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 APPLIED SOLAR ENERGY Vol. 56 No. 1 2020 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 APPLIED SOLAR ENERGY Vol. 56 No. 1 2020 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. APPLIED SOLAR ENERGY Vol. 56 No. 1 2020 ALGORITHM FOR MULTIVARIATE SOLUTION OF MATHEMATICAL MODELS 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 Vol. 56 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. 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Zhu, W., Lu, A., and Jia, S., Estimation of daily maxi­ mum and minimum air temperature using MODIS Translated by L. Solovyova APPLIED SOLAR ENERGY Vol. 56 No. 1 2020