International scientific journal
“Interpretation and researches”
Volume 2 issue 2 (24) | ISSN: 2181-4163 | Impact Factor: 8.2
250
PROBLEMS OF MATHEMATICAL MODELING OF SOLAR STATION
Iroda Tadjibekova
senior teacher of the department “Mechanization of agriculture and automation”
Tashkent State Agrarian University Tashkent
Ulug`bek Ibrokhimov
assistant of the department “Mechanization of agriculture and automation”
Tashkent State Agrarian University Tashkent
Annotation:
The article presents information that the mathematical model of the
operation of a solar power plant of an energy station is simplified by the fact that its
parameters are grouped as if the parameters responsible for the territorial location of
the plant are used as permanent parameters, and the average values of solar insolation
are variable parameters.
Keywords.
Photovoltaic cell, shunt resistance, mathematical model,
photovoltaic panels, solar radiation, temperature, simulation of operating modes,
voltage-ampere and voltage-watt characteristic.
Introduction.
When analyzing domestic and world experience in the field of
solar panel modeling, two approaches can be distinguished. One of them considers
the photovoltaic panel as a separate component of the power supply system and
examines the parameters of the output power depending on various input factors.
Procedure. For this, a classical circuit for replacing a photovoltaic cell with one
diode can be used, shown in Fig. 1.
This photoelectric cell replacement scheme is widely used and is used to assess
the effect of the illumination degree and temperature of the photoelectric cell on its
characteristics. Such a model is used in the calculation of generating plants with both
serial and parallel connection of elements.
Fig. 1 - Single-ion photovoltaic cell replacement scheme
In this scheme, the diode VD and the current Id passing through it are connected
in parallel with a branch containing a current source Iph, this circuit is a p-n transition
model, to which a resistor Rs is connected in series, characterizing the active
International scientific journal
“Interpretation and researches”
Volume 2 issue 2 (24) | ISSN: 2181-4163 | Impact Factor: 8.2
251
resistance of the connecting contacts and the photopanel itself and a branch with a
shunt resistance R
sh
simulates the reverse p-n junction resistance and the resistance
of the conductive films of the element.
Accordingly, the load current equation I can be written as follows:
𝐼 = 𝐼
𝑝ℎ
− 𝐼
𝑠
[
𝑞(𝑈 + 𝐼 × 𝑅
𝑠
)
𝑁
𝑘
𝑇
−
(𝑈 + 𝐼 × 𝑅
𝑠
)
𝑅
𝑠ℎ
]
where:
𝐼𝑝
ℎ is a photocurrent dependent on solar radiation, defined as:
𝐼
𝑝ℎ
= [𝐼
𝑠𝑐
+ 𝐾
𝑖
(𝑇 − 𝑇
𝑟𝑒𝑓
)] ×
𝐵
1000
𝐼𝑠
is the saturation current of the diode, which is expressed by the cubic
dependence on temperature as follows:
𝐼
𝑠
𝑇 = 𝐼
𝑅𝑆
(
𝐼
𝑇
𝑟𝑒𝑓
)
3
× 𝑁 × 𝑁𝑘 (
𝑞𝑈
𝑡
𝐼
𝑇
−
𝐼
𝑇
𝑟𝑒𝑓
)
Here T is the temperature of the photoelectric cell, Isc is the short-circuit
current, k is the Boltzmann constant, Ki is the temperature coefficient of the short-
circuit current, Tref is the reference temperature, B is the solar insolation value in
W/m2, N is the diode ideal coefficient, N is the number of solar panels, Ut is the
diode thermal voltage.
The reverse saturation current of the diode
𝐼
𝑅𝑆
in this relation for standard
conditions can be approximated as follows:
𝐼
𝑅𝑆
=
𝐼
𝑠𝑐
𝑞𝑈
𝑁𝑘𝑇𝑁
𝑖
Analysis.
Due to the dependence of the load current equation on the parameters
Rs and Rsh, the current Ish through the shunt resistance is sometimes neglected due
to the large value of the shunt resistance. The implementation and testing of such a
mathematical model is given in, where for photovoltaic panels, depending on the
level of solar radiation and temperature, modeling of operating modes and further
construction of voltage-ampere and voltage-watt characteristics are performed, and
the following expression is obtained, which determines the relationship of the output
power of the solar panel with the temperature of the photovoltaic module and solar
radiation:
𝑃 = (0,2465G − 0,1378) × (
100 + 0,759 · (25 − t)
100
)
A similar approach using a single-ion substitution scheme has been proposed,
but it should be noted the importance of investigating the photovoltaic power supply
system as a complex consisting of a set of interconnected elements whose main task
International scientific journal
“Interpretation and researches”
Volume 2 issue 2 (24) | ISSN: 2181-4163 | Impact Factor: 8.2
252
is to convert the energy of the sun into electricity. As a rule, such systems are
represented by a photovoltaic installation, an electric energy storage, a controller, an
inverter and a load. In addition, for the photovoltaic panel, the importance of
modeling these components under various solar insolation data and temperature
modes of operation is noted.
Methods.
A model is considered that estimates the voltage and current potential
of a photovoltaic panel under given operating conditions (solar insolation and
temperature) and uses a single-ion replacement scheme, but excludes from it a
parallel branch with shunt resistance and the expression connecting the load current
reduces to:
𝐼 = 𝐼
𝑝ℎ
− 𝐼
𝑠
[
𝑞(𝑈 + 𝐼 × 𝑅
𝑠
)
𝛼
− 1]
where α is the thermal stress measurement completion factor, B.
This method allows you to take into account the intensity of solar radiation,
ambient temperature, technical characteristics and the degree of spatial orientation of
the photoelectric converter.
In operation, the output power of the photovoltaic modules is also associated
with temperature, but the coupling equation is as follows:
𝑃
ФЭС
=
𝐺
𝐹
× 𝑁 × 𝐺 × 𝑙𝑛(10
6
× 𝐺)
𝑇
ФЭС
Here CF - is determined from the passport data of the solar panel, N - the
number of solar panels forming the station, G - the current level of solar radiation for
a given area, TPP - the current temperature of the photoelectric module
When constructing a mathematical model of a photovoltaic installation, it is
necessary to take into account certain components of the power supply complex, such
as the spread of time constants characterizing the speed of processes in them. For
example, small speed values are characteristic of elements with semiconductor
converters, and such elements of the photovoltaic complex, such as load, can
correspond to large values of time constant. This circumstance makes it possible to
justify problems arising during the construction of adequate mathematical models of
components, along with this it can be a confirmation of the need for decomposition
for consideration outside the composition of the generating complex. In this work, the
input parameters of the mathematical model are the coordinates of the terrain, the
reflection coefficient of the earth's surface (albedo), the day per year, the atmospheric
transparency index, the wind speed for a given terrain, the optimum installation
angles of photovoltaic modules, the characteristics of the solar tracker and the
deviation of the average daily ambient air temperature.
International scientific journal
“Interpretation and researches”
Volume 2 issue 2 (24) | ISSN: 2181-4163 | Impact Factor: 8.2
253
Results.
The mathematical model of the operation of a solar power plant of an
energy station is simplified by the fact that its parameters are grouped as follows: the
parameters responsible for the territorial location of the plant are used as permanent
parameters, and the average values of solar insolation are the variables that change
during modeling.
It is necessary to take into account the stochastic nature of meteorological
factors, and therefore the range of received values of the time constant is in a very
large range. This fact leads to the need to use the apparatus of probabilistic
forecasting methods. The parameters of solar panels set forth in the passport data in
real conditions are stochastic in nature, respectively, when determining the output
power of a photovoltaic panel, each of them must be taken into account.
𝑃
𝑑𝑒𝑡.сэс
= (1 − 𝛼𝜂
ФЭС
)𝑃
уст ФЭС
Where
𝜂
ΦES is the efficiency of the photovoltaic panel, respectively - the
installed power of this panel, α the ideal coefficient, defined as:
𝛼 = 𝛼
𝑟𝑒𝑓
(
𝑇
𝑇
𝑟𝑒𝑓
)
An analysis of existing approaches showed that Russian and foreign researchers
in modeling photovoltaic panels operate according to two scenarios. The first of them
involves modeling the panels as a separate component of the generating plant.
Another approach involves building a model from the position of a group of
interconnected elements, where the solar panel is one of the variables affecting the
operation of the generating complex as a whole. Despite the differences in
approaches to the construction of mathematical models, both cases are characterized
by extensive use in the study of a single-iodine substitution scheme in various
variations. This fact, as a rule, is usually associated with the simplicity of this
scheme. In addition, the use of such a model in modeling complex systems allows
you to get the result with the required accuracy for practice and, accordingly, does
not require large computational costs.
Conclusion.
More complex models are also offered that take into account
additional factors. However, the output parameters of these models, which are used in
further calculations of energy transformations in electrical complexes, show that
energy variables are actually used, similar to those obtained from a model based on a
single-ion substitution scheme. The values of these variables, regardless of the
complexity of these models, do not change qualitatively.
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U.
Ibroximov
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ELEKTR
STANSIYALARINING
TARMOQQA
INTEGRATSIYASINI
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International scientific journal
“Interpretation and researches”
Volume 2 issue 2 (24) | ISSN: 2181-4163 | Impact Factor: 8.2
254
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