MSc Dissertation Final
66
In Partial Fulfilment of the Requirements for the Master of Science Degree in New and Renewable Energy Author: Ariel Villalón M. Supervisor: Dr. Katerina Fragaki SCHOOL OF ENGINEERING AND COMPUTING SCIENCES MSc in New and Renewable Energy 15 th August of 2011. Stand Alone Photovoltaic System Modelling
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Transcript of MSc Dissertation Final
In Partial Fulfilment of the Requirements for the Master of Science Degree in New and
Renewable Energy
Author: Ariel Villalón M.
Supervisor: Dr. Katerina Fragaki
SCHOOL OF ENGINEERING AND COMPUTING SCIENCES
MSc in New and Renewable Energy
15th
August of 2011.
Stand Alone Photovoltaic System Modelling
ii
Table of Contents
List of figures ................................................................................................................................... iv
List of tables ...................................................................................................................................... v
Acknowledgements .......................................................................................................................... vi
Nomenclature .................................................................................................................................. vii
1. Introduction ................................................................................................................................... 1
1.1 Background ............................................................................................................................. 1
1.2 Objective ................................................................................................................................. 2
1.3 Dissertation outline .................................................................................................................. 3
2. Literature Review .......................................................................................................................... 4
2.1 Irradiance and solar radiation ................................................................................................... 4
2.2 The solar cell ........................................................................................................................... 4
2.2.1 Structure ........................................................................................................................... 4
2.2.2 Operating principles .......................................................................................................... 5
2.2.3 Equivalent circuit of a solar cell ........................................................................................ 8
2.2.4 Variations from the basic behaviour .................................................................................. 9
2.3 The Photovoltaic generator .................................................................................................... 11
2.4 The Battery ............................................................................................................................ 15
2.5 The Charge regulator ............................................................................................................. 18
2.6 Stand Alone Photovoltaic Systems ......................................................................................... 19
2.6.1 Sizing of stand-alone PV systems .................................................................................... 20
2.6.2 Modelling of stand-alone PV systems .............................................................................. 21
3. Methodology ............................................................................................................................... 25
3.1. Requirements ........................................................................................................................ 25
3.2. Modelling and simulation...................................................................................................... 26
3.2.1 PV module ...................................................................................................................... 26
3.2.2 Battery ............................................................................................................................ 29
3.2.3 Charge controller ............................................................................................................ 31
iii
3.2.4 Load ............................................................................................................................... 32
4. Experimental data ........................................................................................................................ 33
4.1 Stand-alone PV system .......................................................................................................... 33
4.2 Irradiance and temperature data.............................................................................................. 34
5. Modelling of the stand-alone PV system in MATLAB/SIMULINK ............................................. 37
5.1 Implementation in MATLAB/SIMULINK ............................................................................. 37
5.2 The PV Module ..................................................................................................................... 39
5.3 Charge controller ................................................................................................................... 40
5.4 Battery: SIMULINK subsystem diagram ................................................................................ 42
5.5 Data input to the system ......................................................................................................... 43
6. Results and Discussion ................................................................................................................ 44
6.1 Simulations December 3, 2002 ............................................................................................... 44
6.1.1 PV model ........................................................................................................................ 44
6.1.2 Battery model.................................................................................................................. 47
6.2 Simulation for the Battery, December, 2002 ........................................................................... 50
7. Conclusions ................................................................................................................................. 53
8. Further work ................................................................................................................................ 55
Bibliography ................................................................................................................................... 56
iv
List of figures
Figure 1. Structure of a conventional silicon solar cell ....................................................................... 5
Figure 2. Illuminated I-V characteristic .............................................................................................. 6
Figure 3. Maximum-power point and other operating parameters ....................................................... 7
Figure 4. Equivalent circuit of an ideal solar cell ................................................................................ 8
Figure 5. Equivalent circuit of a non-ideal, one diode solar cell .......................................................... 9
Figure 6. Circuit diagram of a photovoltaic generator....................................................................... 11
Figure 7. Equivalent circuit of a battery ........................................................................................... 15
Figure 8. Diagram of series regulator ............................................................................................... 18
Figure 9. Stand-alone photovoltaic system ....................................................................................... 20
Figure 10. Block diagram for the entire system ................................................................................ 25
Figure 11. Newton‟s method ............................................................................................................ 28
Figure 12. Block diagram of charge series regulation ....................................................................... 31
Figure 13. Schematic representation of the stand-alone PV system ................................................... 33
Figure 14. Irradiance versus time data, December 3 2002, Southampton, UK ................................... 34
Figure 15. Temperature versus time data for 3th of December 2002 ................................................. 34
Figure 16. Irradiance and ambient temperature, December 2002 ...................................................... 35
Figure 17. Stand-alone PV system simulation flowchart ................................................................... 37
Figure 18. Stand-alone PV system structure in SIMULINK ............................................................. 38
Figure 19. SIMULINK diagram for the charge controller ................................................................. 41
Figure 20. SIMULINK diagram for the model of the lead-acid battery ............................................. 43
Figure 21. SIMULINK diagram for the subsystem for the input data ................................................ 43
Figure 22.Comparison experimental data with simulated data .......................................................... 44
Figure 23. Voltage from the PV Module for December 3, 2002, Southampton, UK .......................... 45
Figure 24. Comparison of experimental data and the simulated data ................................................. 46
Figure 25. Comparison of experimental data and the simulated data ................................................. 47
Figure 26. Comparison between experimental and simulated data for the SOC of the batteries ......... 48
Figure 27. Comparison between experimental and simulated data for the battery current .................. 49
Figure 28. Simulated Voltage of the battery for December 2002 ...................................................... 50
Figure 29. Comparison between experimental voltage and simulated voltage for December 2002 .... 50
Figure 30. State of charge of the battery, December 2002................................................................. 51
Figure 31. Comparison between experimental and simulated data for the battery current .................. 52
v
List of tables
Table 1. Values from Sharp NT9075 PV module datasheet .............................................................. 26
Table 2. Characteristics battery SunLyte 12-5000X ......................................................................... 29
Table 3. Truth table for switches in charge regulation ...................................................................... 32
vi
Acknowledgements
This dissertation is part of an MSc in New and Renewable Energy programme carried out at the
School and Engineering and Computing Sciences, Durham University.
First of all, I would like to thank to my beloved parents, Ady and Gustavo, that have supported me all
this time far from my mother land and helped to achieve this new challenge in my life. To my brother,
Mauricio, and friends in Chile that trusted me, and always supported me during all those time and
especially during this last year.
To my supervisor, Dr. Katerina Fragaki, for her guidance and support for developing this project. I
have to thank especially to the people of Chile that by the support of CONICYT Becas Chile
Scholarships Programme, have enabled me to come to the United Kingdom to study. This
achievement is by and for you. I am indebted to the great friends that I have made here in UK, that
without expecting any reward in return, have helped me in many ways to success throughout this MSc
programme.
Finally, to my beloved and unforgettable Grandma Elena, that beyond this World, has always lighted
my path.
vii
Nomenclature
AC alternate current
AM relative air mass
C capacity of a battery to the working conditions at certain moment
C1 constant of proportionality between g and ISC (A cm2 mW-1)
C10 capacity of a battery over a 10-hour discharge regime
C2 constant of proportionality between(Tc-Ta) and G (°C cm2 mW-1)
CA normalized capacity of a photovoltaic generator
Co* no-load capacity at 0°C (A-s) in the Jackey battery model (1)
CS normalized capacity of an accumulator
CU useful capacity of a battery
D battery self-discharge rate
DC direct current
DOC depth of charge
e charge of an electron (1.602 x 10-19 C)
EG0 bandgap at zero K
eV electron-volt (1.6 x 10-19 Joules)
f'(xk) derivative of the approximation of the root of an equation
FF fill factor
FF0 fill factor for an ideal solar cell, with Rs = 0
G direct solar irradiance
G0 1000 W/m2
Gd mean value of daily irradiation on the surface of a photovoltaic generator
Gd(0) daily global irradiation on a horizontal surface
I current output of a solar cell
I* nominal battery current
I0 diode saturation current of a solar cell
Ibat current flowing through a battery
Icharge charging current of a battery
Id current through a diode or dark current
Idischarge discharge current of a battery
IG output current of a photovoltaic generator
IL photocurrent or photogenerated current
Im main branch current of a cell of a battery
Imax maximum-power point current of a solar cell
In current output of a photovoltaic module at time n
In+1 current output of a photovoltaic module at time n+1
ISC short-circuit current of a solar cell
ISCG short-circuit current of a photovoltaic generator
ISCG,0 short-circuit current of a photovoltaic module under standard conditions
Iλ spectral irradiance
k Boltzmann's constant (1.3810 x 10-3 JK-1)
K charge/discharge battery efficiency
Kc constant in the Jackey battery model (1)
Kt temperature dependent look up table in the Jackey battery model (1)
L monthly mean of the energy consumed daily by the load of a photovoltaic system
LLP loss of load probability
Ln nocturnal energy consumption by the load of a photovoltaic system
LOE level of energy in a battery
m ideality factor (1 ≤ m ≤ 2)
MOSFET metal-oxide-semiconductor field-effect transistor
MPPT maximum power-point tracking
NCOT nominal cell operating temperature
viii
Np number of cells in parallel in a generator
Ns number of cells in series in a generator
ns number of 2V cells in series of a battery
P power output of a solar cell
PDe seasonal depth of discharge of a battery
PDmax maximum depth of discharge of a battery
PDd daily depth of discharge of a battery
PL radiant power incident on a solar cell
Pmax maximum power output of a solar cell under standard conditions
PMAXM maximum power output of a photovoltaic module under standard conditions
q present capacity of a battery
Q amount of current stored in a battery (Ah)
Qe extracted charge from a battery (A-s)
Qe_init initial extracted charge from a battery (A-s)
qmax nominal capacity of a battery
R1 internal resistance of a cell of the modelled battery
RBI internal resistance of a cell of a battery
Rch internal resistance of a battery in charge mode
Rdch internal resistance of a battery in discharge mode
Rp parallel resistance of a solar cell
Rs series resistance of a solar cell
RSG series resistance of a photovoltaic generator
SOC state of charge of a battery
SOC exp experimental value of the state of charge
SOC sim simulated value of the state of charge
SOC(t) instantaneous state of charge
SOC1 initial state of charge
SOCm maximum state of charge
T temperature
t time units
Ta ambient temperature
Tc temperature of a solar cell
Tref reference temperature under standard conditions of a photovoltaic cell
Usc cell threshold voltage in overcharge regulation in a battery
V voltage across a solar cell
V sim simulated value of the voltage
V1 battery voltage
Vbat voltage across the terminals of the cell of a battery
VBI internal voltage across the cell of a battery
Vch voltage of a battery in charge mode
Vdch voltage of a battery in discharge mode
Vexp experimental value of the voltage
Vfc final charging voltage per cell of a battery
VG voltage across a photovoltaic generator
Vg voltage across the cell of a battery at which gassing begins
Vmax maximum-power point voltage of a solar cell
Vmax_off voltage in a cell of a battery at which the battery is disconnected from the generator
Vmin_on voltage in a cell of a battery at which the battery is reconnected from the generator
Voc open-circuit voltage of a solar cell
voc normalized open-circuit voltage (voc = Voc/Vt)
VOCG open-circuit voltage of a photovoltaic generator
VOCG,0 open-circuit voltage of a photovoltaic module under standard conditions
Vt thermal voltage, equal to mkT/e
X illumination level or concentration factor
ix
xk approximation of the root of an equation
xk+1 next approximation of the root of an equation
δ constant in the Jackey battery model (1)
η energy-conversion efficiency of a solar cell
ηc Faraday efficiency of a battery
θ electrolyte temperature (°C)
μm micrometre
τ integration of the variable
τsc time constant for the voltage of battery in overcharge
Ω Ohm
1
1. Introduction
1.1 Background
Stand-alone PV systems are also called autonomous PV systems which are independent Photovoltaic
systems. They are normally used in remote or isolated places where the electric supply from the
power-grid is unavailable or not available at a reasonable cost. Some applications for such an
application are mountain huts or remote cabins, isolated irrigation pumps, emergency telephones,
isolated navigational buoy, traffic signs, boats, camper vans, etc. They are suitable for users with
limited power necessities (2).
In general, the components of a stand-alone photovoltaic system are the storage of energy (batteries),
charge controllers, and converters. About the storage of energy, only some applications using energy
directly from the sun, such as pumping or ventilation, can manage without storing energy (3). As the
merit of a stand-alone photovoltaic system should be judged by how reliably it supplies electricity to
the load (4), it is essential to add the accumulator to the system, to supply the electricity during the
hours that there is no solar irradiation or it is not enough according to the electricity demand for a
certain building or application. The charge controller represents, in a stand-alone PV system, less than
5% of the total cost of the system, which at first sight may suggest that this component is not
important, but its function, actually, is essential as its quality will deeply influence the final cost of the
energy produced by one of these systems (3), by protecting the batteries from overcharge and deep
discharge states.
Stand-alone PV systems often do not require an inverter like the grid-tied systems when being used
for particular cases. Since PV systems, whether grid-tied or stand-alone, produce electricity at the first
hand in the direct current type, they require an inverter if they are needed to be converted into
alternating current for supplying to the grid or running machinery which require AC (2).
As photovoltaics are rapidly becoming a mature industry, system engineering and design are turning
to be more important. In this sense, modelling of photovoltaics systems is becoming important as it is
necessary to predict the performance and behaviour under different ambient conditions. There are
several tools for designing and analyzing of photovoltaics systems that can be sorted in different
categories: feasibility, sizing simulation, and generic use. Some of the main tools that can be
mentioned are PVSYST, ILSE, ASHLING, Hybrid2, RAPSIM, PVSOL. These tools are developed
especially for photovoltaics systems. Additionally, PSpice and MATLAB/SIMULINK can be applied
to several systems and photovoltaics are not the exception (5). Considering this, it is that
2
MATLAB/SIMULINK has become one of the main computer environments for modelling and
simulation of photovoltaic systems.
Modelling and simulation of photovoltaic systems have been proposed in several studies and
references (6) (7) (8) (9), but in general, there is no enough development of specific models to
represent these kind of electric systems, especially considering for stand-alone photovoltaic systems.
(10).
In this project, a stand-alone photovoltaic system has been modelled to develop a simulation under
certain ambient conditions.
1.2 Objective
In this project a model of a stand-alone photovoltaic has been developed. For developing it, the
platform MATLAB/SIMULINK was chosen due to its flexibility and strength when modelling. A
simple model was modelled, including the photovoltaic generator, two batteries in parallel, a charge
controller, and a constant resistive load.
The scope of this project is to develop a model of a simple stand-alone PV system and to compare the
results obtained with previous work, in order to validate the results obtained, taking into account
possible inaccuracies.
The model developed was programmed to get close as much as possible to the real behaviour of these
devices, under certain conditions. To do that, two simulations were carried out, considering real data
for the solar irradiance and the ambient temperature.
The simulations were set to run for data from Southampton in the United Kingdom, during the month
of December of 2002.
The results are compared with experimental data, in order to see how the model simulates the current
from the photovoltaic generator, the voltage in it, the voltage in the batteries, their state of charge, and
the current passing through the batteries.
The results are discussed in order to explain differences between simulated and experimental data.
3
1.3 Dissertation outline
Chapter 1 gives an introduction to the problem and the importance of investigating the behaviour of
stand-alone PV systems by modelling and simulation under certain ambient conditions. Additionally,
in this chapter, the aims of the project are presented.
Chapter 2 provides a detailed review of literature regarding the principles and operation of
photovoltaics, and in specific, stand-alone PV systems and its components. Furthermore, the issues for
modelling stand-alone PV systems and its components are reviewed. Several considerations for
showing simulated behaviour is presented.
Chapter 3 describes the methodology followed in this dissertation, presenting analytically the
theoretical assumptions for modelling the different components of the stand-alone PV system.
Chapter 4 describes the experimental data that was previously obtained in another work, and that is
used to compare and validate the results obtained from the simulation ran using the model developed
in this dissertation. The configuration of the experimental is described and the data of solar irradiance
and ambient temperature that is used for the simulating the stand-alone PV system.
Chapter 5 describes how the system and all its components are modelled using
MATLAB/SIMULINK. The diagram for the whole stand-alone PV system is presented and its
subsystems with their correspondent diagrams and MATLAB codes to represent the charge controller,
batteries, PV module, and the data input subsystem.
Chapter 6 presents the results of the simulations carried out, and the results discussed and compared
with the experimental data available for the day and the month considered.
Chapter 7 the conclusions of the work developed are presented. The necessary improvements as the
strengths of the simulation carried out are presented.
Chapter 8 outlines the additional work that is possible to be developed as a next stage of this work,
but that is beyond the scope of this dissertation.
4
2. Literature Review
2.1 Irradiance and solar radiation
The radiation of the sun reaching the earth, distributed over a range of wavelengths from 300 nm to 4
micron approximately, is partly reflected by the atmosphere and partly transmitted to the earth‟s
surface. Photovoltaic applications used for space have sun availability different from that of PV
applications at the earth‟s surface. The radiation outside the atmosphere is distributed along the
different wavelengths in a similar fashion to the radiation of a „black body‟ following Planck‟s law,
whereas at the surface of the earth the atmosphere selectively absorbs the radiation at certain
wavelengths. Two different sun „spectral distributions‟ can be distinguished (11)
a) AM0 spectrum outside of the atmosphere
b) AM 1.5 G spectrum at sea level at certain standard conditions.
Several important magnitudes can be defined. Spectral irradiance, irradiance and radiation as follow
(11):
Spectral irradiance Iλ: the power received by a unit surface area in a wavelength differential
dλ. The units are W/m2μm.
Irradiance: the integral of the spectral irradiance extended to all wavelengths of interest. The
units are W/m2.
Radiation: the time integral of the irradiance extended over a given period of time, therefore
radiation units are units of energy. It is common to find radiation data in J/m2-day, if a day
integration period of time is used, or most often the energy is given in kWh/m2-day, kWh/m
2-
month or kWh/m2-year depending on the time slot used for the integration of the irradiance.
2.2 The solar cell
2.2.1 Structure
In conventional solar cells, the electrical field is created at the junction between two regions of a
crystalline semiconductor having contrasting types of conductivity. If the semiconductor is silicon,
one of these regions (the n-type) is doped with phosphorus, which has five valence electrons. This
region has a much higher concentration of electrons than of holes. The other region (the p-type) is
doped with boron, having three valence electrons. Here the concentration of holes is greater. The large
difference in concentrations from one region to another causes a permanent electric field directed
from the n-type region towards the p-type region. This is the field responsible for separating the
additional electrons and holes produced when light shines on the cell (4).
5
Figure 1. Structure of a conventional silicon solar cell
In silicon cells, the p-n junction is obtained by diffusing a layer of phosphorus into a wafer of silicon
previously doped with boron. The junction is very shallow, typically only about 0.2 to 0.5 µm deep.
This shallow, diffused layer is commonly called the emitter. The electrical contact with the
illuminated side of the cell (the side where the diffusion is done) has to leave most of the surface
uncovered otherwise light cannot enter the cell. However, the electrical resistance of the contact must
not be too high. The compromise usually adopted is to use contacts with the form of a comb, as seen
in figure 1. In contrast, the electrical contact on the dark side of the cell covers the whole surface of
the cell. Usually, an antireflective coating is applied to the illuminated side to increase the fraction of
incident light absorbed (4).
2.2.2 Operating principles
When a load is connected to an illuminated solar cell the current that flows is the net result of two
counteracting components of internal current: a) the photogenerated current or photocurrent, IL, due to
the generation of carriers by the light; and b) the diode or dark current, Id, due to the recombination of
carriers driven by the external voltage, this latter needed to deliver power to the load. Assuming these
currents can be superimposed linearly, the current in the external circuit can be calculated as the
difference between the two components, as follows (4):
Equation 1
According to equation 1, the current supplied by a solar cell to a load is that given by the difference
between the photocurrent IL and the recombination current Id(V), the latter being due to the bias from
the generated voltage. For the sake of simplicity it can be assumed that the current in the diode can be
expressed by a single exponential, the characteristic equation for the device is (4):
Equation 2
6
Where IL= photocurrent; I0 = diode saturation current; eV = electron-volt (1.6 x 10-19
Joules); m =
ideality factor (1 ≤ m ≤ 2); k = Boltzmann‟s constant (1.381 x 10-3
JK-1
)
So, equation 2 represents the I-V curve for a solar cell or an array. An example of I-V curve is
presented in figure 2:
Figure 2. Illuminated I-V characteristic
From equation 2, the greatest value of current with the cell as an electric source is obtained under
short-circuit conditions, when V = 0. So short-circuit current ISC is (4):
Equation 3
If the device is kept in open-circuit, so that I = 0, it biases itself with a voltage that is the greatest that
can arise in the first quadrant. This is the open-circuit voltage VOC. From equation 2 (4):
Equation 4
These two key parameters used in this project to characterize a PV cell are its short-circuit current and
its open-circuit voltage. These values are generally provided on the manufacturer`s data sheet.
In this way, the output current from the PV cell can be found using the equation:
Equation 5
Where Isc is the short-circuit current that is equal to the photon generated current, and Id is the current
shunted through the intrinsic diode. The diode current is given by the Shockley`s diode equation:
Equation 6
7
Where I0 is the reverse saturation current of the diode (A); q is the electron charge (1.602x10-23
C); Vd
is the voltage across the diode (V); k is the Boltzmann‟s constant (1.381x10-23
J/K); and T is the
junction temperature in Kelvin (K).
Combining the diode current equation with the equation for the output current of the PV cell creates:
Equation 7
Where V is the voltage across the PV cell, and I is the output current.
This simple PV cell models does not account for series resistance, and parallel resistance. Series
resistance accounts for any resistance in the current path through the semiconductor material, the
metal grid, contacts, and current collecting bus. In this way, the value of series resistance is multiplied
by the number of series-connected cells. Parallel resistance (or shunt resistance) is a loss associated
with a slight leakage current through a parallel resistive path to the device. It can be neglected as is
small and not as noticeable as series resistance because the effects are minimal unless a number of PV
modules are connected in parallel for a large system. Recombination in the depletion region of PV
cells provides a non-resistive current path in parallel with the intrinsic PV cell, and can be represented
by a second diode in the equivalent circuit (12).
Figure 3. Maximum-power point and other operating parameters
The region of the curve between ISC and VOC (figure 3) corresponds to operation of the cell as a
generator. If the energy is supplied to a resistive load, as shown in figure 3, the power supplied to the
resistance is given by (4):
Equation 8
8
As can be seen in figure 3, there exists an operating point (Imax , Vmax) at which the power dissipated in
the load is maximum, thus having the maximum-power point (4).
The product Imax Vmax, corresponds to the maximum power that can be delivered to the load, and is
represented in figure 3 by the area of the shaded rectangle, being smaller than the area corresponding
to the product Isc Voc. The more pronounced the elbow of the I-V curve, the closer the two products
come to being equal. Thus the ratio between them is called the fill factor FF (4):
Equation 9
So, the maximum power delivered by the cell to the load can be expressed as (4):
Equation 10
The energy-conversion efficiency of a solar cell is defined as the ratio between the maximum
electrical power that can be delivered to the load and the power PL of the radiation incident on the cell
(4):
Equation 11
Where Area = area of the solar cell; G = direct solar irradiance
2.2.3 Equivalent circuit of a solar cell
It is very useful to describe the behaviour of a solar cell by using circuit elements.
Figure 4. Equivalent circuit of an ideal solar cell
The circuit of figure 4 consists of an ideal p-n junction diode having saturation current I0 and ideality
factor m, and of an ideal current source IL, has the same behaviour as represented by equation 2. So,
this is the equivalent circuit of the ideal device (4).
9
In reality there exist other effects that must be considered, and affecting the external behaviour of the
cell. Those effects are the series resistance and current leaks proportional to the voltage, being the
latter usually characterized by a parallel resistance (4).
Figure 5. Equivalent circuit of a non-ideal, one diode solar cell
So, taking into account these resistances, as shown in figure 5, the following expression can be
obtained (4):
Equation 12
Where I = current output of the solar cell; RS = series resistance of a solar cell; RP = parallel resistance
of a solar cell; m = ideality factor (1 ≤ m ≤ 2)
2.2.4 Variations from the basic behaviour
The effect of temperature
Considering a solar cell with an ideality factor m = 1, thus its characteristic equation is:
Equation 13
The photocurrent IL increases slightly with temperature, but can be neglected as it is small. Therefore,
the change in the characteristic equation of the cell with temperature arises through exponential term
and through I0(T). The dependence of the reverse-bias saturation current on temperature can be
written as follows (4):
Equation 14
Where K and EG0 (the bandgap at 0 K) are both approximately constant with respect to temperature.
The following expression for open-circuit voltage can be deduced from equations 12 and 13 (4):
10
Equation 15
Equation 15 predicts a decrease of VOC with temperature. The variation can be appreciated by the
following coefficient of variation (4):
Equation 16
Equation 16 takes a value of about -2.3 mV/°C for silicon cells at ambient temperature (4).
The fill factor also diminishes as temperature is increased, although this effect is not very appreciable
up to 200°C. This decrease in FF is due to the increase in I0 and to the rounding of the elbow of the I-
V curve, as the effect of the increasing T in the exponential term of equation 12 turns out to be more
evident. The decrease in VOC and FF with temperature more than outweighs the slight increase in IL ,
and there is a marked decrease in the efficiency of a solar cell as temperature increases (4).
The effect of illumination intensity
Over a wide range of operating conditions, the photocurrent of solar cells is directly proportional to
the intensity of the incident radiation. Considering the photocurrent at the level of radiation defined as
unity (normally 1 sun AM1 = 100 mW/cm2) is IL1, the photocurrent at a level of radiation X
(concentration factor: X suns) times greater is IL = XIL1. If VOC1 is the open-circuit voltage at 1 sun, the
voltage at X suns is obtained, applying equation 4, and considering the ideality factor m in its general
way, resulting in (4):
Equation 17
It is assumed that m and I0 do not change appreciably as the level of illumination is increased. The fill
factor of the intrinsic cell (Rs = 0 y Rp = ∞) also increases slightly with the level of illumination (4).
The efficiency of a cell subject to an arbitrary level of illumination defined by P = X PL1, is given by
(4):
Equation 18
Considering equation 16 of the coefficient of variation of the open-circuit voltage, we have:
11
Equation 19
Ignoring small variations in FF, the above expressions predict an increase of efficiency due to the
increase in open-circuit voltage. This voltage varies logarithmically with the level of solar irradiance.
The increase cannot continue indefinitely, due to physical limits. In practice, these theoretical limits
do not manifest themselves at low levels of illumination and we do observe a logarithmic increase in
efficiency. If the intensity of illumination, and hence the photogenerated current, is increased further,
the ohmic losses due to the series resistance of the cell are no longer negligible and become
responsible for a considerable deterioration in the efficiency of the device (4).
2.3 The Photovoltaic generator
Solar cells are normally grouped into modules, which are encapsulated with various materials to
protect the cells and the electrical connectors from the environment. The manufacturers supply PV
cells in modules, consisting of NP parallel branches, each with NS solar cells in series (13).
Figure 6. Circuit diagram of a photovoltaic generator
Considering the equation 12, and recalling the expression for the Thermal Voltage Vt equals to mkT/e
(with m = 1, Vt = 25 mV at 300 K), we get the expression (4):
Equation 20
Where IL, I0, RS and RP are the photogenerated current, the dark current, the series resistance and the
parallel resistance, respectively, as it was mentioned before.
12
As a photovoltaic generator has many solar cells connected electrically, and these cells are not
identical, it is not required being too exact, for applied photovoltaics, and a simple model based on the
following assumptions is adequate: the effect of parallel resistance can be neglected; the
photogenerated current is considered equal to the short-circuit current; the expression exp((V+IRS)/Vt)
>> 1 under all working conditions; all the cells of the generator are identical and function under the
same conditions of illumination and temperature; and voltage drops in the conductors connecting the
cells are negligible (4).
From equation 20, can be obtained the characteristic I-V curve for the PV generator, considering (4):
Equation 21
Equation 22
Where IG and VG are the current and voltage of the PV module, respectively.
By combining equations 17, 18, and 19 and with I = 0, the open-circuit voltage of the module is (4):
Equation 23
Whence:
Equation 24
By rearranging the combination of equations 20, 21, 22, and 24, the PV module‟s current under
arbitrary operating conditions can be obtained (13):
Equation 25
Where ISCG, VOCG and RSG are the module‟s short-circuit current, open-circuit voltage and series
resistance, respectively.
The expression of the PV module‟s current IG is an implicit function, being dependent on (13):
The short-circuit current of the module, which is
The open-circuit voltage of the module, which is
The equivalent serial resistance of the module, which is
The thermal voltage in the semiconductor of a single solar cell, which is
13
The electrical behaviour of a module (I-V curve under certain conditions of illumination and
temperature) can be predicted from the information that the manufacturer normally supplies with the
module. The standards conditions applying to this information are (4):
Irradiance: 100 mW/cm2 (or 1 kW/m
2)
Spectrum: AM 1.5
Normal incidence
Cell temperature: 25°C
Under these standard conditions, at least the following quantities are measured (4):
The maximum power for the module
The short-circuit current for the module
The open-circuit voltage for the module
Characterization of the cell is completed by the nominal cell operating temperature, NCOT, defined
as the temperature reached by the cells when the module is submitted to the following operating
conditions (4):
Irradiance: 80 mW/cm2 (or 800 W/m
2)
Spectrum: AM 1.5
Normal incidence
Ambient temperature: 20°C
Wind speed: 1 m/s
To determine the behaviour of the PV module under arbitrary operating conditions of solar irradiance
G and ambient temperature Ta, some assumptions have to be considered (4):
The short-circuit current of a solar cell depends exclusively and linearly on the irradiance, as
follows (4):
Equation 26
Where the constant C1 has the value:
Equation 27
This assumption neglects the effect on Isc of the temperature and the spectral composition of the
radiation. Under real operating conditions, this implies an error of less than 0.5% (4).
14
The open-circuit voltage of a module depends exclusively on the temperature of the solar cells
Tc. In the range of operating conditions encountered (4):
Equation 28
This assumption ignores the effect of the illumination on Voc. Considering equation 23, the
assumption may seem strange, but the strong dependence on temperature makes the effect of
illumination relatively unimportant (4).
The working temperature of the cells depends exclusively on the irradiance and on the ambient
temperature, according to the linear relation (4):
Equation 29
Where the constant C2 has the value:
Equation 30
This assumption leaves aside the effect of wind velocity on Tc, so, heat dissipation from the cells to
the environment is taken to be dominated by conduction through the encapsulation, rather than
convection from the surface. The values of NCOT for modules on the market varies from about 42 to
46°C, implying thus a value of C2 between 0.27 and 0.32°C/(W/m2). If NCOT is unknown, a value for
C2 = 0.3 °C/(W/m2) is reasonable to be considered (4).
The series resistance is a property of the solar cells, unaffected by the operating conditions, and it
is given by (4):
Equation 31
With FF0 defined as:
Equation 32
And the normalized voltage voc as:
Equation 33
15
2.4 The Battery
Another important element of a stand-alone PV system is the battery. The battery is necessary in such
a system because of the fluctuating nature of the output delivered by the PV arrays. Thus, during the
hours of sunshine, the PV system is directly feeding the load, the excess electrical energy being stored
in the battery. During the night, or during a period of low solar irradiation, energy is supplied to the
load from the battery (13).
The general model of a battery as a voltage source VB in series with an internal resistance RBI is shown
in the following figure:
Figure 7. Equivalent circuit of a battery
Lead-acid batteries are the most commonly used energy storage elements for stand-alone photovoltaic
systems. The batteries have acceptable performance characteristics and lifecycle costs in PV systems.
In some cases, as in PV low-power applications, nickel-cadmium batteries can be a good alternative to
lead-acid batteries despite their higher cost (11).
The battery can operate in two main modes: charge or discharge, depending on the Ibat sign. While in
charge mode, the current Ibat flows into the battery at the positive terminal, and it is well known that
the battery voltage Vbat increases slowly and the charge stored increases. On the contrary, while in
discharge mode, the current flows out of the positive terminal, the battery voltage, Vbat, decreases and
the charge stored decreases supplying charge to the load (11).
Parameters of a battery
Nominal capacity qmax: is the number of ampere-hours (Ah) that can maximally be extracted
from the battery, under predetermined discharge conditions.
State of charge SOC: is the ratio between the present capacity q and the nominal capacity
qmax:
with 0 ≤ SOC ≤ 1. If SOC = 1 the battery is totally charged, otherwise if
SOC = 0 the battery is totally discharged.
Charge (or discharge) regime: is the parameter which reflects the relationship between the
nominal capacity of a battery and the current at which it is charged (or discharged). It is
16
expressed in hours, for example, discharge regime is 30 h for a battery 150 Ah that is
discharged at 5A.
Efficiency: is the ratio of the charge extracted (Ah or energy) during discharge divided by the
amount of charge (Ah or energy) needed to restore the initial state of charge. It is depending
on the state of charge SOC and on the charging and discharging current.
Lifetime: is the number of cycles charge/discharge the battery can sustain before losing 20%
of its nominal capacity.
For stand-alone photovoltaic systems some considerations about the working conditions have to be
taken into account, for instance, the superposition of daily and seasonal cycling. So, the state of
charge of a battery varies over a period of time. So, it is convenient to underline two phenomena (4):
1. Daily cycling due to the continued use of electricity during the night. The depth of discharge
associated with this cycling, PDd, depends only on the ratio between the nocturnal
consumption and the capacity of the battery. In particular, it is independent of the size of the
generator and of the local climate. Clearly,
Equation 34
where Ln is the energy consumed each night.
2. Seasonal cycling associated with the periods of reduced radiation, whose depth PDe and
duration D depend on the daily consumption (including the night), on the size of the generator
and on the local climate. To avoid too much active material being lost in the battery, some
control element is normally included to limit PDe to a certain maximum PDmax. The supply to
the load has to be cut when the limit is reached. The available or useful capacity of the battery
is, therefore, less than the nominal capacity and equal to the product qmaxPDmax.
Mathematical models that simulate battery behaviour study the variation of parameters like SOC,
voltage of the battery Vbat1, the extent of overcharge. A general model based on the observation that
the product qmaxRBI (product between the nominal capacity of a battery and the internal resistance of a
cell of a battery) is very similar from one battery to another. So, the model proposed by Lorenzo et al.
(4) for each cell of the battery is as follow:
For discharge:
Equation 35
1 More precisely, Vbat in this case means voltage across the terminals of the cell of a battery (4).
17
For charge before overcharge,
Equation 36
Where I is the charge or discharge current in amps, C10 is the capacity of a battery over a 10-hour
discharge regime, and ΔT = T(°C) – 25.
When the equations 35 and 36 are applied to the simulation of a photovoltaic system, it is advisable to
calculate SOC at each instant as:
Equation 37
Where Q is the amount of current stored by the battery at each moment, and C is the value of the
capacity corresponding to the working conditions at that moment, calculated from the expression:
Equation 38
Where I10 is the battery current for a 10-hour regime. The author adds that it is interesting notice that
as the discharge current tends to zero, the capacity tends towards a value 67% greater than C10 (4).
For the calculation of Q it may be supposed that the Faraday efficiency2 ηc is dependent on SOC
according to the equation (4):
Equation 39
Finally, the voltage across the cell of a battery at which gassing3 begins Vg, and the final charging
voltage per cell of a battery Vfc depend on the operating regime, according to the equations (4):
Equation 40
2 The Faraday efficiency of a battery in a certain state of charge is defined as the ratio of the charge extracted
(Ah) during discharge divided by the amount of charge needed to restore the initial state of charge (4). 3 The phenomena called gassing occurs when charging of a battery is nearly complete, and the active material
starts becoming scarce and some of the current passing through the battery no longer drives the normal reactions
of the battery. Instead, the current simply electrolyzes the water, decomposing it into oxygen and hydrogen, at
the positive plate and at the negative one, respectively (4).
18
And
Equation 41
The voltage from the battery is governed by the exponential expression (4):
Equation 42
Where Qsc is the amount of charge entering the battery from the instant when Vbat becomes greater
than Vg, and τsc is the time constant of the process, as follows (4):
Equation 43
The process of charging is governed by equation 36 while Vbat < Vg (i.e. until gassing begins) and by
equation 42 once Vbat > Vg (4).
2.5 The Charge regulator
In order to conserve battery life, overcharge and excessive discharge should be avoided. For lead-acid
batteries, there is a direct relation between voltage and the state of charge that makes it easy to detect
whether the battery is in a satisfactory condition. Overcharge is accompanied by an excessively high
voltage. It can be avoided either by incorporating a device to dissipate the excess potential generated
by the modules, or by disconnecting the batteries from the generator. The electronic protection used
consists in a transistor connected in parallel with the photovoltaic generator. The transistor conducts
current when the battery voltage exceeds a certain threshold, USC (4). There are three main groups of
charge controllers for stand-alone PV systems: the series regulators, which include a switch between
the generator and the battery to switch off the charge; the shunt regulators, which short-circuit these
solar generator when the charge is complete; and the MPPT, which uses a special electronic circuit
enabling maximum power to be permanently drawn from the panel array (3).
Figure 8. Diagram of series regulator
19
To prevent the battery from discharging due to failure of the transistor, it is advisable to install a
blocking diode between the transistor and the battery. This type of regulator is called parallel
regulator and it is not very efficient. Parallel regulators are only used in small photovoltaic generators.
For large generators, it is better to disconnect the battery from the generator using a switch having a
hysteresis action. This type of regulators is called series regulators. The switch may either be
electromechanical (relays, contactors, etc.) or solid state (bipolar transistors, MOSFET‟s, etc.).
Electromechanical devices have the advantage of not introducing voltage drops between the generator
and the battery, although its use in dusty and sandy environments may be difficult to keep the contacts
clean. In the case of a transistor, a high-gain bipolar device, or better still a MOSFET, should be used
(4).
There are two main operating modes for the controller (13):
1. Normal operating condition, when the battery voltage fluctuates between maximum and
minimum voltages.
2. Overcharge or over-discharge condition, which occurs when the battery voltage reaches some
critical values.
The PV arrays are disconnected from the system when the terminal voltage increases above a certain
threshold Vmax_off and when the current required by the load is less than the current delivered by the PV
arrays. PV arrays are connected again when the terminal voltage decreases below a certain value
Vmax_on, using a switch with a hysteresis cycle (13).
To protect the battery against excessive discharge, the load is disconnected when the terminal voltage
falls below a certain threshold Vmin_off and when the current required by the load is bigger than the
current delivered by the PV arrays. The load is reconnected to the system when the terminal voltage is
above a certain value Vmin_on, using a switch with a hysteresis cycle (13).
2.6 Stand Alone Photovoltaic Systems
For a stand-alone PV system, in order to have true autonomy without any other energy supply but the
energy produced during the day by the PV array, storage is indispensable if there is to be any
consumption outside daylight hours, as opposed to a grid-connected PV system, which can take
energy from the grid at night. Additionally, to protect the battery, as was mentioned before, it is
necessary to use a charge controller, in this case, a series charge controller (3).
20
Figure 9. Stand-alone photovoltaic system
Additionally, for stand-alone PV systems, converters are used for adapting the DC voltage from the
panels or the batteries to supply appliances working either on a different DC voltage or an AC
voltage. In the case of DC/DC converters, there are two possible types: “upward” converters to
increase voltage and “downward” converters to lower the voltage (3). For DC/AC inverters used for
stand-alone installations, the following types can be found (11): pulse-width modulated (PWM)
inverters; square wave inverters; modified sine wave inverters.
Nevertheless, the model considered to simulate the stand-alone photovoltaic system in this project just
considers a resistive load which is considered as a DC load, so inverters are not considered.
2.6.1 Sizing of stand-alone PV systems
The reliability of a stand-alone PV system to supply electricity to a load is quantified by the Loss of
Load Probability, LLP, as follows (4):
Equation 44
The value of LLP is considered to be always greater than zero (4).
The sizing of the PV system is used to mean the size of both, the PV generator and the accumulator
(battery), both related to the size of the load. Thus, the PV generator capacity, CA, is defined as the
ratio of the average power output of the generator divided by the average consumption of the load.
The accumulator capacity CS, is defined as the maximum energy that can be extracted from the
accumulator divided by the average daily consumption of the load. Thus (4):
Equation 45
21
Equation 46
Where AG and ηG are the area and conversion efficiency of the PV generator, respectively, Gd is the
mean value of the daily irradiation on the surface of the generator, L is the mean value of the daily
energy consumed by the load and CU is the useful capacity of the accumulator. For a given location
and load, there are two ideas that are intuitively apparent: it is possible to find different combinations
of CA and CS that lead to the same value of LLP; the larger the photovoltaic generator, the greater the
cost and the better the reliability, i.e. the lower the value of LLP. The task of sizing a PV system is a
matter of finding the best compromise between economy and reliability. Some frequent values of LLP
are: for domestic illumination 0.01; domestic appliances 0.1; and telecommunications 0.0001 (4).
Since mean values of the daily irradiation are generally available for horizontal surfaces only, it is
useful to use the following parameter (4):
Equation 47
Where Gd (0) is the mean value of the daily irradiation on the horizontal surface.
2.6.2 Modelling of stand-alone PV systems
The strategy of modelling a PV module is not different from modelling a PV cell. It uses the same PV
cell model. The parameters are all the same, but only a voltage parameter (such as the open-circuit
voltage) is different and must be divided by the number of cells (14).
Oi in (6) presents the implementation of a generalized photovoltaic model using
MATLAB/SIMULINK software package, which can be representative of PV cell, module, and array
for easy use on simulation platform. The model is analyzed in conjunction with power electronics for
a maximum power point tracker, taking the effect of sunlight irradiance and cell temperature into
consideration. The outputs of the model are the output current and power characteristics of the PV
model.
In the study done by Walker (15), an electric model with moderate complexity is used, and provides
fairly accurate results. The model consists of a current source, Isc, a diode D, and a series resistance,
Rs. The effect of parallel resistance, Rp, is very small in a single module, thus the model does not
include it. To improve the model, it also includes temperature effects on the short-circuit current, Isc,
and the reverse saturation current of diode, I0. It uses a single diode with the diode ideality factor, n,
set to achieve the best I-V curve match. Since it does not include the effect of parallel resistance (Rp),
it just lets it be infinite; giving the equation that describes the current-voltage relationship of the PV
22
cell, the form of the equation 25, previously shown in this project. To solve this equation, the
Newton‟s method of iterations is used for rapid convergence of the answer, due to the complexity of
solving the solution for the current (15). For solving the equation 25 for the PV module, an embedded
MATLAB function in SIMULINK performs the calculation five times iteratively to ensure
convergence of the results. In this way, the equation 25 now is as follows:
Equation 48
According to Oi (14), his testing result showed that the value of In usually converges within three
iterations and never more than four interactions.
About the modelling of the battery can be said that lead-acid batteries are difficult to model, and the
estimation of the battery state of charge value is recognized as one of the most complex tasks (16)
(17). Jackey (1) considers a structure with two main parts for the equivalent circuit of the battery: a
main branch which approximated the battery dynamics under most conditions, and a parasitic branch
which accounted for the battery behaviour at the end of a charge. For the main branch voltage, it was
considered to vary with electrolyte temperature and state of charge (SOC). For the terminal resistance,
the resistance was assumed constant at all temperatures, and varied with SOC, as the resistance of the
main branch of the equivalent circuit of the battery just depends on the Depth of Charge DOC (1-
SOC). Additionally, in this model the capacitance of the main branch was modelled as a voltage delay
when battery current changes with the time. Other resistances were considered for the main branch
and the parasitic one. The extracted charge from the battery was considered as follows:
Equation 49
Where Qe is the extracted charge in amp-seconds; Qe_init is the initial extracted charge in amp-seconds;
Im is the main branch current in amps; τ is an integration time variable; and t is the simulation time in
seconds.
The same author (1) approximated the capacity of the battery based on discharge current and
electrolyte temperature, considering that the capacity dependence on current was only for discharge,
and during charge mode, the discharge current was set equal to zero. In this way, the relationship is
given by:
Equation 50
23
Where Kc is a constant; C0* is the no-load capacity at 0°C in amp-seconds; Kt is a temperature
dependent look-up table; θ is electrolyte temperature in °C; Idischarge is the discharge current in amps;
I* is the nominal battery current in amps; and δ is a constant.
The same author (1), for the state of charge and the depth of charge considered to these variables to be
dependent on the discharge current. Additionally, in this work a thermal model for the electrolyte
temperature was considered, with variables of ambient temperature, thermal resistance, thermal
capacitances, and power losses.
In the work done by Guasch (5), to model the battery the electric model made by Copetti et al. was
considered (18). This model considers the battery as a sequence of steady states, neglecting the
transient effects and taking the currents and temperatures as constants. This leads to numerical
discontinuities that appear in the transitions between steady stages in dynamical application (19).
So, Guasch to develop the battery model in (5), to the terms battery capacity, charging efficiency, the
current flowing through the battery, the state of charge, adds another parameters as redefines the
parameter SOC as well. The SOC is defined as the relation between energy accepted and the capacity
available at all times; when SOC is unity the battery cannot accept more energy from the system,
because the energy stored fills all the battery capacity, and when SOC is zero the battery has no
energy. The new parameter defined is LOE, the level of energy, and it shows the amount of energy
available in the battery under normal working conditions, and depends only on the constitutive
parameters of the device and the accumulated charge over time, not on the working environment of
the battery. LOE is not limited to the higher limit of unity, but LOE values near or greater than unity
are undesirable in order to avoid damaging the battery (19).
An inherent problem of modelling battery for PV applications is the accuracy of the model parameters
with regard to the quality of the device under characterization. The use of nominal values for a battery
family, or of individually adjusted values from the manufacturer, can introduce an error that may be
important, depending of working conditions and the lifetime. A way to avoid this error is to use a
method for automatic model parameter adjustment that is valid for static tests or for free-running
stand-alone PV systems. So, when the nominal values are known, or in order to estimate their values,
or when they are not fixed anywhere, an automatic parameter extraction method based on the
Levenberg-Marquardt algorithm can be used (19).
The battery in the stand-alone PV system is considered a subsystem with memory to calculate the
voltage as output, as the voltage does not have big variations in its value (5).
For Guasch (5), the modelling of the charge controller is mathematically simple, but it has an
important issue as the continuity of the model is considered: when the switches operate,
24
discontinuities in the voltages and in the currents are produced. The parameters for the charge
controller considered are the battery voltage, the PV panels and load currents, and the outputs of the
model are the battery current, the PV panels and load currents. The disconnection of the solar panels
was simulated turning them to their open-circuit voltage, and the disconnection of the loads forcing
the feeding voltage to zero.
The main difficulty for programming the model of a stand-alone PV system with charge controller is
on the discontinuities introduced by the switches of the charge controller, but there are several
alternatives to solve this problem: the using of discrete numeric calculus algorithms instead the
continues one, linearizing of the switches behaviour, etc. These discrete numeric algorithms have
better accuracy and simplicity to be used for solving the discontinuities when modelling the different
components of the stand-alone PV systems (5).
25
3. Methodology
In this project an approach to a simulation of a stand-alone photovoltaic system is presented, and its
results compared to experimental data previously available. The stand-alone PV system is modelled
using MATLAB/SIMULINK, software that has the characteristics of having several libraries with
specific tools that can be applied to electrical and electronics applications, between others.
The following considerations were taken into account to achieve the aims of this project:
3.1. Requirements
The entire system can be broken up into several smaller subsystems. These subsystems include a
Photovoltaic module, two lead-acid batteries, a charger controller, a load, and several switches. The
block diagram for the stand-alone PV system is provided below:
Figure 10. Block diagram for the entire system
Charge
Regulator Switch
PV
module Load Battery Battery
26
3.2. Modelling and simulation
For this project, a model of a stand-alone PV system is created and two simulations are run and
compared with experimental data to test the model. This is done effectively using the software
SIMULINK, that is an interactive tool for modelling, simulating, and analyzing dynamic systems,
including controls, signal processing, communications, and other complex systems (20).
3.2.1 PV module
An equation that shows the behaviour of the PV panels is as follows:
Equation 51
Where m is known as the ideality factor, and takes a value between one and two.
Finally, the effect of the shunt resistance is minimal for a small number of modules. Therefore, is
assumed that Rp = ∞, simplifying the photon-generated current equation to:
Equation 52
In this way, equation 52 is the one that was considered for modelling of the PV modules
The module considered corresponds to the following details:
Table 1. Values from Sharp NT9075 PV module datasheet
Electrical characteristics SHARP NT9075
Short-circuit current 3.5 A
Open-circuit voltage 21.8 V
Maximum power 51.5 W
Other important parameters that were considered to build the model in MATLAB/SIMULINK are the
following:
The value for the diode ideality factor considered is m = 1.62
The band gap energy for silicon considered is 1.12 eV.
The number of solar cells connected in series is 36.
The temperature coefficient of the short-circuit current is (0.065±0.015)%/°C.
The temperature coefficient of the open-circuit voltage is –(160±20)mV/°C.
The temperature coefficient of the power is –(0.5±0.05)%/°C.
The Nominal Operating Cell Temperature (NOCT) is 47±2°C.
27
These last parameters do not correspond to the PV module Sharp NT9075, as they were not available
at the moment of this project, but they were considered as was mentioned. These parameters
correspond to the model developed by Oi (14), for a BP Solar BPSX 150S photovoltaic module.
These parameters were taken for convenience, as the main structure for developing the PV module in
MATLAB/SIMULINK is based on the model of PV module mentioned.
Isc at Tref is found on the data for the datasheet for the PV module.
Additionally, it is worth to mention that Tref corresponds to the reference temperature under standard
conditions, STC, of the PV cell in Kelvin, usually 298K. Therefore, the photon generated current at
any other irradiance, G (W/m2), is given by (12):
Equation 53
Where G0 = 1000 W/m2, standard conditions.
The reverse saturation current of the diode I0 at the reference temperature is given by:
Equation 54
The reverse saturation current is dependent on (junction) cell temperature, and is given by the
equation:
Equation 55
The diode ideality factor m is a value between one and two (4), and must be estimated. For the
purposes of this project, the value estimated by Oi (14) is used, being m = 1.62 that according the
author, attains the best match with the I-V curve on the datasheet of the PV module considered by Oi.
Similarly in the same work (14), the value of Rs for the BP SX 150 PV module is estimated at 5.1 mΩ.
Equation 52 is solved using Newton‟s method, in order to estimate the roots of the equation by
iteration. So, if xk is an approximation of the root, can be related to the next approximation xk+1 using
the right-angle triangle (21) in the following figure:
28
Figure 11. Newton‟s method
Equation 56
Where f’(x) is df / dx. Solving for xk+1 gives:
Equation 57
Equation 52 is rewritten now as:
Equation 58
Applying the Newton‟s method to the equation 58, can be found:
Equation 59
The embedded MATLAB code in the SIMULINK scheme written to solve this equation is set to
iterate five times to ensure convergence of the results. The MATLAB function is on chapter 5.
29
3.2.2 Battery
The battery considered is a lead-acid battery model SunLyte 12-5000X. In the next table, the
characteristics are presented:
Table 2. Characteristics battery SunLyte 12-5000X
Specifications
Container and cover: Reinforced polypropylene
Separators: Spun glass, microporous matrix
Safety vent: 4 psi nominal, self resealing
Self discharge: 0.5-1.0% per week
Terminals: Heavy duty copper
Charge voltage: 2.25 -2.35 VPC @ 25°C (15 amp max current)
Positive plate: Patented MFX alloy
Negative plate: Lead tin
Estimated cycle life: 8 hour rate to 1.75 VPC @ 25°C
300 cycles @ 80% DOD
600 cycles @ 50% DOD
1000 cycles @ 20% DOD
Physical Characteristics
Length: 307mm / 12.07"
Width: 175mm / 6.87"
Height: 221mm / 8.69"
Weight: 30 kgs / 66 lbs
Electrical Performance
Type 12-5000X - VRLA
6 Cells - 12Volt Nominal
Ah Capacity to 1.75VPC @ 25°C:
1 hour - 54Ah, 5 hours - 72Ah, 8 hours - 86Ah
24 hours - 93Ah, 48 hours - 96Ah, 100 hours - 1OOAh
The battery model is based on a lead-acid battery PSpice model (11). Lead-acid batteries are formed
by two plates, positive and negative, immersed in a dilute sulphuric acid solution. The positive plate,
or anode, is made of lead dioxide (PbO2) and the negative plate, or cathode, is made of lead (Pb).
The battery operates in two main modes: charge and discharge. The battery is in charge mode when
the current into the battery is positive, and discharge mode when the current is negative.
The battery model considered has the following input parameters:
Initial state of charge: SOC1 (%), indicating available charge.
Maximum state of charge: SOCm (Wh), maximum battery capacity.
Number of 2V cells in series: ns
Charge/discharge battery efficiency: K (unitless)
Battery self-discharge rate: D(h-1
)
The terminal voltage of the battery is given by (11):
30
Equation 60
Where V1 and R1 are the battery voltage and the internal resistance, respectively, and are governed by
a different set of equations depending on which mode of operation the battery is in. It is necessary to
state that the values for the battery current (Ibat) are positive when the battery is in charge mode and
negative when the battery is in discharge mode (11).
So, the set of equations for the charge mode are:
Equation 61
Equation 62
With SOC(t) as the instantaneous state of charge (%).
The set of equations for the discharge mode are:
Equation 63
Equation 64
A very important part of modelling the battery is the estimation of the instantaneous value of the
SOC(t). The estimation is performed as described by the following equation (11):
Equation 65
Equation 66 basically is the energy balance equation computing the value of the SOC increment as the
energy increment in a differential of time taking into account self-discharge and charge discharge
efficiency. As time has units of seconds, some terms are divided by 3600 so SOC is in Wh (11).
Equation 66 can be simplified substituting Vbat as a function of V1 (11):
Equation 66
And finally, integrating for solving for SOC(t) (11):
Equation 67
31
With t as the number of time units. Therefore, SOC(t) can be found if one knows the previous
condition. Since SOC(0) = SOC1 = initial state of charge, SOC(1) can be found, and by looping the
result with SIMULINK, the current value for SOC(t) for any t can be estimated.
In order to compare with the experimental data available, the initial states of charge SOC of both in
parallel batteries are set to the experimental values of SOC for the simulations considered. The values
are as follows:
December 3, 2002: SOC1 = 0.9618
Entire month of December 2002: SOC1 = 0.901
3.2.3 Charge controller
The controller is necessary to keep the battery from being overcharged or undercharged, either of
which may reduce the battery‟s life. Typically, a deep-cycle battery should not be discharged past
20% or charged past 100%. The charge controller considered in this project is based on that used in
the PSpice photovoltaic model (11).
The charge controller consists of one switch, on one side of the battery‟s positive terminal. Switch A,
on the PV module side, is opened if the battery voltage becomes larger than 14.4V and will remain
open until the battery voltage has dropped to 12.9V. There is no switch for the load side as one of the
assumptions for modelling the stand-alone PV system is that the accumulator (two batteries in
parallel) is large enough to not have loss of load situation.
Figure 12. Block diagram of charge series regulation
Switch A
Battery PV
Module
Load
32
The truth table for the switches is presented below:
Table 3. Truth table for switches in charge regulation
Condition A' A
V ≥ 14.4 0 0
V ≥ 14.4 1 0
12.9 < V < 14.4 0 0
12.9 < V < 14.4 1 1
V ≤ 12.9 0 1
V ≤ 12.9 1 1
Where: closed switch = 1 (true); open switch = 0 (false); A‟ is the previous state of switch A; V is the
battery voltage.
So, summarizing, the battery voltage of 14.4V, disconnects the PV module from the batteries to avoid
overcharge, and a value of 12.9V of the battery voltage, will reconnect the PV module to the battery
to avoid undercharge of the batteries.
3.2.4 Load
As load, a single resistor is used to represent the load. Its resistance is considered as 40 Ω.
33
4. Experimental data
4.1 Stand-alone PV system
The experimental data used for comparing the simulation results comes from a previous work by
Fragaki (22).
The panels of the stand-alone photovoltaic system have a tilt angle of 66°, with them facing south
over a roof of a building in Southampton, United Kingdom (22).
The model of the PV module is Sharp NT9075 with 36 cells in series, with a resistor at 40 ohms. The
system includes two low maintenance sealed valve regulated lead-acid batteries, SunLyte 12-5000X,
of 100 Ah capacity each and nominal voltage of 12 V, connected in parallel (22).
The system has a simple on/off shunt charge regulator SOLLATEK SPCC10E, with Low Voltage
Disconnect and temperature compensation of -3mV/°C/cell. It includes a diode which protects the
regulator against reversed polarity (22).
The experimental data was obtained with a datalogger connected to the stand-alone system. To
compare with the simulation results, the photovoltaic current, the voltage in the PV module, the power
generated, the battery voltage, the current passing through the battery, and the state of charge were
considered.
Figure 13. Schematic representation of the stand-alone PV system
34
4.2 Irradiance and temperature data
For this project, irradiance data for a specific location over a 24 hour period of time is used. The data
is from Southampton, United Kingdom. The data used corresponds to December, 2002, for the day
December 3, 2002, with maximum value of irradiance of 915 W/m2. Additionally, a simulation for the
complete month of December of 2002 is carried out, considering the values of irradiance and ambient
temperature.
Figure 14. Irradiance versus time data, December 3 2002, Southampton, UK
Figure 15. Temperature versus time data for 3th of December 2002
Simulations are run for scenarios, the 24 day and the complete month as inputs for the model the
ambient temperature and the irradiance data.
0
100
200
300
400
500
600
700
800
900
1000
0
40
12
0
20
0
24
0
32
0
40
0
44
0
52
0
60
0
64
0
72
0
80
0
84
0
92
0
10
00
10
40
11
20
12
00
12
40
13
20
14
00
14
40
15
20
16
00
16
40
17
20
18
00
18
40
19
20
20
00
20
40
21
20
22
00
22
40
23
20
Irra
dia
nce (W
/m^
2)
Time of Day
Irradiance for Southampton, UK - December 3, 2002
0
5
10
15
5
50
13
5
22
0
30
5
35
0
43
5
52
0
60
5
65
0
73
5
82
0
90
5
95
0
10
35
11
20
12
05
12
50
13
35
14
20
15
05
15
50
16
35
17
20
18
05
18
50
19
35
20
20
21
05
21
50
22
35
23
20
Tem
per
atu
re (d
egre
es
C)
Time of Day
Temperature data for 3th December, 2002, Southampton, UK
35
For the purposes of this project, for the simulation the initial state of charge of both batteries is set at
0.9618, for December 3, 2002, in order to compare with the experimental data available. The initial
state of charge for the simulation of the complete month of December 2002, for the 0:00 hr of the 1th
of December 2002 is 0.901.
The simulations are run to begin at 0:00 am of the day corresponding to its data, and to end at 23:55
pm of the same day. The data of irradiance is in five minute time steps, so each simulation uses 288
steps of five minutes for a complete day and 8,925 steps of five minutes for the complete month of
December of 2002.
The data of temperature is available for the same days but just one value per hour, so the same value
is assumed for the complete hour of the days considered for the simulation, as can be seen in the
graph from figure 15.
Summarizing, the two simulations are listed as follows:
Simulation 1: winter day conditions and constant load resistance, 3th of December 2002.
Simulation 2: for the complete month of December of 2002, considering the values for the
battery model (battery voltage and state of charge of the battery).
Additionally, the parameters voltage and state of charge SOC of the battery are simulated for the
month of December 2002, in order to show the accuracy of the model developed. The values
considered for the irradiance and the ambient temperature are shown in the following figure:
Figure 16. Irradiance and ambient temperature, December 2002
The values obtained by simulation for the battery voltage and the SOC of the battery are compared
with the experimental data previously available, in order to analyze the accuracy of the simulation.
0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
0
100
200
300
400
500
600
700
800
900
1000
Tem
peratu
re (°C
)
Irrad
ian
ce (
W/m
2)
Time
Irradiance and Temperature December 2002, Southampton, UK
Irradiance
Temperature
36
The models are aimed at reproducing the behaviour of the system under realistic operation conditions.
In a real PV system, the irradiance and temperature values evolve during the observation time
according to meteorological conditions and site location. Although the irradiance changes are not very
sudden in general, they are of random nature and differ from the laboratory conditions under which
the parameters of the PV components have been measured. This raises the question as to whether the
performance of the stand-alone PV system‟s different components‟ models will accurately reproduce
the dynamics of the system operation (11). So, the results of the simulation for different parameters
(outputs) of the model developed are compared with the experimental data previously taken under real
conditions for the days considered for the study. These outputs considered are as follows:
PV current
PV voltage
PV power
Battery voltage
Battery current
State of Charge of the Battery (the initial state of charge was considered equal for both in
parallel batteries)
37
5. Modelling of the stand-alone PV system in MATLAB/SIMULINK
5.1 Implementation in MATLAB/SIMULINK
The simulation can be summarized in the following flowchart of the processes carried out by the
model developed in MATLAB/SIMULINK.
Figure 17. Stand-alone PV system simulation flowchart
38
The simulation is carried out according to the flowchart, and for the charge controller, considering the
values of the battery voltage from the table 3, to open or close the switch for the PV generator.
The model of the stand-alone PV system as presented in figure 10 is built in SIMULINK as shown in
figure 18.
The total system has thus as inputs the irradiation and the ambient temperature data for the days
considered. These are used in the PV module together with the voltage from the controller to generate
the PV current. In this way, the charge controller receives the information that comes from the
batteries, and the PV module. Based on these inputs and on the conditions shown in table 3, the
charge controller transmits control signals back to the PV module, and the batteries.
Figure 18. Stand-alone PV system structure in SIMULINK
The advantage of using MATLAB/SIMULINK is the possibility of building hierarchical models,
namely to have the possibility to view the system at different levels.
In this way, different subsystems were built to reflect the different components of the simulated stand-
alone PV system. Some of the subsystems are embedded as MATLAB functions into the SIMULINK
module and the rest as SIMULINK subsystem. The following components are made as subsystems:
Photovoltaic generator
Charge controller (SW control1 block in the figure 18)
Batteries
Additionally, a subsystem in the SIMULINK block model is included to read the solar irradiance and
ambient temperature data from their correspondent MAT files.
39
5.2 The PV Module
To implement the photovoltaic generator in SIMULINK, an embedded MATLAB function was
developed. The theoretical assumptions used to build the code for the PV generator are explained in
the chapter 3 Methodology, in details. For modelling, the following MATLAB code was
implemented, based in some parts on the work developed by Oi (14):
function [Ia,Vmp] = PVmpp01(G,TaC) % Ia_new = 0; Va = 21.8; k = 1.381e-23; % Boltzmann’s constant q = 1.602e-19; % Electron charge % Following constants are taken from the datasheet of PV module and % curve fitting of I-V character (Use data for 1000W/m^2) m = 1.62; % Diode ideality factor (m), % 1 (ideal diode) < n < 2 Eg = 1.12; % Band gap energy; 1.12eV (Si), 1.42 (GaAs), % 1.5 (CdTe), 1.75 (amorphous Si) Ns = 36; % # of series connected cells (Sharp NT9075, 36 cells) TrK = 298; % Reference temperature (25C) in Kelvin Voc_TrK =21.8 /Ns; % Voc (open circuit voltage per cell) @ temp TrK Isc_TrK =3.5; % Isc (short circuit current per cell) @ temp TrKz a = 0.65e-3; % Temperature coefficient of Isc (0.065%/C) % Define variables TaK = 273 + TaC; % Module temperature in Kelvin Vc = Va / Ns; % Cell voltage % Calculate short-circuit current for TaK Isc = Isc_TrK * (1 + (a * (TaK - TrK))); % Calculate photon generated current @ given irradiance Iph = G.*Isc/1000; % Define thermal potential (Vt) at temp TrK Vt_TrK = m * k * TrK / q; % Define b = Eg * q/(m*k); b = Eg * q /(m * k); % Calculate reverse saturation current for given temperature Ir_TrK = Isc_TrK / (exp(Voc_TrK / Vt_TrK) -1); Ir = Ir_TrK*(TaK/TrK).^(3/m)*exp(-b*(1/TaK -1/TrK)); % Calculate series resistance per cell (Rs = 5.1mOhm) dVdI_Voc = -1.0/Ns; % Take dV/dI @ Voc from I-V curve of datasheet Xv = Ir_TrK / Vt_TrK * exp(Voc_TrK / Vt_TrK); Rs = - dVdI_Voc - 1/Xv; % Define thermal potential (Vt) at temp Ta Vt_Ta = m * k * TaK / q; % Ia = Iph - Ir * (exp((Vc + Ia * Rs) / Vt_Ta) -1) % f(Ia) = Iph - Ia - Ir * ( exp((Vc + Ia * Rs) / Vt_Ta) -1) = 0 % Solve for Ia by Newton's method: Ia2 = Ia1 - f(Ia1)/f'(Ia1) Ia=zeros(size(Vc)); % Initialize Ia with zeros % Perform 5 iterations
for j=1:5 Ia = Ia - (Iph - Ia - Ir .* ( exp((Vc + Ia .* Rs) ./ Vt_Ta) -1))... ./ (-1 - Ir.* (Rs ./ Vt_Ta) .* exp((Vc + Ia .* Rs) ./ Vt_Ta)) ; end % C = 0.1; % Step size for ref voltage change (V) % % Define variables with initial conditions % Va = 24.0; % PV voltage % Pa = Va*Ia; % PV output power
40
% Vref_new = Va + C; % New reference voltage % % performs 100 iterations % for m =1:100; % Va_new = Vref_new; % % TaK = 273 + TaC; % Module temperature in Kelvin % Vc = Va_new / Ns; % Cell voltage % Isc = Isc_TrK * (1 + (a * (TaK - TrK))); % Iph = G.*Isc/1000; % Vt_TrK = m * k * TrK / q; % b = Eg * q /(m * k); % Ir_TrK = Isc_TrK / (exp(Voc_TrK / Vt_TrK) -1); % Ir = Ir_TrK*(TaK/TrK)^(3/m)* exp(-b* (1/ TaK -1/TrK)); % dVdI_Voc = -1.0/Ns; % Take dV/dI @ Voc from I-V curve of datasheet % Xv = Ir_TrK / Vt_TrK * exp(Voc_TrK / Vt_TrK); % Rs = - dVdI_Voc - 1/Xv; % Vt_Ta = m * k * TaK / q; % Ia=zeros(size(Vc)); % Initialize Ia with zeros % for j=1:5 % Ia_new = Ia - (Iph - Ia - Ir .* ( exp((Vc + Ia .* Rs) ./ Vt_Ta) -
1))... % ./ (-1 - Ir.* (Rs ./ Vt_Ta) .* exp((Vc + Ia .* Rs) ./ Vt_Ta)); % end % % % Pa_new = Va_new*Ia_new; % deltaPa = Pa_new-Pa; % % P&O Algorithm starts here % if deltaPa > 0 % if Va_new >Va % Vref_new = Va_new + C; % Increase Vref % else % Vref_new = Va_new - C; % Decrease Vref % end % elseif deltaPa < 0 % if Va_new > Va % Vref_new = Va_new - C; % Decrease Vref % else % Vref_new = Va_new + C; % Increase Vref % end % else % Vref_new = Va_new; % No change % end % % Update history % Va = Va_new; % Pa = Pa_new; % end % Ia_new = Ia; Vmp = Va; % Ia = Ia_new; % % disp(Ia);
5.3 Charge controller
The modelled charge controller consists in a subsystem that is embedded to the main SIMULINK
diagram for the complete stand-alone PV system. The model for the controller has as input the battery
voltage and the PV voltage, to, with this information, open or close the switch A, as explained in the
41
chapter 3 Methodology. The switch is modelled as an embedded MATLAB function that its code is
presented after the SIMULINK diagram of the figure 19.
Figure 19. SIMULINK diagram for the charge controller
Charge controller switch A: MATLAB code
function A1=SwitchA(V,A,Solar)
A1 = A;
if (Solar == 1)
if(V>14.4 && A == 0)
A1 = 0;A = 0;
end
if(V>14.4 && A == 1)
A1 = 0;A = 0;
end if (V>12.9 && V<14.4 ) if (A == 1) A1 = 1;A = 1; end end
if (V>12.9 && V<14.4 )
42
if(A==0) A1 = 0; A = 0; end end
if (V < 12.9 && A == 0)
A1 = 1; A = 1; end
if (V < 12.9 && A == 1)
A1 = 1; A = 1;
end
else A1 = 0;
end
5.4 Battery: SIMULINK subsystem diagram
The battery was modelled with the technical characteristics presented in table 2. The lead-acid battery
was modelled in a SIMULINK diagram as a subsystem embedded to the main system of the stand-
alone PV system.
The parameters V1 and R1 are considered as functions blocks for the charge mode (equations 62 and
63) and for the discharge mode (equations 64 and 65). The input is the PV current and varies as the
solar irradiance and temperature change throughout the simulation period. Another relevant parameter
is the SOC1, SOCm (for this battery 1200 Wh), ns (number of 2V cells in series: 6), K (0.8), and D
(1x10-5
h-1
). The initial state of charge SOC1 was set to the same value as for the available
experimental data, being set in the integrator block of the SIMULINK diagram (figure 20).
43
Figure 20. SIMULINK diagram for the model of the lead-acid battery
As was mentioned before, the model does not consider overcharge mode, for simplicity reasons.
5.5 Data input to the system
The data for the solar irradiance and for the ambient temperature was converted to MAT files to be
read by MATLAB/SIMULINK, by a subsystem embedded to the main system of the stand-alone PV
system.
Figure 21. SIMULINK diagram for the subsystem for the input data
44
6. Results and Discussion
The results of the simulations carried out are shown in this section of the project, and discussed the
results obtained. Additionally, the results are compared with the experimental data previously
available.
6.1 Simulations December 3, 2002
6.1.1 PV model
The plot for the simulated current values for the stand-alone Photovoltaic system and compared to the
experimental data is presented in figure 22.
Figure 22.Comparison experimental data with simulated data
As can be seen in figure 22, during the hours of no sunlight, the model shows zero values for the
current from the PV generator.
As can be seen, when the values simulated are compared to the experimental data, there are
differences, showing differences in the peaks shown (figure 22).
Considering the values during the time there is solar irradiance for the current in the PV generator
shown in figure 22, the difference of the simulated values from the experimental data may due to:
Difference in the values considered for the construction of the model in
MATLAB/SIMULINK, especially considering that for the model of PV Module considered
(Sharp NT9075) there may be different values of temperature coefficient of the short-circuit
current and of the open-circuit voltage.
0
0.5
1
1.5
2
2.5
3
3.5
0
40
12
0
20
0
24
0
32
0
40
0
44
0
52
0
60
0
64
0
72
0
80
0
84
0
92
0
10
00
10
40
11
20
12
00
12
40
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20
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00
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40
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00
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00
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20
20
00
20
40
21
20
22
00
22
40
23
20
Cu
rren
t (A
mp
s)
Time of the Day (time steps of 5 minutes)
PV Current: December 3, 2002, Southampton, UK, Experimental Data and
Simulation Data
PV Current
Sim PV
Current
45
Though the controller considered in the modelling of the whole system controls the load,
depending on the voltage on the battery, this last value never reaches the threshold considered
in the controller to disconnect the load from the system, so the load remains connected during
the whole day. According to Hansen et al. (13), this effect may be observed further as
changes in the battery voltage as while the load is disconnected; the PV current is used to
charge the battery, but not in all its magnitude it reaches the batteries.
The plot for the simulated voltage values for the stand-alone Photovoltaic system are presented in
figure 23.
Figure 23. Voltage from the PV Module for December 3, 2002, Southampton, UK
The peaks in the voltage of the PV module coincide with the peaks in the values of the solar
irradiance for the day simulated.
The comparison between the experimental data and the data obtained through the simulation for the
power that the PV generator is able to generate, is shown in the next figure.
46
Figure 24. Comparison of experimental data and the simulated data
As can be seen in figure 24, the simulated values (Sim PV power in the graph) present their peaks
closely to the peaks of solar irradiation of the day, but there are several values with zero as value,
showing that there are some inaccuracies in the model built in MATLAB/SIMULINK. For the
simulated peaks, can be seen that they are bigger than the ones for the experimental data, fact that
may due to the differences presented for the simulation of the PV current; being those ones to not
having all the details from the manufacturers of the PV module. Additionally, can be seen that for the
values of power for values of no solar irradiance (G = 0 W/m2) the values are zero, according to the
experimental data available.
Hansen et al. (13) mention that possible reasons for differences between simulated data and
experimental data for the PV generator are:
Measurements are acquired in realistic weather conditions, i.e. with panels subjected to dust,
etc.
The simulation of the module is performed based on the rated data of the PV module
(supplied by the manufacturer), and not on the specific measured data for the module.
PV model‟s uncertainty.
Measurements‟ uncertainty.
Ageing of the cells.
The cell temperature is approximated and not measured directly.
0
10
20
30
40
50
600
40
12
0
20
0
24
0
32
0
40
0
44
0
52
0
60
0
64
0
72
0
80
0
84
0
92
0
10
00
10
40
11
20
12
00
12
40
13
20
14
00
14
40
15
20
16
00
16
40
17
20
18
00
18
40
19
20
20
00
20
40
21
20
22
00
22
40
23
20
Watt
s
Time of the Day (time steps of 5 minutes)
PV Power: December 3, 2002, Southampton, UK, Experimental Data and
Simulation Data
PV Power
Sim PV
Power
47
6.1.2 Battery model
The plot of the simulated values of the voltage on the batteries compared to the experimental data is
presented in figure 25.
Figure 25. Comparison of experimental data and the simulated data
As can be seen in figure 25, the voltage of the battery (there are two batteries in parallel that have,
indeed, the same values throughout the whole day simulated) goes from a value close to 12.27V to a
value close to 12.92V for the simulated values. The big risings in the voltage that can be seen in the
figure 25, may be due to the current at this point is positive, meaning that the batteries are in charge
mode, and dropping again when the batteries are in discharge mode, to finally achieve an almost
constant value of voltage.
Considering that the charge controller defined for the model as a PV module/battery switching has
values of the battery voltage Vbat for disconnecting the PV module of the batteries of 14.4V and for
reconnecting the PV module with the batteries of 12.9V, the effect of the charge controller cannot be
seen in the results shown in figure 25, as for the simulated values, the range of values for the battery
voltage is between 12.27 and 12.92 volts. What can be said about this is that none of the batteries
reach during the simulated day the threshold value of 14.4V to disconnect the PV module. The same
conclusion can be deducted from the profile during the day for the experimental values, as the values
of the battery voltage do not reach further than 13.7V (very far from 14.4V). In this way, the PV
module is never disconnected from the batteries, fact that is more evident when the minimum values
of the battery voltage for the simulation are in the range 12.26-12.28 volts. This fact can be seen as
well for the experimental data, but the minimum values of battery voltage are between 12.54-12.84
for the hours with low or null sunlight, and around 13V for the hours with higher sunlight.
11.5
12
12.5
13
13.5
14
0
40
12
0
20
0
24
0
32
0
40
0
44
0
52
0
60
0
64
0
72
0
80
0
84
0
92
0
10
00
10
40
11
20
12
00
12
40
13
20
14
00
14
40
15
20
16
00
16
40
17
20
18
00
18
40
19
20
20
00
20
40
21
20
22
00
22
40
23
20
Volt
age
Time of the Day (time steps of 5 minutes)
Batteries Voltage: December 3, 2002, Southampton, UK, Experimental Data and Simulation
Data
Battery
Voltage
Sim
Battery
Voltage
48
The simulation carried out for the day 3th of December of 2002, accordingly to the experimental data,
has an initial state of charge of the battery of 0.962, the final SOC, after the end of the day simulated
is 0.9637 (blue curve in figure 26), and the experimental value for the SOC after the day considered is
0.961 approximately. The comparison served in figure 26 where the curve Sim SOC (in red)
represents the simulation carried out in MATLAB/SIMULINK and the curve SOC (in blue) represents
the experimental data, shows a good fit of the simulated values with the experimental data. The
simulation (red curve in the graph of the figure 25) shows an increase in the values of the voltage in
the batteries at around 9:20 hrs of the day, time when the solar irradiance is 62.23 W/m2. The
difference with the experimental data (blue curve in the graph) is considerable, and definitively, the
battery voltage is underestimated by the simulation, mainly due to the arbitrary parameters that are
considered for the modelling of the battery, though model for lead-acid batteries was considered (11).
Figure 26. Comparison between experimental and simulated data for the SOC of the batteries
The decreasing of the SOC of the battery shown in the graph is because there is no current flowing
from the PV generator to the batteries, so the latter ones are in discharge mode as they are delivering
current to the constant load connected to the stand-alone PV system. The simulated SOC shown in
figure 26 has an increase some time after the sunlight is available to be used by the PV generator.
Comparing the values obtained with the simulation (Sim SOC curve in the graph of the figure 26), the
model foresees an almost constant discharge mode of the batteries through the hours without solar
irradiance, but when the sun starts shining, the PV generator starts to delivers current to the load and
the batteries. In the other hand, the experimental data shows clearly that the batteries during the time
there is no solar irradiance, are in discharge mode and once the sun starts shining, the batteries
increase their inner voltage, thus, having in this charge mode, positive current on the batteries. As the
battery model considered for this project is based mainly in the PSpice model proposed by Castañer
and Silvestre (11) for lead-acid batteries, many of the constants used in the battery block of the model
0.93
0.935
0.94
0.945
0.95
0.955
0.96
0.965
0.97
0.975
0.98
0
40
12
0
20
0
24
0
32
0
40
0
44
0
52
0
60
0
64
0
72
0
80
0
84
0
92
0
10
00
10
40
11
20
12
00
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40
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20
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00
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40
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00
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00
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00
20
40
21
20
22
00
22
40
23
20
SO
C
Time of the Day (time steps of 5 minutes)
State of Charge of the Batteries : December 3, 2002, Southampton, UK,
Experimental Data and Simulation Data
SOC
Sim
SOC
49
developed in SIMULINK were arbitrarily chosen, though the main parameters for the model of the
battery were considered, as its maximum capacity, self discharge rate, and maximum current.
Nevertheless, the model developed in SIMULINK adjusts properly to the experimental data, as can be
seen. Additionally, the inaccuracies may be explained to the fact that the battery model does not
account for overcharge mode and the effect of temperature over the electrolyte, though the levels of
battery voltage never reach the threshold of 14.4V, to the PV module to be disconnected, as can be
seen in the figure 25.
Figure 27. Comparison between experimental and simulated data for the battery current
In figure 27, the experimental current that flows through the battery and the simulated (Sim Battery
Current curve) are compared. There are discrepancies because the model overestimates the values of
the current entering the batteries, but the trend of the curve coincides with the experimental values.
Additionally, can be seen that during the hours that there are no solar irradiance, the current is being
delivered to the load, thus having negative values for the current, and during the hours with solar
irradiance, the battery is on charge mode (positive values of the current).
The battery current with positive values means that is the current coming from the PV module,
charging the batteries, and with negative values is current going out the battery.
In addition, several authors (5) (11) (6) (4) agree in the point that the most difficult task for modelling
in a stand-alone PV system is the battery, as it has several technical difficulties for modelling its
behaviour. Between them, can be mentioned the following:
The complex task of modelling the state of charge of a battery.
The estimation of the internal resistance of a battery.
The effect of temperature on the value of voltage at which overcharge phenomena begins.
-1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
0
40
12
0
20
0
24
0
32
0
40
0
44
0
52
0
60
0
64
0
72
0
80
0
84
0
92
0
10
00
10
40
11
20
12
00
12
40
13
20
14
00
14
40
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40
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20
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00
22
40
23
20
Cu
rren
t (a
mp
s)
Time of the Day (time steps of 5 minutes)
Battery current: December 3, 2002, Southampton, UK, Experimental Data and
Simulation Data
Battery
current
Sim
Battery
current
50
6.2 Simulation for the Battery, December, 2002
As for the day simulated, 3th of December of 2002, for the state of charge of the battery SOC the
results are close to the experimental data, the battery model for the stand-alone photovoltaic system
for the complete month of December of 2002 was analyzed (31 days of the month simulated). So, the
voltage in the batteries and their state of charge was simulated and compared to the real data.
Figure 28. Simulated Voltage of the battery for December 2002
The simulated voltage of the battery is shown in figure 28, where its trend shows that the voltage
decreases as the days pass throughout the month of December of 2002. The peaks that can be seen
correspond to the hours of sun light of everyday as the PV generator produces current that is sent to
the load (constant throughout the month) and to the batteries.
Figure 29. Comparison between experimental voltage and simulated voltage for December 2002
The real battery voltage and the simulated battery voltage are compared in the graph of the figure 29.
The model developed underestimates the values of the voltage, but can be seen that the trend for the
peaks goes with the peaks shown for the curve of the real battery voltage values.
11.8
12.3
12.8
13.3
13.8
14.3
01
-Dec
-02
02
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03
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04
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05
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11
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14
-Dec
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15
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16
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28
-Dec
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29
-Dec
-02
30
-Dec
-02
31
-Dec
-02
Volt
age
Day
Voltage of the Batteries: December, 2002, Southampton, UK. Experimental Data and
Simulation Data
V exp
V sim
51
As can be seen in figure 29, and taking into account the threshold values for PV module/battery
switching device, the voltage in the batteries at no moment during the month reaches the value 14.4V
to make the PV module disconnect of the batteries in parallel. This fact happens for the simulated
values and for the experimental values of the battery voltage throughout the month of December
2002. With the levels of solar irradiance for the entire month, the voltage has a decreasing trend,
aspect that is direct linked with the state of charge of the batteries (4).
As the simulation of the state of charge of the batteries carried out for the day 3th of December of
2002 was close to the real values, the simulation for the entire month of December is shown in the
figure 30. The SOC of the battery (the stand-alone PV system has two batteries in parallel that were
modelled in the same way) diminishes as the days pass, as the sunlight decreases into the winter
season.
Figure 30. State of charge of the battery, December 2002.
The state of charge of the battery SOC for the entire month was simulated. The initial state of charge
considered was 0.901, and throughout the month the simulated values decrease as there are less day
with peak levels of solar irradiance above 500 W/m2, especially at mid of the month, with lower
ambient temperatures as well (graph of the figure 16). In figure 30, it can be seen that the battery is
discharging continuously, recovering part of the energy in the hours of maximum solar irradiance.
This was previously stated as it was shown that the battery voltage has a decreasing trend during the
month simulated.
52
Figure 31. Comparison between experimental and simulated data for the battery current
The simulated values compared with the experimental values for the SOC of the battery have a
difference, with the simulated values (SOC sim in the figure 31) with lower values. The explanation
for this may be that the battery model considered does not work very well for SOC values over 0.7-
0.8, because overcharge is not considered by the model, as the model considered is based on the one
used for developing the PSpice model by Castañer and Silvestre (11). Nevertheless, it can be seen that
the model predicts the trend of the behaviour of the SOC but with the differences mentioned before.
Additionally, the differences can be related to that the model developed considers that the cell of the
battery has a higher internal resistance compared to the resistance that the real battery has (22),
according to its SOC pattern throughout the month shown in figure 31.
The profile of the SOC during the month simulated shows a pattern as a sawtooth, that according to
Fragaki (22), may happen when the battery stands idle after discharge, with certain chemical and
physical changes taking place, which can result in a recovery of battery voltage, so the battery voltage
which has dropped during heavy discharge will rise after a rest period.
As was mentioned, the model of the battery for experimental values of SOC bigger than 0.8 does not
work very well as it does not consider overcharge of the battery. Additionally, the differences between
the simulated and the experimental SOC that show up in figure 31 get bigger as the battery is
discharging throughout the month.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
01
-Dec
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02
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03
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29
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30
-Dec
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31
-Dec
-02
SO
C
Day
State of Charge of the Batteries : December, 2002, Southampton, UK. Experimental Data
and Simulation Data
SOC exp
SOC sim
53
7. Conclusions
In order to represent each of the components, it was necessary to develop their models in SIMULINK,
platform that allows to connect each of them considering different subsystems and the possibility to
create tailored blocks as embedded MATLAB functions to the necessity of modelling the expected
behaviour of these devices, when exposed to different ambient conditions considering different
amounts of solar irradiation and ambient temperature.
The system developed in this project to represent a stand-alone photovoltaic system consists in a
generator, accumulator (batteries in parallel), charge controller, and a resistive constant load.
As was shown in the results and discussion, the model developed has problems about some
magnitudes, overestimating values of currents and underestimating values of voltages. This is evident
when comparing with the experimental data available for this project, some discrepancies are shown.
Specifically, and for the simulated day December 3th, 2002, the discrepancies for the values of the PV
current were found for the hours with sunlight, as for hours with no sunlight the values simulated
fitted to zero as the experimental values. As can be expected, the discrepancies shown for the PV
power values coincide with the hours with sunlight during the day. The discrepancies are due to the
realistic ambient conditions may not all be considered in the model; the specific parameters of the PV
generators were not all considered for the model; the difference between the measured temperature of
the cell, as it is more likely to have it as an approximation; there might be uncertainties in the
measurements of the experimental data. All those reasons led to have uncertainties in the developed
model for the PV module. Considering the model of the battery, uncertainties in the model developed,
for the values of the voltage in the battery, though there was a good fit between experimental and
simulated values for the state of charge. These good results obtained when simulating the values for
the state of charge of the battery throughout the day considered, and that the model, in general,
coincides with the trend in the curves for the current, voltage, power, indicate that the model
developed has some strengths. In the other hand, for the simulation of the state of charge of the
battery for a longer period of time (in this case, one month), the model showed that for lower values
of state of charge, the differences with the experimental data started to increase, aspect that is
undesirable for modelling or predicting the behaviour of the stand-alone photovoltaic system. The
discrepancies in the modelling of the battery are found for the hours with sunlight. Some reasons
wielded for this are the discrepancies that might be considering the internal resistance of the battery,
and the possible effect of temperature on the voltage value of the overcharge state.
The modelling of the charge regulator as a switch that disconnects and connects the photovoltaic
generator from the batteries according to two certain levels of charge (battery voltage), could not be
tested because the data used for doing the simulation and its comparison did not have levels of input
54
data to the system (solar irradiance and ambient temperature) that allowed to have higher values of
battery voltage over or under the threshold values for disconnecting or reconnecting the photovoltaic
module, respectively. December of 2002, for the day and the month simulation, was not the ideal data
to show the performance of the model of the charge regulator.
As was mentioned, the modelling of the battery was one of the hardest tasks, according to what
several authors indicates with the exceptions commented earlier.
The usefulness of having a model developed in SIMULINK should be significant. It is not difficult to
simulate a variety of different ambient conditions, taking into account the ambient temperature and
the solar irradiance.
Additionally, it is easy to make changes to the parameters of the PV system, in order to represent
different photovoltaic devices, with different characteristics, considering the deficiencies of the
model.
There are a variety of possible improvements that could be made to the model developed in this
project. Many of the constants used in the battery block were arbitrarily chosen based on values for
the PSpice model, and should be adjusted to represent better the behaviour of the battery, especially to
show the profile of the current passing through it (charge mode or discharge mode).
About the simulation correlation between the battery voltage and the state of charge, can be said that
it was not quite as predicted. As other models, in the model applied in this project there may be an
effect of the charge controller on this correlation, allowing some differences that inside the scope of
this project, are not quite as easy to understand and explain completely.
Finally, it is evident that the model can be improved over the time to show in a better way the real
behaviour of the whole stand-alone photovoltaic system.
55
8. Further work
As was mentioned in the conclusions, the model developed in this project can be improved, there is
necessity to validate the results properly, comparing against real and experimental data, fact that was
made in this project though it is necessary to make it statistically, to get, for example, the mean error
of the simulation, index that was not calculated in this project.
The model needs to be improved, especially related with some inaccuracies about overestimations and
underestimations of values of voltages and currents. In this way, to have a better performance, it is
necessary to run more simulations, and additionally, over longer periods of time. In this way, deeper
and longer simulations are needed, as instance, considering a complete year. With this, it may be
easier to model the behaviour of the stand-alone photovoltaic system.
As was mentioned, modelling in MATLAB/SIMULINK is quite time demanding, but allows high
flexibility with different hierarchical systems and subsystems that can be applied. Considering this, it
is real necessary to add other components to this simple model of stand-alone photovoltaic system, as
maximum power point tracker (MPPT), inverter (DC/AC), DC/DC converter, and dynamic load over
the day, to reflect closer to the real behaviour of electricity demand of a dwelling or other building.
56
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