AN IMPROVED MAXIMUM POWER POINT TRACKING ...

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

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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

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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 1000th

second) .................. 40


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

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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

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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 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 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 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 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 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 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 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 depth-

recycled 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 𝑚2. Throughout calculation, the size of a solar

array with a 240V rated voltage is about 108m2. 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 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].

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Figure 1-7 A grid-connect photovoltaic system with micro inverters

Figure 1-8 A grid-connected photovoltaic system with power optimizers

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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

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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 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 4𝑐𝑚2 to

225𝑐𝑚2. The nominal power of a solar cell, under standard test condition (STC),

is less than 4 watts. The STC means an irradiation of 1000W/m2 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 inter-

connected 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.

15 15

2.2 PV Modelling and Simulation

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 (𝐼𝑠𝑐), the open-circuit voltage (𝑉𝑜𝑐), 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 𝑅𝑠ℎ and a series resistance 𝑅𝑠. 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 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].

I= 𝐼𝑠𝑐 − 𝐼𝑜 (𝑒(𝑞𝑉+𝐼𝑅𝑠𝑎𝐾𝑇

) − 1) − (𝑉+𝐼𝑅𝑠

𝑅𝑝) (2.1)

Where

I is the photovoltaic current (A)

𝐼𝑠𝑐 is the short circuit current (A)

𝐼𝑜 is the reverse saturation current (A)

V is the photovoltaic voltage (V)

17 17

q is the electron charge (1.6 × 10−19𝐶)

k is the Boltzmann’s constant (1.381 × 10−23 𝐽/𝐾)

T is the junction temperature (K)

𝑎 is the diode ideality factor

𝑅𝑠 is the series resistance (Ω)

𝑅𝑝 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 (𝐼𝑠𝑐) 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].

𝐼𝑠𝑐 = (𝐼𝑠𝑐_𝑆𝑇𝐶 + 𝐶𝑖∆𝑇)𝐺

𝐺𝑆𝑇𝐶 (2.2)

∆𝑇= T - 𝑇𝑆𝑇𝐶 (K) (2.3)

Where,

𝐼𝑠𝑐_𝑆𝑇𝐷 is the short-circuit current (STC)

∆𝑇 is the temperature error (K)

𝐶𝑖 is the short circuit coefficient(A/)

18 18

𝐺𝑆𝑇𝐶 is the STC irradiation (1000W/𝑚2)

𝑇𝑆𝑇𝐶 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 (𝑉𝑜𝑐)

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, 𝐼𝑜 is rarely provided by the

manufactures. However, it may be approximated by using equation (2.4) [17].

𝐼𝑜 = (𝐼𝑠𝑐𝑆𝑇𝐶+𝐶𝑖∆𝑇)

𝑒(𝑞𝑉𝑆𝐶_𝑆𝑇𝐶+𝐶𝑣∆𝑇

𝑎𝐾𝑇)−1

(2.4)

Where 𝐶𝑣 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 𝑅𝑠 and 𝑅𝑝. The accuracy of

these two parameters determines the similarity between simulated I-V and

19 19

experimentally measured I-V curves provided by manufactures. Fortunately, [18]

provides a reasonable approach to compute 𝑅𝑠 and 𝑅𝑝. The core concept of [18] is

to keep increasing the value for 𝑅𝑠, while simultaneously calculate the value for

𝑅𝑝 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].

𝑅𝑝 = 𝑉𝑚𝑝𝑝(𝑉𝑚𝑝𝑝+𝐼𝑚𝑝𝑝𝑅𝑠 )

[𝑉𝑚𝑝𝑝(𝐼𝑠𝑐−𝐼𝑑)]−𝑃𝑚𝑝𝑝 (2.5)

Where

𝑃𝑚𝑝𝑝 is the maximum power point power

𝐼𝑚𝑝𝑝 is the maximum power point current

𝑉𝑚𝑝𝑝 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,

𝐼𝑚𝑝𝑝 and 𝑉𝑚𝑝𝑝 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 inter-

connected 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), 𝑁𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 is the number of columns shown in Figure 2-7,

while 𝑁𝑠𝑒𝑟𝑖𝑒𝑠 is the number of rows.

20 20

Figure 2-7 The equivalent circuit model of a photovoltaic matrix

𝐼𝑝𝑣= 𝐼𝑠𝑐𝑁𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 − 𝐼𝑜𝑁𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙[𝑒

(𝑞

𝑉+𝐼𝑅𝑠(𝑁𝑠𝑒𝑟𝑖𝑒𝑠𝑁𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙

)

𝑎𝐾𝑇𝑁𝑠𝑒𝑟𝑖𝑒𝑠)

− 1] − (𝑉+𝐼𝑅𝑠

𝑁𝑠𝑒𝑟𝑖𝑒𝑠𝑁𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙

𝑅𝑝(𝑁𝑠𝑒𝑟𝑖𝑒𝑠𝑁𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙

)) (2.6)

By reviewing the equation (2.5), which is provided by [18], the relationship

between the 𝑅𝑠and 𝑅𝑝in 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 𝑅𝑝 can be computed by using equation (2.7).

𝑅𝑝 = 𝑉𝑚𝑝𝑝(𝑉𝑚𝑝𝑝+𝐼𝑚𝑝𝑝𝑅𝑠(

𝑁𝑠𝑒𝑟𝑖𝑒𝑠𝑁𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙

))

(𝑁𝑠𝑒𝑟𝑖𝑒𝑠𝑁𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙

)[𝑉𝑚𝑝𝑝(𝐼𝑠𝑐−𝐼𝑑)]−𝑃𝑚𝑝𝑝

(2.7)

Throughout equation (2.7), for any given pair of 𝑉𝑚𝑝𝑝, 𝐼𝑚𝑝𝑝, according to

an arbitrary value for 𝑅𝑠, there must be an unique solution for 𝑅𝑝. By substituting

any solution pair of 𝑅𝑝, 𝑅𝑠 and a given 𝑉𝑚𝑝𝑝 into equation (2.6), a

corresponding value for 𝐼𝑝𝑣 can be calculated. However, the product of the

calculated 𝐼𝑝𝑣 and 𝑉𝑚𝑝𝑝 may not optimally match to the experimental maximum

21 21

power provided by manufactures. Therefore, the significant step for modelling

PVs is to duplicate the above calculations for finding a solution pair of 𝑅𝑝, 𝑅𝑠,

which can best match the calculated maximum power to the provided 𝑃𝑚𝑝𝑝. After

𝑅𝑝 and 𝑅𝑠 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

22 22

demonstrates two series of parameters that present the characteristics of the SW-

260-mono operating under two different testing conditions. 𝑅𝑠 and 𝑅𝑝 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/𝑚2, 25

Mono-Crystalline

SW-260-mono-silver

Test condition

800𝑊/𝑚2, 25

𝑃𝑚𝑎𝑥 260W 194.2W

𝑉𝑜𝑐 38.9V 35.6V

𝑉𝑚𝑝𝑝 30.7V 28.1V

𝐼𝑠𝑐 9.18A 7.42A

𝐼𝑚𝑝𝑝 8.47A 6.92A

𝐶𝑖 0.004%/K 0.004%K

𝐶𝑣 -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/𝑚2, 25

Mono-Crystalline

SW-260-mono-silver

Test condition

800𝑊/𝑚2, 25

𝑃𝑚𝑎𝑥 260.2W 194W

𝑉𝑜𝑐 38.9V 35.6V

𝑉𝑚𝑝𝑝 30.7V 28.1V

𝐼𝑠𝑐 9.18A 7.42A

𝐼𝑚𝑝𝑝 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.

23 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 24

By observing Figures 2-9 and 2-10, decreasing irradiation obviously leads

𝑉𝑚𝑝𝑝 and 𝐼𝑚𝑝𝑝 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/𝑚2.The various

temperature conditions do not heavily change the 𝐼𝑚𝑝𝑝 of the SW-260-mono,

while they obviously affect the 𝑉𝑚𝑝𝑝 and 𝑃𝑚𝑝𝑝.

Figure 2-11 The simulated I-V curves of the SW-260-mono operating at different

temperature conditions.

25 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 𝑉𝑝𝑣/𝐼𝑝𝑣. To extract the potential

maximum power from PV cells, the load impedance is supposed to be 𝑉𝑚𝑝𝑝 𝐼𝑚𝑝𝑝⁄ .

Throughout a simple experiment, this concept can be easily validated. As the load

impedance linearly increases from 0 ohms to 𝑉𝑚𝑝𝑝 𝐼𝑚𝑝𝑝⁄ , the photovoltaic voltage

will nonlinearly increase from 0 volts to 𝑉𝑚𝑝𝑝, and the photovoltaic current will

nonlinearly decrease from the short circuit current to 𝐼𝑚𝑝𝑝 while the photovoltaic

power will nonlinearly increase from 0 watts to 𝑃𝑚𝑝𝑝. As the load impedance

increases from 𝑉𝑚𝑝𝑝 𝐼𝑚𝑝𝑝⁄ to a sufficient large value, such as hundreds ohms or

thousands ohms , the photovoltaic current will nonlinearly decrease from 𝐼𝑚𝑝𝑝 to 0

amps, and the photovoltaic voltage will nonlinearly increase from 𝑉𝑚𝑝𝑝to the open

circuit voltage while the photovoltaic power will nonlinearly decrease from 𝑃𝑚𝑝𝑝

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 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 27

in a steady-state of the continuous conduction mode, its input dynamic resistance

will equal (1 − 𝑑)2𝑅𝑙𝑜𝑎𝑑 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.

28 28

Table 2-3 illustrates the maximum power point internal resistance (𝑅𝑚𝑝𝑝)

of the SW-260-mono, which operates under variable irradiation conditions and the

invariant 25 temperature condition. Table 2-4 illustrates the 𝑅𝑚𝑝𝑝 of the SW-

260-mono, which operates under variable temperature conditions and the invariant

1000 W/𝑚2 irradiation.

Table 2-3 The 𝑅𝑚𝑝𝑝 of the SW-260-mono under Different Irradiation Conditions

Irradiation condition The 𝑅𝑚𝑝𝑝 of the SW-260-mono

1000 W/𝑚2 3.62 Ω

800 W/𝑚2 4.07 Ω

600 W/𝑚2 6.00 Ω

400 W/𝑚2 9.41 Ω

200 W/𝑚2 16.91 Ω

Table 2-4 The 𝑅𝑚𝑝𝑝 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 𝑅𝑚𝑝𝑝 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.

29 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.

31 31

Table 3-1 Parameters of Photovoltaics

Short circuit current (𝐼𝑠𝑐) 4.75 (A)

Open circuit voltage (𝑉𝑜𝑐) 27.03 (V)

Maximum power point (𝑃𝑚𝑝𝑝) 98.23 (W)

Maximum power point voltage (𝑉𝑚𝑝𝑝) 22.37 (V)

Maximum power point current (𝐼𝑚𝑝𝑝) 4.39 (A)

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 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 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 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 =

𝜕(𝑉×𝐼)

𝜕𝑉= 𝐼 + 𝑉 ×

𝜕𝐼

𝜕𝑉 (3.0)

𝐼 = 𝐼𝑠𝑐 − 𝐼𝑜 (𝑒(𝑞

𝑉

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

𝜕𝑉 = −𝐼𝑜 ×

𝑞

𝑎𝐾𝑇× 𝑒(𝑞

𝑉

𝑎𝐾𝑇) (3.2)

𝜕𝑃

𝜕𝑉 = 𝐼 + 𝑉 ×

𝜕𝐼

𝜕𝑉= 𝐼𝑠𝑐 − 𝐼𝑜 (𝑒

(𝑞𝑉

𝑎𝐾𝑇) − 1) − 𝐼𝑜 ×

𝑞𝑉


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.

𝜕2𝑃

𝜕2𝑉= −𝐼𝑜 ×

𝑞

𝑎𝐾𝑇(𝑒(𝑞

𝑉

𝑎𝐾𝑇) − 1) − 𝐼𝑜 ×

𝑞


𝑉

𝑎𝐾𝑇) (3.4)

−𝐼𝑜 × (𝑞

𝑎𝐾𝑇)2 × 𝑉 × 𝑒(𝑞

𝑉

𝑎𝐾𝑇)

35 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]

36 36

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, 𝑒𝑡 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

dV= 0 (3.5)

|dP

dV| ≤ 𝑒𝑡 (3.6)

|𝐼 + 𝑉 ×𝜕𝐼

𝜕𝑉| ≤ 𝑒𝑡 ↔ |

𝜕𝐼

𝜕𝑉| ≤ |

𝑒𝑡∓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.

37 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 (𝑉𝑐𝑝) can be obtained by using equation (3.8).

𝑉𝑐𝑝 = 𝛽𝑉𝑜𝑐 (3.8)

The value of 𝛽 is normally selected in range from 70% to 80%.The constant

voltage method is derived from experimental experiences: 𝑉𝑚𝑝𝑝 generally is

located within the range from 70% to 80% of 𝑉𝑜𝑐[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 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 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 40

The MPPT efficiency can be calculated by using equation (3.9) [30].

η𝑀𝑃𝑃𝑇= 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.


second)

41 41


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 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 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, 𝜇𝐴(𝑥))|𝑥 ∈ 𝑈 (3.12)

In equation (3.12), 𝜇𝐴(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 𝜇𝐴(𝑥)

44 44

𝜇𝐴(𝑥) =

0 𝑖𝑓 𝑥 < 3 1 𝑖𝑓 3 ≤ 𝑥 < 5

𝑥−5

3 𝑖𝑓 5 ≤ 𝑥 ≤ 8

0 𝑖𝑓 𝑥 > 8

(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 𝑉𝑝𝑣 ∈ (0,18). The slope of the

PV curve nonlinearly decreases towards zero in the range of 𝑉𝑝𝑣 ∈ (18,23) and

deviates from zero towards -78 in the range of 𝑉𝑝𝑣 ∈ (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

45 45

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.

46 46

Figure 3-14 The membership function E

Figure 3-15 The membership function CE

Figure 3-16 The membership function PT

47 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 𝑥1is 𝐴1and 𝑥2is 𝐴2 and… and 𝑥𝑛is 𝐴𝑛, THEN y is 𝐵𝑗

Where 𝐴𝑖 and 𝐵𝑗 are fuzzy sets in 𝑈𝑖 and V, respectively, and 𝑈𝑖 consists of

any possible value for the input variable, 𝑥𝑖. 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 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 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 = ∑𝜇𝐴𝑖

(𝑥)×𝐵𝑖

∑𝜇𝐴𝑖(𝑥) (3.14)

Where 𝜇𝐴𝑖(𝑥) is the membership function of the output variable fuzzy set

and 𝐵𝑖 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, 𝜇𝐴1(𝑥), 𝜇𝐴2(𝑥) and 𝜇𝐴3(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% (3.15)

= 2.22

50 50

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, 𝐼𝑠𝑐 4.75 (A)

Open circuit voltage, 𝑉𝑜𝑐 27.03(V)

Maximum power , 𝑃𝑚𝑝𝑝 98.23 (W)

The parameters related to the algorithms and the simulation configurations

Initial operation point, (V,P) (0 V, 0 W)

Perturbation intensity of P&O 0.1 V

Perturbation frequency 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 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 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 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 3𝑡ℎ 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 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 6th

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.

55 55

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

irradiation condition

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-signal-

model 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 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 switch-

mode 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 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 𝑉𝑖𝑛 and

𝑉𝑜𝑢𝑡.

Figure 4-2 The equivalent circuit during switching on periods

Figure 4-3 Inductor current and voltage during switching on periods

59 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.

𝑉𝑖𝑛 × 𝑡𝑜𝑛 = (𝑉𝑜𝑢𝑡 − 𝑉𝑖𝑛) × 𝑡𝑜𝑓𝑓 (4.1)

𝐿𝑑𝑖𝐿

𝑑𝑡= 𝑉𝐿 (4.2)

The average approximation of equation (4.2) is written as:

𝐿∆𝑖𝐿

∆𝑡= 𝑉𝐿 (4.3)

Where ∆𝑖𝐿 is the magnitude of the ripple component of the inductor current.

𝑉𝐿 equals 𝑉𝑖𝑛in switching-on periods, while it equals the (𝑉𝑖𝑛 − 𝑉𝑜𝑢𝑡) in switching-

off periods. Therefore, equation (4.4) can be obtained.

60 60

∆𝑖𝐿 = 1

𝐿𝑉𝑖𝑛 × 𝑡𝑜𝑛 =

1

𝐿(𝑉𝑜𝑢𝑡 − 𝑉𝑖𝑛) × 𝑡𝑜𝑓𝑓 (4.4)

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].

𝑡𝑠 = 𝑡𝑜𝑛 + 𝑡𝑜𝑓𝑓 (4.5)

𝑡𝑠 = 𝑑𝑡𝑠 + (1 − 𝑑)𝑡𝑠 (4.6)

𝑉𝑜𝑢𝑡

𝑉𝑖𝑛=

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. 𝑉𝑜𝑢𝑡 of a boost DC-DC converter

operating in DCM can be calculated by using equations (4.7) and (4.8) [21]:

𝑉𝑜𝑢𝑡 =𝑉𝑖𝑛

2(1+√1 + 4𝑀) (4.7)

M = (𝑅

2𝐿𝑓𝑠)D2 (4.8)

Where

R is the load impedance of the boost DC-DC converter.

𝑓𝑠 is the switching frequency.

D is the duty ratio of the switching signal.

61 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 𝐺𝑣(𝑠)/𝑑(𝑠)

62 62

and 𝐺𝑣(𝑠)/𝐺𝑖(𝑠) can be derived. In this thesis, the related control design only

requires the knowledge of 𝐺𝑣(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, , in swiching signal has been injected to the

boost circuit. Then, equation (4.8) and (4.9) can be satisfied.

𝐼𝑜𝑢𝑡 =𝑉𝑜𝑢𝑡

𝑅 (4.8)

𝑉𝑖𝑛 × 𝐼𝑖𝑛 = 𝑉𝑜𝑢𝑡 × 𝐼𝑜𝑢𝑡 (4.9)

By ignoring the current through the input capacitor, the average inductor

current is assumed to equal the input current.

𝐼𝐿 = 𝐼𝑖𝑛 = 𝑉𝑜𝑢𝑡×𝐼𝑜𝑢𝑡

𝑉𝑖𝑛=

𝑉𝑜𝑢𝑡2

𝑉𝑖𝑛×𝑅 (4.10)

By applying the Norton’s theorem, the procedure of simplifying the

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 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, 𝐺𝑣𝑜𝑢𝑡(𝑠)/𝑑(𝑠) is written as [21]:

𝑉𝑜𝑢𝑡(𝑠)

(𝑠) =

𝑉𝑖𝑛

(1−𝐷)2(1 −

𝑠𝐿

(1−𝐷)2𝑅)

1+𝑠𝑟𝐶𝑜𝑢𝑡𝐿𝐶𝑜𝑢𝑡(1−𝐷)2

[𝑠2+𝑠(1

𝑅𝐶𝑜𝑢𝑡+𝑟(1−𝐷)2

𝐿)+

(1−𝐷)2

𝐿𝐶𝑜𝑢𝑡)]

(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 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 is the load’s capacitance (F)

Q is the charged coulomb (A∙s)

U 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

65 65

The 𝐼𝑝𝑣, 𝑉𝑝𝑣, and 𝑟𝑝𝑣 denote the photovoltaic current, photovoltaic voltage,

and internal resistance, respectively. 𝑖𝐿 and 𝑖𝐶denote the inductor current and

capacitor current, respectively. 𝑟𝐶 and 𝑟𝐿 present parasitic resistance for the input

capacitor and the input inductor. The 𝑉𝑝𝑟is the equivalent voltage on the primary

side of the transformer.

A perturbation, , 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:

𝑉𝑝𝑟 = × 𝑉𝑏𝑎𝑡 (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.

𝑑

𝑑𝑡⌈ 𝐼𝑉𝑝⌉ = ⌈

−𝑅𝐿𝐿

−𝑅𝑝𝑣(1−𝑅𝐿𝑅𝑐)

𝐿𝐶(1−𝑅𝑝𝑣)

1

𝐿

𝐶𝑅𝐿𝑅𝑐−1

𝐿𝐶(1−𝑅𝑝𝑣)

⌉ ⌈ 𝐼𝑉𝑝⌉ + ⌈

−𝑉𝑏𝑎𝑡𝐿

𝑅𝑐𝑅𝑝𝑣𝑉𝑏𝑎𝑡𝐿𝐶(1−𝑅𝑝𝑣)

⌉ (4.14)

= [0 1] ⌈𝐼𝑉𝑝⌉ (4.15)

𝐺𝑣𝑑(𝑠) = 𝐴

𝑠2+𝑁𝑠+𝑀 (4.16)

𝐴 = −𝑉𝑏𝑎𝑡(1

𝐿2+

𝑅𝑐2𝑅𝑝𝑣

2−𝑅𝑐𝑅𝑝𝑣

𝐿2𝐶(1−𝑅𝑝𝑣)2 −

𝑅𝑐𝑅𝑝𝑣

𝐿(1−𝑅𝑝𝑣)𝑠) (4.17)

𝑁= (𝑅𝐿

𝐿−

𝑅𝑐𝑅𝑝𝑣−1

𝐿𝐶(1−𝑅𝑝𝑣)) (4.18)

𝑀= 𝑅𝐿𝑅𝑐𝑅𝑝𝑣−𝑅𝐿+𝑅𝑝𝑣𝐿−𝐶𝑅𝐿

2𝑅𝑐𝑅𝑝𝑣

𝐿2𝐶(1−𝑅𝑝𝑣) (4.19)

The purpose of analyzing the linear approximation is to design a voltage

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 66

such a nonlinear system shown in Figure 4-10 change with 𝑅𝑝𝑣, 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 Parameters

𝑉𝑏𝑎𝑡 26(V)

𝑅𝐶𝑖𝑛 0.8(Ω)

𝑅𝐿 0.2(Ω)

L 12(mH)

C 210(uF)

𝑅𝑝𝑣 75~150(Ω)

The only variable parameter in the linear approximation is 𝑅𝑝𝑣 .Therefore,

to analyze the system’s behavior with respect to 𝑅𝑝𝑣, four transfer functions with

four different 𝑅𝑝𝑣 are presented. Four values for 𝑅𝑝𝑣, 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 𝑅𝑝𝑣.

Given Table 4-2 and Figure 4-9 (p. 64), the variable 𝑅𝑝𝑣 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 67

Figure 4-11 Bode plot of the variable-parameters system

Table 4-2 Linear Approximations with Different values of 𝑅𝑝𝑣

𝑅𝑝𝑣 Linear approximation Damping

ratio

Natural

frequency

75 (Ω) 5.56𝑒8(1 +

𝑠3.165𝑒5

)

3.8𝑠2 + 1.202𝑒6𝑠 + 2.156𝑒5 3.4062 238.19(rad/s)

100 (Ω) 5.546𝑒8(1 +

𝑠3.1676𝑒5

)

3.8𝑠2 + 1.203𝑒6𝑠 + 2.157𝑒5 3.4082 238.25(rad/s)

125 (Ω) 5.5386𝑒8(1 +

𝑠3.1692𝑒5

)

3.8𝑠2 + 1.204𝑒6𝑠 + 2.158𝑒5 3.4103 238.31(rad/s)

150 (Ω) 5.532𝑒8(1 +

𝑠3.1703𝑒5

)

3.8𝑠2 + 1.204𝑒6𝑠 + 2.159𝑒5 3.4095 238.36(rad/s)

Where 𝑒5, 𝑒6 and 𝑒8 denote 105, 106 and 108, respectively.

68 68

4.2.3 Voltage Regulation

𝐺𝑐(𝑠) = 𝐾𝑝 + 𝐾𝑖1

𝑠 (4.20)

The general equation of a PI controller is given by equation (4.20). To

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

time

Steady-state

error Stability

𝐾𝑝 Decrease Increase Small

change Decrease Degrade

𝐾𝑖 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:

𝐺𝑐(𝑠) = 0.1 +2.2

𝑠 (4.21)

The purpose of tuning the PI controller is to thoroughly eliminate the

potential overshoot of the closed-loop compensated system in order to protect the

input power source. In Table 4-4, 𝐺75, 𝐺100, 𝐺125and 𝐺150 denotes the closed-loop

compensated systems with the corresponding values for 𝑅𝑝𝑣. 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.

69 69

Table 4-4 Step Response of the Closed-Loop Compensated System

Models Rise time Settling Time Overshoot

𝐺75 40.8(ms) 61.6 (ms) 0%

𝐺100 40.9(ms) 61.7 (ms) 0%

𝐺125 41.0(ms) 61.8 (ms) 0%

𝐺150 41.0(ms) 61.9 (ms) 0%

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 𝑅𝑝𝑣, 𝑅𝑐𝑖𝑛, and 𝑅𝐿. 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

71 71

power tracking performance of two MPPT control strategies is provided in the last

section.

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


Solar Panel Boulder 15W

𝐶𝑖𝑛 35V/210uF

𝑅𝐶𝑖𝑛 0.8 Ω

L 12 mH

𝑅𝐿 0.2 Ω

𝐶𝑜𝑢𝑡 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 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

voltage 𝑉𝑚𝑝𝑝 12~17 (V)

Nominal maximum power point

current 𝐼𝑚𝑝𝑝 0.166~0.176(A)

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 =𝑉𝑖𝑛(𝑉𝑜𝑢𝑡−𝑉𝑖𝑛)

2𝐿𝑉𝑜𝑢𝑡𝑓𝑠 (5.1)

73 73

Notice that the parameter, 𝑉𝑜𝑢𝑡 is supposed to be selected before analyze

the ripple current. 𝑉𝑜𝑢𝑡 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.

𝑓𝑠 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 =𝑉𝑖𝑛(26−𝑉𝑖𝑛)

2𝐿∗26∗25000 (5.2)

The last predictable parameter in equation 5.2 is the photovoltaic voltage,

𝑉𝑖𝑛. 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 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 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 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 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 peak-

peak 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 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.

79 79

The switching noise on the 15-volts DC-bus can be suppressed by applying

the same solution, which is shown in Figure 5-4.

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 gate-

source 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 drain-

source 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 falling-

edge 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).

𝐶𝑠>> 2𝐶𝑜𝑠𝑠 (5.3)

𝐶𝑠 is the capacitance of a RC snubber. 𝐶𝑜𝑠𝑠 is the output capacitance of the

MOSFET. Referring to the IRFP460A specification, the value of 𝐶𝑜𝑠𝑠 varies

around 6000pF when 𝑉𝐷𝑆 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 𝐶𝑜𝑠𝑠, equation (5.3) can be rewritten as (5.4).

80 80

𝐶𝑠>> 2𝐶𝑜𝑠𝑠= 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).

𝑅𝑠 ≤ 𝑉𝑜𝑢𝑡

𝐼𝑜𝑢𝑡 =26𝑉

0.2𝐴 = 130 Ω (5.5)

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).

𝑃𝑑𝑖𝑠𝑠 ≤ 𝐶𝑠𝑉𝑜𝑢𝑡2𝑓𝑠 = 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 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 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

83 83

Table 5-3 Parameters of the Voltage Divider


R1 20 kΩ

R2 3 kΩ

C1 0.01 uF

D 1N4007

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) =𝐷𝐶𝑔𝑎𝑖𝑛

1+𝑠

𝑓𝑐𝑢𝑡

(5.7)

The 𝑓𝑐𝑢𝑡 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

(5.8)

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

84 84

250 kHz sampling frequency and is calculated by applying the Tustin mapping

theory.

𝐺𝐿𝑃𝐹(𝑧) =6.663∗10−5+6.663∗10−5𝑧−1

1−0.999𝑧−1 (5.9)

𝑦[𝑛] − 0.999𝑦[𝑛 − 1] = 6.663 ∗ 10−5𝑥[𝑛] + 6.663 ∗ 10−5𝑥[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 85

Figure 5-12 The low pass filter for voltage sensing

𝐺𝐿𝑃𝐹(𝑠) = 𝑅2 𝑅1⁄

𝑅2𝐶1𝑠+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 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. 𝑉𝑑𝑖𝑓is 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 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.

88 88

Table 5-5 Parameters of the Current Sensing Circuit

Component Parameter

𝑅𝑠ℎ𝑢𝑛𝑡 1 Ω

R1 10 kΩ

R2 100 kΩ

C1 0.394 uF

D 1N4007

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 trade-

off 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 89

Figure 5-16 The layout of the MPPT system.

Figure 5-17 The topology of the MPPT system.

90 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

𝑠=

1

1−𝑍−1 (5.12)

The continuous time PI controller discussed in chapter 4 can be written as:

𝐺𝑃𝐼(𝑧) = 0.1 +2.2

1−𝑧−1 (5.13)

Given the experimental results, the final parameters of the digital PI

controller is tuned as:

𝐺𝑃𝐼(𝑧) = 0.1 +0.05

1−𝑧−1 (5.14)

𝑦[𝑛] − 𝑦[𝑛 − 1] = 0.15𝑥[𝑛] − 0.1𝑥[𝑛 − 1] (5.15)

91 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 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 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 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 𝐾𝑡ℎ

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 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.

96 96

5.4 Implementations of the MPPT Algorithm

5.4.1 The Conventional P&O Algorithm

The proposed MPPT algorithm, the MPPT system is compared with the

conventional P&O algorithm. In terms of the stability of the PWM signal and

system safety, the initialized photovoltaic voltage of the Boulder 15W is set to 6.0

volts. As mentioned above, the control frequency of the top control layer is set to

5-Hz for successfully covering the settling time of the current sensing circuit. The

minimum perturbation intensity should be greater than 200mV for enhancing the

identification of every photovoltaic voltage perturbation. The peak-peak voltage of

the circuit noise is controlled around 100mV. To evaluate the system, the fixed

perturbation intensity of the conventional P&O algorithm is set to 500 mV. The

performance of the conventional P&O algorithm is shown in Figure 5-25.

Figure 5-25 The performance of the conventional P&O algorithm

97 97

The conventional P&O algorithm with 0.5 volts perturbation intensity can

extract average 2.5 watts power from the solar panel, while the tracking time is

4.32 seconds. Under test conditions, the potential maximum power is 3.1 watts so

that the MPPT efficiency of this P&O algorithm is 80.64%.

5.4.2 The Improved MPPT Algorithm using Fuzzy Logic Controller

Prior to implement the proposed algorithm, the parameters of the

membership functions are to be adjusted a bit by using the same methodology

presented in chapter 3. As the tuning result, the perturbation intensity generated by

the Fuzzy Logic Controller varies from 0.2 volts to 3 volts. Due to the complexity

of the proposed MPPT algorithm, the proposed FLC is directly programed as the

embedded codes. The performance of the proposed FLC is shown in Figure 5-26.

The tracking time of the proposed MPPT algorithm is 1.81seconds, while the

average generated power is about 3.0 watts. Given Table 5-2, the MPPT efficiency

of the improved MPPT algorithm with the Fuzzy Logic Controller is 96.78%,

while that of the conventional P&O algorithm is only 80.64%. Therefore,

throughout the experimental validations, the proposed MPPT algorithm using FLC

shows shorter tracking time and higher MPPT efficiency, compared to that of the

conventional P&O algorithm.

98 98

Figure 5-26 The performance of the improved MPPT algorithm using FLC

6 CONCLUSION

In this thesis, an improved MPPT algorithm using Fuzzy Logic Controller

has been investigated by referring to the mechanisms of the conventional P&O

algorithm and nonlinear characteristics of photovoltaics. The strategy that is to

create adaptive perturbation intensities for accelerating the MPPT velocity and

improving the MPPT efficiency was validated by both of the simulation and

experimental results. In addition, this thesis also emphasizes the hardware

implementation for proving the effectiveness of the proposed MPPT algorithm. To

provide enough theoretical tools for the voltage regulation of photovoltaic cells, a

small signal model of the input terminal of a photovoltaic boost DC-DC converter

was introduced and used for deriving the linear approximations without ignoring

any parasitic resistance of the input capacitor and inductor. The effectiveness of

the linear approximation was confirmed by the experimental results. The concepts

of the presented input terminal small signal modelling may be helpful for further

investigations to be conducted related to the renewable energy applications such as

wind energy and ocean wave energy applications. To improve the overall

performance of the MPPT system, the efforts of noise filtering and noise

suppression were taken in this design, as well as the DCM detection mechanism.

Eventually, the designed PV system successfully implemented the conventional

P&O algorithm and the proposed algorithm using FLC.

In this thesis, a single phase PWM control for a photovoltaic boost DC-DC

converter is presented. To reduce the ripple current, two approaches are mentioned

in chapter 5. The first approach is to enlarge the input inductance, while the circuit

may become bulky. The second method is to increase the switching frequency.

However, due to the limitations of the hardware system, the switching frequency is

100 100

limited to 25 kHz. In fact, there exists another efficient strategy, the Multi-Phase

Sliding Mode Control, which can contribute to suppress the ripple current. In a

geometric sense, the AC components of inductor currents can cancel out each

other, if the phase error between each waveform is properly selected. For instance,

if a photovoltaic converter has two phases, and the duty ratio of each PWM signal

is 50% (where the corresponding inductor current waveform is symmetry), then

keeping a 180 degree phase error between the two inductor currents can cancel

away the ripple component of the input current. In other words, the photovoltaic

current will only contain a DC component, and the ripple currents can be

suppressed at both of the input terminal and output terminal of the system. For the

future research, the multi-phase sliding mode control strategy for PV systems

should be investigated.

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APPENDICES

APPENDIX A: MATLAB CODE

107 107

Functions

The following functions were developed using MATLAB to compute various

functions or logic needed to model the photovoltaics ,to linearize the small signal

model of the input terminal of a boost DC-DC converter, and to find the proposed

parameters for building the hardware system.

Implementation of the Newton Raphson Method % this function is design to solve the nonliear equation related to % the photovoltaic current and photovoltaic voltage, by applying the % Newton Method. function [Iout]=cmp_out_current1000(voltage,R_s,R_p) Ns=60; % setting the number of resistors connected in series Np=1;% setting the number of resistors connected in parallel Voc=38.9;% open-circuit voltage Isc=9.18;% short-circuit current Pmpp=260;% provided maximum power Ci=0.004e-2;%current coefficient Cv=-0.3e-2;%voltage coefficient Cp=-0.45e-2;%power coefficient q=1.6e-19;% constant parameters k=1.381e-23; Vmpp=30.7;% maximum power point voltage Impp=8.56;% maximum power point current Tstc=273.15+25; % for Celsius C= K-273.15 VT=q/k/Tstc;% simplfy the computation a=1.5;% ideal diode factor Io= Isc/(exp(Voc*VT/a/60)-1);% compute the saturation current x0=0.0001;% set the inital guess value for solve the nonlinear x1=0;% equation T=Tstc; for n=1:1:200 fnt= x0-Isc+Io*(exp(q/a/k/T*(voltage+x0*R_s*60)/60)-

1)+(voltage+x0*R_s*60)/R_p/60; fnt_dot=

1+Io*exp(q/a/k/T*(voltage+x0*R_s*60)/60)*q/a/k/T*R_s+R_s/R_p; x1=x0-fnt/fnt_dot; x0=x1; end Iout=x1;%generate the final solution end

108 108

Finding the series resistance and parallel resistance of the SW-260-mono clear clc % modelling for SW-260-mono Ns=60;% set the number of resistors connected in series Np=1;% set the number of resisitors connected in parallel Voc=38.9;% setting the open-circuit voltage Isc=9.18;% setting the short-circuit current Pmpp=260;% setting the provided maixmum power Ci=0.004e-2;%short-circuit current coefficient Cv=-0.3e-2;%open-cirucit voltage coefficient Cp=-0.45e-2;%maixmum power coefficient q=1.6e-19;% constant parameter setting k=1.381e-23; Vmpp=30.7;% maximum power point voltage Impp=8.56;% maximum power point current Tstc=273.15+25; % for Celsius C= K-273.15 VT=q/k/Tstc; % simplfy the parameter for further computation a=1.5;% ideal diode factor Io= Isc/(exp(Voc*VT/a/60)-1);% saturation current computation Rs=0.1;% setting the inital value for sereis resistor for finding the

reasonable Rp = Vmpp*(Vmpp+Impp*Rs)/(Vmpp*Isc-

Vmpp*Io*exp((Vmpp+Impp*Rs)*VT/a/60)+Vmpp*Io-Pmpp); p_db=[];% solutaion paramet Rs, Rp for Rs=0:0.0001:300; % begin calculation Rp = Vmpp*(Vmpp+Impp*Rs*60)/(Vmpp*Isc-

Vmpp*Io*exp((Vmpp+Impp*Rs*60)*VT/a/60)+Vmpp*Io-Pmpp)/60; if Rp>0 P0=0; P1=cmp_out_current(24,Rs,Rp)*24; for v=25:0.01:35; P0 = cmp_out_current(v,Rs,Rp)*v; P1=max(P0,P1); end p_db=[p_db;[Rs Rp P1]]; end end

109 109

Computation of the linear approximations clear clc close all %% System parameter L=12e-3; %inductance of the input inductor Cin=210e-6; %capacitance of the input capacitor RL=0.2; %parasitic resistance of the input inductor Vbat=26; %DC bus voltage Rc=0.8; %parasitic resistance of the input capacitor %% Transfer function parameter computation area % should be noticed, the gain of the following transfer function is % negative Rpv=75; % the lower limit for internal resisitance of the Boulder 15W num=[-Vbat*Rc*Rpv/L/(1-Rpv) Vbat*((Cin*(1-Rpv)^2+(Rc*Rpv)^2-

Rc*Rpv)/L/L/Cin/(1-Rpv)^2)]; den=[1 RL/L-(Rc*Rpv-1)/L/Cin/(1-Rpv) -(RL*Rc*Rpv-RL+Rpv*L-

RL*Rc*Cin*Rpv)/L/L/Cin/(1-Rpv)]; Converter_sys_Rmpp75=tf(num,den)/3.8;%building linear approximation

with 75 ohm %% Rpv=100 Rpv=100; num=[-Vbat*Rc*Rpv/L/(1-Rpv) Vbat*((Cin*(1-Rpv)^2+(Rc*Rpv)^2-



with 100 ohm %% Rpv=125 Rpv=125; num=[-Vbat*Rc*Rpv/L/(1-Rpv) Vbat*((Cin*(1-Rpv)^2+(Rc*Rpv)^2-



with 125 ohm %% Rpv=150 Rpv=150;% the upper limit for the internal resistance of the Boulder

15W num=[-Vbat*Rc*Rpv/L/(1-Rpv) Vbat*((Cin*(1-Rpv)^2+(Rc*Rpv)^2-



with 150 ohm %% drawing graphs figure(1) %% bode plots of the four transfer functions bode(Converter_sys_Rmpp75,'-k',Converter_sys_Rmpp100,'--

k',Converter_sys_Rmpp125,'-.k',Converter_sys_Rmpp150,':k');grid legend('Rpv=75ohm','Rpv=100ohm','Rpv=125ohm','Rpv=150ohm') figure(2) margin(Converter_sys_Rmpp75)% check the phase margin of the original

system figure(3)

110 110

Gc=tf([0.1 1.9],[1 0]);% building the PI controller Gcl_cs75=feedback(Gc*Converter_sys_Rmpp75,1);%compute the closed-loop

tf Gcl_cs100=feedback(Gc*Converter_sys_Rmpp100,1); Gcl_cs125=feedback(Gc*Converter_sys_Rmpp125,1); Gcl_cs150=feedback(Gc*Converter_sys_Rmpp150,1); t=0:0.001:0.2;% creating the time vector stepinfo(Gcl_cs75)% gethering the information of the step responses stepinfo(Gcl_cs100)% of the four closed-loop compensated system stepinfo(Gcl_cs125) stepinfo(Gcl_cs150) y75=step(t,Gcl_cs75); y100=step(t,Gcl_cs100);%draw the step response y125=step(t,Gcl_cs125); y150=step(t,Gcl_cs150); plot(t,y75,'-k',t,y100,'--k',t,y100,'-.k',t,y125,':k');grid xlabel('time(sec)')% mark the axises and title ylabel('Output') title('Step resonse of closed-loop compensated system'); legend('Rpv=75ohm','Rpv=100ohm','Rpv=125ohm','Rpv=150ohm');

Finding the proper inductor for suppressing the ripple component L1=0.5e-3;%set inductance L2=1e-3; L3=12e-3; ripple1=[];%create vectors for recording data ripple2=[]; ripple3=[]; for n=1:0.1:18 % create for-loop to automatically compute ripple1=[ripple1 0.5*n*(26-n)/26/25000/L1];% the possible amplitude of ripple2=[ripple2 0.5*n*(26-n)/26/25000/L2];% ripple component ripple3=[ripple3 0.5*n*(26-n)/26/25000/L3]; end figure(1) n=1:0.1:18; plot(n,ripple1,'k-',n,ripple2,'k-.',n,ripple3,'k:'); xlabel('Photovoltaic voltage(volts)'); ylabel('Amplitude of the ripple current(amps)'); legend('L=0.5mH','L=1mH','L=12mH');

APPENDIX B: SYSTEM SCHEMATICS

112 112

Schematics of the Hardware System

The following schematics demonstrate the main power electronic circuit,

peripheral circuits, and spots for signal measurements.

Boost DC-DC converter powered by the solar panel, Boulder 15W

Voltage sensing circuit for supporting the secondary control layer

113 113

Voltage sensing circuit for supporting the top control layer

Current sensing circuit for supporting the top control layer

Signal connections of TI F28035

114 114

PWM driving circuit

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Date

Pengyuan Chen

May 19, 2015

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AN IMPROVED MAXIMUM POWER POINT TRACKING ...

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