.Vé— . l. ‘v‘ .-
.. AMA-

.fi. u
:2.

‘ n. .o ‘ h o 5
bravunfimmm 35::
. A .
afifinfitfi «an?! .L
iii»... 3.: S.
nvvt‘dn
v...)-
..-
turd
1'...
E.

I...
3.:

‘ .‘
is.

x .u :54 as:
13—. its? .Anwltflu

Ii?
sifpzu.

 

I ; I
3.3.1:}...11
‘4I.!Ir\...

 

 

 

 

J.
S 2..
.

 

 

This is to certify that the
dissertation entitled

MODELING AND ANALYSIS OF SOLAR
DISTRIBUTED GENERATION

presented by
Eduardo Ivan Ortiz Rivera

has been accepted towards fulfillment
of the requirements for the

Ph. D. degree in Electrical Engineerigq

[FT/$771K "2

7 Major‘Professor’sSiénature
May 11, 2006

Date

MSU is an Affirmative Action/Equal Opportunity Institution

 

 

LIBRARY
Michigan State
University

 

 

 

I
|

PLACE IN RETURN BOX to remove this checkout from your record.
TO AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE DATE DUE DATE DUE

 

NOV 2 6 2008
121808

 

 

{3:31 2,7 2005i
111709

 

 

 

 

 

 

 

 

 

 

 

 

2/05 p:/ClRC/DaIeDue.indd.p.1

 

 

MODELING AND ANALYSIS OF SOLAR
DISTRIBUTED GENERATION

By

Eduardo Ivan Ortiz Rivera

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of
DOCTOR OF PHILOSOPHY
Department of Electrical and Computer Engineering

2006

ABSTRACT

MODELING AND ANALYSIS OF SOLAR
DISTRIBUTED GENERATION

By

Eduardo Ivan Ortiz Rivera

Recent changes in the global economy are creating a big impact in our daily life.
The price of oil is increasing and the number of reserves are less every day. Also, dra-
matic demographic changes are impacting the viability of the electric infrastructure
and ultimately the economic future of the industry. These are some of the reasons
that many countries are looking for alternative energy to produce electric energy. The
most common form of green energy in our daily life is solar energy. To convert solar
energy into electrical energy is required solar panels, dc-dc converters, power control,
sensors, and inverters.

In this work, a photovoltaic module, PVM, model using the electrical character-
istics provided by the manufacturer data sheet is presented for power system appli-
cations. Experimental results from testing are showed, verifying the proposed PVM
model. Also in this work, three maximum power point tracker, MPPT, algorithms
would be presented to obtain the maximum power from a PVM. The first MPPT al-
gorithm is a method based on the Rolle’s and Lagrange’s Theorems and can provide
at least an approximate answer to a family of transcendental functions that cannot
be solved using differential calculus. The second MPPT algorithm is based on the

approximation of the proposed PVM model using fractional polynomials where the

shape, boundary conditions and performance of the proposed PVM model are satis-
fied. The third MPPT algorithm is based in the determination of the optimal duty
cycle for a dc—dc converter and the previous knowledge of the load or load matching
conditions.

Also, four algorithms to calculate the effective irradiance level and temperature
over a photovoltaic module are presented in this work. The main reasons to develop
these algorithms are for monitoring climate conditions, the elimination of temperature
and solar irradiance sensors, reductions in cost for a photovoltaic inverter system,
and development of new algorithms to be integrated with maximum power point
tracking algorithms. Finally, several PV power applications will be presented like
circuit analysis for a load connected to two different PV arrays, speed control for a
dc motor connected to a PVM, and a novel single phase photovoltaic inverter system

using the Z-source converter.

Copyright © by
Eduardo Ivan Ortiz Rivera
2006

To my parents, Eduardo and Haydee,

to my sister, Enid, and to my wife, Yulia

 

ti

 

Rx;

GK 7‘

 

ACKNOWLEDGMENTS

First, all the glory to God! This dissertation would not be possible without
several contributions. It is a pleasure to thank my advisor, Professor Fang Z. Peng,
for his guidance, assistance, support, and for providing me with a pleasant working
environment. I would also like to thank Professors Steven Shaw, Hassan K. Khalil,
Elias Strangas, and Robert Schlueter for their time and effort in being part of my
committee.

I would like to extend special thanks to Professors Percy Pierre, Barbara O’Kelly,
the National Consortium for Graduate Degrees for Minorities in Engineering and Sci-
ence and the Alfred P. Sloan Foundation for providing the funding which made this
work possible and for all their support during my stay at Michigan State Univer-
sity. I am very grateful to Professor Wesley G. Zanardelli, my friend and colleague,
for his interest in my research, encouragement, friendly discussions, and excellent
suggestions.

Special words of thanks go to my lab colleagues Alan Joseph, Jin Wang, Miaosen
Shen, Fan Zhang, Yi Huang, Joel Anderson, Richard Badin, Honnyong Cha, Lihua
Chen, Irving Balaguer, Zhiguo Pan, Varun Chengalvala and Kent Holland for all of
their support. I would also like to thank the members of the Department of Electrical
Engineering and College of Engineering whose help was much appreciated, including
Roxanne Peacock, Sheryl Hulet and Professor Drew Kim. I am appreciative for the
extended use of computer resources at Michigan State University.

A special note of thanks is extended to my parents Eduardo and Haydee for

vi

it

Lin
3ft i

56w

 

their constant support, encouragement and unconditional love, without which I would
never have accomplished this goal; to my sister Enid for always believing in me;
to my lovely wife Yulia 'I‘rukhina for her support, love and care, for being a fine
example of an excellent scientist herself, making me proud of her and wanting to
live up to her standards; to my good friends Miguel Figueroa, Nelson Sepulveda, Jeff
Ahrens, Uchechukwu Wejinya, Ana Becerril, Sandra Soto, and Pedro Meléndez for
being supportive and always ready to help. I would like to thank all my other friends
for making my life in East Lansing, Michigan enjoyable and meaningful.

Finally, I would like to thank Professor Gerson Beauchamp from University of
Puerto Rico, Mayagiiez for all his support before, during and after this journey. I
would like to thank Professor Sonia Diaz for driving my curiosity. I would like to
thank Professor R. L. Tummala who prepared me during my previous degree for
this challenge. I would like to thank Professor Claudio Rivetta from the Stanford
Linear Accelerator Center for his hospitality during the summers of the 2001 and
2002 at Fermi National Accelerator Laboratory where I got a better understanding of
several aspects of my research. I would also like to express my gratitude and thanks
to Professor Raul Marrero for his guidance, and his professional as well as personal
example as a mentor during all his life at Barranquitas, Puerto Rico where I was born

and grew up.

vii

TABLE OF CONTENTS

LIST OF TABLES

LIST OF FIGURES

1 Introduction

2 Photovoltaic Module Model

2.1
2.2
2.3
2.4
2.5
2.6
2.7

Methods Proposed in the Past ......................
Typical Requirements for a PVM Data Sheet ..............
Proposed Photovoltaic Module Model ..................
Dynamic PVM model and the internal PVM capacitance .......
How to Calculate the Characteristic Constant? .............
Relationship Between the PVM Performance and the Fill Factor . . .
PVM Model Verification .........................

3 Linear Reoriented Coordinates Method

3.1
3.2
3.3

3.4
3.5
3.6

4.1
4.2
4.3

Introduction ................................
Rolle’s and Lagrange’s Theorems ....................
Linear Reoriented Coordinates Method .................
3.3.1 Description for the LRCM ....................
3.3.2 Conditions for the LRCM ....................
3.3.3 Approximation for 2:0,, and fmar .................
3.3.4 Validation for the LRCM .....................
LRCM as a MPPT Algorithm ......................
LRCM Results for a PVIS with MPPT .................
Additional Examples using the LRCM .................

Fractional Polynomial Method

Introduction ................................
Fractional Polynomial Method ......................
Integer Polynomial Approximation Method ...............

viii

xi

1

9

9
12
12
17
19
20
22

30
31
32
35
35
36
36
37
38
42
47

54
54
56
58

4.4 Examples using FPM and IPAM ..................... 62
5 Fixed Point Algorithms to Estimate T and EN over a PVM 66
5.1 Introduction ................................ 66
5.2 Algorithms to Estimate the Effective Irradiance Level and Temperature
over a PVM ................................ 68
5.3 Experimental Results using the Proposed Algorithms ......... 72
6 Proposed PV Power Applications 74
6.1 Introduction ................................ 75
6.2 LRCM and F PM applied to commercial PV modules ......... 76
6.3 PV modules connected to resistive loads ................ 80
6.4 PVM connected to a RLC Load ..................... 81
6.5 Optimal Duty Ratio for a dc-dc Converter for PV Applications . . . . 86
6.6 Algorithm and Simulations for a dc-dc Converter using Load Matching 88
6.7 PVM connected to a dc motor ...................... 91
6.8 Z-Source Converter ............................ 98
6.9 Proposed PVIS using the Z-Source Converter and Load Matching Control 99
7 Conclusions 103
7.1 Summary ................................. 103
BIBLIOGRAPHY 108

ix

 

 

 

 

 

l‘\') [0
ix.) H

 

3.1

5.1
5.2

5.3
5.4

2.1
2.2

3.1

4.1
4.2

5.1
5.2
5.3
5.4

6.1
6.2
6.3
6.4

LIST OF TABLES

Photovoltaic Module Specifications ...................
Photovoltaic Module Specifications (cont.) ...............

Comparison for LRCM Results and Optimal Values ..........

Conditions satisfied by the proposed fractional approximation method
PVM parameter approximation using the FPM under STC ......

Measured Values for Algorithm 5.1 ...................
Calculated Values using Algorithm 5.1 .................
Measured Values for Algorithm 5.2 ...................
Calculated Values using Algorithm 5.2 .................

Electrical specifications for commercial PV modules under STC

PVM parameter approximation using the LRCM under ST C .....
PVM parameter approximation using the FPM under STC ......
Optimal Duty Ratio for Different dc-dc Converters for Load Matching

24
24

52

58
63

73
73
73
73

77
78
79
88

 

 

 

1.3

2.1

2.2

 

«1630!

F0 .‘J to to N (J to
7

p—0 h—J \‘I’ ) ( )
A.

1.1
1.2

1.3

2.1

2.2

2.3

2.4

2.5
2.6
2.7
2.8
2.9
2.10
2.11

2.12

2.13

2.14

2.15
2.16

3.1

LIST OF FIGURES

Irradiance level (W/mz) at Austin, Texas during March 16, 2006 [1]..
Irradiance level (W/mz) in different regions around the state of Georgia
during April 18, 2006 [2] ..........................
Daily temperature data (°C) from morning to sunset at Barranquitas,
PR during February 16, 2005. ......................

I-V Curve for a single Photovoltaic Module (note: I :1: is I,C and V2: is
V0c under STC) ...............................
P-V Curve for a single Photovoltaic Module (note: I :1: is I,C and V2: is
V0c under STC) ...............................
R-V Curve for a single Photovoltaic Module (note: I :c is I“ and V2: is
Vac under STC) ...............................
Schematic for a dynamic PVM model including the internal capaci-
tance, Cx. .................................
Proposed method to approximate the internal resistance Ca: ......
Outputs for V(t) and VR(t) after the switch is close at time t = 0. . .
Non-Iterative method using the PVM I-V Curve to calculate b. . . . .
I-V Curves for the SX-lO and SX-5 under different temperatures.
P-V Curves for the SX-10 module under different temperatures.

R-V Curves for the SX-lO module under different temperatures. . . .
I-V Curves for the SX-lO and SX-5 modules under different effective
irradiance levels. .............................
P-V Curves for the SX-lO module under different effective irradiance
levels. ...................................
R-V Curves for the SX—lO module under different effective irradiance
levels. ...................................
Experimental measures and estimation for the SA—05 I—V Curve. . . .
Experimental measures and estimation for the SA-05 P-V Curve.
Experimental measures and estimation for the SA-05 R-V Curve.

Linear Reoriented Coordinates Method (LRCM). ...........

xi

15

16

17

18
18
19
2O
25
25
26

26

27

27
28
28
29

34

 

 

 

;.- ..- ;: lg.

 

6.5

3.2
3.3
3.4
3.5
3.6
3.7
3.8

3.9

3.10

3.11
3.12

4.1
4.2
4.3
4.4

4.5

5.1
5.2

5.3

6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10

Rolle’s Theorem. ............................. 34

Lagrange’s Theorem. ........................... 35
P-V and I-V Characteristics for different intensities of light. ..... 39
Relationship between the P-V curve, I-V curve and the LRCM. . . . 40
I-V Curves with estimated knee points .................. 43
P-V Curves with estimated maximum points for the Pm“ curve. . . . 44
Error Curve for Pm“ and Pm” estimated under different normalized

optimum voltage. ............................. 44
P-V curve showing the closeness between Pm” and Pm” approximated
by Pap .................................... 45
Maximum percentage of error curve for Pm” versus the characteristic

constant, b. ................................ 46
Profit vs # of employees contracted by the company X ......... 49
V-1 Characteristic Curve for a Fuel Cell. ................ 51
PVM connected to a RC Load. ' ..................... 61
I-V curves and the approximations for a PVM SX—5 under STC. . . . 64
Error curves between I ( V) and the approximations of I ( V) ....... 64
I-V Curves for a PVM SLK6OM6 under different cell temperature and

irradiance .................................. 65
P-V Curves for a PVM SLK6OM6 under different cell temperature and

irradiance .................................. 65
Flowchart for Algorithm 5.1 to calculate T and E,. .......... 69
Integrated PVM converter system, using a DSP Board programmed

with the Algorithm 5.1 ........................... 70
Flowchart for Algorithm 5.3 to calculate I x and V112, integrated with

Algorithm 5.2 ................................ 71
PV Inverter System for utility applications ................ 76
PVM connected to an incandescent light bulb .............. 80
Two distinct PV Arrays connected in series to a 150W load. ..... 81
PVM connected to a RLC load. ..................... 82
PVM SX-lO supplied voltage, V vs. t. ................. 83
PVM SX-lO supplied current, I vs. t ................... 84
PVM SX—10 supplied power, I - V vs. t. ................ 84
VC Load Voltage, VC vs. t ......................... 85
IL Load Current, I L vs. t. ........................ 85
IL . V0 Load Power, IL - V0 vs. t. .................... 86

xii

6.11
6.12

6.13
6.14
6.15
6.16
6.17
6.18
6.19
6.20
6.21
6.22
6.23
6.24
6.25

Algorithm to calculate the Optimal duty ratio given E, and T. . . . . 89
Integrated PV power system using load matching and the optimal duty

ration given E,- and T. .......................... 89
PVM power and voltage with respect to the time. ........... 90
Load power and voltage with respect to the time. ........... 90
PVM connected directly to a dc motor .................. 91
Results of SX-20 PV array connected to a dc motor ........... 92
PVM connected to a buck-boost converter and a dc motor. ...... 93
Algorithm to calculate the fixed duty ratio, D, using the FPT ..... 94
Expected transition between the PVM and the dc motor to produce wr. 96

Results of a PVM connected to buck-boost converter and a dc motor. 97
Z—Source configuration ........................... 99
Proposed single phase PVIS using the Z-Source converter. ...... 100
Diagram for the phase shifting control with a frequency of 50Hz. . . 101
Inverter voltage output, Vd and current output Id. .......... 102
Induction motor input voltage, V3, with a frequency of 50H 2. . . . . 102

xiii

 

 

 

Int

The \n

 

megaw;
States ;
Depart
over 1.3
the f‘lt‘
that a;
cousin
and pi;
the otl
{Ellen}
grim

in ma:

enl'lfi}?

CHAPTER 1

Introduction

The worldwide electric utility generation is estimated at over 3,000,000 installed
megawatts (MW) and is growing by more than 80,000MW per year. In the United
States alone, the electric utility generation is estimated at 722,200MW and the US.
Department of Energy (DOE) forecasts that, for the coming decade, an average of
over 15,000MW per year of new generation facilities will be added in order to supply
the electricity growth and to replace the estimated 6,000MW per year of old plants
that are expected to be retired. To solve this problem, the typical solution is the
construction of a large central power station, more transmission lines, transformers,
and poles to deliver the power to the end-user, often hundreds of miles away [3]. On
the other hand, another alternative solution for providing power has been the use of
renewable energies in distributed generation applications. The use of renewable and
green energies (i.e. solar energy, wind energy, geothermic energy, etc.) is growing
in many countries and the contribution to reduce global warming and protect the
environment is increasingly important [4, 5, 6, 7, 8, 9]. The most common of these
green energies in our daily life is solar energy.

Since the last three decades the interest to use solar energy in applications of
distributed generation are growing very fast. Applications for the solar energy are

in urban areas, motor drives, race vehicles, satellites, etc [5, 10, 11, 12, 13, 14, 15,

 

 

i511

loca

ene
USU

for

1111

16, 17, 18]. For some applications where small amounts of electricity are required,
like emergency call boxes, PV systems are often justified even when grid electricity
is not very far away. When applications require larger amounts of electricity and are
located away from existing power lines, photovoltaic systems can in many cases offer
the least expensive, most viable option [15]. Today, solar energy is considered as a real
alternative resource of energy to be used for production of electrical energy around
the world [19, 20, 21]. The key component to convert solar energy into electrical
energy is the photovoltaic module, PVM, also known as a solar panel. Sometimes the
use of photovoltaic modules (PVM’s) can be more practical than the typical solutions
for power generation.

An example of the last statement is that solar panels can supply power for the
electronic equipment aboard a satellite over a long period of time, which is a distinct
advantage over batteriae [18, 22]. Also, it is possible to obtain useful power from the
sun in terrestrial applications using solar panels, even though the atmosphere reduces
the solar intensity [20, 23, 24, 25]. Inclusive for many remote locations, the cost of
a PV generation system is less than the cost of extending the grid to that location
[4, 26].

Unfortunately in PV circuit analysis, often it is assumed that the PVM is working

under the following three assumptions [27]:
1. The environment conditions are constants.
2. The PVM is working under maximum power.

3. The PVM voltage output is constant, hence the PVM can be assumed as a

constant voltage source.

But for practical purposes, these assumptions are not always valid due to the fact
that in a regular day, the temperature, T, and the solar irradiance, Ei, levels are

changing.

 

 

 

 

figur

As
lfiadia
can b,.
the (hf
Findlij
data f
lowest
Mint
4 hon:
the p
Sliarli]

Al

ph‘\ll

Austin, Texas

 

 

 

 

1,200
‘1“:- 1,000
E 800
o
5
_I 600
0
g 400
8
_t:_ 200

0
6:00am 9:00am 12:00pm 3:00pm 6:00pm

Time

Figure 1.1. Irradiance level (W/m2) at Austin, Texas during March 16, 2006 [1].

As examples of the last statement, figure 1.1 shows how in a normal day the
irradiance level can change from OW/m2 up to 700W/m2 [1]. Also, the irradiance level
can be different in a region during the same period of time [2, 28]. Figure 1.2 shows
the different irradiance levels around the state of Georgia during an instance of time.
Finally, the temperature can change during the day; figure 1.3 shows the temperature
data from a normal day at Barranquitas, Puerto Rico, where at 5:00am (5:00), the
lowest temperature point is 16°C and at 3:00pm (15:00) the highest temperature
point is 28°C. In a typical day, the temperature can change up to 10°C in less than
4 hours. Hence, it is clear that these parameters will affect the maximum power and
the PVM output voltage. Rapid changes in the temperature and the influence of
shading will affect the maximum power supplied by the PVM [29, 30].

Also, the actual models to describe solar panel performance are more related to

physics, electronics, and semiconductors than to power systems and these models do

 

 

 

Figtu

d‘drir;

 

 

Figure 1.2. Irradiance level (W/mz) in different regions around the state of Georgia
during April 18, 2006 [2].

Temperature Data - Wednesday, February 15. 21115

 

 

 

 

28 T. 1 ‘1’ V l
E E E 0 E
25 ------------ ------ e ------------ -------
i 5 E E o
o 24 """"""" """""" 1 """""" :' """""" """"
D I I I I
2 : : : :
3 I I I I
E 22 ------------ E ----------- {-e --------- E- ----------- I- ------ o
2. : : : :
E : : : :
a) I I I I
r— 213-----.-----: ------------ : ------------ : ------------ :- -------
<> 5 E E 5
1a ------------ g -------- e-g ------------ jr ----------- g -------
o i 5 s i
15 ' e i i i
0 3T 10 15 20

Hourly trend view for Barranquitas, Puerto Rico from morning to sunset

Figure 1.3. Daily temperature data (°C) from morning to sunset at Barranquitas,
PR during February 16, 2005.

not necessarily consider the effects of the temperature and effective irradiance level
[22, 23, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44]. Some of the models
require several parameters such as the temperature coefficients, photon current, open
circuit voltage, series/ shunt resistance of the device, etc. Also some of the required
parameters in those models are not available by the manufacturer data sheets so it is
necessary to find the information in other sources. At the same time, these models
can be impractical and too complex for common tasks in power systems such as
power flow, harmonic analysis, sensitivity analysis, load matching for maximum power
transferred from the source to the load. To solve these problems and to maximize the
use of information provided by the manufacturer data sheets, Chapter 2 proposes a
photovoltaic model based on the electrical characteristics, Standard Test Conditions
(STC), and I-V Curves. This model will be more beneficial and practical for power

system analysis [45].

 

 

In
Pl. p1
Track
the IT
track'
prowl

lr

58W)
are

anal
an (I
user

100

by 1
tho}

thiS

In addition to the PVM, to convert solar energy into electrical energy, a basic
PV power generation system includes inverters, control (i.e. Maximum Point Power
Tracking) and sensors. As a fact, a PVM will Operate at the highest efficiency when
the maximum power is supplied from the PVM [46, 47]. The maximum power point
tracker, MPPT, is the typical algorithm to calculate the maximum power, Pm“,
provided by a PVM [46, 47, 48, 49].

In the past, many authors described different variations of the MPPT algorithm
[46, 47, 48, 49, 50, 51, 52, 53, 54, 55] and their applications to control dc-dc converters
in energy conversion [51, 52, 53, 54, 55, 56, 57]. Unfortunately, most of the existing
MPPT methods to estimate the maximum power are based on trial and error algo-
rithms where the voltage is increased until the maximum power is achieved, better
known as the hill—climbing method [46, 48, 49]. Other MPPT algorithms compare the
last sampled voltage and the current to the presently sampled voltage and the current
to see which state will produce the maximum power [52]. Additionally, the literature
offers other types of MPPT algorithms such that rippled based method [50], look-up
table methods, [53] and fuzzy logic [54].

Disadvantages with these MPPT algorithms are that discrete algorithms require
several iterations to calculate the optimal steady-state duty ratio [56]. Some of them
are not designed for quick changes in the weather conditions [49, 58]. Also, for non-
analytical methods, the time for the iterations will depend on the initial conditions
and can create bifurcation problems [57, 59, 60]. Additionally, the dynamic models
used to describe the interaction of solar panels, MPPT control, and converter are
too complicated, with a lot of required parameters and the models cannot produce
an analytical solution to obtain the optimal voltage and maximum power produced
by the PV inverter creating the necessity to use long and tedious iterations. Also,
these results are not very practical for straightforward power flow analysis. To solve

this problem, Chapter 3 proposes an analytical method using the proposed PVM in

 

 

 

 

 

list
for l
Sdlll‘
and
that
Valur
OYCI
appli
501111
A
nomi:
P\'.\l
tive c

P

"IQ-I"
like rr
factor
fiher
defilgr
Tl
DUIUII
dram]
the flu
Ollr (la

ap Prog-

Chain]

Chapter 2 to obtain a close approximation of the optimal voltage that will produce
the maximum power. The name of the method is the Linear Reoriented Coordinates
Method, LRCM, [61]. The LRCM is very useful as an alternative MPPT algorithm
for power and utility applications [62]. The LRCM is a simple method which uses the
same variables as the prOposed dynamic model and is a time saver for calculating V0,,
and Pm reducing the long and tedious iterations. The simulated results will show
that the proposed technique is very effective, giving a small error between the real
values and estimated values, even if the effective intensity of light is changing rapidly
over the PVM. In addition, other applications for LRCM will be shown including
applications for fuel cells, economics, space optimization for floorplan design, and
solving transcendental functions [63].

An additional MPPT algorithm is presented in Chapter 4. It is Fractional Poly-
nomial Method, FPM. The main idea of the FPM is to approximate, the proposed
PVM model given in Chapter 2, using fractional polynomials then using the deriva-
tive of the approximation of P(V) is possible to approximate analytically V0,, and
Pm”. The literature provides several examples using fractional polynomials in areas
like medicine (e.g. mathematical modeling of breast cancer [64, 65], modeling of risk
factors in epidemiology [66]), signal processing and pattern recognition (e.g. digital
filter design [67, 68, 69] and face recognition [70]), control systems (e.g. observer
design [71]), biological and agricultural models [72, 73], etc.

The main advantage of a fractional polynomial is the reduction of high order
polynomial models for curve fitting [73] and increased accuracy of the approximation
describing more adequately a physical model. Also, fractional polynomials belong to
the family of fractals. It is important to understand that the fractals can be found in
our daily life, in the nature and physical world [74, 75] doing fractional polynomial
approximation a useful tool to describe the PVM physical behavior [76]. Additionally,

Chapter 4 provides an approximation method for converting fractional polynomials

 

to the

model 1

Che
and ten
algoriil'
the 0pc-
these a
Pyrono
in the 1
WM 51
measur.
algoritl.
Additio
PTOgTari

Cha
110m Of

andap

for a P\
COHSldrxr

Chaplet

to the closest integer polynomials, keeping similar properties as the proposed PVM
model in Chapter 2. These two non-traditional methods are fully explained in Chapter
4

Chapter 5 presents several algorithms to estimate the effective irradiance level, E,,
and temperature, T, over a PVM using the PVM model proposed in Chapter 2. These
algorithms are based on the Fixed Point Theorem [77] and the online measurements of
the Open circuit voltage and short circuit current. The main purpose for developing
these algorithms is to eliminate the use of pyronometers and thermocouples [78].
Pyronometers usually are very expensive and need calibration after sudden changes
in the irradiation. The thermocouples usually are cheap but direct contact to the
PVM surface is required and several thermocouples are required to obtain an accurate
measure and Often calibration is required. From the economic point of view, these
algorithms can reduce the cost for a PV power system eliminating extra components.
Additionally, these algorithms are very accurate, easy to understand, and can be
programmed in a microprocessor or DSP board.

Chapter 6 presents the following sections: circuit analysis using PVM’s, calcula-
tions of the Optimal duty ratio to obtain Pm“. for different types of dc—dc converters
and a proposed PVM transformerless inverter using a resonant Z-source converter
[79]. It is important to understand that for each analysis the dynamic equations
for a PVM were used under variable environmental conditions. This chapter can be
considered as a tool for power systems analysis using photovoltaic modules. Finally,

Chapter 7 presents a summary Of the results and plans for future work.

CHAPTER 2

Photovoltaic Module Model

This chapter prOposes an analytical model for the performance Of photovoltaic mod-
ules to be used in distributed power generation. The proposed photovoltaic module
(PVM) model uses the electrical characteristics provided by the manufacturer data
sheet. The required characteristics are short-circuit current (In), open-circuit volt-
age (Voc) and the temperature coefficients of I,c and Vac. The proposed model takes
into consideration the nominal values provided by the manufacturer data sheet un-
der Standard Test Conditions (STC). Also, the proposed PVM model considers the
changes and effects of the temperature and the effective irradiance levels over the
PVM. Finally, simulations about V-I and P-V curves under different irradiance levels

and temperatures are provided for different solar panel modules, data sheets.

2.1 Methods Proposed in the Past

Different conversion methods have been proposed in the past, some of them work
point by point and others model the solar cell performance with analytical equations.

Some of the models and the conversion equations are made up of the following:

1) Method of Anderson [34]:

1 _ Ez-Il
2_E1+E1-TCi-(T1—T2)

 

 

V2: q'Vl
[1+TCV-(T1-T2ll' [4+k'T'1n(%)l

2) Method of Bleasser [35]:

E
1

k-T (E2
q E1

V2=V1—Rs-(I2—I1)+—-ln —) +TCV-(T2-T1)
3) IEC-891 procedure [41]:

I
12=II+ISCI-(iii—1)+TCi-(T2—T1)

V2=V1—R3-(lg—11)—K-12-(T2—T1)+TCV-(T2—T1)

4) Photovoltaic Utility Scale Application (PVUSA) model [43]:
P=E,-(a+b-E,-+c-T+d-WS)

5) Solar Cell, Semiconductor (one diode) Model [22]:

(IV
IV 21 I —I - —
I) Ph+ SR SR €XP(k.T)
6) Two-Exponential Model [37]:
V+I°Rs
I(V) = Iph+151—151°8Xp(————)
771°VTh

10

(2-1)

(2.2)

(2.3)

(2.4)

(2.5)

(2.6)

(2.7)

 

 

f‘n

Hid

um.

I' s ' s
V+ R)_V+IR (2.9)

I —I -
+ 52 52 9XP( 772 . VTh RP

The variables P, I and V are the photovoltaic module output power, current and
voltage. Index 1 indicates measured data, index 2 labels the new temperature T
and irradiation E,- and the conversion results for I and V [38]. Iph, 1501 and 153
are photocurrent, saturation and reverse saturation current. VT}, is the thermical
voltage. Rs, RP are the series/shunt resistance. k is Boltzmann constant, 1.38 x
10‘23J/Kelvin. TCi is the temperature coefficient of I“, (A/°C). T CV is the
temperature coeflicient of VOC, (V/°C). K is the curve correction factor. n is the
ideality factor. a, b, c and d are regression coefficients. q is the charge Of electron, 1.6 x
10‘19As. WS is the wind speed, (m/s). The first three methods are working point
by point [34, 35, 41], the PVUSA method is based on continuous data collection and
regression model [43] and the last two methods are analytical equations to describe
the performance of a solar cell, [22, 37].

All the given methods require several parameters that can be Obtained from the
manufacturer data sheet, such as the temperature coefficients, short circuit current
under STC, and open circuit voltage under STC, etc. Unfortunately, some of the re-
quired parameters for these models cannot be found in the manufacturer data sheet,
such that the photon current, the series/ shunt resistance, thermal voltage, the ideal-
ity factor, the diode reverse saturation current, Boltzmann’s constant, band gap for
the material, etc. Also, the IEC-891 uses a fourth parameter curve correction factor
K. The Two-Exponential model requires a curve-fitting and simulation computer
program. Limitations with the PVUSA are: poor model performance at low irradi-
ance, and requires sufficient data to calculate the regression coefficients a, b, c and
d to perform the curve fitting. Also, three the methods consider the PVM internal
resistance as a constant value where in reality it is not true due the changes in volt—

age of operation. Additionally, the literature offers other complex methods Of PVM

11

modeling using fuzzy logic, look-up tables or learning algorithms more suitable for
physics but not necessary for power applications [36, 39]. In general, these models
of the PV panels required additional parameter data not given by the manufacturer
data sheets; it is difficult to Obtain the needed data, and most of them are static

models not designed for distributed power applications analysis [45].

2.2 Typical Requirements for a PVM Data Sheet

The Underwriters Laboratories has developed a sample of information requirements
for photovoltaic modules [80]. A photovoltaic module datasheet should include the
ratings Of the short circuit current (I 3c), Open circuit voltage (Vac), Optimal voltage
(V0?) and Optimal current (10p) of an individual module when Operating at maximum
power (Pmax). These ratings are to be based upon Standard Test Conditions (STC).
The STC (also known as SRC or Standard Reporting Conditions) is defined with
nominal cell temperature 25°C, nominal irradiance level 1000W/m2 at spectral dis-
tribution of Air Mass 1.5 solar spectral content. Additionally, the ratings are tested
at Nominal Operating Cell Temperature (N OCT). NOCT is defined as 20°C ambient
air temperature, 800W/m2 irradiance with a 1m / 3 wind across the module from side
to side. Finally, it is the purpose of the next section to propose to the reader a pho-
tovoltaic module model more suitable and useful for power system applications using

the valuable information required from the manufactures.

2.3 Proposed Photovoltaic Module Model

The proposed model for the photovoltaic module (PVM) takes into consideration the
relationship of the current with respect to the voltage, effective irradiance level, E,-
and temperature, T of operation for the PVM, the characteristic constant for the

I-V curves, the short—circuit current and the Open—circuit voltage [45]. The prOposed

12

PVM model is described by (2.10), (2.11), (2.12) and (2.13). The main advantage of
the proposed PVM model is that for any photovoltaic module, it can be described
in terms of the values provided by the manufacturer data sheet and the standard
test conditions [45]. Also, the proposed PVM model is continuous and differentiable

with respect to the voltage giving a unique relationship between voltage, current and

 

 

 

 

power.
Ix V 1
I V = - — —— 2.
( ) 1 — exp (:bl) [1 exp (b-Vx b)] ( 10)
E1
V17 = 8' 'TCV'(T—TN)+3'Vma:r
EiN
Ei Vmax "’ Vac
"—3 ° (Vmaa: '7 Vmin) ' 81p (E:— ' In (Vmaz _ Vmin)) (2'11)
E1 .
Ixzp-T-[ISC+TC2~(T—TN)] (2.12)
iN

The power produced by a PVM is described in (2.13) and is calculated multiplying
(2.10) by the voltage, V.

 

P(V) = 1 luff?) - [1 — exp (2)va — 3] (2.13)

The variables P, I and V are the photovoltaic module output power, current and
voltage. In is the short-circuit current at 25°C and 1000W/m2. V0c is the Open-circuit
voltage at 25°C and 1000W/m2. Vmal. is the Open-circuit voltage at 25°C and more
than 1, 200W/m2, (usually, Vmam is close to 1.03 - Vac). Vmin is the open-circuit voltage
at 25°C and less than 2OOW/m2, (usually, Vmin is close to 0.85 - Vac). T is the solar
panel temperature in °C. E,- is the effective solar irradiation in W/m2. A PVM is
tested under Standard Test Conditions (STC) when the nominal temperature, T N,
is 25°C and the nominal effective solar irradiation, Em, is 1, OOOW/mz. TCi is the

temperature coefficient Of 13,; in A/°C. TCV is the temperature coefficient of VOC in

13

V/°C. Sometimes the manufacturer provides TCV in terms Of (mV/°C) just divide
TCV by 1000 to convert in terms of (V/°C). b is the characteristic constant for the
PVM based on the I-V Curve. The variable 3 is the number of PVM’s with the same
electrical characteristics connected in series and p is the number of PVM’s with the
same electrical characteristics connected in parallel as a note for a single PVM, s and
p are 1. I a: is the short circuit current at any given E,- and T, and it can be calculated
from (2.12) when the voltage, V is zero. Va: is the open circuit voltage at any given
E,- and T, also Vx is the voltage of Operation for the PVM when the current, I is zero
(2.11). The range Of existence Of V will be from 0 to V3: and the range of existence
of I(V) will be from 0 to Ix.

The maximum power, Pm”, produced by a PVM when the PVM is Operating at
the optimal voltage, Vop, is given by (2.14). Chapter 3 and Chapter 4 will be shown
the uniqueness and existence Of the V0,, and how to approximate V0p and Pm” using

several nontraditional methods.

Vop-Ia:

Pm = P(Vop) = v,,, . I(Vop) = 1 _ exp (‘71) . [1 — exp (3%; — 21.)] (2.14)

 

Finally, the PVM internal resistance, Ri, or conductance, Gi, can be calculated
from the prOposed PVM model as given in (2.15). The optimal internal resistance,
Rop, is given by (2.16). Typically, the batteries have an internal resistance between
0.29 to 0.79 [81] and a short circuit could be very dangerous for the battery. Instead,
the PVM internal resistance is much larger and the value depends on the voltage and
power drawn from the PVM. Additionally, a PVM is a current limited system hence
can be short-circuited without damage at difference Of the batteries. Also, if a PVM
connected to a resistive load, R, is operating at V0,, then Ri is equal to the Optimal
resistance Rop, hence if Rop is equal to R then Pm” is transferred to the load, but if

Rap and R are different then the transferred power will be less than Pm“. The value

14

 

 

 

figure ;
under 8'

The I

2-1 slum.

IQPII’SETT?

the PT 011"
panel is
Port-er Is 1

and [lie r.

 

 

PVM I-V Curve

 

 

 

 

 

 

ojtlx—I—fi Knee Point
06 10p ’ Ii 1 .1; i1 I I (VOP, IOP).
s at g, y ,. 1
A - at . ‘
$0.5 .‘ ‘3""‘IH’$‘ -
“' e Ir 4", a e If ’-
c e v * .’ -.#fi,‘ .. '3‘ I.
20.4”]; .d, or _"f"
E 0.3 :ZW» yogic” 4'
° 0-2»; ¢_§*V'Vaw'.'r.sr.l l
., ‘ g '0‘» “I“:
01W. ._.'..** .11 ‘ . ‘
. [It a. 0, q ., 1* ”fig
0 “15* ‘1 *.-"..11=-¥ Xx. VOp Vx
0 5 1o 15 20 25
VoItagaM

Figure 2.1. I—V Curve for a single Photovoltaic Module (note: I a: is I” and VI is Vt,C
under STC).

of Rep will be very useful for PVM systems using load matching applications.

 

 

. 1 V V—V-exp(‘—1)
R =—=—=——° 2.15
2 Ci I(V) Ix—Ix-ezp(fi—%) ( )
Rop=L=i= V031 = V°P_V°P'e°p(Tl) (2.16)
GOP I(VOP) Pm” Ix—Ix-exp(;‘%—%)

The proposed PVM model can show the effects of T and E,- over a PVM. Figure
2.1 shows the I-V characteristics of an illuminated solar panel. The shaded rectangle
represents the maximum power obtained by the solar panel. The knee point is when
the product of the current and the voltage is the maximum power [45]. The solar
panel is working in the optimal current (10p) and voltage (Vop), hence the maximum
power is delivered to the load by the solar panel. Figure 2.2 shows the P-V Curve

and the relationship between Pm” and the knee point.

15

PVM P-V Curve

 

(Pmax
10" \ ‘

 

 

 

 

 

V°P Vx
1/ .

0 5 1 0 1 5 20 25
Voltage(\/)

 

Figure 2.2. P-V Curve for a single Photovoltaic Module (note: I :r is I“ and V2: is
Vac under STC).

Figure 2.3 shows the relationship between the internal impedance and voltage of
Operation for the PVM. The Optimal internal impedance, Rap, has a direct relationship
with the maximum power and is unique. If a resistive load with the same value as the
Optimal internal impedance is connected to a photovoltaic module then the maximum
power is transferred. It is important to note that figure 2.3 can be used to maximize
the efficiency Of a solar power system when load matching is required. Figure 2.3
shows that the resistance is quasi-linear up to the point that the optimal resistance

to produce the maximum power is obtained.

16

 

figur
I¢~u

2.4

TheP'
Capaph
ITIOtlQI
Capaeh
AUG
“lure 7
”0,0ng
knmnlr

Pl’llaIi

 

PVM Ri-V Curves

OU'I

 

names.
0'1

010
T
50
o
'o

 

 

 

 

’u?
E
.C
3
E”
c"
U
5 [ ‘
.12 20
In
3315- /
E10-
0 5- .1
*-' Vop
C
— O I I I j/ l
0 5 10 15 20 25

Voltage(\/)

Figure 2.3. R—V Curve for a single Photovoltaic Module (note: I :r is IsC and Va: is
Vac under STC).

2.4 Dynamic PVM model and the internal PVM
capacitance

The PVM dynamic model is based on the PVM static model and considers the internal
capacitance for the PVM, Czr. Figure 2.4 shows the schematic for the dynamic PVM
model and it is modeled by the differential equation given in (2.17). The internal
capacitance is measured using a capacitance meter.

Another way to measure the internal capacitance is using the time constant T
where 7' is the product of Ca: and a known resistive load, R. Figure 2.5 shows the
proposed method to approximate the internal capacitance connecting the PVM tO a
known resistive load, R. Figure 2.6 shows the measurements of the voltage in the

PVM and the resistive load using an oscilloscope then the required time to Obtain

17

 

 

 

 

 

 

 

 

 

 

 

:" Ito “~. 11

: -—> E ——> '1'

I P‘M I F {It} .
3 l 5 ‘I" IO
5% V —— 1"»-

: CB: 5 "“1? A
E s by p — ; """' 5: D
\pynamic PVM Schematic ,x'

Figure 2.4. Schematic for a dynamic PVM model including the internal capacitance,
Cx.

.............................
\

 

 

 

 

 

 

 

 

 

 

 

 

;' I(V) ": II

I PVM I ‘ =0 ‘

a + 5+ L t j
3% V :: 1 V1' [3 E] R
sbyp — at i— [”19 ”(U—I

Dynamic PVM Schematic}.

Figure 2.5. Proposed method to approximate the internal resistance Cat.

around 0.6321 of the final voltage, V0 is calculated. The calculated time will be
known as the time constant, 7‘. Finally to approximate the C113, just divide r by R.
The typical value for C2: is on the range of 1nF to 10nF but this value can vary

depending the type Of PVM.

93—wJe-Ix'expg—‘gg—a _£
at - 0:17 — Cx-C$°€$p(_T1) Cl:

 

(2.17)

18

 

 

Fign

2.5

Now an

 

perform
Ship wit
the fill it
for any

algorith:
single P‘
Ellen in
Another
Shows 1],.

0.6231 1.1

 

re ..
Vac

V0

 

 

 

 

Figure 2.6. Outputs for V(t) and VR(t) after the switch is close at time t = 0.

2.5 How to Calculate the Characteristic Con-
stant?

Now arise the questions, how the characteristic constant, b, is related to the PVM
performance and how to calculate it? The PVM performance has an inverse relation-
ship with the characteristic constant, b, where the smaller the b the greater will be
the fill factor and the produced power for the PVM [45]. The characteristic constant
for any PVM is positive definite with a typical range for b from 0.01 to 0.18. An
algorithm based on the Fixed Point Theorem, and the electrical characteristics for a
single PVM (i.e. p and s are 1) under STC, is used to calculate b. The algorithm is
given in (2.18). The variable 5 is the maximum allowed error to stop the iteration.
Another way to approximate b without using iterations is given in (2.19). Figure 2.7
shows that V}, is the voltage of operation that will produce the current 1,, which it is

0.6234 by 186.

UJhilelbn+1 — an > 5

V —' oc
11.1.1 = °” V (2.18)

Vm-ln[1-i$°(1-6$P(3—3))l

 

19

PVM I-V Curve

 

 

    

 

 

 

0.7 K
0'6 " lsc T = 25 °C .
o 5__ b = 0.064978 q
A ' Ei = 1,000 Wlm2
§0.4r~‘\' ---------------------------------------- | -
5 lb = Isc-lsc.exp(-1) :
g 0.3- .
O
0.2 r .
0.1 h Vb [l/Vocd
0 + . . \1.
0 5 10 15 20 25
Voltage (V)

Figure 2.7. Non-Iterative method using the PVM I-V Curve to calculate b.

Isc

I,c . (1 — exp(—l)) = 1_ ”PI—Tl) ' [1_ exp (bl/I27; _ 3)]

 

 

2.6 Relationship Between the PVM Performance

and the Fill Factor

The fill factor, (2.20) is a figure Of merit for solar panel design [22]. It is defined as
the rectangular area covered by Pm“ (i.e. 10,, multiplied by Vop) divided by the total
rectangular area produced by I,C and Vac. Using figure 2.2, the inequality (2.21) can

be found for a single PVM where the maximum power will be less than the area Of

20

the V-I Curve and more than a quarter Of the product of I 8c and V0c [62].

 

Pmaa: '

v0c
I,C - V0C >/ I(V)dV > Pm” > i - Isa - V0C (2.21)
0

Before to prove the inequality (2.21), consider the limits for I (V) when b tends to
0 and 00 as presented on (2.22) and (2.23). Now, it is trivial to prove the upper part

of the inequality (2.21) and it can be done by inspection.

 

 

 

1 — exp .V
lim 1., — 1,. . (b?) = I,c (2.22)
b—*" 1 — exp (3)
1 - exp (L)
lim 1,. — 13.. b ‘1’° = 1,. — 1,, - I— (2.23)
1.100 1 — exp (3) Vac

The first part Of the upper inequality is the maximum power for an ideal PVM
when b is equal to 0. This is the ideal maximum power for a PVM with fill factor
equal to one and it can be seen as the maximum rectangular area that can be obtained
between I,c and Vac. The second part in the upper inequality is the integral of (2.10)
evaluated from 0 to V0c under STC hence the total area under the curve of I (V) will

be always more than any rectangular area inside of the curve I (V)

 

 

Voc Vac 1— exp .__‘/_. _ l — . :1
/ I(V)dV =/ 130- (mi, b) dV = 1,C.VOC.1 5+ b ex[31(1)
0 0 1‘ 6331’ (T) 1 - 6X13 (7)
(2.24)

TO prove the lower part of (2.21) consider IL(V). I L(V) is the limit of (2.10)
when b tends to 00 under STC as prove on (2.23). I L(V) is a straight line equation
where any value in I L(V), without include the boundaries i.e. V E (0 Vac) under

STC, will be always less than any value produced by I (V) The inequality for the

21

last statement is given in (2.25).

1 — exp (7“; ) V
Isc — 3c ' _ = -
1 _ exp (i) > I Vac IL(V) (2 25)

 

I(V) = I,c —— 13¢-

The maximum power produced for an ideal PVM modeled by IL(V), i.e. b equal
to 00, is calculated using differential calculus and is shown in (2.26). Now, it is clear
that the maximum power produced by I L(V) will be always less than any maximum

power produced by I (V) more than that the inequality (2.21) is satisfied.

 

v2
P(V) : V'IL(V)=Isc’V—ISC.V
3P V
=> EV-Isc—Z.ISC.-l/:—O
l/oc Isc'l/oc

Additionally after prove (2.21), the inequality for the fill factor is given in (2.27).
Finally, it is proved that the fill factor is more than one quarter and less than the
total area inside of the I—V Curve divided by the short circuit current, Isa, and open

circuit voltage, Vac.

 

V“ I(V)dV 1
1 ' _ _
>/0 Isc'Voc > fillfactor > 4 (2 27)

2.7 PVM Model Verification

The proposed model was tested using different manufacturer data sheets. Tables
2.1 and 2.2 shows the electrical characteristics for the SX-10, SX—5 and other PVM
products. Figures 2.8-2.13 show different simulations for the SX—lO module using the
information provided by the manufacturer SOLAREX. Figures 2.8—2.10 show simu-
lation results for the photovoltaic module under different temperatures of Operation

(i.e. 0°C, 25°C, 50°C and 75°C) with the irradiation level at 1000W/m2. Figures

22

2.11-2.13 Show the simulation results for photovoltaic module SX-lO with the temper-
ature at 25°C and the effective irradiance level changing (i.e. 200W/m2, 400W/m2,
600W/m2, 800W/m2, and 1000W/m2). The effects of change in the irradiance level
are more drastically visible than the effects of temperature over the solar panel. The
changes in temperature can be used tO determine now the photovoltaic modules will
Operate in tropical areas versus non-tropical areas.

Typically, a PVM datasheet includes the I-V curves under changes in the tempera-
ture. Figure 2.8 shows the simulation for the I-V Curves under different temperatures.
The simulated I-V curves are similar to the I-V curves provided by the manufacturer
SOLAREX SX—lO and SX—5. At the same time other plots, not provided by the
manufacturer, can be calculated using the proposed model such as P-V curves, R-V
curves and LP curves. Unfortunately, the manufacturer data sheet does not provide
these figures despite the fact that this information is very important for solar power
systems where the irradiance level changes quickly. An example is when the clouds
are hiding the sun for a period of time, and then the irradiance level increases and
the temperature remains constant. Figure 2.9 shows how the temperature can affect
the maximum power supplied by the photovoltaic module under a constant irradi-
ance level. Figures 2.10 and 2.13 show how the internal resistance Of the photovoltaic
module SX—lO changes when the output voltage changes.

As a final test for verification of the proposed PVM model, the experimental mea-
sures for the voltage and current for the solar panels BP SOLAREX SA-05 [82] were
done at Lansing, Michigan (May 10, 2005) with T = 25C and the sun irradiating
at the maximum intensity light (1:30pm). The test shows how accurate is the pro-
posed PVM model comparing between the measured and estimated data. Figures
2.14, 2.15 and 2.16 show the direct relationship between the experimental measures
and estimation for any of the cases related to the I-V, P-V and R—V curves of the

PVM SA-05. Finally, all Of these curves, equations, and relationships give valuable

23

Table 2.1.

Photovoltaic Module Specifications

 

Datasheet

18C

Voc

10p

V01)

b

 

Siemens SP75
Shell SQ80
SLK60M6

Solarex SA—5

Solarex SX-5

Solarex SX—lO

4.80A
4.85A
7.52A
0.38/1
0.30A
0.65A

21.7V
21.8V
37.2V
25.0V
20.5V
21.0V

4.4OA
4.58A
6.86A
0.34A
0.27A
0.59A

17.0V
17.5V
30.6V
15.0V
16.5V
16.8V

0.08717
0.06829
0.07292
0.13900
0.08474
0.08394

 

Table 2.2. Photovoltaic Module Specifications (cont.)

 

 

Datasheet TCi TCV me V"m
Siemens SP75 2.06mA/°C —77mV/°C 18.45V 22.243V

Shell SQ8O 1.4mA/°C —81mV/°C 20.25V 21.810V

SLK60M6 2.2mA/°C —127mV/°C 32.55V 37.312V
Solarex SA-5 0.3mA/°C —60'mV/°C 21.00V 25.500V
Solarex SX-5 0.2mA/°C —80mV/°C 17.43V 21.115V
Solarex SX-10 0.2mA/°C —80mV/°C 17.85V 21.630V

 

information to be considered for photovoltaic power systems and distributed power

generation design.

24

SX-10 and SX—5 l-V Curve

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0]
05
T=75C
.~°5 T=50c
$04 T=2sc
a - T=OC
2
503
O
02 T=75c
T=50c
0.1 T=25C
T=OC
0
0 5

 

Figure 2.8. I-V Curves for the SX-lO and SX-5 under different temperatures.

SX-1O P-V Curves

 

 

 

 

 

 

12
T=0C
10
T=25c _
g3 T=5oc ‘
“ 6
g T-75c
4 _
2
o

0 5 1 0 1 5 20 25
Voltage (V)

Figure 2.9. P-V Curves for the SX—lO module under different temperatures.

25

SX—1 0 R-V Curves

 

 

 

 

 

 

 

 

 

 

 

A120
3
5,100
8 80 T=0C I;
= ll
g T=2sc
'a 60 l
o T=50C J/
n:
a 40
E T=75C
g 20

0 e

0 5 10 15 20 25

Voltage (V)

Figure 2.10. R—V Curves for the SX-lO module under different temperatures.

 

 

 

 

 

 

 

 

 

SX-1 0 I-V Curve
0.7
0-6 1,000 wrm2 \.
0.5
2 800 Wlm2
I: 0.4
§ 600 wrm2
s 0.3
O
0.2 400 Wlm2
0-1 200 WIm2——\
0

 

 

 

0 5 1 0 1 5 20 25
Voltage (V)

Figure 2.11. I-V Curves for the SX—lO and SX—5 modules under different effective
irradiance levels.

26

SX—1 0 P-V Curves

 

 

 

 

 

 

 

12
10 1,000wrm2 e
:8
5' 6
E
4
2
0 200WIm2—>\ ]
0 5 10 15 20 25

Voltage (V)

Figure 2.12. P—V Curves for the SX-lO module under different effective irradiance
levels.

 

 

 

 

 

      
   
  

SX-10 R-V Curves
A 120 /
3
.5 100
v 200 wrm2
§ 80 400 wrm2 -
‘3 / 7
'fi 60 600 wrm2
m
E
.3 20 800 Wlm2
"' 0 <———1 .000 wrm2

 

 

 

0 5 1 0 15 20 25
Voltage (V)

Figure 2.13. R—V Curves for the SX-lO module under different effective irradiance
levels.

27

l-V Curve

 

 

 

 

 

 

 

 

 

 

0.30 [x be
0P o
0.25
g 0.20
g 0 Measured I(V)
5 0-15 — Estimated I(V)
0.10
0.05
0 VOP VX
0 5 1O 15 20 25
Voltage (V)

Figure 2.14. Experimental measures and estimation for the SA-O5 I—V Curve.

   

 

 

  
  

 

 
 

 

 

 

 

 

 

 

5.0 P-V Curves
Pmax
4.5
4.0
3.5
g 3.0
g 2.5
a, 2.0
1.5
1.0 0 Measured P(V)
—- Estimated P(V)
0.5
0 0/ Vop VX
0 5 10 15 20 25
Voltage (V)

Figure 2.15. Experimental measures and estimation for the SA-05 P-V Curve.

28

R-V Curve

 

 

 

 

 

 

 

 

 

 

 

200
1 80
1 60
0
E 140 0 Measured RN)
.3 120 — Estimated RN)
5? 100
g 80
o
E 60 Rop
40
20
0 Vop Vx
0 5 10 1 5 20 25
Voltage (V)

Figure 2.16. Experimental measures and estimation for the SA-05 R-V Curve.

29

 

 

 

 

COIL-

fling

CHAPTER 3

Linear Reoriented Coordinates

Method

This chapter presents a non-traditional method and algorithm tO calculate the in-
verse solution for a one-dimensional function without the diffeomorphism property.
The proposed method is called the Linear Reoriented Coordinates Method (LRCM).
The LRCM is a very powerful and useful too to calculate the symbolic solutions for
transcendental functions where the inverse function is not possible to calculate using
other traditional methods and only analytic solutions can be calculated but symbolic
solutions are not possible to obtain. The description and conditions for the appli-
cation of the method are presented in the chapter. The main application presented
in the chapter will be to determine the maximum power for a photovoltaic module
(PVM) using the proposed PVM given in the Chapter 2. Additional examples and
simulations for the LRCM related to maximum profit and revenue for a company,
fuel cells and to optimize the maximum rectangular area for a floorplan for an 8-bit
A/ D converter given space constraints, are presented. Finally, the LRCM should be
consider as a method that can provide at least an approximate answer to a family Of

functions that cannot be solved using differential calculus.

30

3. 1 Introduction

For the last several centuries, the solution for transcendental functions has been a
challenge for physics, engineers and mathematicians. A transcendental function is
defined as function which does not satisfy a polynomial equation, whose coefficients
are polynomials themselves, (i.e. F(:r) = aux" + + mm + amt/a,- E if? ). Some
examples for transcendental functions are exponential functions, logarithmic func-
tions, and trigonometric functions [83]. The most useful transcendental functions for
science are exponential functions. They have an incredible number of applications,
but it is not always possible to solve them symbolically. Examples for modeling with
transcendental functions are in RLC circuits [84], fuel cells [85], photovoltaic modules
[45], maximum area for space Optimization given shape constraints [86], [87], [88],
neural networks [89], robotics [90], etc.

Unfortunately, the only way to solve them it is numerically, sometimes with long
and tedious iterations and the use of computers with complex algorithms [90], [91],
[92], [93]. Now, for any kind of function, the traditional and effective way to cal-
culate the maximum or minimum values is using differential calculus. But in many
cases in physical sciences, engineering or math when it is required modeling using
transcendental functions are very complex to work with them.

If a function y = f (x) has the diffeomorphism property then it is possible to
Obtain the maximum value ymax. It is determined when the first derivative of f (2:)
is calculated with respect to 2:, then the function f’ (2:) = 0 is solved with respect
to a: to find the Optimal :1: and gm“. Diffeomorphism is defined as a map between
manifolds which is differentiable and has differentiable inverse. In other words, for a
one-dimensional system, it is a change of coordinates that does not change information
given by the original system [94]. A function f (2:) has the diffeomorphism property
if it is smooth, it has an inverse and the inverse is smooth. If a function has the

diffeomorphism property, then it is possible to find the inverse for the given function.

31

The inverse function is defined as follows.

If f : X —+ Y is 1 -— 1 and onto then the correspondence that goes backwards from
Y to X is also a function and is called f inverse, denoted f‘l. This map is easily
described by f‘1 : Y -—> X and f‘1(y) = 2: if and only if y = f (2:). This relationship
is easy to remember for a real function since switching coordinates of a point in the
plane puts us at the reflection of the original point about the line y = 2:. Thus the
graph of f ’1 must be the reflection of the graph of f about the line y = x. This is a
great help if the graph of f is already known. It’s the 1 - 1 condition that is really
critical for constructing an inverse function. If f is 1 — 1 but not onto we can simply
replace the codomain with the range f (X) so that f : X —> f (X) in then 1 — 1 and
onto so we can talk about an inverse f ‘1 : f (X) —+ X. The domain of the function is
equal to the range of the inverse and the range of the function is equal to the domain
Of the inverse. Finally, a unique inverse only will exist in 1— 1 functions or the unique
inverse will exists only over the restricted domain [83].

Unfortunately, it is not always possible to find the symbolic inverse for a given
function, a: = f‘1(y), [95]. But then the question arises, is it at least possible to
approximate the inverse of one-dimensional function and how good it is this approxi-
mation? To answer these questions, this paper proposes a non-traditional method to
approximate the symbolic inverse for one-dimension transcendental functions. Also,
the paper provides the different conditions where the method can be applied and

which type of functions can be satisfied.

3.2 Rolle’s and Lagrange’s Theorems

The main idea for the LRCM is based in the Rolle’s and Lagrange’s Theorems (Mean
Value Theorem or Fundamental Theorem Calculus) as shown in figure 3.1; and it is

valid in any domain [a b] but first we need to understand if it is possible to approx-

32

imate the inverse of a one-dimensional functiOn. The Lagrange Inversion Theorem
(LIT) [83] determines the Taylor series expansion of the inverse function of analytic
function. Consider the function, y = f (2:), where if f is analytic at a point :00 and
f’ (20) 79 0. Then it is possible to invert or solve the equation for y, :c = f "1 (y) = h(y)
where h is analytic at the point yo = f (000). The reversion of series is given by the

series expansion of h(y) in (3.1).

 

 

(3.1)

 

00 '1 k—l k
My) = 2:0 + Z (I! Isl/0) . tick-1 ((;:$;f20)k) rm
This equation will give the inverse function h(y), but unfortunately it is required
to do long calculations. Depending the type of functions (or the use of computers),
the result most of the time will be an infinite series polynomial (Taylor series). In
the case of transcendental functions, it will be required to take into consideration
the restrictions on the domain making it difficult to calculate the inverse. But how
can these problems be solved and how can an approximate inverse function be found
without the use of Taylor series, long iterations and be a good approximation? The
Linear Reoriented Coordinates Method (LRCM) can be a solution for these problems
for at least a family of functions!
Theorem 3.1 (Rolle’s Theorem, Fig. 3.2). If f (3:) is differentiable on (a, b),
continuous on [a, b] and f (a) = f (b), then 3 c-value in (a, b) such that f’(c) = 0.
Corollary 3.1 (Modified Rolle’s Th.). If for f (2:) El! maximum value fmax then
3! :r(f’(:r:op) = O) in 52 x [0 22mm].
Theorem 3.2 (Lagrange’s Theorem, Fig. 3.3). If g is continuous and differentiable
on [a, b], then 3 c-value in [a, b] such that, g’(c) = (g(b) — g(a))/(b — a).
Corollary 3.2. If f (2:) = x - g(x) and f (map) = 2:0,, - g(xop) = fmax then g’ (map) =
‘9 ($012” 5501)-

Theorem 3.3 (Cauchy Mean Value Theorem). If g and f are continuous and

33

 

 

 

 

 

 

 

 

0.8 [K ........ I
0'6 " f(Xop) , [fmax
3 0.4 1 :
T 0.2 - =
O L I . xop] xmax
1.0 [Kan-«Juan.---.'-_-._,,,___' .......... I
. ' f]
.. 9°21) -9<xop>=xop a 1x...) 1 ‘ *
X .
‘5 05
0 I l . XOP] Xmait
0 0.2 0.4 0.6 0.8 1

X

Figure 3.1. Linear Reoriented Coordinates Method (LRCM).

 

 

 

 

0.7 . 1

06 I 1 1‘10)

0.5 . max ------------- 5 ........... J
8 0'4 11323.1(?) _____________________________
"- 0.3 - E

0.2 r E

0.1 .

0 a. is . b

Figure 3.2. Rolle’s Theorem.

34

 

 

1,0 ............ w T s
I

 

 

 

0 8 9(3)“-.-"1 _______ _ _______ 9'(6)=9L'(<=)
' 'gtc)
A 0.6- 9er) ,= mL-x +bL ;.
5 g(b) E
°’ 0.4. """""""""""""""" 1 """""
0.2- .
0 a: 1 9' ,bixmae
o 02 04 06 08 1
X

Figure 3.3. Lagrange’s Theorem.

differentiable on [a, b], then c—value in [a, b] such that, f’ (c)/g’ (c) = (f(b) —
f (a)) / (g(b) - g(a))). The proofs for each theorem and corollary are well known and

are skipped in the paper.

3.3 Linear Reoriented Coordinates Method

3.3.1 Description for the LRCM

The LRCM is a method to find the approximate maximum value for a function
f (2:), where f’ (2:) = r(:r:) = O, which cannot be solved using traditional methods
of differential calculus, [62]. The LRCM can also be seen as a method to find the
approximate symbolic solution :1: for the equation r(:r:) = 0 without symbolic solutions.
The function f (2:) is defined as f (2:) = 2: - g(x) and the maximum value of f (2:) is
defined as fm where fmax = 2:0,, - g(zop) and 2:0,, is the Optimal value for fm.
The main idea for the LRCM is to find the Optimal points to calculate fm. These

points are (zap, g(rrop)) and are calculated using g’ (z) and the linear slope ml of g(rr)

35

evaluated at the point rap.

3.3.2 Conditions for the LRCM

The necessary conditions for the application of the LRCM to calculate the maximum

value fmam and the approximate optimal :0, map for a function f (2:), are:
1. f(x) = :1: -g(:1:) in 3? x [0 23mm]
2. f e 010R x [0 xmael)
3. g 6 01(5)? x [0 xmeel)
4. g’(a:) < 0 in at x [0 zmae]
5. g”(:r) S 0 in if? x [0 50mm]
6. Corollary 1 is satisfied in {:0 6 ER x [O xmax]}
7- g’(=rop) = -9($op)/$op

8° fmax = mop ' g(xOP)

3.3.3 Approximation for map and fmax

Now, consider a function, f (2:), that satisfies the conditions for the LRCM hence it
is desire to approximate map. The first step is tO use the straight line given by (3.2)
where gl(:r) is always positive in {2: E 3? I [0 :L‘maxl}. Thederivative of gl(:r) with
respect to x is always negative and unique in {2: E if? I [0 :cmax]}. The derivatives
of gl(z) and g(x) can be intersected in the point rap where it is the Optimal point
500,, plus an error, e, as given in (3.3). For an small 5, the optimal value for 2:0,, is

approximated by (3.4), if e is 0 then (3.4) is the solution for map.

gl(:r) = bl + ml - :r = 9(0) — w a (3.2)

xmax

36

are) = m2 = — 9‘0) = ere...) = 911230;» + e) (3.3)

xmax

 

2:0,, z $0,, + 6 = 9”1 (:19!) (3.4)

xmaz
The approximation of 2:0,, is substituted in f (2:) to approximate fmax as given in

(3.5). Finally, the error for the approximation of fmax is given by (3.6).

 

f(xap) = zap ' g(xap) = fap z fmaz (3.5)
_ _ f(xOP) _ f(xap)
Error — 100 f (170p) (3.6)

3.3.4 Validation for the LRCM

Consider f (x) = :r: - 9(2), and the derivative Of f (2:) with respect to 3:, f’ (2:) =
g(x) +50. g’(:1:) where g(x) has the diffeomorphism property. Now using the Lagrange’s
Theorem and the Cauchy’s Mean Value Theorem to find the optimal value 000,, that
it will produce the maximum value of f (2:) ==> fm = 2:0,, - g(xop) = f (map) in the
domain [0 xmax] (Rolle’s Thm.). Let’s apply the Cauchy’s Mean Value Theorem to

f (2:) and g(x) where both functions have the diffeomorphism property to solve for

 

 

zap-
f’(xop) : f(T: : :::ax) ___ T :31“ (37)
g’(xop) : 9(73 : :(xmax) : T 32:) (38)

r = if)- : ————fl($0p) - ——g($0p) + 11:0,, (3.9)

9(7) 91%) — g’($op)
Using the Corollary 3.2, if r = 0 then the approximation for 2:01,, is given by (3.10)

and the approximation error is 0.

1:0,, = g’-1 (fl) (3.10)

$1710.12

37

Now, if f (2:) does not have the diffeomorphism property then 1:0,, can not be
solved (i.e. 1120,, = f"1(0) is not possible to solve). Now, consider the function g(x) to
determine :rop, instead to use f (3) because f’(:1:) = g(x) + a: - g’(a:). There is a linear
slope (mL) with the same value as g’(xop) to find fm, mL 2 g’ (atop) (Lagrange’s
Thm.). Using Lagrange’s Theorem, there is a function gl(:r) = ml . a: + bl, where
91(0) = 9(0), glam”) = g(xmm.) = O and gl’(:cap) = g’(:1:ap), as given in (3.11) and
(3.12).

 

mL = g'(:z:) z gl’(:1:) = -;g(0) (3.11)
zap z :rop => crap = g"1 (Si—£2) (3.12)

Now, the approximate 10,, can be calculated using (3.12)! Finally, an approximate
f mag: is calculated using map, fmax z f (map) = zap -g(:1:ap). The error of angle 5 for map

and fmax will be calculated using (3.13),
5 = tan-1 (g(xap) + map ' 9’($ap)) (3-13)

If E = 0, then fmax is found, g’(:rop) = gl(:1:op), map = 2:0,, and the inverse map of

the derivative of f (2:) is found.

3.4 LRCM as a MPPT Algorithm

Figure 3.4 shows that at any particular intensity of light, there is a unique point for
the maximum power; this value is named the maximum power point (MPP). The
MPP is calculated exactly by solving for the voltage when (3.14) is equal to zero
then this voltage (Vop) is substituted in (2.13) to obtain the MPP. Unfortunately,
it is not possible to find a symbolic solution hence the only way to solve (3.14) is
numerically and this solution requiras long and tedious iterations, making the solution

not practical.

38

N
01

 

 

 

 

20

$510094. g
g 30% / *5
gm 60% // (‘5;
D.

40% ////

5 2096/

o 50 100 150 200 250
Voltage (V)

 

 

 

Figure 3.4. P-V and I-V Characteristics for different intensities of light.

 

(3.14)

Chapter 1 shows the traditional MPPT algorithms given by the literature. These
MPPT algorithms are versions of the numerical algorithm that relates the derivative
of the current with respect to the voltage equaled to the negative of the current divided
by the voltage, as given in (3.15). Unfortunately, a general (symbolic) solution cannot
be found using these algorithms, most of them depends on the record of previous
conditions and it is not guarantee that these algorithms can work properly under

nonconstant weather conditions.

3P a] BI I
5V=I+V-5V—O=>a—V=—V (315)

In the other hand, the Linear Reoriented Coordinates Method (LRCM) can be an

39

 

 

__——————o

  

Current (A)
ES

 
 

Pmax = lop-Vop

 

 

 

 

 

 

 

o VOID»l vu

a so 100 150i l 250
g 25 L i
8 20 Pmax=|op-Vop | I

|

T. 15 l I
g 10 pm | ‘I
‘L 5 L/Vop l Vx

0

0 50 100 150 200 250

VoltageM

Figure 3.5. Relationship between the P-V curve, I-V curve and the LRCM.

useful tool to approximate the solution of (3.14) with respect to V and be used as
an MPPT algorithm. For PV applications, the main idea for the LRCM will be to
find the I-V curve knee point as seen in Figure 3.5. The I-V curve knee point is the
optimal current (Iop) and the optimal voltage (Vop) that produces Pm. Using the
boundaries of the I-V Curve i.e. initial and final values, a linear current equation,
I L(V) can be determined as given in (3.16). Also, I L(V) can be considered as the
limit of (2.10) when b tends to infinite. The current equation, (2.10) and the linear
current equation, (3.16) are differentiated and set equal to each other to solve for V,
this solution will be known as Vop. The derivatives of I (V) and I L(V) with respect
to V are given by (3.18) and (3.17). It is important to remember that the slope of
the I-V Curve at the knee point is approximated by the slope of the linear current

equation, (3.19) hence the solution V6,, is a close approximation of Vop.

40

 

 

V
V . l—exp b'voc V

.IL(‘/):I(O)-_'.I(())°V;=bE»--I.rlm [lsc—lsc. 1—ex£(l)):|=13c_lsc°v— (3.16)
b 06

61L(V) _ Ia:

 

 

 

8V — _72; (3.17)
8V b-Vx—b-Vx-exp(—b-)
BIL(V) z BI(V) : __£r_ z —I:z: - exp (b—l}; — %)_1 (3.19)
8V 8V Va: b-Vm-b-Vm-exp(-b—)

Now, the equation of the approximate optimal voltage, V0,, is given in (3.20). To
prove that V0,, will be always equal to or more than V0,, for any given b more than
zero, V0,, is substituted into (3.14) resulting in (3.21), where (3.21) is more than zero

for any given b more than zero.

 

Vap=Vx+b-Vx-ln (b—b-exp (——)) SVOP (3.20)
9}:— Ir- [ln(b—b-exp(;b1-))-(b-exp('Tl)-b)+(b+1)-exp(Tl)-b] >0
BV‘ l—wma) -
(3.21)

Now, let’s substitute Vop into (2.10) to obtain Iup then to approximate Pmax mul-
tiply Vap by [a,, as given in (3.23). If Vap solves (3.14) equal to zero hence we found
the exact solutions for Pmax, 10,, and Vop. Also, Rap can be approximated by (3.24).

It is important to note that (3.25) always will be true under any value of b.

1—b+b-e:cp(‘Tl)

Ia” = Ix. l-exp(%1)

 

(3.22)

[1—b+b-e:rp(:bl)] - [1+b-ln(b—b~exp(—%))]

Papzlx-Vzr- 1—exp(‘Tl)

 

(3.23)

‘41

Vx. [1+b°ln(b—b-exp(—%))] . [1—e$p(:b‘1)l

Ruiz—I; 1—b+b-ea:p(——b—1)

(3.24)

 

PM = Vop-10,, 2 V0,, - [a,, = P0,, (3.25)

Additionally after proving (2.21) and (2.27) in the Chapter 2, the whole inequality

for the maximum power and fill factor using the LRCM are given by (3.26) and (3.27).

 

VOC
I,C - Vac > / I(V)dV > Pmax Z Pap > i .186 ' VIE (3.26)
o .
V
0c I(V)dV . Iap ' pr 1
> —— — .
1 >/0 1804/06 > lelfactor _ I“ . Vac > 4 (3 27)

Finally for PVM applications, the LRCM is a simple method where, instead of cal-
culating the optimal voltage (rated voltage) and maximum power solutions using the
power equation, the solutions are obtained using the current equation and the linear
current equation to obtain the approximations of 10p, V0,, and Pm. Also, the LRCM
has the advantage of giving an approximated symbolic solution for Vop, lap, and Pmax
under any T or E). The LRCM can produce the same results as other methods that
use Taylor series, continuous fraction expansion, iterations or other approximations,
and it is more practical for simulations and power flow analysis providing symbolic so—
lutions. The following results will show that the proposed technique is very effective,
giving a small error between the actual values and estimated values, even when the

effective intensity of light is changing over the photovoltaic modulas or solar panels.

3.5 LRCM Results for a PVIS with MPPT

Figures 3.6 and 3.7 show the simulation results for a PVM with the estimated curve

for Pmax and the knee points. The parameters for the simulation results are, T is T N,

42

l-V Curves for different b

- . _ Knee
, j point

 

 

 

Current (A)

 

 

 

 

 

 

 

O 50 100 150 O 250
Voltage (V)

Figure 3.6. I—V Curves with estimated knee points.

V2: is 208V, I,C is 15A, b is between 0.08 to 0.4 and E,- is given by (3.28).

Figure 3.6 shows the I-V curves for different characteristic constants and the
estimated curve for Pm“. The characteristic constant will determine I x and the
location of the knee point. The Pmax will be more for small characteristic constant;
hence an I-V curve with b equal to 0.1 produces a bigger Pm“ than an I-V curve
with b equal to 0.3. Figure 3.7 shows the P-V Curve for different characteristic
constants and the estimated curve for Pmax. The approximation of Pm is very close
to Pm“. Figure 3.8 illustrates the estimated error using the LRCM to approximate
Pm versus the normalized voltage, i.e. V0p divided by V2: with the maximum error
is approximately 0.3%. Figure 3.9 shows the P—V curve when T is TN , Va: is 208V,
I” is 15A, b is 0.08 and E,- is 900W/m2. It is shown that the approximation of Vop,

Vap, gives Pap where it has a similar value for Pmax produced by Vop.

43

P-V Curves for different b
Pmax

Pmax

 

0 50 100 150 200 250
Voltage (V)

Figure 3.7. P-V Curves with estimated maximum points for the Pm curve.

Different Characteristics Constants, b

    

% of error for the maximum power

0
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9
Normalized Optimum Voltage, VopNx

Figure 3.8. Error Curve for Pm and Pm estimated under different normalized
optimum voltage.

44

Approximated Power vs Maximum Power

(Pmax
2000 - Iap-Vap

 

 

    

 

 

 

 

Vat?“ Vop . Voc

0 50 1 00 1 50 200 250
Voltage (V)

 

 

Figure 3.9. P-V curve showing the closeness between Pmax and Pmax approximated
by Pap.

Finally, to prove how good is the range of our approximation for Pmmt using the
LRCM as a general case for any type of PVM consider the functions (3.29) and (3.30).
X, Y and Z are the normalize current, normalize voltage and normalize power for any
PVM. These variables describe a normalize PVM where X is V/Va: and Y is I / I :r.
The range of existence for X and Y is from 0 to 1. Using the LRCM, it possible to
approximate the optimal normalize current Xap as given in (3.31). Xap is substituted
in (3.30) then the approximate maximum normalize power, Zap, produced by any
PVM is given by (3.32). To calculate the approximate maximum power, Pap, just

multiply Zup by I :2: and Vzr.

 

 

 

I 1—e$p(£—l)
Y=—= b b
12: l—exp '71) (3'29)
_ V I _ _X—X-exp(lbf-—%)
_ V3312; _ 'Y _ 1 —e:cp (—Tl) (3.30)

45

ll

 

 

O'aoff>65% Maximum Percentage of Error *

  
  

0.15 - Ess%>r.r.>4o%§ .

  

0.10-

 

fill factor less than 40%

d

0.05 g

% of Error between Pmax and Vaplap

‘1‘ u
r V rL‘

l '.| 1 1

O 1 L -
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Characteristic Constant. b

 

 

1 l

 

Figure 3.10. Maximum percentage of error curve for Pmax versus the characteristic
constant, b.

Xap= 1+b-ln (b—b-exp (—Tl)) (3.31)

Z = [1+b-ln(b—b-exp(’Tl))]-[1—b+b-exp(-‘;1-)]
up 1—exp(—’51-

 

(3.32)

Figure 3.10 shows the maximum percentage of error between Pm and Pop for
any PVM where b changes from 0.001 to 1. Pm was calculated using Matlab for any
given b. The typical values of b for any PVM are from 0.01 to 0.18 hence the error for
the approximation using the LRCM will be from 0.01% to 0.25% for any given PVM.
An excellent result for the LRCM considering that there is no analytical solution for
(3.14). Finally, the LRCM has the advantage to guarantee an approximate symbolic
solution for the PVM exponential functions (2.10), (2.13) and (3.14) without symbolic
solutions. It has been proved that the LRCM has a maximum error for the estimation

of Pmax near to 0.3% where b is changing to obtain different V-I characteristic curves.

46

3.6 Additional Examples using the LRCM

Example 3.1: Consider the function f (11:) in {:0 E 3? | [0 r]} given by (3.33) with the
diffeomorphism property to find the maximum value fmax using differential calculus.
The derivative of f (:c) is given by (3.34) hence the Operation points 230,, and fmax are

given by (3.35).

f(x) = A ~11: - (r2 — x2)0'5 (3.33)

f’(:c) = A - (r2 — $2)0'5 -—- A - 11:2 - (r2 — r2)0’5 = 0 (3.34)
r A - r2

(x0? = $2 f(xop) = T = fmax) (335)

Now let’s find the maximum value for the same function f (11:) using LRCM.

1) Calculate g’(:c) using g(zr) where g(r) is A . r and 9(0) is 0.
g(x) = A - (r2 - 11:2)0'5 (3.36)

g’(x) = —A - x- (r2 — 3:2)”5 (3.37)

2) Calculate gl(:r) using (3.2) then calculate gl’(a:)
gl(:1:) = A - r — A - :1: ==> gl’(:i:) = —A (3.38)

3) Calculate 2:0,, using the LRCM hence g’(x) z gl’(:r)

T

 

map = 1:0,, = 72 (3.39)
4) To approximate fmax, 2:0,, is substituted in f (:0)
A - r2
f(xap) : 2 = fmaa: (340)

47

5) Finally, 5 is the final angle error for the approximation with e = 0° i.e. 0% of
error for the approximation of xop. Both results 300,, and fmaz can be solved and a
symbolic solution is obtained with angle error of 0° i.e. f’(xap) = 0.

Example 3.2: A basic principle in microeconomics is to obtain the maximum profit
and maximum revenues with the minimum costs [96]. Consider the function (3.41)
that describes the profit for the company X given the number of employees, n. Figure
3.11 shows the profit versus the number of employees contracted by the company X.
The variable m is the maximum number of employees to be contracted that will not
create a deficit to the company X, and k is a factor that relates the rate of profit
per employee. It is desired to maximize the profits for a company only contracting
the number of employees necessary to maximize the profit. Unfortunately, (3.41)
does not have the diffeomorphism property. Now, if (3.41) is divided by n, (3.42) is
obtained and has the diffeomorphism property that satisfies the conditions to apply
the LRCM. The derivative of (3.42) given by (3.43) and the boundaries of (3.42) can
be used to calculate the optimal number of employees, n,c to provide the maximum
profit for the company X. Using the LRCM, 71; is calculated using (3.44) where m is
52 and k is 10.06784 hence n1, is 36 with a profit of 284, 600$.

n

 

 

 

Profit(n) = n - k — n . (k — 1) . (79—5—1) ; (3.41)
rate = k _ (k _1) . (Ff—i)— (3.42)
agfe=1;k.ln(kf1)-(E§I)fi (3.43)

ln(k — 1) + ln[ln(k — 1) — ln(k)]
ln(k — 1) — ln(k)

 

(3.44)

711-:

Example 3.3: The next example is to determine the inverse of a function f (:0)

without diffeomorphism. The main goal is to determine the maximum rectangular

48

 

 

 

 

300 . 4 .
250- -
200 .° °. 4
150, .° . -
100- .' . -

f

Profit in thousands $

01
O

 

 

1

010 20 30 40 50 60
# of Employees

 

o
:0
0

Figure 3.11. Profit vs # of employees contracted by the company X.

area inside of the function g(x). g(x) describes the shape constraint relation for a
floorplan for an 8-bit A/ D converter and it is required to maximize the rectangular
area inside of g(x). Floorplan design is the first task in VLSI layout and perhaps the
most important one [88]. In practical designs, the dimensions of some modules are
restricted by physical designs and therefore can not be varied continuously [87]. f (x)
in {x 6 ER | [0 25]} represents the rectangular area occupied by a floorplan for an

8-bit A/ D converter.
_ x - 25 - tan—1(25 — x)
_ tan-1(25)

 

f (:13) (3.45)

Using differential calculus, it can be calculated f’(x) and simplified to be solved

by :1: but it cannot be solve for x, as given in (3.46).

 

tan—1(25 —- x) — 1+ (25 _ x)2 = 0 (3.46)

Consider the LRCM using the following steps:

49

1) Calculate g’(x) using g(x), g(25) is 0 and 9(0) is 25

__ 25 - tan-1(25 — x)

 

 

g(x) — tan_,(25) (3.47)
, _ —25
g (x) _ tan-1(25) + tan'1(25) ' (25 — x)2 (3'48)
2) Calculate gl(x) using (3.2) then calculate 91’ (x)
gl(x) = 25 — :0 => gl'(x) = —1 (3.49)

3) Calculate the approximate value of xop using the LRCM hence g’ (x) z gl’(x)

 

25
z 0 =25— ———1 .
x0, x, \/tan_,(25) (3 50)

4) To approximate fmax, xap is substituted in f (x) hence fmax m f (xap).

25 _, 25 25

5) The percentage of error for the approximation of fm is less than 2.3% and

 

 

was calculated using (3.6). This final result proved how good is the approximation
for fmax considering that there is not analytical solution for (3.46).
Finally, the dimensions for the maximum rectangular area for a floorplan for an
8-bit A/ D converter are for x-axis is 21.0845 units and for g(x)-axis is 21.5693 units.
Example 3.4: Figure 3.12 shows the characteristic curve for a Fuel Cell [85] where
voltage output (V) versus the current density (A/cmz) relationship with an area for
the reactor of lcmz. The voltage, V, and the power, P, in terms of the current, I,

are described by (3.52) and (3.53). To obtain the maximum power, Pmax, is required

50

 

1 OTypical Fuel Cell Polarization Curve

Voltage (V)

 

 

 

I
L

0 0.5 1 1 .5
Current Density (Alcmz)

0.2

 

Figure 3.12. V—I Characteristic Curve for a Fuel Cell.

to solve the derivative of the power with respect to the current equal to zero.

_ 0.7 _1 I
V(I) — 0.3 + 7r cos (0.7 1) (3.52)
P(I)=I°V(I)=0.3-I+9Z-I-cos'l(L—l) (3.53)
7r 0.7
319(1) 07 I I I 2 "0'5
__ : _'_ . -1 __ _ _ _. __ _ ._ _
81 0.3 + 7r cos (0.7 1) 7r [1 (0.7 1) ] (3 54)

Unfortunately, it is not possible to solve (3.54) with respect to 1 due the absence
of the diffeomorphism property. The LRCM can provide a good approximation for
Pmax-

 

avu) _ 1 I 2 ‘0'5
__BI___;T.. 1_(a_7._1)] (3.55)
Vl(I)=1—£=>§/—Q=—l (3.56)

2 BI 2

51

Table 3.1. Comparison for LRCM Results and Optimal Values

 

 

Voltage Current Power

Optimal 0.4902 V 1.1602 A 0.5687 W
Approx. 0.4538 V 1.2398 A 0.5626 W
Error 7.44 % 6.88 % 1.07 ‘70

 

 

After use (3.55) and (3.56), it is possible to solve for the approximate optimal current
(Iap) given by (3.57).
4
[a,, = 0.7 + 0.7- 1 - — (3.57)

7r2
Finally, [0? can be substituted in the voltage and power equations, (3.53) and (3.52).
Table 3.1 shows the results of the LRCM for the voltage, current and power. The row
with the approximation error values for each variable was calculated using (3.6).
Example 3.5 : Consider the function g(x) described by (3.58). It is desire to
calculate the maximum rectangular area inside of g(x) V x in {x E 3? I [0 4]}. The
rectangular area inside of g(x) can be calculated using f (x) = x-g(x), the derivative of
f (x) with respect to x is given by (3.59). Unfortunately is not possible to solve (3.59)
equal to 0 but using the LRCM is possible to approximate the maximum rectangular

area inside of g(x).
g(x) = exp(8) - exp(4) + exp(x) — exp(2 - x) (3.58)

f(x) = exp(8) — exp(4)+ (1+ x) ~exp(x) — (1+ 2 . x) ~exp(2 - x) (3.59)

1. Calculate the linear equation gl(x) using the boundaries of g(x) where g(O) is

exp(4) + exp(8) and 9(4) is 0,

91(2) = (exp(4) + exp(8)) . (1 — E) (3.60)

52

2. Determine g'(x) and gl’ (x)

g'(x) = exp(x) — 2 - exp(2 - x) (3.61)
are) = —§ - (exp(4) + exp(8)) (3.62)

3. Substitute y = exp(x) on g’(x)
g'(x) = exp(x) — 2 - exp(2 - x) = y — 2 - y2 (3.63)
4. Using g’(x) and gl'(x) solve for y

9'6...) x gm...) => .4 — 2 - 42 z 71,- - (exp(8) + exp(4)) (3.64)

 

+ . \/1+ 2 - exp(4) + 2 - exp(8) = 19.7309 (3.65)

ul>|l-‘
fill—i

5. Calculate xap then approximate the maximum area using xap,

xap = ln(y) = 2.96411 => f(xap) == xap -g(xap) = 7, 618.51 (3.66)

Finally, fmax is 7,631.62 hence the percentage of error for the approximation
f (xap) using (3.6) is 0.171524%. Again, f’ (x) = 0 is not possible to solve with respect
to a: due the absence of the diffeomorphism prOperty in f (x) but using the LRCM at

least, it is possible to estimate the optimal value for x with small percentage of error!

53

CHAPTER 4

Fractional Polynomial Method

This chapter presents a non-traditional method for the approximation of the pho-
tovoltaic module, PVM, exponential model using fractional polynomials where the
shape, boundary conditions and performance of the original system are satisfied. The
use of fractional polynomials will provide an analytical solution to determine the 0p-
timal voltage, Vop, Optimal current, 10?, and maximum power, Pm for the PVM
operation. An additional method to calculate a sufficiently close integer polynomial
is given in the chapter using the information obtained from the Fractional Polynomial
Method, FPM. Examples and simulations to validate the proposed methods are given
in the chapter using data sheets for different types of PVM’s. Finally, the proposed
methods are excellent in approximating the PVM exponential model and provide a
different way to approximate exponential functions that are not possible to solve using

differential calculus.

4. 1 Introduction

In engineering and sciences, an accurate mathematical modeling for a physical sys-
tem, object, event or pattern can determine the behavior and characteristics of the

proposed design—saving time, space, money and materials. Examples of mathematical

54

modeling and simulations are in circuit analysis, design of mechanical systems, nu-
clear explosion simulation, power grid simulations, etc. An inaccurate mathematical
model can result in serious problems not expected in the final design of the system.
The performance and behavior of the system can be diminished because of inaccurate
modeling. One of the most dramatic examples is the Tacoma Narrows Bridge, USA,
in 1940, where the natural resonance of the bridge coincided with the frequency of
the wind creating a collapse of the bridge, an effect not considered in the original
design [97].

At the same time, a very complex mathematical model can be hard to analyze
and impractical. So a compromise should be taken between the complexity and the
number of parameters used to describe a physical system [98]. If the correct assump—
tions are made, an approximation of the mathematical model that keeps the main
properties of the physical system can be obtained. An example is the mathematical
model for a resistance in circuit analysis where the temperature effect is neglected on
the nominal value for the resistance.

Chapter 2 proposes a PVM model based on the manufacturer data sheet. Unfor-
tunately, the proposed PVM model cannot be programmed into a microchip because
most of the Arithmetic Logic Units (ALU) will only perform arithmetic operations or
it cannot be used as a PV source simulator in programs like Saber or P-Spice. This is
because often these sources are simulated using polynomials instead of exponentials.
To solve this problem, a method to approximate the photovoltaic module model using
fractional exponents and polynomials is presented in this chapter.

The chapter describes how the proposed PVM model can be approximated by
fractals and polynomials. The obtained polynomial keeps the properties of the given
exponential function and can be used in programs like Saber and Pspice. Also, the
chapter describes the relationship of exponential functions, with fractal functions and

how it can be approximated by polynomials. Additionally, the fractional polynomial

55

that describes the power for the PVM can be used to estimate the maximum power
at any temperature or irradiance level. Finally, the proposed Fractional Polynomial

Method, FPM can be applied to other types of transcendental functions.

4.2 Fractional Polynomial Method

In this section, a method for the approximation of exponential functions in the range
of existence is described. The idea is to approximate the exponential functions as
described in and using fractional polynomials. These fractional polynomials should
keep the same boundaries, shape and performance of and . The question of how to
approximate I (V) as a fractional polynomial, keeping the boundary conditions and
properties of I (V) using the data provided by the PVM manufacturer data sheet, is
addressed in this chapter. The Fractional Polynomial Method, FPM, is also useful in
obtaining analytically the optimal current and voltage and at the same time able to
provide Pm.

Now, consider the fractional polynomials (4.1) and (4.2) that satisfy the same
boundary conditions of and , where n is a positive integer number and q is a non-

integer number greater than or equal to 0 but less than 1 i.e. 0 S q < 1.

I,(V) = Ix — Ix - (%)W (4.1)
P,(V) = v.1,(V) = V-Ix—Von- (72)“ng (4.2)

The derivatives of (4.1) and (4.2) with respect to V are given in (4.3) and (4.4).
To approximate the variables V0,, and Iop, (4.4) is set equal to 0 then solve for V, the
approximation of V0,, will be given by Vop, then substitute Vopf in (4.1) to approximate
[0,, given by Iopf. Finally, Pmax is approximated multiplying Vop, by 10p), as described

in (4.7). It is important to note that (3.14) cannot be solved with respect to V when

56

it is equal to 0 but on the other hand (4.4) can be solved with respect to V giving a

close approximation for Vop, [0,, and Pmax.

 

 

aI,(V) "”‘1
=_ . . __ < .
av 1:1: (n+q) (W) _0 (43)
6P,(V)_ V "+"
8V —IfB-I$'(n+q+1)°(7£) (4-4)
V —Vx ——1 "i9 (45)
°”’— n+q+1 °
n+q
10,, = “5' (am) (4'6)
1
n+q 1 "*1
Pmaf=Vopf-Iopf=Vx-Ix-(———n+q+1)-(——n+q+1) (4.7)

To find the relationship between (2.10) and (4.1), both functions are evaluated
under Standard Test Conditions and set equal to each other as given in (4.8) then

solve for n + q as given in (4.9).

Vop-I V 1 V "+4
= 3" . 1- _£L_- = . . _ LP .
Pm” 1—exp(:.%) l ”(b-v... bll V” I“ l1 (0.) l (48)

) .111 [1 _ 6x1) (b—VVL)] (4.9)

 

 

n+q=

OC

The next three points summarize the proposed FPM for the approximation of
a photovoltaic module model using fractional polynomials. It can be used as an
analytical method to approximate Pmax and satisfy the boundary conditions which
are necessary to provide the best approximation of I (V) and P(V).

1- The boundary conditions are satisfied in I I(V) and Pf(V). Additional condi-
tions are given in Table 4.1.

2- For any value of V, more than 0 and less or equal than Vx, n-derivatives of

57

Table 4.1. Conditions satisfied by the proposed fractional approximation method

 

 

 

V P(V) 135,2 I(V) 9%,?
V=0 P(O)=O %)_l>0 1(0):” 349230

V=Vx P(Vx)=0 i§§fl<0 I(Vx)=0 flagko

v = V0,, P(Vop) = Pm 9% = Pm, I(Vop) = 10,, 64,99 < 0

 

If(V) with respect to V are less than 0 where k = 1, 2, 3, ..., n.

8"I(V) --Ix V 1 1 81(V)
akV _ bk _ b36331) (:51) -exp (—b-Vx — b) — bk-l. 8V < 0 (4.10)

 

 

 

3- Analytical solution to solve for the maximum power, Pmax.

310 (V ) -lap (0)

In the next section, approximation of the fractional polynomial to a close integer

polynomial that keeps most of the properties of I f(V) is discussed.

4.3 Integer Polynomial Approximation Method

Some disadvantages of the fractional polynomial are that it cannot be programmed in
an Arithmetic Logic Unit, it is not easy to handle for Lyapunov analysis and cannot
be used as a custom voltage-current source for simulators like Pspice or Saber. So the
purpose of this section is to provide an additional method to approximate a fractional
polynomial using a close approximation for an integer polynomial where the boundary
conditions are satisfied. The idea of the Integer Polynomial Approximation Method,
IPAM, is to linearize only the non-integer part of (4.1) in the point of reference Vx

where the non-integer part of (4.1) is given by (4.12) then (4.12) is evaluated on V3: as

58

 

given in (4.13). The function yl (V) is an straight line calculated by the linearization
of (4.12) in the point of reference Vx. The straight line parameters m and b are

calculated by (4.14) and (4.15) then are substituted in (4.16).

 

00/) = (7",?) (4.12)
y(Vx) = yl(Vx) = (g) = 1 (4.13)
b=y(Vx)—m-%V_v =1—q (4.15)

 

Finally, yl(V) is the linearization of g(V) and is given by (4.16). Using (4.16),
the integer polynomial Ip(V) that approximate the fractional polynomial I I(V) is
obtained and given by (4.17). Now, it is possible to program Ip(V) in an ALU or

used as a custom source in programs like Saber or Pspice.

V V
y(V)—b+m-V—x—1-q+q-‘—/; (4.16)

V n V n V
Ip(V)—Ix—Ix-(V;) -yl—Ix—Ix-(V—$) -(1—q+q-V;) (4.17)

To calculate the power just multiply 1,,(V) by V as given in (4.18). The maximum
power is calculate by taking the derivative of Pp(V) with respect to V as given on
(4.19) then set (4.19) equal to zero and solve for V, is substituted in (4.18) to find

the maximum power.

Pp(V)=V-IP(V)=V-Ix—V-Ix-(%)n- (l—q+q-LI) (4.13)

%¥—)=Ix- 1—(1—q)-(n+1)-(Ly-9402+?)-(I—YH] (4-19)

59

It is important to note that (4.17) and (4.18) satisfy the famous Weierstrass ap-
proximation theorem [99], therefore the maximum approximation errors for (4.17)
and (4.18) are sufficiently close to the maximum errors for the approximations of
(4.1) and (4.2) using fractional polynomials. Unfortunately if n is more than three,
there is no general solution for (4.19) when it is equal to 0. A similar statement was
proved first by Paolo Ruflini and Neils Henrik Abel. The theorem is known as the
Abel-Ruffini theorem and it was published in the year 1813 [99]. For a polynomial
like (4.17) with n more than three, at thought we can determine the limits for roots
using Maclaurin’s theorem [100] but this does not mean that the solutions (4.19) can
be found; it means that only the range of existence for the solutions of V can be
found. Also, if (4.19) can be solved, V will have 17. solutions making n — 1 impractical
and there is only one useful solution for V which is a unique positive real value in the
range of existence from 0 to Vx.

On the other hand, Ip(V) can be very useful for Lyapunov analysis. Consider a
PVM connected in parallel to a capacitor, C, and a resistance, R, as shown in figure
4.1. It is desired to prove that the voltage, V, is asymptotically under any value of

R. The dynamic function for the voltage is given by (4.20) where V E [0 Vx].

QV_IP(V)_ V __I_x_(1—q)-Ix-V"_q-Ix-V"+1_ V (420)
8t_ C C-R_C C-Vx" C-Vx”+1 C-R '

 

 

 

The equilibrium point of (4.20) is given by (4.21) where V E (0 Vx) hence the
normalized equation that shift the equilibrium to zero is given by (4.22) where V =

V—V.

R-(1—q)-Ix-V"_R-q~Ix-V"+1
Vx" Vx"+1

 

 

V = R - 1,,(V) = R . Ix — (4.21)

:21; = 96:59; . (V. _ (17 16)") .3333 (1761 _ (7 my“) (4.22)

60

'(Vl—> L

l l
5571“

V

 

 

 

 

Figure 4.1. PVM connected to a RC Load.

Now, the Binomial Theorem [99] defined in (4.23) can be used to simplify (4.22) as
demonstrated by (4.24). Due that the range of existence for V is from zero to Vx,

the functions 9,,(V) and gn+1(V) are always positive functions.

~ _ n+1 n+1 ~ —n _ —n n+1 ~ —-n _
(v+v) =Z(2+1)-V"-V +1"=v +Z(g+1)-V"-V ”’1 " (4.23)
k=0 k=1
8‘7 (1—Q)'I$ n n ~k —n—k 9'13 "+1 n+1 ~k —n+1-k
5.: = ‘W';(k)°V 'V ——C,Vxn+1‘k2:(k )‘V 'V
=1 =1
(l—q ~Ix ~ q-Ix ~
_ c.1317: '9"(V)"C.vzn—+1'9n+1(vl (4'24)

Let’s apply the Lyapunov function, S2 = 0.5 - V2, to check the stability of V using
(4.24). The derivative of (2 clearly shows that V is asymptotically stable for any value
of R as given in (4.25).

3V (1—q)-Ix ~ ~ q-Ix ~ ~

=V-—=——.——-g.(V)-V—W-gn+1(V)-vso (4.25)

61

4.4 Examples using FPM and IPAM

In this section, Tables 2.1 and 2.2 will be used to compare the relationships between
the proposed PVM model in Chapter 2 and the fractional approximation method.
The first example will show how to apply the proposed method to approximate the
performance for a PVM SX-5. The first step is to calculate variables n and q using

(4.9), the calculations are given by (4.26) where n is 10 and q is 0.6078.

1 _ 16.5
n + q = ——]fi- -ln [ exp (0'084714‘20'5)] = 10.6078 (4.26)
“1(263) 1‘ 8"P (m)

 

Consider Ix equal to Isc and Vx equal to Vac, the approximations of the optimal

current and Optimal voltage are given by (4.27) and (4.28).

 

 

n + q 10 + 0.6078
1 =1 - —— =03 4.27
”I” 1” (n+q+1) (10+O.6078+1) ( )
V _ Vx 1 3:3 _ 20 5 1 10+0.6078 (4 28)
‘4’" n+q+l _ ' 10+0.6078+1 '

The approximation of the maximum power is the multiplication of (4.27) by (4.28)
and it is given by (4.29). The fractional polynomial and integer polynomial, that
describes the PVM SX—5 under STC, are given by (4.30) and (4.31). Table 4.2 shows
the results for four PVM’s given on Tables 2.1 and 2.2. Figure 4.2 shows the I-V
Curves for a Solarex SX-5 with their fractional and integer polynomial approximations
under STC. The fractional and integer approximations of I (V) are very close to the
I-V Curve. Figure 4.3 shows how good I I(V) and 1,,(V) are when the maximum error

for the approximations of I (V) for the SX-5 is 7.5mA.

Pm, = 10,, - V0,, = 0.27A - 16.23V = 4.46W (4.29)

62

Table 4.2. PVM parameter approximation using the F PM under STC

 

 

PVM Model 10p,(A) Vopf(V) R0,,(o) Pm, (W) n q

 

Solarex SX-5 0.27 16.23 60.11 4.46 10 0.6078
Solarex SX—10 0.60 16.77 27.95 10.00 11 0.0869
SLK60M6 6.96 30.19 4.34 210.10 12 0.4576
Siemens SP75 4.37 17.12 3.92 74.82 10 0.1799
Shell SQ80 4.51 17.82 3.95 80.32 13 0.1461

 

V n+q V 10.6078

V 10 V 11
Ip(V) = 0.3- [I — 0.3922- (35.5) — 0.6078- (2—05) :|

(4.30)

(4.31)

The second example uses the I-V Characteristics given by the data sheet for a

PVM SLK60M6 under different temperature and irradiance levels. It is desire to

approximate the I-V Curves and P-V Curves using fractional polynomial and integer

polynomial approximations. First the variables Ix, Vx, n and q are calculated using

(2.11), (2.12), and (4.9) then these variables are substituted in (4.1), (4.2), (4.17),

and (4.18) then I f(V), Pf(V), Ip(V) and Pp(V) are simulated. Figures 4.4 and 4.5

show the I-V Curves, P-V Curves and their approximations for a PVM SLK60M6

under different cell temperatures and radiations. Clearly, it can be seen that the I-V

Curves, P-V Curves and their approximations are very close to the results given by

the SLK60M6 data sheet.

63

 

0.25 '

I(V)
.0
N
o

0.15 -
0.10 .
0.054

Current

 

 

 

 

0 . L . - 4 1 . . .1
0 2 4 6 8101214161820
Voltage,V

Figure 4.2. I-V curves and the approximations for a PVM SX-5 under STC.

X110.

 

 

|f(V)-|p(V)

oawmemmum

 

 

-|f
. . .'<Vl'P.(V):-». . .
0 2 4 6 8101214161820
Voltage,V

[13.3

 

 

 

Relative Error between I(V)
and the approximations of I(V)

Figure 4.3. Error curves between I ( V) and the approximations of I ( V).

64

 

T

 

OWImfa 0"

 

 

 

 

 

9

l: 2, IPM]

1,000w1m2at 25 °C 1 "M

:6 e— e— )1" (”M
gs? 800WIm2at43°C -
9244 e
8 3~ 500W/m1’atzsoc’v

2.

1L

0 . . . . 1 . '

0 51015 20 25 3035 40

Figure 4.4. I-V Curves for a PVM SLK60M6 under different cell temperature and

irradiance.

Figure 4.5. P-V Curves for a PVM SLK60M6 under different cell temperature and

irradiance.

Voltage, V

 

-t1,100w7m2a'150°b ‘ -

.1,000Wlm2 at 25 ° -
- 800WIm2 at 43 ° 9

 
   

 

Pf(V) _
.P A ‘ - .
. / ' '«_500WIm2at25°C I

 

Voltage, V

65

0 5101520 25 3035

 

CHAPTER 5

Fixed Point Algorithms to

Estimate T and E, N over a PVM

The purpose of this chapter is to present four algorithms to calculate the effective
irradiance level, E,- and temperature, T, of Operation for a photovoltaic module, PVM.
The main reasons to develop these algorithms are for monitoring climate conditions,
the elimination of temperature and solar irradiance sensors, reductions in cost for a
photovoltaic inverter system, and development of new algorithms to be integrated
with maximum power point tracking algorithms. The first three algorithms use only
the short circuit current, open circuit voltage, the operating current and voltage for
the PVM, avoiding the use of pyranometers and thermocouples. The last algorithm
can estimate the irradiance level using only the open circuit voltage and the PVM
temperature of operation. Finally, simulations and experimental results are presented

in the chapter.

5. 1 Introduction

The environmental conditions are an important factor in the performance of any

photovoltaic module, PVM. An accurate measurement of the temperature, T and

66

effective irradiance level, E,- is needed to improve the design of PV power systems
and maximum power point tracking, MPPT algorithms. Also, the measured data
is useful for online PV system characterization [101, 102], reliability of the weather
conditions and meteorological data collection on a long term basis [103]. Also, with
the measured data, it will be possible to determine if a PV system is cost—effective
[4] and to predict the annual energy production for a PVM in a specific geographic
region [104].

The typical sensor used to measure the solar irradiance over a PVM is a pyra-
nometer [105]. A pyranometer is defined as an instrument for measuring the so-
lar radiation and diffuse sky radiation, i.e. effective irradiance on a plane sur-
face [106]. Typically, pyronometers are used in terrestrial and space applications
[4, 1, 2, 102, 103, 104, 105, 106, 107, 108, 109, 110]. Usually, for a low cost pyra-
nometer a reasonable accuracy should be 35% and for a high cost pyranometer a
reasonable accuracy should be :l:2% [107, 108]. Disadvantages with a pyranometer
are that usually the price can be between 300 US. dollars and 1,800 US. dollars [108],
the sensitivity may change with time and exposure to radiation [104], long periods
of high temperature (> 50°C) can damage the accuracy of the instrument [106] and
often the pyranometers need to be calibrated every day whenever there is significant
change in weather conditions [106].

In addition to the solar irradiance, the temperature can affect the output of a
PVM. The average temperature for a PVM should be measured using multiple ther-
mocouples attached to the rear surface [111]. An advantage, thermocouples can mea-
sure a wide range of temperatures and are cheap and standard devices in the industry
[112]. Thermocouples in photovoltaic applications are used mainly for safety reasons
monitoring the average temperature variations in a PVM [106]. The main limitations
using thermocouples are limitations in the range of accuracy, noise, connection prob-

lems, decalibration [112] and the positioning over the surface of the PVM, where it

67

can affect the PVM performance, or under the PVM, where inaccurate measurements
of the PVM temperature could be obtained [111].

To avoid the use of sensors and to solve the problems exposed before, this chapter
proposes several Fixed-Point Iteration, F PI, algorithms using voltage and current
measurements to calculate T and E,- over a PVM. These algorithms can be integrated
with other algorithms related to MPPT (e.g. Linear Reoriented Coordinates Method,
LRCM [62]) or to monitor the PVM performance [101]. The PVM mathematical
model is described in the chapter, and it is based on the manufacturer data sheets
[102]. Finally, this chapter describes the algorithms and compares the algorithm

results to the measured results.

5.2 Algorithms to Estimate the Effective Irradi-
ance Level and Temperature over a PVM

To understand the proposed algorithms and their validity, the following paragraphs
will explain the definition and theorems related to Fixed-Point Iteration, FPI and
their relationship with the PVM mathematical model. A fixed point is defined as a
number a: such that a: is the solution of :1: = g(x) [77]. Theorem 5.1 and Theorem
5.2 are the basis for the conditions of existence and uniqueness for the prOposed
algorithms.

Theorem 5.1 (F ired Point Existence): Assume that g(x) is continuous on [a, b],
and that a S g(x) S b \7’ a: E [a, b] then 3 a fixed-point c in [a, b]. The proof can
be found in [77].

Theorem 5.2 (F ired Point Uniqueness): Assume that g(x) satisfies Theorem 5.1,
Bg(a:)/6:c is continuous on (a, b) and El a positive constant P < 1 where |g’(:1:)| _<_ P,
then g(x) has a unique fixed point c on (a, b). The proof is in [77]. Theorem 5.2 is

also known as the Contraction Mapping Theorem.

68

 

 

Figure 5.1. Flowchart for Algorithm 5.] to calculate T and E,.

For additional FPI theorems, definitions and applications please refer to [77],[113]
and [114]. Now the proposed algorithms will be presented with their descriptions and
applications.

Algorithm 5.1: Fixed-Point Iteration to calculate T and E,- given Vrr, V1 and [1.
The algorithm considers the data provided by the PVM data sheet. Figure 5.1 shows
the flowchart for Algorithm 5.1. The first step is to calculate I :1: using (5.1). The
second step is to iterate (5.2) and (5.3) to calculate T and E,- using TN and EN as

initial conditions.
I — I - e2: —‘1
I :1: — l 1 p( b )

_1-exp(rV‘h-%)

(5.1)

E,(n) - (V11: — Vm) E,(n)
TCV - Em TCV - Em

Ei Vmax _ Vac

T(n+1) = TN'l‘

 

III: ' EiN
E4“ 1’ = W “’3’

Figure 5.2 presents an integrated PVM converter system using a DSP Board to con-
trol the maximum power to the load and to calculate T and EN without pyranometers

or thermocouples. Algorithm 5.1 is programmed to the DSP Board. Finally, the pro-

69

l

 

Sun Light 0, Bi)

 

 

SHELL SQ80
11 = 3.62 A
V] = 16.0 v
Vx = 20.0 v

 

 

 

 

 

Figure 5.2. Integrated PVM converter system, using a DSP Board programmed with
the Algorithm 5.1.

posed algorithm is able to find a unique solution for the effective irradiance level and
temperature of operation over a PVM because (5.2) and (5.3) satisfy Theorem 5.1
and Theorem 5.2.

Algorithm 5.2: This fixed iteration algorithm considers the use of Ia: and Vx to
calculate T and EN. First, Algorithm 5.2 reads I x and V0: then iterates (5.2) and
(5.3) as presented on the Algorithm 5.1 description.

Algorithm 5.3: Fixed-Point Iteration to Calculate T and E,- given V1, V2, II and
12. Algorithm 5.3 is designed for a variable load with faster dynamics than T and
E, dynamics. The basic principle for Algorithm 5.3 is the following: if the power
in the load is changing but T and E, are constants then the new operation point
(V2, 12) will remain in the same I—V curve as the old operation point (V1, [1); hence,
it is possible to calculate T and E,. Figure 5.3 shows the flowchart for Algorithm 5.3
where the first step is to read V1, Vg, I1 and 12, as an initial value, Va:(1) is equal to

V1 then iterate (5.4) and (5.5) to calculate I :c and Van. Finally, V2: and I :1: are sent

70

 

 

Figure 5.3. Flowchart for Algorithm 5.3 to calculate Ia: and Vx, integrated with
Algorithm 5.2.

to Algorithm 5.2 to calculate T and Eg.

I — I - -—1
Ia:(n + 1) = W (5.4)
1 _ 617]) (b-Vz(n) - b)
Vz(n + 1) = V‘ (5.5)

 

1+b-ln [1—%+%-exp(b—v‘;3w—)—%)]
Algorithm 5.4: Fixed—Point Iteration to calculate E,- and I a; given Vx, and T. Algo-
rithm 5.4 is designed using the fact that the thermocouples are cheap. Hence using
one sensor for the open circuit voltage, it is possible to calculate E,. The algorithm
reads T and Vi: then iterates (5.6) to find E,.

__ -T 'Ei
E,(n+1)= (T T”) CV N W, (5.6)

Vx — Vm + (Vm _ V,,,,-,,) - (lunch

Vmaz ‘Vmin

 

Finally, the proposed algorithms are valid to calculate T and E, because Theorem

5.1 and Theorem 5.2 are satisfied due the continuity of the functions and partial

71

derivatives of (5.2)-(5.6). As an advantage, the proposed algorithms can be integrated
with other algorithms or methods with MPPT without affecting the performance of
the PVM.

5.3 Experimental Results using the Proposed Al-
gorithms

The electric specifications for four PVM (Table 2.1 and Table 2.2) were used to vali-
date and test the proposed algorithms. Figure 5.2 shows an integrated PV converter
system where Algorithm 5.1 and Algorithm 5.2 were simulated. Table 5.1 and Table
5.3 show the measured and expected parameters for the four PVM’s using Algorithm
5.1 and Algorithm 5.2 respectively. The results for the Algorithm 5.1 and Algorithm
5.2 are given in the Table 5.2 and Table 5.4 respectively. The number of iterations
required to calculate T and E,- were less than 5 for both algorithms. The maximum
relative error to approximate E,- is less than 3% and the maximum absolute error
between the measured T and the calculated T was only i6°C showing a good per-
formance. Also, the algorithms converge very fast with a good performance with the
uniqueness property presented in Theorem 5. 2. The LRCM [62] was integrated with
Algorithm 5.2 to approximate the maximum power produced by the PVM’S on real
time conditions, Pap as shown in Table 5.4. Finally, these algorithms can track the
meteorological conditions for a long term because the collected data can be stored

and recorded without interfering with the PVM performance.

72

Table 5.1. Measured Values for Algorithm 5.1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Datasheet [1 V1 V2: E,- T
Siemens SP75 3.00A 18.0V 19.8V 1, 000W/m2 45°C
Shell SQSO 3.62A 16.0V 20.0V 800W/m2 46°C
SLK60M6 8.20A 10.0V 35.0V 1, 100W/m2 50°C
Solarex SA-5 0.28A 19.5V 25.3V 1, OOOW/m2 20°C
Table 5.2. Calculated Values using Algorithm 5.1
Datasheet I terations(n) E,(Appr.) T(Appr.)
Siemens SP75 5 955.7W/m2 47.976°C
Shell SQ80 4 785.7W/m2 42.271°C
SLK60M6 4 1, 084W/m2 44.045°C
Solarex SA-5 4 966.5W/m2 19.552°C
Table 5.3. Measured Values for Algorithm 5.2
Datasheet [1 V1 V2: E,- T
Siemens SP75 3.00A 18.0V 19.8V 1, 000W/m2 45°C
Shell SQ80 3.62A 16.0V 20.0V 800W/m2 46°C
SLK60M6 8.20A 10.0V 35.0V 1, 100W/m2 50°C
Solarex SA-5 0.28A 19.5V 25.3V l, 000W/m2 20°C
Table 5.4. Calculated Values using Algorithm 5.2
Datasheet I terations(n) E,(Appr.) T(Appr.) Pap
Siemens SP75 5 795.4W/m2 44.980°C 64.7W
Shell SQ80 4 810.3W/m2 42.845°C 60.9W
SLK60M6 4 794.7W/m2 39.204°C 162W
Solarex SA-5 4 1, 015W/m2 72.656°C 5.12W

 

73

CHAPTER 6

Proposed PV Power Applications

In this chapter, several PV applications will be shown like a PVM connected to
different loads, an additional MPPT algorithm and other PV applications. The first
sections show how to analyze a PV circuit using the PVM model given in Chapter
2. The MPPT algorithm proposed in this chapter is based in the control of the
Optimal duty cycle for a dc-dc converter and the previous knowledge of the load or
load matching conditions. The procedure to calculate the Optimal duty ratio for a
buck, boost and buck—boost converters, to transfer the maximum power or required
power, from a PVM to a load is presented in this chapter. Additionally, the existence
and uniqueness of the optimal internal impedance, to transfer the maximum power
from a PVM using load matching and how to obtain it using the Optimal duty ratio,
is shown. Finally, a Photovoltaic Inverter System, PVIS, is proposed for single—phase
power applications. The proposed PVIS has three stages, a photovoltaic module
connected to a buck-boost converter and a resonant Z—source converter. The PVIS
takes into consideration changes in temperature and irradiance level, the dynamic
model for a PVM, buck-boost converter model, Z-source converter operation principle

in resonance to provide a frequency of 50Hz and voltage output (rms) of 120V.

74

6.1 Introduction

In the previous chapters, solutions were provided to many problems in the area of
PV power systems, which includes better modeling to describe a PVM, improved
algorithms to track the maximum power from a PVM, better design and control PV
inverter systems, more accurate methods to estimate the temperature and effective
irradiance level over the PVM, etc.

It is the purpose of this chapter to provide several PV applications related to the
area of power systems. The first sections should be consider as a guidance for PV
circuit analysis some of the examples are a PVM connected to a resistance, RLC
load. Also, it is shown that the analysis for two PV arrays, with different electrical
characteristics, connected in series to a power load. The following section will show
how to calculate the optimal duty ratio for a dc—dc converter using the PVM electrical
characteristics and the load matching conditions.

Finally, the last section of this chapter will be propose a transformer-less pho-
tovoltaic inverter system for single phase applications. Figure 6.1 shows the typical
configuration for a photovoltaic inverter system. The main components are a PV
array, a dc-dc converter to keep the PVM operating at the maximum power, an in-
verter to match the required frequency and to convert the dc voltage to ac voltage,
a transformer to amplify and keep the desired output voltage and filters to clean the
noise and reduction of harmonics.

Disadvantages with this configuration are the use of transformers, which are usu-
ally expensive, will decrease the efficiency, heavy and physically large [115]! To achieve
good performance with this configuration, several sensors and a control design which
takes into consideration the synchronization between the different current, voltages,
maximum power and effects of the environment over the PVM are required. To avoid
the use of transformers, minimize the number of required components, and to operate

the PVM in the optimal performance under changes in T and E,, it is proposed in

75

 

 

 

 

Tel/5

» /
mild gt}

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

—» M“
+ DC DC a 0
+l
v% I, ] V0 A
— DC T AC " D
PVM DC Bus Batery Transformern.
&Filter

Figure 6.1. PV Inverter System for utility applications.

this chapter a dynamic photovoltaic inverter system using a resonant Z-source con-
verter. The PV array will have the function to supply power to the load and to charge
the batteries. The resonant Z-source converter will have two functions to reduce or

eliminate the harmonics, and to amplify the ac voltage to the required rated voltage.

6.2 LRCM and FPM applied to commercial PV
modules

This section details additional commercial PV modules not presented in the previous
chapters. These commercial PV modules were added as a reference material. Some
of the PVM manufacturers are UniSolar (US), SunWize (OEM, SW), BP Solar (BP),
GE Photovoltaic (GEPV), Sanyo (ND, NE), Sharp (HIP, PC). The electrical speci-
fications of each PVM under STC were evaluated using the Linear Reoriented Coor-
dinates Method (LRCM) and Fractional Polynomial Method (FPM) to approximate
the Optimal current, the optimal voltage, the Optimal resistance, and the maximum
power.

Table 6.1 shows the electrical specifications for additional PV modules under STC.
Tables 6.2 and 6.3 show the approximations of the Optimal current, 10?, the Optimal

voltage, Vop, the Optimal resistance, R0,, and the maximum power, Pmax.

76

Table 6.1. Electrical specifications for commercial PV modules under STC

 

 

 

 

PVM Model I..(A) V0,,(V) 10,,(4) V0,,(V) R0,,(0) Pm, (W) b
US—3 0.40 12.0 0.33 8.1 24.55 2.67 0.1890
US-5 0.37 23.8 0.30 16.5 55.00 4.95 0.1864
OEM5 0.38 20.5 0.31 16.4 52.90 5.08 0.1183
SWPV-IO 0.66 21.0 0.59 16.8 28.47 9.91 0.0891
OEMlO 0.70 21.0 0.61 16.4 26.89 10.00 0.1068
US-11 0.78 23.8 0.62 16.5 26.61 10.23 0.1966
OEM20 1.38 21.0 1.22 16.5 13.52 20.13 0.0995

SWPv.20 1.21 21.0 1.19 16.8 14.12 19.99 0.0487
sx—20 1.29 21.0 1.19 16.8 14.12 19.99 0.0782
US-21 1.59 23.8 1.27 16.5 12.99 20.96 0.1941
SX—30 1.94 21.0 1.78 16.8 9.44 29.90 0.0802
US-32 2.40 23.8 1.27 16.5 8.51 32.01 0.1880
BP34O 2.54 21.8 2.31 17.3 7.49 39.96 0.0859
OEM40 2.68 21.0 2.40 16.7 5.96 40.08 0.0907
US-42 3.17 23.8 2.54 16.5 6.50 41.91 0.1925
BP350 3.17 21.8 2.89 17.3 5.99 50.00 0.0851
SW50 3.40 21.0 3.05 16.4 5.38 50.02 0.0964

GEPV-050 3.30 22.0 2.90 17.3 5.97 50.17 0.1013
SW55 3.65 21.0 3.30 16.7 5.06 55.11 0.0873
SW60 3.95 21.0 3.60 16.7 4.64 60.12 0.0845
US—64 4.80 23.8 3.88 16.5 4.25 64.02 0.1880
BP365 3.99 22.1 3.69 17.6 4.77 64.94 0.0787

GEPV-O72 4.80 21.0 4.40 17.0 3.86 74.80 0.0767
BP375 4.75 21.8 4.35 17.3 3.98 75.25 0.0834
NE-80U1 5.30 21.3 4.67 17.1 3.66 79.86 0.0926
BP375 4.80 22.1 4.55 17.6 3.87 80.08 0.0689
SW85 5.70 21.4 4.88 17.4 3.57 84.91 0.0964
SW90 5.90 21.4 5.17 17.4 3.37 89.96 0.0895
SW100 6.70 21.0 6.00 16.7 2.78 100.20 0.0907
SW115 7.70 21.0 6.89 16.7 2.42 115.06 0.0909
US—116 4.80 43.2 3.88 30.0 7.73 116.40 0.1872
SW120 8.00 21.0 7.18 16.7 2.33 119.91 0.0899

ND-L3EIU 8.10 21.3 7.16 17.2 2.40 123.15 0.0894
BP3160 4.80 44.2 4.55 35.1 7.71 159.71 0.0697
165—PC 5.40 44.5 4.72 35.0 7.42 165.20 0.1031
175—PC 5.43 44.6 4.95 35.4 7.15 175.23 0.0850

HIP-IQOBA3 3.75 67.5 3.47 54.8 15.79 190.16 0.0725

 

77

Table 6.2. PVM parameter approximation using the LRCM under STC

 

 

 

PVM Model 1,,(4) V,,,(V) R,,,(O) Pa, (W)
US—3 0.3264 8.2100 25.1514 2.6799
US—5 0.3028 16.3261 53.9246 4.9428
OEM5 0.3351 15.3231 45.7229 5.1352
SWPV—lO 0.6012 16.4744 27.4037 9.9040
OEMlO 0.6253 15.9827 25.5605 9.9938
US—ll 0.6315 16.1608 25.5898 10.2060
OEM20 1.2428 16.1791 13.0183 20.1073
SWPV—20 1.1510 17.9072 15.5578 20.6115
32520 1.1891 16.8145 14.1404 19.9943
US-21 1.2907 16.2004 12.5519 20.9095
sx-30 1.7845 16.7519 9.3874 29.8939
US—32 1.9606 16.3000 8.3137 31.9582
BP340 2.3217 17.2020 7.4092 39.9383
OEM4O 2.4371 16.4296 6.7415 40.0402
US-42 2.5776 16.2264 6.2953 41.8244
BP350 2.9004 17.2300 5.9406 49.9734
SW50 3.0725 16.2656 5.2939 49.9760

GEPV—050 2.9660 16.8983 5.6973 50.1209
SW55 3.3313 16.5285 4.9616 55.0605
SW60 3.6163 16.6155 4.5946 60.0863
US-64 3.9212 16.3000 4.1568 63.9164
BP365 3.6761 17.6790 4.8092 64.9890

GEPV-O72 4.4321 16.8655 3.8053 74.7490
BP375 4.3538 17.2828 3.9696 75.2453
NE—80U1 4.8094 16.6070 3.4530 79.8691
BP375 4.4692 18.0263 4.0334 80.5639
SW85 5.1506 16.5738 3.2178 85.3657
SW90 5.3723 16.7788 3.1232 90.1411
SW100 6.0927 16.4296 2.6966 100.1006
SW115 6.9999 16.4215 2.3459 114.9495
US-116 3.9244 29.6099 7.5450 116.2013
SW120 7.2809 16.4520 2.2596 119.7864

ND-L3EIU 7.3761 16.7026 2.2644 123.2011
BP3160 4.4656 35.9961 8.0608 160.7430
165-PC 4.8439 34.0784 7.0354 165.0711
175.120 4.9683 35.2525 7.0955 175.1454

HIP-19OBA3 3.4781 54.6566 15.7146 190.1003

 

78

Table 6.3. PVM parameter approximation using the FPM under STC

 

 

 

PVM Model Iopf(A) Vopf(V) Rap, (fl) Pmaf (W) n q
US-3 0.3264 8.1922 25.0989 2.6739 4 0.4346
US—5 0.3033 16.3270 53.8355 4.9516 4 0.5452
OEM5 0.3357 15.4388 45.9875 5.1831 7 0.5811

SWPV—10 0.6003 16.5362 27.5466 9.9267 10 0.0552
OEMIO 0.6247 16.0512 25.6940 10.0272 8 0.2966
US-ll 0.6335 16.1670 25.5200 10.2418 4 0.3244
OEM20 1.2411 16.2410 13.0861 20.1566 8 0.9345
SWPV-20 1.1476 17.8728 15.5743 20.5105 18 0.3857
SX-20 1.1865 16.8510 14.2026 19.9931 11 0.4600
US—21 1.2943 16.2054 12.5210 20.9740 4 0.3764
SX-30 1.7808 16.7929 9.4302 29.9040 11 0.1824
US-32 1.9644 16.3018 8.2986 32.0232 4 0.5096
BP340 2.3170 17.2489 7.4446 39.9650 10 0.3884
OEM40 2.4332 16.4876 6.7761 40.1176 9 0.8588
US-42 2.5841 16.2306 6.2809 41.9419 4 0.4107
BP350 2.8943 17.2748 5.9687 49.9976 10 0.4959
SW50 3.0665 16.3139 5.3200 50.0272 9 0.1959

GEPV-050 2.9626 16.9680 5.7274 50.2691 8 0.7803
SW55 3.3251 16.5792 4.9861 55.1268 10 0.2331
SW60 3.6088 16.6597 4.6164 60.1221 10 0.5779
US-64 3.9288 16.3018 4.1493 64.0465 4 0.5096
BP365 3.6673 17.7131 4.8300 64.9600 11 0.3659

GEPV—072 4.4238 16.9115 3.8228 74.8133 11 0.7596
BP375 4.3441 17.3237 3.9879 75.2558 10 0.7024

NE-80U1 4.8045 16.6816 3.4721 80.1473 9 0.6970
BP375 4.4566 18.0357 4.0470 80.3783 12 0.9784
SW85 5.1504 16.6728 3.2372 85.8709 9 0.3703
SW90 5.3684 16.8620 3.1410 90.5222 10 0.0989
SW100 6.0830 16.4876 2.7104 100.2939 9 0.8588
SW115 6.9889 16.4801 2.3580 115.1785 9 0.8289
US-116 3.9321 29.6166 7.5320 116.4548 4 0.5305
SW120 7.2689 16.5084 2.2711 119.9976 9 0.9422

ND-L3EIU 7.3685 16.7770 2.2768 123.6220 10 0.0737
BP3160 4.4526 36.0123 8.0879 160.3497 12 0.8183
165-PC 4.8392 34.2271 7.0729 165.6308 8 0.6285
175—PC 4.9579 35.3444 7.1290 175.2325 10 0.5008

HIP-19GBA3 3.4712 54.7818 15.7820 190.1561 12 0.4484

 

79

 

 

 

 

Figure 6.2. PVM connected to an incandescent light bulb.

6.3 PV modules connected to resistive loads

Figure 6.2 shows a PVM connected to a light bulb. It is desired to calculate the power,
current, and voltage supplied by the PVM to a 3509 light bulb under STC conditions.
The PVM parameters are I,C is 0.0182A, V0C is 8.0V and b is 0.10. Using Kirchoff’s
Voltage Law, it is possible to set the relationship between the current supplied by the
PVM and received by the light bulb as given in (6.1).

I(V) = 0.0185 — 0.0185 - exp (1.25 - V — 10) = 31% (6.1)

The supplied voltage is calculated using numerical analysis where V = 5.9177V, V is
then substituted in (6.1) where I (5.9177) = 0.0169A. Finally, the power supplied to
the load is 0.1W.

Figure 6.3 shows a 150W load connected to two different PV arrays at 25°C and
900W/m2. Remembering the definition for a PV array, the interconnection of two or
more PV modules with the same electrical characteristics in series (3) or parallel (p),
the dimension for the PV array is given by (s x p) and the total amount of PV modules
that form the PV array can be calculated multiplying s by p. The array PVl has 9
(i.e. 3 x 3) SX-lO PV modules connected in 3 series (3 = 3) and 3 parallel (p = 3)

80

 

 

Figure 6.3. Two distinct PV Arrays connected in series to a 150W load.

for each series. The array PV2 has 18 (i.e. 3 x 6) SA-5 PV modules connected in 3
series (3 = 3) and 6 parallel (p = 6) for each series. The electrical characteristics for
the PV modules SA-5 and SX-10 are give by the Table 2.1. The relationship between
the power supplied by the two PV arrays connected in series and the 150W load, is
given by (6.2).

P(V)=50-I—5--Iln(1—0.5-I)+57-I—5.7-Iln(1—0.556-I) =150W (6.2)

Using numerical analysis, it can be found that the current supplied by both PV
arrays is I = 1.4874A, the voltage supplied by the array PV1 is 49.018V and for the
array PV2 is 51.832V. Finally, the power produced by the array PV1 is 72.905W
and by the array PV2 is 77.095W. It is important to notice that both arrays are
Operating at the combined power required by the load and not operating at their

maximum power levels.

6.4 PVM connected to a RLC Load

Figure 6.4 shows a PVM, Solarex SX-IO, under STC connected to a transmission line,

L,, to supply 7.5W to a RLC load. The dynamic equations for the PVM connected

81

   

1 _
vc::cm>R
(T,Ei) - - 1 T

 

 

 

Figure 6.4. PVM connected to a RLC load.

to a RLC load are given by (6.3) — (6.5) with state variables V0, I L and I . The voltage
produced by the PVM (6.6) is calculated using the inverse of (2.10) with respect to
the current, I where the internal resistance, Ca: is zero. The supplied apparent power

5' (t) is calculated by multiplying (6.6) and the solution of (6.5) with respect to the

 

 

 

current.

BVC __ I — 1,,

52‘ 7 a (6'3)
8h_%—Rh

8t _ L (6.4)

(91 V — VC
5? _ L9 (6.5)
V-Vrc+b-V:c-ln1—i+—I—- —1 (66)

‘ Ir [1: °Xp b ‘

so) = v .1 = P(t) + j . on) (6.7)

Simulations were done using Simulink to observe the performance of a PVM con-
nected to an RLC load. The parameters for the SX-10 are I 2: = 0.65A, Va: = 21.0V,

= 0.8394, the transmission line L3, is 160nH, and the load parameters are L is

160,uH, C is 1000/1F and R is 509.

Figures 6.5 - 6.10 show the simulation waveforms of how the power, voltage, and

82

Iv

  

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

Figure 6.5. PVM SX—lO supplied voltage, V vs. t.

current are changing over time for the PVM connected to an RLC load. Figures 6.5
and 6.8 show the supplied voltage by the PVM and the voltage for the RL load. It
is clear that the supplied voltage waveforms have harmonics injected to the PVM
by the load. Also, the voltage in the RL load is a smooth waveform without ripple.
Figures 6.6 and 6.9 show the supplied current and the load current where a small
ripple produced by the change of the voltage in the load current.

The supplied and output power are shown in figures 6.7 and 6.10. Figure 6.10
shows the supplied power reaching the maximum power and then stabilizing to the
requiered power for the load. The power in the load is increased and stabilized up to
the required level with a small ripple. Finally, thaee simulations are consistent with
the theoretical analysis and, at the same time, prove the effect of nonlinear loads in

a photovoltaic module.

83

 

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

Figure 6.6. PVM SX—lO supplied current, I vs. t.

 

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

Figure 6.7. PVM SX—10 supplied power, I - V vs. t.

84

 

20

 

. . ' I . .
.
. ' , o
. .
. 0 ° , .
. 0 ' ' .
. e
. I ' .
p. ................................... . ............................................................ —1
. r . _ .
. e
. , . '
. ' ,
. . u '
. 0 .
, .
o
, . . . . .
e
_ . - . , .
o
_ u _ . . .
l . o
p ............................................... _ ................................................. n1
_ . ~ . .
. .
e
, . . . . .
. ' ' _ . ' '
. . ‘ . .
, n
. - ‘ , .
. - '
u - . , .
. . ' .
. . . . .
y. ................................................................................................ J
. 6 ' .
. . . '
e ' ' . b '
.
. I ‘ _ .
. u

c . . a
' l
.
u ' ' . o '
, n
. - . , .
. . -
, . - . . .
............................................................................................
o . . I
_ - ‘ . I .
, e
o ' . . .
. .
. ' , .
_ .

‘ .
- . . .
. ' '
. ' 5
, o
. . ' . .
. I I
---------------------------------------------------------------------------------------------
N , 0
, e ‘ .
. . u '
. ' . u '
-
u - ' . .
, .
. ' .
. -

 

 

1 1 i

 

0.40

0.35

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

Figure 6.8. VC Load Voltage, Vc vs. t.

 

 

l

 

 

 

l

 

0 0.02 0.04

0.06 0.08 0.1 0.12 0.14 0.16

Figure 6.9. IL Load Current, 1,, vs. t.

85

 

ll

 

 

 

 

’ p. oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo d
6 ............................................................................................... ..
g p ............................................................................................... q
..
4 - ............................................................................................... cl
3 ................................................................................................ ..
z y. .............................................................................................. q
.
............................................................................................... ..[
.
.
.
.
, .
. .
0 1 1 1 i 1

 

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

Figure 6.10. IL . VC Load Power, IL . VC vs. t.

6.5 Optimal Duty Ratio for a dc-dc Converter for
PV Applications

Consider a PVM connected to a buck-boost converter to supply power to a resistive
load. The objective is to calculate the Optimal duty ratio, D, so the PVM will supply
Pmax. The analysis will be done using the steady-state conditions for a buck-boost
converter, where all the components are ideal, the inductor current is continuous, the
capacitor is large enough to assume a constant output voltage and the switch is closed
for time D/ f and open for (1 — D) / f , where f is the frequency. An advantage of
the buck-boost converter is that the magnitude Of the output voltage can be either
greater than or less than the source voltage, depending on the duty ratio Of the switch
[116], making it excellent for photovoltaic applications where the weather conditions

are changing very fast. The only minor disadvantage for the buck boost converter is

86

the polarity reversal on the output.

The first step for load matching will be done using the relationship between the
voltage input and output for a buck-boost converter relationship. The load resistance
R, can be seen as voltage output, V2,, divided by current output, Io. Using this infor-
mation, the relationship between the input resistance, 12,, and the output resistance,
R0, is given by (6.8). If V is Vop, hence R,- is Rap, the optimal duty cycle, D, can
be solved. The optimal duty ratio, D, is Obtained and only depends on R, and Rap.
Switching at the Optimal duty ratio guarantees that the power supplied to load is
Pmax.

Vo -D - V.- 02 - vi 132 - R:

= —— (6.8)

R°=Z=(1_D).1,=(1_D)2.1, (1—D)'~’

 

Using (6.8), the optimal duty ratio, D, as a relationship of the optimal resistance,
Rap, and output resistance, R0, can be solved and is given in (6.9). Additionally,
if the power input and the power output are both Pmax i.e. Pi = P0 = Pmax, D
can be expressed as a relationship between the Optimal voltage, Vop, and the output

voltage, V0 as given in (6.10).

JR;

 

 

D = 6.9
m4. fizz; ( )

V0
D = v, + V0,, (6.10)

For design purposes, the minimum inductance me for the buck-boost converter
to preserve the continuous current mode using the optimal duty cycle is given in

(6.11). The voltage output ripple using the optimal duty cycle is given in (6.12).

 

 

 

 

R-(l—D)2 170.30,, R,
2f 2-f-(\/R_o+\/I2;)2<2'f ( )
D 1 (6.12)

Vori e:—_=
”0’ f-C-Ro C.f-(R,+,/Ro-'R—,,,)

87

The same type of procedure is done to calculate the duty cycle for the buck con-
verter or boost converter. Table 6.4 shows the conditions and Optimal duty ratio for
a buck converter, boost converter and buck—boost converter. The only disadvantage
of using a buck or boost converter is the restriction in the values of R0,, and R0 for

both cases.

Table 6.4. Optimal Duty Ratio for Different dc—dc Converters for Load Matching

 

 

Converter for Po 3 Pmax for P,- = P0 = Pm,“c Conditions

 

_ = 313., = V,z
Buck Boost D «Tr—0+ R0,, D Vo+vop none
Boost D=1—,/%f- D=1—Kv°5 R,>R,,,
Buck D: %f 13:15; R0,,>R,

 

Finally, this method for load matching can be integrated to other algorithms such
that the linear reoriented coordinates method, LRCM, which was described in details
in Chapter 3. Using the LRCM, the Optimal resistance, Rap, is calculated under any
changes in T or E,. Also, R0,, can be calculated using I a: and V2: as given in Chapter
4 using fractional polynomials, then the Optimal duty ratio is calculated using the
Table 6.4 to control the dc-dc converter and transfer the desired P from the PVM to
the load.

6.6 Algorithm and Simulations for a dc—dc Con-
verter using Load Matching

Figure 6.11 shows a proposed algorithm measuring E, and T to obtain the Optimal

duty cycle for load matching. Figure 6.12 Shows a photovoltaic system with a dc-dc

88

 

. I Calculate Calculate
Read E1 ' T Rap uringll 16) D from #12 Table 61 F6

Figure 6.11. Algorithm to calculate the Optimal duty ratio given E,- and T.

 

Sun Light (1, Bi)

11111111
11111111
........

      
  

T = 30C T hermacouples .
Ei=1100 W/nf .4 ’
Ix=0.75A ‘

Pyranometer

Figure 6.12. Integrated PV power system using load matching and the optimal duty
ration given E,- and T.

converter to supply power to a load, using a pyranometer to measure the irradiance
level and thermocouples to measure the temperature over the PVM surface. The
photovoltaic system has a Sharp ND—208U1 PVM with P”m is 208W, R0, is 2.659, V0,,
is 23.48V, I a: is 0.75A, V2 is 30V and b is 0.1, connected to a dc bus with capacitance
400pF. The dc-dc converter is a 50kHz buck-boost converter with inductance 10011H
and capacitance 40011F, and the resistive load is 0.759. Figures 6.13 and 6.14 show
the transient results simulations for the photovoltaic system and the dc—dc converter
connected to a load. The simulations were done using Simulink. These results show
how effective the proposed method can be to calculate the optimal duty ratio to

deliver the required power (e.g. Pm) using load matching.

89

h 250

$203
3150

N
0'!

PVM Voltage

Figure 6.13. PVM power and voltage with respect to the time.

,_ 250

$200-
3150’
100-

50
0

0

PVM

Load Voltage

-15

Figure 6.14. Load power and voltage with respect to the time.

A...”
0010010

—'-s .
o 0'

 

T

 

Pmax = 208W

 

 

 

 

 

Vop = 23.48V

 

 

 

0 0.005 0.01 0.015 0.02 0.025 0.03
time (s)

 

Pmax = 208W

 

 

 

 

Vo = -12.49V

 

 

 

0 0.005 0.01 0.015 0.02 0.025 0.03
time (s)

90

 

 

 

 

 

 

Figure 6.15. PVM connected directly to a dc motor.

6.7 PVM connected to a dc motor

Figure 6.15 shows a PVM connected to a dc motor. Using figure 6.15, the dynamic
equations for the system are given by (6.13), (6.15), and (6.14) where Ii is equal to
IL and Vrn is equal to V. The variable Lm is the armature inductance (H), Rm is
armature resistance (9), w is the rotor speed (rad/s), Vm is the dc motor terminal
voltage (V), I Lm is the dc motor armature current (A), TL is the load torque (N - m),
J is the rotor inertia (N /m2), K is the torque and back emf constant (NmA‘l),
d is the damping constant (N ms). Now, consider a dc motor with the following
parameters Lm is 55mH, Rm is 7.569, J is 0.068N/m2, d is 0.03475N ms, TL is zero,
and K is 3.475N mA'1 connected to a PV array of 16 SX-20 PVM’s under STC with
parameters Ia: is 1.29A, V2: is 21V, b is 00782,}? is 4 and s is 4. Simulink was used
to simulate the dynamic equations (6.13), (6.15), and (6.14). The results are shown
in the figure 6.16. At steady state, the dc motor will be running at 62rad/s with
a supplied PV power of 285W and voltage operation of 89V. Unfortunately, a PV
array connected directly to a dc motor cannot be set to a desired speed electronically.
The only way to control the speed is by increasing or decreasing the temperature or

effective irradiance level making this type of PV system very impractical.

 

 

 

8V III: V 1 I,-
— = - 1— — — — .1
at C2: — Ca: - exp (‘71) [ exp (b- V1: b)[ C2: (6 3)

91

4,35
4 4.90
f 4 75
S

0 4.70

PVM Voltage. V

0* 4.85
4 .60

 

2 4 6 8101214
limb)

Speed (rad/s)

 

0246 8101214
time(s)

 

 

a — J ‘ T ' 7 (614)
BILm_Vm Rm-ILm Kw
8t ' Lm Lm _ Lm (6'15)

Figure 6.17 shows a PVM connected to a buck—boost converter and a dc motor. Using
a buck-boost converter, a dc motor can be controlled to achieve a desired speed. It
should be noted that, the maximum speed for a dc motor will depend in the maximum
power provided by the PV array. The dynamic equations that describe the figure 6.17
are given by (6.13), (6.15), (6.14), (6.16), (6.17) and (6.18).

I,- = 5 ~ IL (6.16)

92

 

 

 

 

 

 

 

Figure 6.17. PVM connected to a buck-boost converter and a dc motor.

BIL S-l S
‘57-‘74”? (618)

The speed tracking control design is based on the fixed duty ratio, D, to control the
buck-boost converter. The fixed duty ratio, D, is calculated using Table 6.4 and the
steady state performance of the dc motor as given by (6.19), (6.20) and (6.21). The
variable I Lm, is the steady state armature current, wr is the steady state speed, Vm*
is the steady state terminal voltage for the dc motor, P* is the power that should be
supplied from the PVM to the dc motor for speed wr. Also, the the internal steady

state resistance for the dc motor, R*, is given by (6.22).

 

 

d'QU’l" TL
1171*: — -
L K +K (619)
Vm*=Rm-1Lm4+K-wr=(R72.d+K)-wr+BI—?-TL (6.20)

P* = ILm.-Vm*=Rm-I£m,+K-1Lm. -wr

d2 2-Rm-d Rm

93

 

 

Figure 6.18. Algorithm to calculate the fixed duty ratio, D, using the FPT.

Vm*2_ ((Rm-d+K2)-wr+Rm-TL)2
P* .— (Rm-d2+d-K2)-wr2+(2-Rm-d+K2)~wr-TL+Rm-T§
(6.22)

R*=

 

It is assumed that the buck—boost converter does not lose power, hence the PVM
. power is transferred directly to the dc motor. Using the Fixed Point Theorem, it is
possible to calculate the voltage of operation, V, to produce the required power for
the dc motor, P*. The algorithm to calculate V is given by (6.23). The variable 5 is
the maximum allowed error to stop the iteration. Table 6.4 provides the equation to
calculate the duty ratio to transfer desired power using a buck—boost converter. Fi-
nally, the fixed duty ratio, D, is calculated by (6.24). As a summary, figure 6.18 show
the algorithm to calculate the fixed duty ratio, D, using the Fixed Point Theorem,
FPT.

while|V(n +1) — V(n)| S e
P :1: —P >1: exp (—%)

 

 

V(n + 1) = (6.23)
Iz—Im-exp(:%}—%)
D— ”R" — 1 — 1 (624)
“mm 1+(/% 1+\/%’:.—?¥

94

As a note, if TL is zero then P25, Vm*, and R* are simplified to (6.25), (6.26), and

(6.27) respectively. Also, Ra: will be a constant value that will not depend on wr.

cl2
P* = (:K—z- - Rm + d) -wr2 (6.25)
Vm*=Rm-ILm4+K-wr= (2%9-+K)-wr (6.26)

Vm*2 (Rm - d+ K2)2
R“ _ P»: ‘ (Rm - d2 + d - K2) (6'27)

 

 

The following example will simulate a PVM connected to buck-boost converter and
a dc motor given a variable speed, wr, and TL which is equal to zero. The parameters
for the PVM are the following I a: is 0.3A, V1: is 21V, b is 0.08; for the buck-boost
converter C is 400pF, L is 10011H , the frequency is 20kHz; the parameters for the
dc motor are Rm is 19, Lm is 0.5HJ is 0.01N/m2, dis 0.01Nms, K is 0.1NmA"1.

f

01 Host<a

0.21 if5gt<10;
wr=l 0.15 if10_<_t<15;
0.20 if 15gt<20;

 

[0 if20$t§25.

Figure 6.19 shows the expected transition given the voltage and current for the PVM
to produce the required power for the dc motor to run at the reference speed, wr.
Figure 6.20 shows the tracking performance given the reference speed, wr. Also, it
is shown the changes in the necessary power to produce the required speed. Finally,
the same figure shows the transient for the voltage and current supplied by the PVM.

These simulations were done using Simulink.

95

 

 

 

 

 

 

 

 

 

 

 

 

  
   

 

 

 

 

 

 

 

 

 

 

 

 

 

Current (A), PVM Pm Power (W), P(W
E 25 r " f ‘ E E
. l . l
a. 2“ 2° 0.
Q 15 . ::.".::.';::::::::." 1 1 15 gt
\. I x.
m 10’ 1 l "' ‘10 m
00 _ "+- ————— I' 00
g 5 . 1 ‘l ' 1 5 S
B '- '—"'!-D-. m— I I I ‘6
k 0 . . . 1 I .I I 0 A
0 0.1 0.2 0.3 0 i 2. 4: i 6
4 '5 u
l I 1 l
I I .1 1
0.25 1 - I a I 0.25
1 1 ‘l l
"""" :::::’t::::r::::::: , .
:2} 02 + | ' 02 Q
8015 --5--»--+-— 515‘s
3. E
\H
x 01 --------- -----+--‘ -01 k
a
a
0.05' . 0.05
I30 5 10 15 20 25 0 2 I. 6 U
fime(s) dc Motor Power (110, P*

Figure 6.19. Expected transition between the PVM and the dc motor to produce wr.

96

 

 

T r r 0.35
0.30

3 0.25 - [ [ 1
E 0.20 - -
1 3 0.15 -
0.10 _
0.05 »

20.

 

 

 

 

 

 

 

Voltage (V)

 

 

 

 

 

 

 

 

0510152025 0510152025
limc(s) m6)

 

 

 

 

 

 

     

 

 

 

 

 

 

0 5 10 15 20 25 0 5 10 15 20 25
lime(s) 5105(5)

Figure 6.20. Results of a PVM connected to buck-boost converter and a dc motor.

97

6.8 Z-Source Converter

In the next sections, a photovoltaic system integrated with the Z—source converter for
ac power applications is presented. The Z-source converter is an impedance—source
power converter [117]. The Z-source concept can be applied to all dc-ac, dc—dc, ac-ac,
and ac-dc power conversion [27, 79, 118, 119]. The Z—source is a two-port network
that consists of a splits-inductor L1 and L2 and capacitors Cl and 02 connected in
an X shape. L1 and L2 can be provided through a split inductor or two separate
inductors. The Z—source converter employs a unique impedance network to couple
the converter main circuit to the power source, thus providing unique features that
cannot be obtained in the traditional voltage-source and current-source converters
where a capacitor and inductor are used, respectively [117].

The unique feature of the Z-source connected to an inverter is that the output
voltage can be any value between zero and infinity. The Z-source converter is a buck-
boost converter that has a wide range of obtainable voltage. Figure 6.21 shows the
Z-source and the Z—source dynamic model is described from (6.28) to (6.31). The
voltage and current inputs of the Z-source are given by Vd and Id, and voltage and

current outputs are given by V; and I 8.

BVCI 1

 

 

 

a, = 5—1 . (1L2 — I.) (6.28)
0g? = é . (1,,1 — I.) (6.29)
5%; = .LLl . (vd — Va) (630)
35:2 ——— g - (vd — Vol) 16.31)

V, = VC1 + VC2 — V, (6.32)

Id = [1,1 + 1,,2 -— I, (6.33)

98

1
ll

 

 

 

+01 +

Vd Vs

L2

Figure 6.21. Z-Source configuration.

6.9 Proposed PVIS using the Z-Source Converter

and Load Matching Control

Figure 6.22 shows the proposed PV inverter system. The PVM is connected to a
buck-boost converter, to control the voltage, Vi, of the PVM to it’s Optimal value,
Vop. The maximum power for the PVM will result from the optimal voltage, Vop.
The dc output voltage of the buck-boost converter will charge the battery, Vb, and it
is connected to an inverter and the Z-source working under resonance where the dc
voltage of the battery will be boosted to the desired ac voltage output. The dynamic
equations (6.34) and (6.35) describe a buck-boost converter. The dynamic model for
the battery connected to an inductance is given by (6.37) and (6.36). The inverter
model is described by (6.38) and (6.39). The load is an induction motor described by
(6.40). The proposed PVIS dynamic model is described using (6.28) to (6.41).

%_S-IL—IL—Iz
815— C

 

(6.34)

%_S'Vi+VC—S-VC
at — L

 

(6.35)

99

 

 

 

033.07—

 

 

 

 

 

 

 

% = 315 . (Vc _ Vb) (6.36)
1,, = I“, + I. (6.37)

Vd = (S2c + S, — 1) - Vc (6.38)
I2 = (S; + S, — 1) - Id (6.39)
if; = 31;, ~1v. — Rm .1.) (6.40)
D: “,wa (a...)

The buck-boost converter is controlled to transfer PM to the battery using the 0p-
timal duty ratio, D, given by (6.41). This switching, based on the optimal duty ratio
will result in the maximum power and the optimal voltage from the PVM as given
in Table 6.4. The components for the proposed PVIS are 13 PVM’s SQ80 connected
in parallel with an internal capacitance Ca: of 39nF. The electrical characteristics
for a PVM SQ8O are given in Table 2.1. Using these values, the maximum power
provided by the PV array is around 1040W and the optimal voltage is 18.1V under
STC. The buck—boost converter has an inductance, L, of 0.1mH and capacitance, C,
of 0.4mF. The battery should be charged and kept to the rated voltage of 36V. Also,
the bank of batteries is connected to an inductance, Lb, of 0.1mH. The Z-source was

designed to work in resonance to eliminate the use of transformers and filters. The

100

t4 (1,0)

(SX.Sy) Sx 8y Duration Vd
t1 1 8.1057ms-Vb

1
13(00) 0 1111.1) t2 0 1 1.8943ms 0
-Vdc \ Vdc t3 0 O 8.1057ms Vb
t4 1 0 1.8943ms 0
12(01)

Figure 6.23. Diagram for the phase shifting control with a frequency of 50H 2.

parameters for the Z-source are: L1 is 2mH, L2 is 5mH, C 1 is lmF and CZ is 6mF.
The load is a single phase induction motor described by a resistance, Rm, of 1552 and
an inductance, Lm, of 0.039H. The power required for the load is 960W with a rated
voltage of 120V(rms) and a frequency of 50Hz.

The single phase inverter is controlled using the basic technique of phase shifting
control. Figure 6.23 shows the diagram for the phase shifting control with a frequency
of 50Hz and the duration of time for each state. The inverter voltage output enters the
resonant Z-source converter and is amplified 4.7 times. Also, most of the harmonics
are eliminated with a THD of less than 3%. Figure 6.24 shows the output voltage,
Vd, and output current, Id for the inverter using phase shifting control. Figure
6.25 shows the performance of the resonant Z-source output voltage for the proposed
PVIS; hence, the induction motor is receiving a quasi-sinusoidal voltage waveform and
sinusoidal current. Finally, advantages of the proposed PVIS are that the use of a
transformer is eliminated, harmonics are reduced, and the maximum power and rated

output voltage are supplied to the load which in this case is the induction motor.

101

 

400 -

300

200

100 -

-100

-200 -

-300

-400

 

 

 

 

9.91 9.92 9.93 9.94 9.95 9.96 9.97 9.98 9.99 10

 

150

100

50

 

 

Iiii'll'L

i
9.90 9.91 9.92 9.93 9.94 9.95 9.96 9.97 9.98 9.99 10

 

 

Figure 6.25. Induction motor input voltage, V3, with a frequency of 50Hz.

102

CHAPTER 7

Conclusions

7 .1 Summary

This work presented the modeling and analysis of solar distributed generation. All the
areas that constitute solar distributed generation were covered in this work, including
the modeling and mathematical approximation for a photovoltaic module, algorithms
for maximum power point tracking, dc-dc converter control, PV circuit analysis, es-
timation of the climate conditions, inverter design for residential applications. Each
chapter is summarized in the next paragraphs.

Chapter 2 showed an analytical model for a photovoltaic module. The PVM
model takes into consideration the manufacturer data sheet, the temperature, and the
irradiation level. This model has been verified using different types of PVM’s giving
excellent results to simulate the I-V and P-V curves. Also, this model can be used to
calculate the internal resistance of operation for a PVM. The proposed model has the
advantage of producing, not only the I-V curves provided by the manufacturer, but
also the P-V curves, R-V curves, P-I curves under changes in the temperature and
effective irradiance level. Unfortunately, the proposed PVM cannot provide direct
analytical solutions do not exist to calculate the optimal voltage, current and the

maximum power using differential calculus. This PVM model was used in other

103

chapters to develop MPPT algorithms, irradiance level and temperature estimation
algorithms, PV applications in the area of power systems, etc.

Chapter 3 presented a method called Linear Reoriented Coordinates Method
(LRCM). The LRCM is a nontraditional method to be applied for functions without
the diffeomorphism property. With the use of the LRCM, solutions to obtain the
approximate maximum value fm for a function, f (:6) = a: - g(x), will be obtained
using g(x) and the linear equation, 91(17). Another application for the LRCM is the
approximation for the symbolic solutions of Pm, V0,, and L,p for a PVM hence the
LRCM can be considered as a MPPT algorithm. Also for PVM applications, the
LRCM is more practical for simulations due to the symbolic solutions. Additionally,
the LRCM can be integrated into other Optimization methods. The LRCM may be
applied to other fields like math, geology, civil engineering, economy and mechanical
engineering.

Chapter 4 described the Fractional Polynomial Method (FPM) to approximate the
exponential functions that describe the performance for a PVM. Also, this chapter
described a second method to approximate a fractional polynomial by a sufficiently
close integer polynomial. This method is called the Integer Polynomial Approxima-
tion Method (IPAM). Several examples were shown and verified using the data of
different PVM’s. These results proved that the proposed methods FPM and IPAM
were excellent to approximate the PVM exponential model and could be applied
to other systems. The proposed methods are very useful as tools to solve and ap—
proximate certain types of exponential functions keeping the boundary conditions,
shape and performance of the original exponential model. The following paragraphs
summarize the FPM and the IPAM with the advantages and disadvantages of each
approximation or model.

The FPM assumes the same boundary conditions, shape and similar performance

as the original PVM model. The F PM is excellent to estimate the analytical solutions

104

for the optimal voltage, current and the maximum power. The parameters n and q
are unique values with a direct relationship to the characteristic constant, optimal
voltage and Open circuit voltage. Also, this method can be considered as an additional
alternative Maximum Power Point Tracker, (MPPT) due to the fact that n and q are
non-variant. An approximate solution of Vop can be Obtained measuring the online
signals [:6 and V2: and used as a reference for maximum power tracking purposes.

The IPAM is an approximation Of the FPM. The performance for the integer
polynomial is very close to the fractional approximation and uses the parameters 71
and q previously calculated. The integer polynomial method is useful for Lyapunov
analysis, custom polynomial PV source simulation and for ALU programming.

Chapter 5 described four algorithms, which are capable Of calculating the effective
irradiance level and temperature over a PVM. The prOposed algorithms eliminate
the use Of thermocouples and pyranometers, reducing the cost and complexity Of a
PV power system. Thase algorithms have several advantages such as being easy to
execute and very efficient at using the data provided by the PVM and having very fast
convergence. Three Of the algorithms use only the data provided by the voltage sensor
and current sensor, and are excellent at tracking changes in the temperature and
irradiance level in a geographic region over long periods Of time. These algorithms can
be integrated into MPPT and other monitoring algorithms, and can be implemented
in RT Linux or a fast controller like a DSP. The algorithms have high accuracy (3%),
working as effectively a high cost pyranometer without the high price. They also work
like an integrated thermocouple without affecting PV power system performance.
Finally, these algorithms are excellent for monitoring in remote areas

Chapter 6 presented new contributions to the field Of solar energy conversion.
One of the contributions was circuit analysis for a PVM connected to different loads.
Another Of the contributions is the determination Of the Optimal duty ratio for load

matching to transfer the maximum power from a PVM to a load. Also, the optimal

105

duty ratios for different types of dc-dc converters for PV applications were derived.
The equations for the Optimal duty ratios can be integrated with other algorithms
such as LRCM to calculate the PVM internal resistance.

Also, in this chapter was presented a novel single phase photovoltaic inverter
system for power applications. The proposed PVIS has the advantages that it takes
into consideration the dynamic model for a PVM, it has an Optimal duty cycle for
the buck-boost converter to transfer the maximum power to the load, and it uses
the Z—source converter in resonance to eliminate the use Of transformers and filters
to the load. The dynamic PVM model is excellent for power applications due to the
fact that the model considers the temperature and effective irradiance level and it is
useful for calculating the Optimal duty ratio for a buck-boost converter. Finally, with
the use Of a resonant Z-source converter, the desired voltage output can be achieved
with minimal harmonics.

Finally, there are some issues that could be explored in the future related to this
work. One possibility for future work could be the physical residential implementa-
tion Of the prOposed photovoltaic inverter system using the Z-source converter which
can be compared with the traditional photovoltaic inverter system. Research in the
area Of bifurcations and chaotic behavior in photovoltaic systems could be explored
in the future as well. It is possible that the typical problems found in power systems
like voltage collapse or instability can be predicted in photovoltaic systems using the
proposed PVM model. Future work could be done in applying the techniques devel-
Oped in this work to a problem where several alternative systems are interconnected
and supplying power to different loads. In this type Of problem, issues as choosing the
correct model to describe sources like wind turbines, fuel cells, geothermic systems
could be explore as well as consequences arising from having these alternative power
systems connected to the utility grid could be explored. Also, how economically viable

they are to produce power and invest in them.

106

BIBLIOGRAPHY

107

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

BIBLIOGRAPHY

University of Texas. Today’s Solar Radiation for Austin, Texas.
http://ccwf.cc.utexas.edu/ solar/today.html, 2006. cited March 16, 2006.

University of Georgia. Georgia Solar Radia-
tion Database. http://www.griffin.uga.edu/aemn/cgi-
bin/AEMN.pl?site=AAAA&report=XO, 2006. cited April 16, 2006.

F. Z. Peng. Editorial Special Issue on Distributed Power Generation. IEEE
Trans. Power Electronics, 19:1157—1158, September 2004.

G. Batsukh, D. Ochirbaani, C. Lkhagvajav, N. Enebish, B. Ganbat,
T. Baatarchuluun, K. Otani, and K. Sakuta. Evaluation Of solar energy po-
tentials in Gobi desert area of Mongolia. In Proc. World Conference on Photo-
voltaic Energy Conversion, pages 2262—2264, 2003.

R. Hoffrnann and P. Mutschler. The influence Of control strategies on the energy
capture of wind turbines. In Proc. IEEE Industry Applications Conference,
volume 2, pages 886—893, Oct. 2000.

P. Mutschler and R. Hoffmann. Comparison Of wind turbines regarding their
energy generation. In Proc. IEEE Power Electronics Specialists Conference,
volume 1, pages 6—11, June 2002.

D. Pruitt. The simulation Of building integrated photovoltaics in commercial
Office buildings. In Proc. International Building Performance Simulation Assoc.
Conf, pages 685—688, 2001.

J. Wang, F.Z. Peng, J. Anderson, A. Joseph, and R. Buffenbarger. Low cost

fuel cell converter system for residential power generation. IEEE Trans. Power
Electronics, 19(5):1315—1322, Sept. 2004.

J. Wang, F.Z. Peng, J. Anderson, A. Joseph, and R. Buffenbarger. Low cost
fuel cell inverter system for residential power generation. In Proc. IEEE Applied
Power Electronics Conference and Exposition, volume 1, pages 367-373, 2004.

108

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]

K. Chomsuwan, P. Prisuwanna, and V. Monyakul. Photovoltaic grid-connected
inverter using two-switch buck-boost converter. In Proc. IEEE Photovoltaic
Specialists Conference, pages 1527—1530, May 2002.

T.L Dinwoodie and BS. Shugar. Optimizing roof-integrated photovoltaics: a
case study Of the PowerGuard roofing tile. In Proc. IEEE Photovoltaic Special-
ists Conference, volume 1, pages 1004—1007, Dec. 1994.

D. Cruz-Martins, R. Demonti, and R. Ruther. Analysis Of utility interactive
photovoltaic generation system using a single power static inverter. In Proc.
IEEE Photovoltaic Specialists Conference, pages 1719-1722, Sept. 2000.

J.M.A. Myrzik and M. Calais. String and module integrated inverters for single—
phase grid connected photovoltaic systems - a review. In Proc. IEEE Power
Tech Conf, volume 2, pages 23—26, 2003.

S. Saba and VP. Sundarsingh. Novel grid-connected photovoltaic inverter. IEE
Proceedings Generation, Transmission and Distribution, 143(2):219-224, March
1996.

State Energy Conservation Office. www.1nfinitePower.org. cited on March 17,
2006.

T. Shimizu, O. Hashimoto, and G. Kimura. A novel high-performance utility-
interactive photovoltaic inverter system. IEEE Trans. Power Electronics,
18(2):704—711, March 2003.

CR. Sullivan and M.J. Powers. A high-efficiency maximum power point tracker
for photovoltaic arrays in solar-powered race vehicle. In Proc. IEEE Power
Electronics Specialists Conference, pages 574—580, June 1993.

F.Y. Wang, 0. Kwan, and TY. Yi. Dynamic modeling Of rotating flexible
platforms. In Proc. IEEE Conf. on Decision and Control, pages 1315—1316,
Dec. 1992.

J .M.A. Myrzik M. Calais. String and Module Integrated Inverters for Single-
Phase Grid Connected Photovoltaic Systems A Review. In Proc. IEEE Power
Tech Conf, 2003.

J. Schmid. PV in commercial buildings. In Proc. IEE Colloquium on Electricity
Generation by Commercial Customers Using Photovoltaics, pages 6/ 1—6/ 5, Oct.
1994.

109

[21] S. Xiaofeng, W. Weiyang, W. Baocheng, and W. Xiaogui. A research on photo-
voltaic energy controlling system. In Proc. IEEE International Conference on
Electrical Machines and Systems, volume 1, pages 542—545, Aug. 2001.

[22] B.G. Streetman and S. Baerjee. Solid State Electronic Devices. Prentice Hall,
5th edition, 1999.

[23] B. Marion and J. Adelstein. Long-term Performance Of the SERF PV Systems.
In Proc. NCPV and Solar Program Review Meeting, pages 199-201. National
Renewable Energy Laboratory, 2003.

[24] C. Prapanavarat, M. Barnes, and N. Jenkins. Investigation of the performance
Of a photovoltaic AC module. IEE Proceedings Generation, Transmission and
Distribution, 149(4):472—478, July 2002.

[25] A. Hussein, K. Hirasawa, H. Jinglu, and J. Murata. The dynamic performance
Of photovoltaic supplied dc motor fed from DC—DC converter and controlled
by neural networks. In Proc. IEEE International Joint Conference on Neural
Networks, volume 1, pages 607—612, May 2002.

[26] K. Khouzarn and K. Hoffman. Technical guidelines for the interconnection Of
PV generation systems in Queensland. In Proc. IEEE Photovoltaic Specialists
Conference, pages 1349—1352, Oct. 1997.

[27] Y. Huang, M. Shen, and F. Z. Peng. A Z-Source Inverter for Residential Pho-
tovoltaic Systems. In Proc. International Power Electronics Conference, April
2005.

[28] Resource Assessment Program. Average daily solar radiation per month in usa.
http://www.nrel.gov/solar. cited April 21, 2006.

[29] T. Markvart and R.J. Arnold. Integration Of photovoltaic converters into the
public electricity supply network. In Proc. IEE Colloquium on Power Electron-
ics for Renewable Energy, pages 4/1—4/4, June 1997 .

[30] V. Quaschning and R. Hanitsch. Influence Of shading on electrical parameters of
solar cells. In Proc. IEEE Photovoltaic Specialists Conference, pages 1287-1290,
May 1996.

[31] W. Xiao, W.G. Dunford, and A. Capel. A novel modeling method for photo-
voltaic cells. In Proc. IEEE Power Electronics Specialists Conference, volume 3,
pages 1950—1956, June 2004.

110

[32]

[33]

[34]

[35]

[36]

[37]

[38]

[39]

[40]

[41]

[42]

[43]

L. Shengyi and R. A. Dougal. Dynamic multiphysics model for solar array.
IEEE Trans. Energy Conversion, 17(2):285—294, June 2002.

T. Markvart. Principles Of the radiation damage in solar cells. In Proc. IEE
Colloquium on Solar Cells for Space Applications, pages 3/1—3/1, Nov. 1988.

A. J. Anderson. Final Report for Task 2.0. subcontract TAD-4-14166—01, NREL,
Oak Leaf Place, 1995.

G. Blaesser. PV Array Data Translation Procedure. In Proc. European Photo-
voltaic Solar Energy Conference, 1995.

C.-T Chiang, T.-S. Chiang, and H.-S. Huang. Modeling a photovoltaic power
system by CMAC—GBF Photovoltaic Energy Conversion. In Proc. World Con-
ference on Photovoltaic Energy Conversion, volume 3 Of 3rd, pages 2431—2434,
May 2003.

S. Coors and M. Bohm. Application Of the Two-Exponential Model to Cor-

rection Procedures for Silicon Solar Cells. In Proc. EuroSun, pages 614—619,
1996.

S. Coors and M. Bohm. Validation and comparison Of curve correction prO-
cedures for silicon solar cells. In Proc. European Photovoltaic Solar Energy
Conference, pages 220—223, 1997.

Th. F. Elshandatter, M. T. Elhagry, E. M. Abou—Elzahab, and A. A. T. Elk-
Ousy. Fuzzy modeling Of photovoltaic panel equivalent circuit. In Proc. IEEE
Photovoltaic Specialists Conference, pages 1656—1659, Sept. 2000.

EM. Gaddy, R. Cikoski, and D. Mekadenaumporn. Curve fitting solar cell
degradation due tO hard particle radiation. In Proc. World Conference on
Photovoltaic Energy Conversion, volume 1, pages 723-725, May 2003.

IEC 891 International Electrotechnical Commission. Procedures for temperature
and irradiance corrections to measured I—V characteristics of crystalline silicon
photovoltaic devices, 1987.

IEC 904 International Electrotechnical Commission. Photovoltaic devices. Part

1: Measurement of photovoltaic current-voltage characteristics, 1987.

D.L. King, J .K. Dudley, and WE. Boyson. PVSIM: a simulation program for
photovoltaic cells, modules, and arrays. In Proc. IEEE Photovoltaic Specialists
Conference, pages 1295—1297, 1996.

111

[44] C. M. Whitaker, T. U. Townsend, H. J. Wenger, A. Iliceto, G. Chimento, and
F. Paletta. Effects of irradiance and other factors on PV temperature coef-

ficients. In Proc. IEEE Photovoltaic Specialists Conference, volume 1, pages
608-613, Oct. 1991.

[45] E. I. Ortiz-Rivera and F.Z. Peng. Analytical Model for a Photovoltaic Mod-
ule using the Electrical Characteristics provided by the Manufacturer Data

Sheet. In Proc. IEEE Power Electronics Specialists Conference, pages 2087—
2091. IEEE, June 2005.

[46] S. Jain and V. Agarwal. A new algorithm for rapid tracking Of approximate
maximum power point in photovoltaic systems. IEEE Power Electronics Let-
ters, 2(1):16—19, March 2004.

[47] Y.-Ch. Kuo, T.-J. Liang, and J .-F. Chen. Novel maximum power-point-tracking
controller for photovoltaic energy conversion system. IEEE Trans. Industrial
Electronics, 48(3):594—601, June 2001.

[48] Y. H. Lim and D. C. Hamill. Simple maximum power point tracker for photo-
voltaic arrays. IEEE Power Electronics Letters, 36(11):997—999, May 2000.

[49] W. Swiegers and J .H.R. Enslin. An integrated maximum power point tracker
for photovoltaic panels. In Proc. IEEE International Symposium on Industrial
Electronics, volume 1, pages 40—44, July 1998.

[50] J .A. Jiang, T.L. Huang, Y.T. Hsiao, and CH. Chen. Maximum Power Ti‘acking
for Photovoltaic Power System. Tamkang Journal of Science and Engineering,
8(2):147—153, 2005.

[51] Ch. Hua and Ch. Shen. Study Of maximum power tracking techniques and
control Of DC/ DC converters for photovoltaic power system. In Proc. IEEE
Power Electronics Specialists Conference, volume 1, pages 86-93, 1998.

[52] Ch. Hua, J. Lin, and Ch. Shen. Implementation Of a DSP-controlled photo-
voltaic system with peak power tracking. IEEE Trans. Industrial Electronics,
45(1):99—107, Feb. 1998.

[53] T. Hiyama and K. Kitabayashi. Neural network based estimation of maximum
power generation from PV module using environmental information. IEEE
Trans. Energy Conversion, l2(3):241—247, Sept. 1997.

112

[54]

[55]

[56]

[57]

[58]

[59]

[60]

[61]

[62]

[63}

A. Moreno, J. Julve, S. Silvestre, and L. Castaner. A fuzzy logic controller for
stand alone PV systems. In Proc. IEEE Photovoltaic Specialists Conference,
pages 1618—1621, 2000.

T.J. Liang, J.F. Chen, TO. M, Y.C. Kuo, and CA. Cheng. Study and Im-
plementation Of DSP-based Photovoltaic Energy Conversion System. In Proc.

IEEE Power Electronics and Drive Systems Conference, volume 1, pages 807—
810,2001.

M.A.S. Masoum, H. Dehbonei, and BF. Fuchs. Theoretical and experimen-
tal analyses Of photovoltaic systems with voltage and current-based maximum
power-point tracking. IEEE Trans. Energy Conversion, 17(4):514—522, Dec.
2002.

Y.H. Lim and DC. Hamill. Nonlinear phenomena in a model spacecraft power
system. In Proc. Workshop on Computers in Power Electronics, pages 169—175,
July 1998.

J.H.R. Enslin, M.S. Wolf, D.B. Snyman, and W. Swiegers. Integrated pho-
tovoltaic maximum power point tracking converter. IEEE Trans. Industrial
Electronics, 44:769—773, Dec. 1997.

O. Dranga, C.K. Tse, and H.H.C. Iu. Bifurcation behavior Of a power-factor-
correction boost converter. International Journal of Bifurcation and Chaos,
13(10):3107—3114, 2003.

Y.H. Lim and DC. Hamill. Chaos in spacecraft power systems. IEEE Power
Electronics Letters, 35(6):510—511, March 1999.

EL Ortiz-Rivera and F.Z. Peng. Linear Reoriented Coordinates Method. In
Proc. IEEE Electra/Information Technology Conference. IEEE, May 2006.

EL Ortiz-Rivera and F.Z. Peng. A novel method to estimate the maximum
power for a photovoltaic inverter system. In Proc. IEEE Power Electronics
Specialists Conference, volume 3, pages 2065-2069. IEEE, 2004.

El. Ortiz-Rivera. A Novel Method Of Area Optimization for a System Modeled
with Transcendental Functions. In Proc. AIAA International Energy Conver-

sion Engineering Conference. American Institute Of Aeronautics and Astronau-
tics, Aug. 2005.

113

[64]

[65]

[56]

[67]

[68]

[69]

[70]

[71]

[72]

[73]

[74]

[75]

P. Royston and W. Sauerbrei. Stability of multivariable fractional polynomial
models with selection Of variables and transformations: a bootstrap investiga-
tion. Journal Statistics in Medicine, 22(4):639—659, Jan. 2003.

G. Ambler and P. Royston. Fractional polynomial model selection procedures:
investigation Of Type I error rate. Journal of Statistical Simulation and Com-
putation, 69:89—108, 2001.

P. Royston, G. Ambler, and W. Sauerbrei. The use Of fractional polynomials
to model continuous risk variables in epidemiology. International Journal of
Epidemiology, 28:964—974, 1999.

K. Galkowski and A. Kummert. Fractional polynomials and nD systems. In

Proc. International Symposium on Circuit and Systems, volume 3, pages 2040—
2043, May 2005.

G. Maione. Laguerre approximation Of fractional systems. IEEE Power Elec-
tronics Letters, 38(20):l234—1236, Sept. 2002.

S. Samadi, M. Omair-Ahmad, and M.N.S. Swamy. Exact Fractional-Order Dif-
ferentiators for Polynomial Signals. IEEE Power Electronics Letters, 11(6):529—
532, June 2004.

Ch. Liu. Gabor based kernel PCA with fractional power polynomial models
for face recognition. IEEE Trans. Pattern Analysis and Machine Intelligence,
25(5):572—591, May 2004.

D. Matignon and B. d’Andrea Novel. Observer-based controllers for fractional
differential systems. In Proc. IEEE Conf. on Decision and Control, pages 4967-
4972, Dec. 1997.

SP. Ellner and J. Guckenheimer. Dynamic Models in Biology. Princeton Uni-
versity Press, lst 2006.

SC. Gilmour and LA. Trinca. Fractional Polynomial Response Surface Mod-
els. Journal of Agricultural, Biological, and Eviromental Statistics, 10(1):50—60,
2005.

Hans Lauwerier. Fractals Endlessly Repeated Geometrical Figures. Princeton
University Press, 1991.

BB. Mandelbrot. The Fractal Geometry of Nature. W. H. Freeman, 1982.

114

[76]

[77]

[78]

[79]

[80]

[81]

[82]

[83]

[84]

[85]

[86]

[87]

El. Ortiz—Rivera, U.C. Wejinya, and F.Z. Peng. PVM model approximation
using fractional polynomials. In Proc. IEEE Electro/Information Technology
Conference, Aug. 2006.

D. R. Smart. F ixed Point Theorems. Cambridge University Press, 1974.

EL Ortiz-Rivera and F.Z. Peng. Algorithms to estimate the temperature and
effective irradiance over a photovoltaic module using the Fixed Point Theorem.
In Proc. IEEE Power Electronics Specialists Conference, 2006.

El. Ortiz-Rivera and F.Z. Peng. A Dynamic Photovolotaic System Using the Z-
Source Converter. In Proc. AIAA International Energy Conversion Engineering
Conference. American Institute of Aeronautics and Astronautics, Aug. 2005.

Underwriters Laboratories (UL) . PVM Specifications.
www.ul.com/dge/photovoltaics.

D. Li and RH. Chou. Maximizing Efficiency Of Solar-Powered Systems by Load
Matching. In Proc. IEEE International Symposium on Low Power Electronics
and Design, pages 162—167, Aug. 2004.

SOLAREX. Sa-05 solar panel datasheet.
http: / / www.southwestpv.com / Catalog / PDF / BP cited January 16, 2005.

R. Larson. Calculus With Analytic Geometry. D C Heath & CO, 1994.

MC. Potter. Principles 69' Practice of Electrical Engineering: The Most Efli-
cient and Authoritative Review Book for the PE License Exam. Great Lakes
Press, lst edition, 1998.

K. Holland and F.Z. Peng. Control Strategy for Fuel Cell Vehicle Traction
Drive Systems Using the Z-Source Inverter. In Proc. IEEE Vehicle Power and
Propulsion Conference. IEEE, Sept. 2005.

S. P. Fekete and W. R. Pulleyblank. Area Optimization Of Simple Polygons. In
Proc. Annual Computational Geometry, pages 173—182. Association for Com-
puting Machinery, 1993.

K. Wang and W. K. Chen. Floorplan area Optimization using network analogous

approach. In Proc. International Symposium on Circuit and Systems, volume 1,
pages 167—170. IEEE, April 28 May 3 1995.

115

[88] C.-H. Chen and LG. Tollis. Efficient area Optimization for multi—level spiral
floorplans. In Proc. International Symposium on Circuit and Systems, volume 1,
pages 25—28. IEEE, May 1992.

[89] T. Kim and T. Adali. Complex backpropagation neural network using elemen-
tary transcendental activation functions. In Proc. IEEE International Confer-
ence on Acoustics, Speech, and Signal, volume 2, pages 1281—1284. IEEE, May
2001.

[90] R. Balaniuk, E. Mazer, and P. Bessiere. Fast direct and inverse model acqui-
sition by function decomposition. In Proc. IEEE International Conference on
Robotics and Automation, volume 2, pages 1535—1540. IEEE, May 1995.

[91] R. Sang-Moon, P. Dong-JO, and O. SeOk-Yong. Inverse mapping using FLS and
its application to control. In Proc. IEEE International Conf. on Systems, Man
and Cybernetics, volume 2, pages 1245—1249. IEEE, October 1995.

[92] M. D. Ercegovac, T. Lang, J .-M. Muller, and A. Tisserand. Reciprocation,
square root, inverse square root, and some elementary functions using small
multipliers. IEEE Trans. on Computers, 49(7):628-637, July 2000.

[93] A. M. Eydgahi and S. Ganesan. Genetic-based fuzzy model for inverse kine
matics solution Of robotic manipulators. In Proc. IEEE International Conf.
on Systems, Man and Cybernetics, volume 3, pages 2196—2202. IEEE, October
1998.

[94] H. K. Khalil. Nonlinear Systems. Prentice Hall, 3rd edition, 2001.

[95] F. L. Lewis and V. L. Syrmos. Optimal Control. Wiley-Interscience, 2nd edition,
1995.

[96] Vesna M. Sekulic Miroljub D. Milojevic. Some comparative considerations Of
revenue,economy, profit, and profitability functions. Journal Facta Universitatis
Series: Economics and Organization, 2:81—91, 2004.

[97] H. Petroski. Design Paradigms: Case Histories of Error and Judgment in
Engineering. Cambridge University Press, lst edition edition, 1994.

[98] T. J. Rivlin. An Introduction to the approximation of Functions. Dover Publi-
cations, 1981.

[99] M. Gianquinta and G. MOdica. Mathematical Analysis Aprorimation and Dis-
crete Processes. Birkhauser, 2004.

116

[100]

[101]

[102]

[103]

[104]

[105]

[106]

[107]

[103]

[109]

[110]

V. L. Zaguskin. Handbook of numerical methods for the solution of algebraic
and transcendental equations. Pergamon Press, 1961.

T.F. E1 Shatter and MT. Elhagry. Sensitivity analysis Of the photovoltaic model

parameters. In Proc. IEEE Midwest Symposium on Circuits and Systems, pages
914—917, 1999.

R. Gottschalg, M. Rommel, D.G. Infield, and M. J. Kearney. The influence of
the measurement environment on the accuracy Of the extraction of the physical

parameters Of solar cells. Journal Measurement Science and Technology, 1:797—
804, 1999.

D.L. King, W.E. Boyson, and P. Kratochvil. Analysis of factors influencing the
annual energy production Of photovoltaic systems. In Proc. IEEE Photovoltaic
Specialists Conference, pages 1356—1361, 2002.

M.W. Davis, A.H. Fanney, and RP. Dougherty. Evaluating building integrated
photovoltaic performance models. In Proc. IEEE Photovoltaic Specialists Con-
ference, pages 1642—1645, 2002.

MA. Abella, E. Lorenzo, and F. Chenlo. Effective irradiance estimation for PV

applications. In Proc. World Conference on Photovoltaic Energy Conversion,
pages 2085—2089, 2003.

Bureau Of Meteorology Australian Government. Solar Radiation Definitions.
http: / / www.bom. gov.au / sat / glossary, 2006.

F. Chenlo, N. Vela, and J. Olivares. Comparison between pyranometers and
encapsulated solar cells as reference PV sensors-outdoor measurements in real
conditions. In Proc. IEEE Photovoltaic Specialists Conference, pages 1099—
1104,1990.

D.L. King, W.E. Boyson, B.R. Hansen, and W.I. Bower. Improved accuracy
for low-cost solar irradiance sensors. In Proc. IEEE Photovoltaic Solar Energy
Conversion, pages 1947—1952, 1998.

A. Hunter Fanney, B.P. Dougherty, and M.W. Davis. Performance and char-
acterization Of building integrated photovoltaic panels. In Proc. IEEE Photo-
voltaic Specialists Conference, pages 1493—1496, 2002.

OJ. Van Dan B08 and A. Van Dan Bos. Solar radiation sensors: applications,
new detector development, characterization and classification according to ISO

117

9060. In Proc. IEEE Instrumentation and Measurement Technology Conference,
pages 175—178, 1995.

[111] D.L. King, J .A. Kratochvil, and WE. Boyson. Temperature coeflicients for PV
modules and arrays: measurement methods, difliculties, and results. In Proc.
IEEE Photovoltaic Specialists Conference, pages 1183—1186, April 1999.

[112] Pico Technology Limited. Thermocouple Application Note.
http: / / www.picotech.com / applications / thermocouplehtml. cited Febru-
ary 10, 2006.

[113] W. Rakowski. The fixed point theorem in computing magnetostatic fields in a
nonlinear medium. IEEE Trans. Magnetics, 182391—392, March 1982.

[114] V. Mainkar and KS. 'Ifivedi. Sufficient conditions for existence of a fixed
point in stochasticreward net-based iterative models. IEEE Trans. Software
Engineering, 22(9):640—653, September 1996.

[115] S.J. Chiang and CM. Liaw. Single-Phase T hree-Wire Transformerless Inverter.
In IEE Proc. on Electric Power Applications, volume 141, pages 197—205. IEE,
July 1994.

[116] D.W. Hart. Introduction to Power Electronics. Prentice Hall, lst edition, 1996.

[117] F.Z. Peng. Z-source inverter. IEEE Trans. Industry Applications, 39(2):504—
510, Apr. 2003.

[118] F.Z. Peng, Y. Xiaoming, F. Xupeng, and Q. Zhaoming. Z-source inverter for
adjustable speed drives. IEEE Power Electronics Letters, 1(2):33—35, June 2003.

[119] F.Z. Peng, A. Joseph, J. Wang, M. Shen, L. Chen, Z. Pan, E.I. Ortiz-Rivera, and
Y. Huang. Z-source inverter for motor drives. IEEE Trans. Power Electronics,
20(4):857—863, July 2005.

118