbation frequency 0.1 V 100 Hz Temperature condition, T 25 The MATLAB/Simulink block for simulating the performance of the FLC is shown in Figure 3-20. The main three MATLAB function blocks are used to handle:. Setting the short circuit current with respect to time.. Calculating parameters of the solar panel. Implementing the proposed Fuzzy Logic Controller.

51 Figure 3-20 The Simulink block diagram of the Fuzzy Logic Controller Given the specified membership functions shown in Figures 3-14, 3-15, and 3-16 (p.47), the proposed Fuzzy Logic Controller cannot be implemented via the Fuzzy Logic Toolbox, which is provided by MATLAB, due to the numerical unions are extremely close to zero. Therefore, the proposed FLC is realized by MATLAB/function blocks. For testing the robustness of above MPPT algorithms, irradiation variations is coded into the simulation. Irradiation variations are represented by the changing in PV short-circuit current. The time dependent short circuit current is shown in Figure 3-21. Figure 3-21 The variable short circuit current of the simulated solar panel.

52 As discussed in chapter 2, if the temperature condition is invariant, the irradiation condition will solely dominate the short-circuit current, and other characteristics of photovoltaics will be determined by the short-circuit current. As the irradiation varies, the difference in terms of MPPT transient responses conducted by the two MPPT strategies is evident. Another objective of the proposed FLC is to make transitions in photovoltaic power smooth and fast when environmental variations occur. Figure 3-22 illustrates the full-scope view of the performances of the two MPPT strategies. In the time interval, (0s,3s), the proposed algorithm shows a short rising time, compared to that of the P&O algorithm. By zooming in this time interval, Figure 3-23 demonstrates the detailed tracking time. Figure 3-22 The MPPT traces of the two MPPT strategies

53 Figure 3-23 MPPT traces of two MPPT strategies in the time interval (0s,3s) Given Figures 2-9 and 2-10 (p. 23), the irradiation drop leads PV cells to change their I-V and P-V curves so that the corresponding MPP will jump to an unpredictable position. Thus, a transition in PV power will occur if the irradiation condition varies. After the 3 th second (irradiation drops), the FLC and P&O algorithm track the new MPP with different transition periods, which are shown in Figure 3-24. Figure 3-24 MPPT traces of the two MPPT strategies during the decrease of the irradiation

54 After the FLC detects a large-scale reduction in power due to the rapidly decreasing irradiation condition, the rule IF CE is NZ and E is NB, then the PT is NB is activated. The decisions of the FLC in the time interval (3.0s, 3.16s) are shown in Figure 3-25. Figure 3-25 The decisions of FLC in the transition period (3.0s, 3.16s) As illustrated in Figure 3-26, after the irradiation increases at the 6 th second, for tracking the new MPP, the FLC spends 0.08 seconds, while the P&O algorithm uses 0.13 seconds. Given the above patterns, compared to the performance of the P&O algorithm with the perturbation intensity of 0.1 volts, the proposed FLC shows the better performance during the transitions. In this chapter, the origination of the conventional MPPT algorithm and the methodology for designing a proper FLC were discussed. The advantages of the proposed algorithm were validated by the simulation results. Given the practical requirements, the parameters of the proposed FLC should be tuned. In chapter 4, the plant of a photovoltaic system, the boost DC-DC converter is discussed.

Figure 3-26 The responses of the two strategies during the change in the irradiation condition 55

4 BOOST DC-DC CONVERTER A DC-DC converter is an essential element in a stand-alone PV tracking system. Note that the voltage level of PV cells is variable due to the location of the operation point so that directly supplying the DC photovoltaic power to the electric load may be inappropriate. In a MPPT system, a DC-DC converter is used to convert an irregular input power into a regulated one with a desired voltage level. Switch-mode DC-DC converters are currently popular for their advantages in terms of small volume-size and high controllability. In a switch-mode DC-DC converter, the MOSFET/IGBT and inductor performs as a transformer with a programmable factor. To rebuild the factor between input DC voltage level and output DC-Link, one only needs to change the duty ratio of the switching signal. Associating with a digital signal processor, a switch-mode converter can provide numerous functions. In the first section of this chapter, the basic topology of a typical boost DC-DC converter is addressed. In section 4.2, the small-signalmodel of the output terminal of a typical DC-DC boost converter, which supplies a resistive load, is briefly reviewed. The theoretical analysis of the voltage regulation for photovoltaics and the small signal model related to the input terminal of a boost converter is presented in section 4.3 in greater detail. 4.1 Topology of the Typical Boost DC-DC Converter The function of a boost DC-DC converter is used to step-up the voltage level of the input DC power. As shown in Figure 4-1, a typical boost DC-DC converter consists of a switching device, an inductor, an input capacitor, a diode, an output capacitor, and an electric load.

57 Figure 4-1 The typical topology of a boost DC-DC converter. The definitions of each denotations shown in Figure 4-1 are that: - V_DC is the input linear DC power supplier, which can be treated as a voltage source with a stabilized voltage level regardless of load impedance. - Cin is the input capacitor. The input capacitor is normally used for suppressing the harmonics within the input DC power. In fact, the ripple current caused by the switching-pattern, which is the nature of switchmode circuits, will flow through the input capacitor and eventually flow to the ground. Therefore, the input capacitor can be also used to protect the input power source from the ripple current. - L is the inductor, which is a medium energy storage device. By being charged and by being discharged, the input inductor maintains the voltage level of the output terminal, and transfers the input DC power to the electric load. - S is the switching device. The switching device is the trigger to charge and to discharge the input inductor. Currently, the popular switching devices involve IGBTs, MOSFETs, etc. MOSFETs can generally switch at relatively higher frequency, compared to IGBTs. - D is the diode. It regulates the electric current direction in order to regulate the system structure during switch-on and switch-off periods.

58 - Rload is the electric load. The electric load can be resistive and capacitive. In this section, the electric load is presumed as a resistor. To supply a resistive load, a boost DC-DC converter is typically controlled by a feedback loop for regulating the output voltage level. 4.1.1 Switching States Given the binary switching states, a typical boost converter shows two structures in each switching period. As shown in Figure 4-2, the input DC source charges the inductor when the switch is turned on. As shown in Figure 4-3, in a switching-on period, the inductor current linearly increases. The electric energy temporarily accumulates in the inductor. On the contrary, as shown in Figures 4-4 and 4-5, in switching-off periods, the inductor serves as the secondary power source to supply the load, and compensates the voltage drop between the V in and V out. Figure 4-2 The equivalent circuit during switching on periods Figure 4-3 Inductor current and voltage during switching on periods

59 Figure 4-4 The equivalent circuit during switching off periods Figure 4-5 Inductor current and voltage during switching off periods Assuming that the boost converter operates in the dc steady state, the average inductor voltage is supposed to be zero so that equation (4.1) could be obtained [21]. Moreover, the ripple component of the inductor current should be periodical, and the average inductor current equals the input current. V in t on = (V out V in ) t off (4.1) L dil dt L (4.2) The average approximation of equation (4.2) is written as: L il t = V L (4.3) Where il is the magnitude of the ripple component of the inductor current. V L equals V in in switching-on periods, while it equals the (V in V out ) in switchingoff periods. Therefore, equation (4.4) can be obtained.

il = 1 L V in t on = 1 L (V out V in ) t off (4.4) 60 Assuming that the boost DC-DC converter operates in Continuous Conduction Mode (CCM), which means that inductor current is continuous in every switching period. By introducing the concept of duty ratio, substituting equations (4.5) and (4.6) into (4.4), the mathematical expression related to the input voltage and output voltage of a typical boost DC-DC converter is written as equation (4.7) [21]. t s = t on + t off (4.5) t s = dt s + (1 d)t s (4.6) V out V in = 1 1 d (4.7) 4.1.2 Discontinuous Conduction Mode (DCM) System dynamics of a boost DC-DC converter can be predicted by using the above expressions if the converter operates in Continuous Conduction Mode (CCM). In CCM, the inductor current never falls to zero in any switching period and there exists no such time intervals where the inductor voltage stays on zero volts. If the maximum input current of a boost converter is less than the amplitude of the calculated ripple current, which is derived via equation (4.4), then the boost DC-DC converter definitely operates in DCM. V out of a boost DC-DC converter operating in DCM can be calculated by using equations (4.7) and (4.8) [21]: Where R f s D V out = V in (1+ 1 + 4M) 2 (4.7) M = ( R )D 2 2Lf s (4.8) is the load impedance of the boost DC-DC converter. is the switching frequency. is the duty ratio of the switching signal.

61 If the boost DC-DC converter operates in DCM, the regulation of system dynamics will be relatively difficult because the linear approximation of the system is not straightforward. Therefore, the inductor size and switching frequency are supposed to be chosen carefully for avoiding the converter operating in DCM. 4.2 Small Signal Model For regulating the output voltage of a typical boost DC-DC converter, the transfer function related to the output voltage and duty ratio is obtained by applying the small signal model analysis, which is normally used to approximate behaviors of nonlinear devices with linear equations [21]. In a typical boost DC-DC converter, the inductor and MOSFET perform as a traditional transformer, which handles the electric energy transition. The equivalent circuit is illustrated by Figure 4-6. Figure 4-6 The average dynamic model of a boost DC-DC converter Assuming that a small scale perturbation is added into the control signal, the system characteristics, such as output voltage and current, instantly change. By viewing the consequences of the perturbation, transfer functions, like G v (s)/d(s)

62 and G v (s)/g i (s) can be derived. In this thesis, the related control design only requires the knowledge of G v (s)/d(s). 4.2.1 Output Terminal Small Signal Modelling Assume that: an ideal boost DC-DC converter shown in Figure 4-6 operates in CCM; the efficiency of the power conversion is 100%; the electric load is resistive; a small scale perturbation, d, in swiching signal has been injected to the boost circuit. Then, equation (4.8) and (4.9) can be satisfied. I out = V out (4.8) R V in I in = V out I out (4.9) By ignoring the current through the input capacitor, the average inductor current is assumed to equal the input current. I L = I in = V out I out V in = V 2 out V in R By applying the Norton s theorem, the procedure of simplifying the (4.10) equivalent circuit of the small signal model is illustrated by Figure 4-7, by Figure 4-8, and by Figure 4-9 [21]. Figure 4-7 The equivalent circuit of small signal model (a) [21]

63 Figure 4-8 The equivalent circuit of small signal model (b) [21] Figure 4-9 Small signal model: output terminal of a boost DC-DC converter [21] Given the equivalent circuit shown in Figure 4-9, by applying the Kirchhoff Voltage s Law, the transfer function, G vout (s)/d(s) is written as [21]: V out (s) d (s) = V in (1 D) 2 (1 sl (1 D) 2 R ) 1+srC out LC out 1 (1 D) 2[s2 +s( + r(1 D)2 )+ (1 D)2 )] RC out L LC out (4.11) Given equation (4.11), the RHP zero varies with the duty ratio. Note that to realize MPPT control, the duty ratio of a boost DC-DC converter should be changeable in order to perturb the internal impedance of PV cells. Thus, due to the variation of the duty ratio, the damping ratio, natural frequency and RHP zero in equation (4.11) are variable. In this case, it may be impossible to design a compensator for stabilizing the output voltage and for regulating the phase of the system. Therefore, the traditional topology shown in Figure 4-1 and its small signal model will not be adopted for designing a photovoltaic system.

64 To regulate an irregular power source and to stabilize the output voltage level, the proposed topology and its small signal model is investigated. 4.2.2 Input Terminal Small Signal Modelling If a boost DC-DC converter contains medium of the energy storage such as batteries and ultra-capacitors, or if its load is totally capacitive like grid and DC- Link, given equation (4.12), the output voltage level can be stabilized at an invariant level if the capacitance of the load is sufficiently large. C = Q U (4.12) Where C Q U is the load s capacitance (F) is the charged coulomb (A s) is the voltage across the load (V). With a stabilized output voltage level, in order to control such a boost DC- DC converter supplied by an irregular power source, the knowledge of the linearized approximation at the input terminal is required. Applying the Norton s theorems to simplify the input terminal of the boost converter, the equivalent circuit presenting the input terminal of a boost converter is shown in Figure 4-10. Figure 4-10 Equivalent circuit of a boost converter with irregular input source

The I pv, V pv, and r pv denote the photovoltaic current, photovoltaic voltage, and internal resistance, respectively. i L and i C denote the inductor current and capacitor current, respectively. r C and r L present parasitic resistance for the input capacitor and the input inductor. The V pr is the equivalent voltage on the primary side of the transformer. A perturbation, d, in the control signal instantly results in a voltage drop/increase on the primary side of the transformer shown in Figure 4-10. The voltage variation caused by the duty ratio perturbation is that: V pr = d V bat (4.13) Based on the equivalent circuit shown in Figure 4-10, the linearized state space equations are derived by considering the parasitic resistance of the input capacitor and input inductor. 65 d I L dt V pv = R L L R pv(1 R L Rc) LC(1 Rpv) 1 L CR L Rc 1 LC(1 Rpv) Vbat I L + L V pv RcRpvV bat LC(1 Rpv) d (4.14) y = [0 1] I L V (4.15) pv G vd (s) = A s 2 +Ns+M (4.16) A = V bat ( 1 + R c 2 R 2 pv R c R pv R cr pv s) (4.17) L 2 L 2 C(1 R pv ) 2 L(1 R pv ) N= ( R L L R cr pv 1 LC(1 R pv ) ) (4.18) M= R LR c R pv R L +R pv L CR L 2 R c R pv L 2 C(1 R pv ) The purpose of analyzing the linear approximation is to design a voltage (4.19) controller in order to regulate the photovoltaic voltage level. The topology of the voltage regulation is illustrated by Figure 1-4 (p. 6). Notice that the parameters of

66 such a nonlinear system shown in Figure 4-10 change with R pv, which is the internal resistance of photovoltaics. Therefore, the presented linear approximation is only a basic reference for the controller design. The final parameters and structure of the voltage controller must be tuned by referring to the related experimental results. The parameters of the system are shown in Table 4-1. The range of the internal resistance of the adopted solar panel is concluded by the experimental tests. Table 4-1 Parameters of the Designed PV System Components V bat R Cin R L L C R pv Parameters 26(V) 0.8(Ω) 0.2(Ω) 12(mH) 210(uF) 75~150(Ω) The only variable parameter in the linear approximation is R pv.therefore, to analyze the system s behavior with respect to R pv, four transfer functions with four different R pv are presented. Four values for R pv, are 75 Ω,100 Ω,125 Ω and 150 Ω, respectively. The bode plot of four transfer functions are shown in Figure 4-11. Table 4-2 shows the linear approximations with different values of R pv. Given Table 4-2 and Figure 4-9 (p. 64), the variable R pv does not heavily affect the system behaviors. The system shows the over-damped characteristic and its phase never goes below -90 degree at low frequency. Additionally, due to the locations of poles and zeros, the system behaves as a first order system. Therefore, in this case, a PI controller is applicable for compensating the system.

67 Figure 4-11 Bode plot of the variable-parameters system Table 4-2 Linear Approximations with Different values of R pv Damping R pv Linear approximation ratio s 5.56e8(1 + 75 (Ω) 3.165e5 ) 3.8s 2 + 1.202e6s + 2.156e5 s 5.546e8(1 + 100 (Ω) 3.1676e5 ) 3.8s 2 + 1.203e6s + 2.157e5 s 5.5386e8(1 + 125 (Ω) 3.1692e5 ) 3.8s 2 + 1.204e6s + 2.158e5 s 5.532e8(1 + 150 (Ω) 3.1703e5 ) 3.8s 2 + 1.204e6s + 2.159e5 Where e5, e6 and e8 denote 10 5, 10 6 and 10 8, respectively. Natural frequency 3.4062 238.19(rad/s) 3.4082 238.25(rad/s) 3.4103 238.31(rad/s) 3.4095 238.36(rad/s)

4.2.3 Voltage Regulation G c (s) = K p + K i 1 The general equation of a PI controller is given by equation (4.20). To s (4.20) obtain the desired step response of the closed-loop system, tuning parameters can refer to bode plots of compensated systems. The fundamental tuning principles are shown in the Table 4-3[22]. The controller design does not involve the derivative operation. This is because the switching devices inevitably inject plenty of noise to the voltage and current signals. Additionally, the derivative operation may boost noise level and affect the performance of the controller. Table 4-3 Effects of Independently Increasing a Parameter in a PI Controller [22] Parameter Rise Time Overshoot Settling Steady-state time error Stability K p Decrease Increase Small change Decrease Degrade K i Decrease Increase Increase Eliminate Degrade By observing the step responses of the closed-loop compensated systems, the proportional gain and the integral gain are selected as 0.1 and 2.2, respectively. The continuous-time transfer function of the PI controller is that: G c (s) = 0.1 + 2.2 s The purpose of tuning the PI controller is to thoroughly eliminate the 68 (4.21) potential overshoot of the closed-loop compensated system in order to protect the input power source. In Table 4-4, G 75, G 100, G 125 and G 150 denotes the closed-loop compensated systems with the corresponding values for R pv. The simulated step responses of the system operating at different operating points are given by Figure 4-12. Under different operating points, the rising time of the step response of the closed-loop compensated system is about 60ms. And the system is left with a bit damping characteristic. The MATLAB code for simulating the system dynamics can be found in Appendix A.

Table 4-4 Step Response of the Closed-Loop Compensated System Models Rise time Settling Time Overshoot G 75 40.8(ms) 61.6 (ms) 0% G 100 40.9(ms) 61.7 (ms) 0% G 125 41.0(ms) 61.8 (ms) 0% G 150 41.0(ms) 61.9 (ms) 0% 69 Figure 4-12 Step response of the closed-loop compensated system. In this section, the system behaviors are discussed with numerical parameters. The internal resistance of the adopted solar panel, does not heavily affect the damping ratio and natural frequency of the linear approximation so that the original system behaves as a linear invariant system. In fact, behaviors of photovoltaic power converters, highly depend on R pv, R cin, and R L. Hence, given different internal resistance of PV cells, the switching-mode converter may become a slightly damped system, which is a difficult control problem. In such case, the controller design will be challenged in terms of balancing the phase margin and stability of the compensated system operating at different operating points. In the next chapter, the hardware and software fulfillment of the proposed MPPT system will be discussed.

5 PROTOTYPE IMPLEMENTATION The photovoltaic MPPT system is designed to implement the proposed algorithm. The system consists of a power electronic system and a signal process system. The power electronic system is controlled by the signal process system via a Pulse-Width-Modulation signal. The core element of the power electronic system is a boost DC-DC converter which deals with energy transmission and perturbing the PV operation point. A functioning switching-mode converter generates plenty of noise and ringing, which can be harmful for system control and system stability. Hence, the practical solutions for the noise and ringing suppression are presented in this chapter. The signal process system, which possesses two control layers, is embedded into a microcontroller, TI F28035. In the top control layer, the proposed FLC continuously sets new photovoltaic voltage references and send them to the secondary control layer. Additionally, the top control layer is enhanced by a DCM detection mechanism to guarantee that the MPPT system always operates in a controllable region. The proposed PI controller dominates the secondary control layer. As mentioned in chapter 4, the PI controller continuously perturbs the duty ratio of the PWM signal to change the system dynamics until the photovoltaic voltage converges to its reference. To obtain a reasonable internal cooperation, the two control layers operate with different control intervals, according to the settling time of the voltage regulation loop and voltage/current measurement loop. In the first section of this chapter, the main parameters of the photovoltaic boost converter are explained. In section 5.2, peripheral circuits are introduced. In section 5.3, the detail of the signal process system is addressed. The practical

power tracking performance of two MPPT control strategies is provided in the last section. 71 5.1 Parameters of the Boost DC-DC Circuit In this section, how to decide parameters of the boost DC-DC converter is discussed. The proposed topology of the PV boost DC-DC converter is illustrated by Figure 5-1. The parameters of the circuit components are listed in Table 5-1. Figure 5-1 The proposed topology of the PV boost DC-DC circuit Table 5-1. Parameters of Components of the Boost Circuit Components Parameters Solar Panel Boulder 15W C in 35V/210uF R Cin 0.8 Ω L 12 mh R L 0.2 Ω C out 100V/1000uF MOSFET IRFP460A Diode HFA50PA60C DC bus 26 V For safety, the solar panel, Boulder 15W is adopted as the input power source of the MPPT system. The light source for simulating the sun light is two

72 250W generic electrical light bulbs. The experimental parameters of the solar panel irradiated by the two 250W light bulbs are shown in Table 5-2 Table 5-2 Parameters of the Solar Panel under Testing Conditions Electronic characteristics Parameters Open circuit voltage 17.5~21.4 (V) Short circuit current 0.18~0.22 (A) Nominal maximum power point 12~17 (V) voltage V mpp Nominal maximum power point 0.166~0.176(A) current I mpp Maximum power 2.0~ 3.1 (W) The parameters shown in Table 5-2 have variation ranges because as the time increases, the surface temperature of the irradiated solar panel gradually increases. As discussed in chapter 2, increasing temperature reduces the potential maximum photovoltaic power. In consequence, variation ranges of photovoltaic current and voltage are suppressed so that the photovoltaic current may not always keep system operating in CCM at any operating point. If the boost DC-DC converter operates in DCM, the linear approximation discussed in chapter 5 will become invalid. Therefore, the circuit s parameters are selected for the worst case. Three factors must be taken into account: - The switching frequency - The minimal photovoltaic current - The voltage level of the DC bus Assuming that the boost converter operates in CCM, equation (5.1), which is derived from equation (4.4) and (4.7), shows the amplitude of the ripple current, A IL. A IL = V in(v out V in ) 2LV out f s (5.1)

73 Notice that the parameter, V out is supposed to be selected before analyze the ripple current. V out is the voltage level of the DC-Link. Given principles of a typical boost DC-DC converter, the voltage level of the output terminal must be greater than the open circuit voltage of the Boulder 15W. To avoid EMI issues, the voltage drop between the input and output of a boost DC-DC converter should be relatively small. Therefore, the voltage level of the DC bus is selected to 26 volts. f s is the switching frequency of the PWM signal. Under an ideal condition, the switching frequency is supposed to be as high as possible for suppressing the ripple current. In fact, the practical switching frequency is generally limited by the following factors: - The resolution of the duty ratio of the PWM signal - The bandwidth of the gate driver chip - The electronic characteristics of the switching device In this design, the switching frequency is mainly fixed by the bandwidth of the gate driver chip, IR2110. Under experimental conditions, the IR2110 does not respond for a PWM signal with a high frequency over 30-kHz. Additionally, increasing the switching frequency will heavily increase the level of voltage spikes on the drain-source voltage of the MOSFET. In this design, the gate resistance of the MOSFET has to be increased to suppress the voltage spikes. In consequence, the resolution of the switching signal will be decreased such that the linear approximation may become inaccurate. By considering such three factors, the final switching frequency is determined as 25-kHz, and equation (5.1) can be written as: A IL = V in(26 V in ) (5.2) 2L 26 25000 The last predictable parameter in equation 5.2 is the photovoltaic voltage, V in. Given the parameters shown in Table 5-2, the photovoltaic voltage varies from 0 volts to 21.4 volts. In fact, by taking into account the effect of the MPPT

74 algorithm, the range of photovoltaic voltage can be further refined. Note that the MPPT algorithm will force the operation point to oscillate around the MPP. Furthermore, assuming that the perturbation intensity of the conventional P&O algorithm is 1 volts, a more reasonable variation range of the photovoltaic voltage is from 1 volts to 18 volts. By attempting different values for the input inductor, several approximations for the amplitude of ripple current versus photovoltaic voltage are presented by Figure 5-2. Figure 5-2 Amplitude of the ripple current versus photovoltaic voltage As shown in Figure 5-2, a larger inductor can suppress the amplitude of the ripple current. Note that under the experimental conditions, the output current of the Boulder 15W will be lower than 220 ma. To keep the boost circuit operating in CCM, the amplitude of the ripple current should be lower than 100mA. Therefore, a 12-mH inductor is adopted for building the system.

75 An electrolytic capacitor is placed between the PV module and the boost DC-DC converter so that this capacitor can protect the input power from the ripple components. The Figure 5-3 illustrates current waveforms of the photovoltaic current, input inductor current, and input capacitor current. Figure 5-3 Current waveforms of the PV model, inductor and input capacitor The phase error between the input capacitor current and the ripple component of the inductor current is 180 degree. In an ideal case, the input current only has a DC component. This will save considerable works related to the noise filtering for the input current measurement. A 35V/210uF electrolytic capacitor with 0.8-ohm parasitic resistance is adopted. This is because a larger ESR can contribute more damping to the system dynamics so that the potential ringing and overshoot may be suppressed [11]. 5.2 Peripheral Circuits The peripheral circuits consist of the gate driver circuit, analog low pass filters, voltage dividers and RC-snubbers. The gate driver circuit is used to enhance the power of the PWM signal generated by the TI F28035 and to protect

76 the DSP board from over-current and over-voltage. Two snubber circuits are considered to suppress the voltage spikes and ringing on the drain-source voltage of the MOSFET. Given the nature of a switching mode circuit, the switching devices will inject noise into the circuit. The noise diminishes the accuracy of signal measurements so that the performance of the controller will be affected. Therefore, low pass filters are designed to eliminate the noise on the measurement signals. 5.2.1 Gate Driver Circuit To drive the MOSFET, a gate driver chip and a digital inverter are used to build the driver circuit. As the first layer protection for the DSP board, the digital inverter, CD74HC04E inverts the TTL voltage of the original PWM signal and sends the inverted PWM signal to the gate driver chip, IR2110. IR2110 will output the enhanced switching signal to the gate lead of the MOSFET, IRFP460A. The schematics of the gate driver circuit is shown in Figure 5-4. Figure 5-4 The gate drive circuit Given the experimental observations, both of the digital inverter and the gate driver chip inject the switching noise to the 5-volts DC bus and 15-volts DC

77 bus. Figures 5-5 illustrates the peak-peak voltage of the switching noise on 5-volts DC bus. The peak-peak voltage of the switching noise is 814.5 mv. Figure 5-6 illustrates the fundamental frequency of the switching noise. The fundamental frequency of the switching noise is around 25-kHz which is the exact frequency of the PWM signal. With the peak-peak voltage level, the switching noise will heavily reduce the accuracy of the voltage and current measurement. This is because the core elements of the voltage and current measurement circuits, i.e., OP-Amplifiers are supplied by the noised 5-volts DC bus. To solve this issue, multiple filtering capacitors (0.1uF, 10uF, and 100uF) are connected between the 5-volts DC bus and ground. The improvement is seen in Figure 5-7. The peakpeak voltage of the suppressed switching noise on the 5-volts DC bus is less than 145 mv. Figure 5-5 The peak-peak voltage of the noise on the 5 volts DC bus (without filtering capacitor)

78 Figure 5-6 The fundamental frequency of the noise on the 5 volts DC bus (without filtering capacitor) Figure 5-7 The suppressed switching noise.

The switching noise on the 15-volts DC-bus can be suppressed by applying the same solution, which is shown in Figure 5-4. 79 5.2.2 Eliminating Voltage Spikes on the Drain-Source Voltage MOSFET switches have parasitic output capacitance and layout capacitance. The diode shown in Figure 5-1 (p.72) has a forward recovery time. When the MOSFET is fully turned off, voltages may accumulate across the gatesource capacitor while the diode attempts to conduct in the forward direction. If the forward-conduction time is longer than the turn-off time of the MOSFET, voltage spikes can be seen at the drain-source voltage of the MOSFET [16]. For suppressing the spikes and ringing on the falling edges of the drainsource voltage of the MOSFET, a conventional solution is to add a resistor in series with the MOSFET gate lead for prolonging the turn-on time so that the drain-source voltage can have a relatively smooth falling-edge. And the fallingedge ringing can be suppressed. To eliminate the voltage spikes and ringing on the rising-edge of the drain-source voltage, a conventional approach is to build a RC snubber circuit to consume the power accumulated at the drain of the MOSFET before the diode is fully forward-conduction. The general equations for calculating the parameters of a RC snubber are given by equations (5.3) through (5.6). C s >> 2C oss (5.3) C s is the capacitance of a RC snubber. C oss is the output capacitance of the MOSFET. Referring to the IRFP460A specification, the value of C oss varies around 6000pF when V DS is less than 10 volts. Remind that the photovoltaic voltage varies from 0 volts to 21.4 volts. Therefore, by selecting 6000pF as the value for C oss, equation (5.3) can be rewritten as (5.4).

C s >> 2C oss = 2 6000pF = 1.2 nf (5.4) The resistance of a RC sunbber is calculated by using equation (5.5), which considers the worst case. The worst case is that the voltage across the resistor of a RC snubber may be the exact output voltage of the boost DC-DC circuit. Hence, a power resistor with a 50V tolerance is adopted in this design. The resistance does not heavily affect the performance of a RC snubber, as long as it is less than its theoretical value given by equation (5.5). 80 R s V out = 26V = 130 Ω (5.5) I out 0.2A Given the computational and experimental results, a 18.3-nF capacitor and a 10-ohm resistor are used for building the RC snuber. A 200 ohm resistor is selected as the gate resistor. Notice that the maximum photovoltaic power is about 3.1 watts. Therefore the power dissipation on the designed RC snubber should be considered. The power dissipated on a RC snubber can be calculated by using equation (5.6). P diss C s V 2 out f s = 18.3nF 26V 2 25000Hz = 0.3 watts (5.6) By applying equation (5.6), the maximum power dissipated on the RC snubber is less or equal 0.3 watts, which is acceptable. As shown in Figure 5-8, the drain-source voltage of IRFP460A has a 11.5V voltage spike when the gate resistor and RC snubber are not connected. Figure 5-9 illustrates the drain-source voltage of IRFP460A after the gate resistor and RC snubber are connected. The suppressed voltage spikes reduce to 2.8 Volts.

81 Figure 5-8 The drain-source voltage of the IRFP460A (without gate resistor and RC snubber circuit) Figure 5-9 The drain-source voltage of the IRFP460A (with gate resistor and RC snubber circuit)

82 5.2.3 Voltage Sensing The MPPT system consists of two types of voltage sensing circuit. As shown in Figure 5-10, the first type is a straightforward voltage divider with an overvoltage protection for the ADC channels of the TI F28035. The parameters of the first type of voltage sensing circuit are listed in Table 5-3. The second type of voltage sensing circuit is an analog low pass filter with a DC gain. Remind that the proposed signal process system has two control layers. The first type of voltage sensing circuit is used to provide instant values of the photovoltaic voltage to the secondary control layer for the voltage regulation. The second type of voltage sensing circuit is used to send the filtered voltage signal to the top control layer for the MPPT control. The control interval of the top control layer should cover the settling time of the low pass filter. Those are because the proposed MPPT algorithm involves derivative operations so that the noise level on input signals must be suppressed again. The trade-off is the time. Figure 5-10 The topology of the voltage divider

Table 5-3 Parameters of the Voltage Divider Components R1 R2 C1 D Parameters 20 kω 3 kω 0.01 uf 1N4007 83 The photovoltaic voltage varies from 0 to 21.4 volts. The input voltage range of the ADC channels is from 0 to 3.3 volts. Therefore, the proportional gain is set to 0.13 by adopting a 20 kω resistor and a 3 kω resistor. The schematics of the low pass filter can be found in Appendix B. Given that the fundamental frequency of the switching noise is 25-kHz, the cut-off frequency of the low pass filter is supposed to be below 25-kHz. In this design, a first order low pass filter is adopted. The general transfer function of a first order low pass filter is given by equation (5.7). G(s) = DC gain 1+ s f cut (5.7) The f cut is the cut-off frequency which will mainly determine the magnitude of the fundamental harmonic of the switching noise. Weighing the magnitude response against the settling time of the low pass filter, the transfer function of the voltage sensing circuit is decided as: G(s) = 0.13 1+ s 256 The settling time of the low pass filter is around 15.2 ms. The magnitude response at the 25-kHz frequency is about -57.5 db as shown in Figure 5-11. There are two approaches to implement the low pass filter. Equation (5.9) and (5.10) introduce the z-domain transfer function of (5.8). The digital filter has (5.8)

84 250 khz sampling frequency and is calculated by applying the Tustin mapping theory. G LPF (z) = 6.663 10 5 +6.663 10 5 z 1 1 0.999z 1 (5.9) y[n] 0.999y[n 1] = 6.663 10 5 x[n] + 6.663 10 5 x[n 1] (5.10) Figure 5-11 The bode plot of the proposed low pass filter However, to implement a digital low pass filter in the DSP TI F28035, an additional control layer is required for the implementation of the high sampling frequency and utilizations of interruptions. Given the Round-Robin sampling mechanism of the TI F28035 [32], the 250-kHz sampling rate may adversely affect other low sampling rate functionalities. To avoid the above issue, the analog low pass filter is adopted. The topology of the low pass filter is shown in Figure 5-12.

85 Figure 5-12 The low pass filter for voltage sensing G LPF (s) = R 2 R 1 R 2 C 1 s+1 (5.11) Equation (5.11) illustrates the transfer function of the analog circuit shown in Figure 5-12. The parameters of the analog low pass filter are listed in Table 5-4. Table 5-4 Parameters of the Analog Low Pass Filter for Voltage Measurement Component Parameter R1 76 kω R2 10 kω C1 0.39 uf D 1N4007 5.2.4 Current Sensing In this design, the current sensing has been a big issue because the photovoltaic current is so small that it is easily disturbed by the switching noise. Note that the maximum photovoltaic current is 220mA under test conditions. Additionally, the lower limit of input current of commercial current sensors is normally greater than 1A. Hence, the current sensing circuit is designed

86 independently. There are two general approaches to sense the current. The first approach is the High Side Current Sensing shown in Figure 5-13. V dif is the differential voltage signal which is the voltage across the shunt resistance. The connected Op-Amplifier circuit can be used to provide a proportional gain for the converted current-signal. The output signal of the OP-Amplifier circuit is a voltage signal, which is proportional to the value of the current flowing through the shunt resistor. Figure 5-13 The topology of High-Side Current Sensing The advantages of the High-Side Current Sensing involve: 1) isolation from the ground disturbance; 2) easy implementation. However, to convert the differential signal shown in Figure 5-13, the OP-Amplifier circuit must be supplied with a specified voltage level which is higher than the voltage across the load. If so, an extra 24-volts DC bus is needed, whereas this extra requirement can be avoided by adopting the Low-Side Current Sensing. The Figure 5-14 illustrates the topology of Low-Side Current Sensing, which is adopted for this design.

87 Figure 5-14 The topology of Low-Side Current Sensing By using this approach, the Op-Amplifier circuit can be supplied by the 5- volts DC bus. However, the differential signal shown in Figure 5-15 may contain the ground noise. Therefore, an analog low pass filter should be designed for filtering the noised current measurement signal. The Figure 5-15 illustrates the schematics of the current sensing circuit. The parameters are listed in Table 5-5. Figure 5-15 The Low-Side Current Sensing circuit.

Table 5-5 Parameters of the Current Sensing Circuit Component R shunt R1 R2 C1 D Parameter 1 Ω 10 kω 100 kω 0.394 uf 1N4007 88 The transfer function of the current sensing circuit is similar to equation (5.11). Increasing the value for R2 is to suppress harmonics within the current signal. This is because the proportional gain of the current sensing circuit is 10, which will enhance the identification of any DC change in photovoltaic current and simultaneously boost the noise level. Thus, the cut-off frequency is supposed to be chosen as low as possible for neutralizing the boosted noise level. The tradeoff of this strategy is the enlarged settling time. Throughout simple calculation, the settling time of the current sensing circuit is about 152 ms. This is the main reason why the control interval of the top control layer is set to 200 ms. 5.3 Signal Process System As mentioned at the beginning of this chapter, the signal process system has two control layers. The top control layer is designed to realize the MPPT algorithm. The secondary control layer is designed to realize the voltage regulation of photovoltaics. The layout of the whole system is illustrated by the Figure 5-16. The detailed topology of the MPPT system is illustrated by Figure 5-17. The picture of the MPPT system is shown in Figure 5-18.

89 Figure 5-16 The layout of the MPPT system. Figure 5-17 The topology of the MPPT system.

90 Figure 5-18 The designed MPPT system 5.3.1 Voltage Regulation of Photovoltaics As discussed in chapter 4, the core element that fulfills the voltage regulation is a PI controller. To implement the control strategy in the signal process system, a digital PI controller is needed. Based on the impulse mapping method, the discrete time integrator can be written as: 1 = 1 s 1 Z 1 (5.12) The continuous time PI controller discussed in chapter 4 can be written as: G PI (z) = 0.1 + 2.2 (5.13) 1 z 1 Given the experimental results, the final parameters of the digital PI controller is tuned as: G PI (z) = 0.1 + 0.05 (5.14) 1 z 1 y[n] y[n 1] = 0.15x[n] 0.1x[n 1] (5.15)

91 The partial embedded code related to the digital PI controller is generated by using MATLAB/Simulink Embedded Coder Toolbox. The Simulink diagram is shown in Figure 5-19. Figure 5-19 Simulink block of the digital PI controller Figure 5-20 illustrates the step response of the secondary control layer, which is designed for the voltage regulation of photovoltaics. The final value of the input step function is 500 mv. The rising time of the step response of the secondary control layer is around 55.44ms. The ripple on the voltage waveform shown in Figure 5-20 is due to the unideal switching device and duty ratio resolution. In the microcontroller, the duty ratio is represented by a 12-bit Hex number, which means that the precision of the duty ratio is about 0.024 percentage. Hence, the practical duty ratio may not exactly converge to its reference value so that it keeps fluctuating with a small scale offset.

92 Figure 5-20 The voltage regulation of photovoltaics 5.3.2 DCM Detection Mechanism To realize MPPT algorithms, the boost DC-DC converter must operate in CCM for maintaining the validity of the linear approximation discussed in chapter 4. Therefore, a DCM detection mechanism should be designed and embedded into the top control layer. Before design the detection mechanism, an indicator that can indicate the system s status, should be selected. Figure 5-21 demonstrates that the inductor voltage waveform of the boost converter after the photovoltaic voltage reference is set to 14 volts. It is obvious that the boost converter operates in CCM. The error between the actual photovoltaic voltage and its reference value is about 0.1 volts.

93 Figure 5-21 The inductor voltage waveform (CCM) Figure 5-22 demonstrates the inductor voltage of the boost converter, which operates in DCM. Setting the voltage reference to 20 volts will lead the boost converter to operate in DCM. The error between the voltage reference and actual photovoltaic voltage is about 1.4 volts. Figure 5-22 The inductor voltage waveform (DCM)

94 Given the patterns shown in Figure 5-21 and 5-22, the proper indicator for distinguishing the conduction mode of the PV boost converter is the steady-state error of the PI controller. According to the experimental observation, if the system operates in DCM mode, the steady-state error will be greater than 150 mv. The DCM status of the system also means that in every switching period the inductor current reaches zero due to the weak photovoltaic current. Note that the I-V curve is a monotonously decreasing curve. Therefore, a proper solution for recovering the conduction mode from DCM to CCM is to enhance the photovoltaic current. In other words, the PV voltage reference has to be decreased. Figure 5-23 illustrates an example for the DCM detection mechanism. Figure 5-23 The illustration of DCM detection mechanism For example, during the procedure of the MPPT control, the K th perturbation is to shift the operation point from position A towards B. To achieve the position B, the converter has to operate in DCM because the corresponding

95 photovoltaic current falls below the critical value, which keeps the converter operating in CCM. During this process, the linear approximation discussed in chapter 4 is nullified. Meanwhile, the PI controller is still functioning and pushing the system s operating point to an unpredictable position. Hence, unreasonable steady-state error occurs. When this happens, the proposed DCM detection mechanism will push the system s operating point back to its previous position, A. Additionally, the DCM detection mechanism will further diminish the intensity of the next perturbation. The mechanism will eventually attempt to shift the operation point towards the position C for the further trial. The flow chart of the DCM detection mechanism is illustrated by Figure 5-24. Figure 5-24 The flow chart of the DCM detection mechanism.