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IS S N 0003-70IX, Applied Solar Energy, 2020, Vol. 56, No. 1, pp. 63—69. ©Allerton Press, Inc., 2020.
Russian Text © The Author(s), 2020, published in Geliotekhnika, 2020, No. 1, pp. 88—97.
SOLAR POWER PLANTS
Algorithm for Multivariate Solution of Mathematical M odels
in MATLAB to Create a Database of Environmental Parameters
A. G. Kornilov*
Physical-Technical Institute, Academy of Sciences of the Republic of Uzbekistan, Tashkent, 100084 Uzbekistan
*e-mail: [email protected]
Received August 8, 2019; revised October 1, 2019; accepted November 20, 2019
Abstract—This article proposes an algorithm for solving the modeling problem with numerous variables in the
MATLAB programming language. According to the developed algorithm, the MATLAB model calculates
the values of solar insolation and ambient temperature of a particular location depending on five variables
simultaneously: (1) time of day (x), (2) day of the year (n), (3) tilt angle ((3), (4) azimuth angle (y) and (5)
albedo (p). Meanwhile, the calculation results form a multidimensional database of solar insolation and
ambient temperature of a given location for any values of the above variables and their combinations .An
example of using the created database is given for calculating the annual total photovoltaic panel productivity
and productivity dynamics on days of the year for latitude 41°2995', which corresponds to a location in the
city of Tashkent. The developed algorithm can be used to analyze/forecast the efficiency of solar plants using
parameters from the database and other variables, such as temperature coefficient, dust content, nominal
efficiency, wind speed, humidity, etc., as input data. The availability of data makes it possible to use data for
both one-way and two-way reception of radiation as for bifacial photovoltaic panels. The database created
based on the proposed program model can serve as a basis for comparison with other calculations or mea­
surement databases and for studying or determining other parameters that affect possible differences in the
compared data and creating advanced modeling algorithms for the solar irradiance.
Keywords: MATLAB, algorithm, modeling, environment, daily, monthly, annual, solar radiation, ambient
temperature
DOI: 10.3103/S0003701X20010077
IN T R O D U C T IO N
ods. Empirical models use the same mathematical
expressions, which merely have different regressive con­
stants. However, these constants are completely depen­
dent on the considered location of a solar plant [5].
The fundam ental relationships for determining the
intensity of solar radiation were summarized and pre­
sented by J.A. Duffie and W.A. Beckman [6]. Using
these relationships, the studies were carried out on
solar heat and cold supply [7], modeling of the topo­
graphic variation of solar radiation [8], three-dim en­
sional modeling of the solar energy potential of an
entire city [9], modeling of PY [10] as well as PV /T
[11], calculations of the im pact of solar radiation on
steel sheet structures [12], and the software products
PV*SOL [13], Polysun [14], IN SE L [15], and RETScreen were developed [16].
A variety of m ethods with different degrees of com ­
plexity have been developed to approximate daily tem ­
perature curves. The authors of the study [17] analyzed
previously developed models and proposed a more
accurate model. Em pirical models of daily tem pera­
ture curves of varying complexity are also effectively
used [18—20], just like A N N models are [21—23].
There are models that correlate other environmental
Environm ental param eters such as tem perature
and humidity, wind speed and direction, as well as
solar radiation have a great influence on all aspects of
life. Based on this, a lot of research is conducted on
the forecasting and influence of these parameters.
M odeling and estimation of the solar radiation
level based on A N N s are performed using geographic
coordinates such as latitude, longitude, altitude, and
meteorological data, as well as air tem perature, rela­
tive humidity, sunny day duration, pressure, etc. [1].
In some cases, A N N models give m ore accurate
results in short-term forecasting [2], using m ore than
one parameter. Meanwhile, in order to obtain an accu­
rate m odel for estimating the m agnitude of solar
energy, it is necessary to have long-term data, which
are not always available due to the high cost of m easur­
ing instrum ents and difficulty of accessing measuring
sites [3], which leads to a high m easurem ent cost [4].
A com parison of the efficiency of different algorithms
for estimating solar radiation has shown that linear
and nonlinear modeling of the solar radiation level
and tem perature that use only one input param eter is
simpler than ANNs and other machine learning m eth­
63
64
KOMILOV
Zenith
Fig. 1. Schematic representation of the parameters used for modeling.
parameters with temperature [24, 25], as well as com­
mercial products/programs [26, 27]. Temperature m od­
els are widely used to provide comfort in buildings and in
urban planning [28, 29]. Certainly, as well as data on the
solar irradiance, temperature curves are very important
in the planning of solar power systems [30—34].
The study considers two environmental param e­
ters: (1) the intensity of solar radiation and (2) tem ­
perature. The MATLAB m odel is proposed, which
calculates the values of solar insolation and ambient
tem perature of a certain location (latitude cp) depend­
ing on five variables simultaneously: (1) time of day
(t), (2) day of the year (я), (3) tilt angle ((3), (4) azi­
m uth angle (y) and (5) albedo (p). Meanwhile, the cal­
culation results are six-dimensional and represent a set
of data for calculating solar insolation and ambient
tem perature of a given location for any values of these
variables and their combinations.
M ETH O D O LO G Y
Program ming was perform ed using the MATLAB,
the m ain feature of which is its wide capabilities for
working with matrices. The program was created on the
basis of the Duffie and Beckman clear sky solar irradi­
ance algorithm presented in 2006, which is calculated in
four main stages. Lirst, the position of the Sun is calcu­
lated depending on the coordinates of a location and
time. Then direct normal radiation is calculated taking
into account the sine of the Sun angle. Based on the posi­
tion vector of the Sun and normal vector to the surface,
the ray incidence angle is calculated (Fig. 1), and then
hourly direct, diffuse, and reflected radiations are esti­
mated. Diffuse radiation is calculated using an isotropic
model.
Figure 1 shows the following parameters: (3-slope is
the angle between the plane of the surface under con­
sideration and the horizontal; 0Q< (3 < 180° ((3 > 90°
means that the surface has a downward facing com po­
nent); y-azim uth angle of the surface is the deviation
of the projection onto the horizontal plane of the no r­
mal to the surface from the local m eridian with zero,
negative, and positive value being equal to the south,
east, and west, respectively, —180° < у < 180°; 0Zis the
zenith angle, the angle between the vertical and the
line to the Sun, i.e., the angle of ray incidence on the
horizontal surface; a s is the Sun height angle, the
angle between the horizontal and line to the Sun, i.e.,
an addition of the zenith angle 90°— y; ys is the azi­
m uthal angle of the Sun, the angular displacement of the
radiation projection from the south onto the horizontal
plane. Displacements from east to south and from west to
south are negative and positive, respectively.
The reflected radiation depends on the albedo
value and can theoretically vary from zero for a com ­
pletely black surface to unity for a surface that com ­
pletely reflects electromagnetic waves. The albedos of
some natural surfaces are shown in Table 1 [35].
The daytime tem perature curves are not symm etri­
cal, and their shape can vary significantly depending
on the weather conditions of a location and time of the
year. The study applies the presented m odel using the
average values of the m aximum and m inim um daily
tem peratures for the last five years (Fig. 2), which were
measured within the framework of the project [36].
A program has been developed for calculating the
ambient tem perature Ta and global solar radiation I G,
which consists of direct I B, diffuse I D and reflected I R
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ALGORITHM FOR MULTIVARIATE SOLUTION OF MATHEMATICAL MODELS
65
Table 1. Albedo of some natural surfaces
Surface type
Dark soils
Coniferous forests
Wet gray soils
Rye and wheat fields
Deciduous forests
Albedo
Surface type
Albedo
0.05-0.15
0.10-0.15
0.10-0.20
0.10-0.25
0.15-0.20
Potato fields
Meadows
Cotton fields
Dry steppe
Dry clayey or gray soils
0.15-0.25
0.15-0.25
0.20-0.25
0.20-0.30
0.20-0.35
components. These param eters are dependent on the
above variables:
Ta(h,D),
I G(,nT
yap ),
and I R(a,x,n,p) and I G(a,y,x, n, p)
= I B(a, y, x, n) + I D(а, x, л) + 7^(a, x, n, p).
I B(a
(1)
Here, x (0:24) is the time of day, n (1 : 365) is the day
of the year, (3 (0 : 180) is the tilt angle, у (—180 : 180) is
the azim uth angle, and p (0 : 1) is the albedo. The flow
chart of the program is shown in Fig. 3.
As an example, the calculations were m ade for lat­
itude 41°2995', which corresponds to terrain in the ter­
ritory of Tashkent. W hen calculations are made for
each hour of day for 365 days of the year, Ta(x, n) will
have 8760 (24 x 365) datapoints. The dynamics of the
efficiency of solar plants are calculated using a per-second change in the parameters, and in this case 7^(x, n)
will have 22.776 m in (24 x 3600 x 365) datapoints. In
the same way, the data volume of each param eter
depends on the increm ent of each variable.
Our calculations were m ade using the variables
with the following increments: х = 0 : 1 : 2 4 , я = 1 : 1 :
365,(3 = 0 :1 :1 8 0 , y = -180 :10:180 and p = 0 :0.05 :1.
In these conditions, the program needed 53 m in to
complete the calculations on a com puter with 8 GB of
RAM. The size of the file saved in mat. format was
1.04 GB, and the am ount of data stored was about
2.57 GB according to Table 2.
The calculation error is determ ined by the sensitiv­
ity of the “algorithm ” (Fig. 3) to the variables. Thus,
the ‘algorithm ’ m ust be developed determining the
m aximum values of the increments o f the variables,
which ensure the m inim um or permissible errors. In
this case, it is necessary to take into account the
assignm ent/representation of the variables in the
MATLAB (“Class” in Table 2), which helps to speed
up the calculation, but also conceals the possibility of
errors, both because of the m antissa and because of
possible rounding, which leads to further errors in case
of num erous repetitive calculations.
Since the m ain task was to dem onstrate the algo­
rithm for solving the modeling problem with num er­
ous variables, “m axim a” for increments of the param ­
eters of the presented “algorithm ” were not calculated
in this paper, but the “Class” o f variables was assigned
based on their value during the initial start of the algo­
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Surface type
Sea ice
Dry light sandy soils
Contaminated snow
Clean wet snow
Fresh dry snow
Albedo
0.30-0.40
0.35-0.40
0.40-0.50
0.60-0.70
0.85-0.95
rithm with assigning all variables with the binary m an­
tissa (’double').
RESULTS A N D D ISC U SSIO N
The database data can make it possible to calculate
the energy efficiency of solar plants and buildings,
algorithms for tracking the Sun and conduct num eri­
cal experiments to identify other optim al param eters
of solar plants, which depend on these parameters.
Figure 4 shows the daily changes in ambient tem ­
perature over the year.
Using the simple empirical formula from [37],
Tc =1.4117; +6.414,
(2)
it is possible to determine the tem perature of a photo­
voltaic solar (PV) or photovoltaic system (PVS) (Tc)
and calculate the efficiency with the help of the wellknown formula
л = Л о (1 - с г ( 7 ; - г 0)),
(3)
where r|0 is the nom inal efficiency of the panel under
standard testing conditions (STCs) at a tem perature
7 q, Ct is the tem perature coefficient of the efficiency
of the PV, which ranges from 0.0001 —0.007 [1/°C] for
various technologies [38].
The productivity of the PV can be calculated as
P = r\IG(a,y,x,n,p), (4)
Fig. 2. Average values of maximum and minimum daily
temperatures for the last five years for the city of Tashkent.
KOMILOV
66
and, summing up the values of productivity for the day
and for the year, we can obtain the values of the annual
total productivity depending on the tilt angle, azimuth
angle, and albedo. The ratios of the tilt angle and azimuth
angle for the maximum annual productivity can be deter-
Table 2. The result of the “whos” command for the con­
tents of the saved file
Name
Size
Bytes
I
5-D
2566625250
Intl6
h
4-D
122220250
intl6
Ir
4-D
69368250
intl6
25 x 365
3303250
intl6
25 x 365
36500
Ta
Class
Single
Albedo
1 x 21
168
Double
Angle
1 x 91
1448
double
Azimangle
1 x 31
248
double
Days
1 x 365
2920
double
Time
1 x 25
200
double
mined in the diagram from the value of the annual total
productivity for a certain albedo value (Fig. 5).
Figure 5 makes it possible to estimate that the m ax­
im um values of the annual total productivity are
achieved within a tilt angle of 35—50 degrees and an
azim uth angle of —30 to 30 degrees for an albedo of
0.2 (a). The calculations show that the values of
annual total productivity differ within these limits by
no m ore than 10%. For an albedo of 0.6 (b), the m ax­
im um values of annual total productivity are achieved
within a tilt angle of 40—60 degrees and an azim uth
angle of —40 to 40 degrees. It is natural that the m axi­
m um values of annual total productivity with an
albedo of 0.6 (b) are higher due to the larger propor­
tion of reflected radiation. The daily or annual
dynamics of the PV productivity can be obtained for
any value in Fig. 5 (Fig. 6).
Figure 6 shows that the times of productivity peaks
change throughout the year, and their maximums
occur in the cool season, which is due to the influence
of a decrease in tem perature. The data shown in Fig. 6
will be useful in predicting the im pact of PPSs on the
electric network or planning the volume of storage
capacities.
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67
35
30
25
20
15
10
5
0
-5
Fig. 4. The calculated values of the daily changes in ambient temperature throughout the year.
450
400
350
300
250
200
150
100
50
500
450
400
350
300
250
200
150
100
50
Tilt angle
Fig. 5. The calculated values of the annual total productivity for p = 0.2 (a) and p = 0.6 (b) at r|0 = 0.18, T0 = 25°C and CT =
0.005°C_1.
CO N CLU SIO N S
The MATLAB m odel has been developed, which
calculates the values of solar insolation and ambient
tem perature of a particular location simultaneously
depending on the time of day, days of the year, tilt
APPLIED SOLAR ENERGY
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No. 1
2020
angle, azim uth, and albedo. Moreover, the calculation
results are six-dimensional and represent a set of data
for calculating solar insolation and ambient tem pera­
ture of a given location for any values of these variables
and their combinations.
KOMILOV
68
Fig. 6. Dynamics of the PV productivity during the days of the year for p = 0.2, a = 40° and у = 0° at r|0 = 0.18, T0 = 25°C and
CT = 0.005°C_1.
To give an example, the calculations of PV produc­
tivity with the use of the generated database are pre­
sented. The calculated values of the annual total PV
productivity and productivity dynamics during the
days of the year are demonstrated. This makes it pos­
sible to estimate the optim al tilt and azim uth angles for
the m axim um values of the annual total productivity.
The developed algorithm can be used to analyze/forecast the efficiency of solar plants using the
param eters from the database and other variables such
as tem perature coefficient, dust content, nom inal effi­
ciency, wind speed, humidity, etc., as input data.
The availability of data with different albedos for all
tilt angles with respect to the position of the Sun makes it
possible to use the data for both one-way and two-way
reception of radiation, for example, as for bilateral PVs.
The database created based on the proposed pro ­
gram m odel can serve as a basis for com parison with
other calculations or m easurem ent databases and for
studying or determining other param eters that affect
possible differences in the compared data and creating
m ore advanced modeling algorithms.
FUNDING
This study was financially supported by the Ministry of
Innovation Development of the Republic of Uzbekistan
within the OT-Atech-2018 517 + 513 + 362 research project
“Integration of Photovoltaic Systems into the Central Elec­
tric Network.”
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Translated by L. Solovyova
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