System for maintaining optimal temperature modes of solar power plants

Objectives : To ensure eﬃcient operation of solar modules and temperature mode in accordance with standard test conditions range within 20-30 ◦ C. Methods : A mathematical model was developed using the laws of photo-electricity, heat and mass transfer, which provides improvement in the eﬃciency of solar power plants. Findings: This study presents the theoretical analysis of diﬀerent embodiment for systems of solar modules using physical and mathematical models. The main design and functional characteristics of solar modules were calculated: temperature and eﬃciency depending on internal parameters and ambient environment. Dependence between temperature and eﬃciency of solar modules for diﬀerent values of air temperature, emissivity factor of solar cells, and eﬃciency of solar power plants, were determined and presented in the form of diagrams. It was found that solar modules have the highest heating temperature at solar irradiance of Еc=1200 W/m 2 , air temperature of Т a =50 ◦ C, and that the solar modules have the lowest heating temperature at Т a =30 ◦ C and e =0.8, with a decrease in the heating temperature from 111 to 38 o С, the eﬃciency decreases at an air temperature of 50 o С and at the emissivity factor of e =0.8 and 0.3 and makes from 4.2 to 10.3%. The dependences were calculated for diﬀerent latitudes φ : 56 o (Moscow), 45 o (Krasnodar), 35.5 o (Eslamshahr, Iran), 31.6 o (Béchar, Algeria), 13.1 o (Chennai, India). Solar modules with cooling devices were installed in the Istra district of the Moscow region and bench tests, to speciﬁc calculated parameters, were carried out under full-scale conditions. Conclusions : It was found that the eﬃciency of solar panels increased signiﬁcantly due to the use of systems with heat-exchange tubes, and the losses, when using an antigravity heat exchanger for cooling photovoltaic cells, reduced by 6-7 times. Novelty : Usage of an anti-gravity heat exchanger to cool the photovoltaic cells, to maintain the optimal operating temperature prevent electrical distortion due to extreme temperatures.


Introduction
Due to the growing environmental crisis and to ensure the energy security of countries, over the past 15-20 years, an almost explosive growth in the use of renewable energy sources has been observed across the globe (RES) (1) . Access to reliable and sustainable energy encourages the inflow of investments, including government investments, to develop a wide range of energy converters, converting energy from renewable sources to energy of consumer-oriented formats in order to fully realize the potential of renewable energy sources in a particular country. Renewable energy systems, included in the centralized energy grids of countries, provide up to 25% of power consumption in some of them.
Systems of stand-alone type are most commonly used in agriculture, both in animal breeding (2) and plant growing (3) . Currently, the development and creation of thermal and photovoltaic modules is one of the key directions in the development of solar energy (4)(5)(6)(7) . The main objective of this work is to increase the efficiency of solar power conversion, by reducing the loss of incoming solar power and, as a result, to reduce the cost of the produced energy (8,9) . The estimated payback period based on the experimental model of photovoltaic modules with a cooling system on areas of more than 8 square meters is 30-40 percent less in comparison with conventional photovoltaic modules. The proposed installation will increase the efficiency of photovoltaic modules and power generation, as well as increase the service life of photovoltaic modules by cooling with heat pipes.
It is known that only 6 -20% of solar radiation, incident on a photocell, is used to generate electricity. The rest of the energy, to a greater extent, is spent on heating the photocell, resulting in a significant increase in its surface temperature, which, in turn, has a negative impact on its operation (10)(11)(12) .
Standard Test Conditions (STC) for solar modules with a capacity of 1 kWp/m 2 are applied for their operating temperature of 25 • C.
With solar irradiance of 1000-1200 W/m 2 , the cells heat up to 60-70 • C, each losing 0.07-0.09 V. This is the main reason for the reduction in the efficiency of solar modules.
The aim of this work is to develop mathematical models, describing the operation of individual units of the system, to study the thermal behavior of a solar module and a system for maintaining optimal temperature modes of photovoltaic cells to improve the efficiency of solar power plants.

Materials and Methods
When using solar panels, the main task is to reduce photovoltaic losses during their operation. To maintain the optimal operating temperature of photovoltaic cells and prevent distortion of electrical characteristics induced by extreme temperatures, the provision is made to cool the photovoltaic detector of a solar module due to heat exchange between the substrate of photovoltaic cells and the lower soil horizon using an antigravity heat exchanger. In the process of heat exchange, heat is transferred by a cooling agent from the evaporation zone of the antigravity device down to the condensation zone, where the coolant is condensed due to transferring the latent heat of vaporization to the lower soil horizon at a depth of 3-5 meters, depending on climatic conditions, wherefrom the coolant, in liquid form, rises along the capillary body up to the evaporator. The coolant regeneration process is repeated cyclically. In this case, the coolant parameters are selected in a manner that the boiling point coincides with the lower limit of the range of temperatures optimal for the operation of photovoltaic cells. In addition, the filling depth of the condenser part of the antigravity heat exchanger is chosen in such a way that the soil temperature ensures the coolant cooling to the temperature optimal for the operation of photovoltaic cells, which makes from 30 to 50% of the annual soil heating depth. In the process of the method implementation, the photovoltaic module is cooled to an optimum temperature of 20-30 • C. This results to an increase in the solar module efficiency.

Analytical methods for calculating the heating temperature of solar panels without accounting for the atmospheric parameters
The temperature of solar modules (SMs), without accounting for the atmospheric parameters, is determined by heat transfer via radiation. The equilibrium operating temperature of SMs is determined by the equation (13) : where a is the integral absorption coefficient of solar radiation (SR) of the surface of solar cells (SCs), f f ill is the filling factor of SCs of the SM panel, https://www.indjst.org/ η is the actual efficiency of SCs, S pp is the area of the SM front surface, Еc is the solar constant, W/m 2 , y is the angle of incidence of solar radiation on the SM, ε is the SR absorption coefficient of the SM front surface, ε is the SR absorption coefficient of the SM dark surface, S t p is the area of the SM dark surface, σ is the Stefan-Boltzmann constant (σ =5.67 • 10 -8 , W/m 2 K 4 ). Heat removal from solar modules occurs due to heat exchange with the ambient air through convection and radiation. Various systems of forced cooling of photovoltaic detectors can be used to increase the efficiency of solar panels (SMs) (14)(15)(16) .
For applications of air-cooling systems, the thermal behavior of a solar module with solar cells is studied when cooled by atmospheric air and via natural heat exchange with the environment. Heat transfer between the environment and the radiator occurs according to Newton's law (17) .
Electric power: where η se is the efficiency of solar cells, p.u., η opt is the optical efficiency of SMs. The efficiency of solar cells depends on temperature (18) : where η o is the efficiency of solar cells at standard temperature of Т o =298 K, T is the temperature of solar cells, K; k is the temperature coefficient (k is less than 0.003). Heat losses to the environment: • Induced by convection: where α is the heat transfer coefficient, determined according to the McAdams formula W/(m 2 K), α = 5.7+3.8V, t c is the mean temperature of the receiver wall, determined by iteration (19) ; F is the module area, m 2 ; V is the airflow rate, m/s. • Induced by radiation: where ε is the emissivity factor; σ is the Stefan-Boltzmann constant, W/m 2 K 4 ; Т is the absolute temperature, K. T c and T a are absolute temperatures of the radiating surface and the environment.
The flux of solar radiation, entering the solar panel surface, is spent on photovoltaic conversion, heating the module, transferring heat to the environment and is determined by the formula: Substituting the values of W el , W conv , W rad , expressed by the set of equations (2) -(6), into equation (7), we obtain a complex dependence of the heating temperature of SMs on the environmental parameters: solar irradiance Е c , air temperature T a , wind speed V and the SM parameters: the emissivity factor ε, the efficiency of solar cells η se , the optical efficiency η opt . The solution of the set of equations was reduced to determining the functional dependence of each component of equation (6) on temperature under the boundary condition: Where ∑W = W el +W conv +W rad https://www.indjst.org/

Analytical methods for calculating the operation of various solar panels
The calculation of the operation of SMs with accurate tracking is based on the known dependence of the hour angle ω of the Sun's motion (from sunrise to sunset), on the declination angle δ and the location latitude φ: The operation time of SMs with tracking, t tr , h, is determined according to the expression: where a=0.2618 rad/h (15 • /h) is the Sun's rate of motion (the Earth's rotation rate). The calculation of the operation time of (planar) stationary SMs is based on the dependence of the hour angle of the SM operation ω on the declination angle δ and the difference ∆s between the latitude φ and the angle of the SM inclination to the horizon s, ∆s=φ-s.
In the summertime sinω corresponds to the obtained expression: The declination angle δ depends on the annual angle w and corresponds to the expression: where δ is the angle between the Earth's rotation axis and the plane of its motion around the Sun (δ =23.5 • ). The declination δ can also be determined using the approximate formula by Cooper. Expressions obtained: The operation time of (planar) stationary SMs in summertime: The operation time of (planar) stationary SMs in wintertime: The angle of the Sun inclination during the day to the perpendicular to the SR input surface of a planar SM -j: where in summer time π/2 ≤ă≤ δ ; in wintertime ω ≤ ă ≤ δ . Daily average cosj cp : For a planar SM: Substituting the values of η opt -optical efficiency, p.u., η f m -FM efficiency, p.u., t sm -SM operation time, h, S sm -SM area, m 2 , Q sm -the SM energy output results in the following expression, W • h: https://www.indjst.org/

Systems of solar panels with antigravity heat exchangers
To maintain the optimal operating temperature of SMs, including solar cells cooling, it is possible to efficiently use of a system that removes thermal energy from heated solar cells by antigravity heat exchangers into the lower soil horizon to a depth equal to 30 to 50% of the annual soil heating depth (20,21) . The considered physical-mathematical model of the solar photovoltaic system consists of two close-fitting plates of solar cells and a substrate, attached to the evaporative part of an antigravity heat exchanger with different temperatures with the boundary conditions of type IV (22) . According to these conditions, at the boundary between these plates, the temperature is set equal to: where Т is the initial temperature of a metal plate, Т se is the initial temperature of silicon solar cells.
The coefficient m is determined by the formula; where λ is the thermal conductivity, c is the heat capacity, ρ is the density, respectively, of silicon solar cells and a metal plate.
η sm = η 0 k se (22) where η o is the initial efficiency of solar cells.     The presented diagram shows that the heating temperature of solar panels increases with an increase in solar irradiance Е c , air temperature Т a and with a decrease in wind speed v.

Calculation of the heating temperature of solar panels without accounting for the atmospheric parameters
The highest heating temperature of solar panels is observed at solar irradiance of Е c =1200 W/m 2 , air temperature of Т a =50 • C and wind speed of v=0, which exceeds the heating temperature in space.
The calculated dependences of the heating temperature of solar panels on the wind speed, at an air temperature of 20 and 30 • C, at the emissivity factor of ε=0.8 and 0.5 and solar irradiance of Е c~1 000 W/m 2 are presented in [ Figure 8].
https://www.indjst.org/ The presented diagram shows that the lowest heating temperature of solar panels is observed at Т a =20 • C and ε=0.8 and with a decrease in the heating temperature from 78 to 30 • C when the airflow rate increases from 1 to 20 m/s. The highest heating temperature of solar panels is observed at Т a =30 • C and ε=0.5 and with a decrease in the heating temperature from 95 to 40 • Cwhen the airflow rate increases from 1 to 20 m/s.  The presented diagram shows that the lowest heating temperature of solar panels is observed at Т a =30 • C and ε=0.8 and with a decrease in the heating temperature from 111 to 38 • C when the airflow rate increases from 0 to 20 m/s. The highest heating temperature of solar panels is observed at Т a =30 • C and ε=0.3 and with a decrease in the heating temperature from 164 to 60 • C when the airflow rate increases from 0 to 20 m/s. https://www.indjst.org/ Thus, to reduce the heating temperature of solar panels, it is necessary to increase the emissivity factor as well as to increase the airflow rate using natural conditions or a forced ventilation system. When using solar panels, the main task is to reduce photovoltaic losses during their operation. Middle latitudes are characterized by solar irradiance Е c~1 000 W/m 2 and summer air temperatures from 20 to 30 o С. The calculated dependences of the efficiency of solar panels at η c =5%, η o =0.8 on the wind speed, at an air temperature of 20 and 30 • C, at the emissivity factor of ε=0.8 and 0.5 and solar irradiance of Е c~1 000 W/m 2 are presented in [ Figure 10]. The presented diagram shows that the lowest heating temperature of solar panels is observed at Т a =30 • C and ε=0.8 and the efficiency increases from 13 to 14.8% with an increase in the airflow rate from 1 to 20 m/s. The efficiency reduction at the airflow rate of 1 m/s makes 20%.
The highest heating temperature of solar panels is observed at Т a =30 • C and ε=0.5 and the efficiency increases from 12.4 to 13.9% with an increase in the airflow rate from 1 to 20 m/s. The efficiency reduction at the airflow rate of 1 m/s makes 21%.
The efficiency reduction at the airflow rate of 20 m/s at an air temperature of 20 and 30 • C and at the emissivity factor of ε=0.8 and 0.5 ranges from 0.7 to 4.2%.
The calculated dependences of the efficiency of solar panels at η c =15%, η o =0.8 on the wind speed, at an air temperature of 20 and 30 • C, at the emissivity factor of ε=0.8 and 0.3 and solar irradiance of Е c~1 200 W/m 2 are presented in [ Figure 11]. The presented diagram shows that the lowest heating temperature of solar panels is observed at Т a =30 • C and ε=0.8 and the efficiency increases from 11.8 to 14.4% with an increase in the airflow rate from 0 to 20 m/s. The efficiency reduction at the airflow rate of 0 m/s makes 27%.
The highest heating temperature of solar panels is observed at Т a =50 • C and ε=0.3 and the efficiency increases from 9.8 to 13.5% with an increase in the airflow rate from 0 to 20 m/s. The efficiency reduction at the airflow rate of 0 m/s makes 53%.
The efficiency reduction at the airflow rate of 20 m/s at an air temperature of 50 and 50 • C and at the emissivity factor of ε=0.8 and 0.3 ranges from 4.2 to 10.3%.
Thus, to increase the efficiency of solar panels, it is necessary to increase the emissivity factor as well as to increase the airflow rate using natural conditions or a forced ventilation system.  As is seen from the diagram, solar panels with accurate tracking generates 1.5-1.7 times more energy per year than stationary planar solar panels (lower curve).

Calculation of the operation of various security systems
Based on the NASA tabular data on the horizontal illumination of the Earth's surface at latitudes 56 • (Moscow), 40.4 • (Madrid -continental region), 23.1 • (Havana -coastal region), the calculation of the decrease in daylight illumination from June to December was performed, with the results presented in [ Table 1]. The greatest decrease in daylight illumination from June to December is observed in northern latitudes (Moscow), the smallest -in southern latitudes in coastal regions (Havana).
Using these coefficients, the dependences of energy generated for stationary solar panels with an area of 10 m 2 and solar panels with accurate tracking on the time of the year (June-December) for different latitudes φ: 56 o (Moscow), 45 o (Krasnodar), 35.5 o (Eslamshahr, Iran), 31.6 o (Béchar, Algeria), 13.1 o (Chennai, India -coastal region) at T a =30 • C, ε=0.8, v=3 m/s, with the initial efficiency of solar cells η ose =15%, η opt =0.9, Е c =1100 W/m 2 , and the efficiency of solar cells η se =12.1% at a heating temperature of T c =72 • С, were (approximately) calculated under completely cloudless weather conditions, with the results presented in [ Figure 13]. Thus, the above calculation methods allow performing a comparative analysis of energy parameters taking into account the location latitude for the modules and installations being developed.

Systems of solar panels with antigravity heat exchangers
The calculated dependences of the efficiency of solar panels at η o =15%, η opt =0.8 on the heating temperature for various substrate materials are presented in Figure 14. The presented diagram shows that a solar panel with a copper substrate has the smallest decrease in the efficiency, from 15 to 13.6%, with an increase in the heating temperature from 30 to 165 • C; and a solar panel with a steel substrate has the greatest decrease in the efficiency, from 15 to 12.1%, with an increase in the heating temperature from 30 to 165 • C.
In case of such a design of the solar panel, the impact of an air temperature Т a , the emissivity factor ε, the airflow rate and solar irradiance Е c is taken into account.
The calculated dependences of the efficiency of solar panels at η o =15%, η opt =0.8 on the wind speed at Т a =20 • C and ε=0.8, Еc= 1000 W/m 2 for different substrate materials are presented in Figure 15. The calculated dependences of the efficiency of solar panels at η o =15%, η opt =0.8 on the wind speed at Т a =30 • C and ε=0.8, Еc=1200 W/m 2 for various substrate materials are presented in Figure 16. The presented diagram shows that a solar panel with a copper substrate has the smallest decrease (6.4%), an increase in the efficiency from 14.2 to 14.8% (5.7%), with an increase in the wind speed from 0 to 20 m/s. A solar panel with a steel substrate has the greatest decrease (14%), an increase in the efficiency from 13.2 to 14.6% (12.1%), with an increase in the wind speed from 0 to 20 m/s.
Having compared the calculated curves, it can be seen that the loss of systems without heat pipes at a minimum wind speed at Т a =50 • C and ε=0.8, Еc=1200 W/m 2 is 34%, and for systems with heat pipes-5.7%. At Т a =30 • C and ε=0.8, Ec=1000 W/m 2 , it makes 20%, and for systems with heat pipes -2.8%.
Thus, to reduce the heating temperature of solar panels, it is necessary to increase the emissivity factor as well as to increase the airflow rate using natural conditions or a forced ventilation system. The efficiency of solar panels increases significantly due to the use of heat pipe systems.

Experimental module
In order to develop and create new thermal and photovoltaic solar modules with antigravity heat exchange systems, solar modules with cooling devices were developed and bench-tested, to specify the calculated parameters, in full-scale conditions at the wind test center of the FSBSI "Federal Scientific Agroengineering Center "VIM", in the Istra district of the Moscow region.
[ Figure 17] shows the solar module (a) attached to the cooled part of the cooling device with a controlled temperature and mounted on a bench (b) with a biaxial solar tracking system. The experimental module operates as follows. Solar radiation, when accurately tracking the Sun, falls perpendicular to the surface of the solar module. The SR receiver is made of a number of solar cells by sequential electrical commutation with a width of h o = cm and a length of L = cm. By adjusting the temperature of the cooled part of the cooling device, it is possible to https://www.indjst.org/ optimize the heating of solar cells, thereby increasing the module efficiency.
Based on the above calculations, depending on natural conditions -solar radiation output, wind speed, ambient temperature; the module design parameters-optical efficiency, the materials used, it is possible to predict the output parameters (thermal and electrical) and the efficiency of the module as a whole.

Conclusion
The place of solar power in energy development of the future is determined by the possibilities of industrial use of new physical principles, techniques, materials and designs of solar cells, modules and power plants. The need to develop power supply based on renewable energy sources (RES), especially in rural areas, is now becoming more and more relevant, due to the presence of large territories, power supply of which, via providing therein traditional energy sources, is very challenging, complicated and unprofitable. It is in the latter case that the use of renewable energy sources becomes expedient and frequently cost-effective.
Having studied the impact of temperature on the efficiency of photovoltaic cells, it was revealed that a decrease in the heating temperature of SMs depends on a decrease in the integral absorption coefficient of SR a of the surface of solar cells, as well as on an increase in the SR absorption coefficient of the front surface of SMs ε and the SR absorption coefficient of the dark surface of SMsέ.
At the same time, the absorption coefficient ε has a smaller effect on reducing the heating temperature of SMs than the integral absorption coefficient of SR a of the surface of solar cells, and when the value of the SR absorption coefficient of the dark surface of SMsέ changes from 0.1 to 1, the heating temperature of SMs decreases by 18%, while at a similar change in the value of the integral absorption coefficient of SR a of the surface of solar cells, the heating temperature of SMs increases by 2.5 times.
For applications of air-cooling systems, the thermal behavior of a solar module with solar cells is studied when cooled by atmospheric air and via natural heat exchange with the environment. Based on the developed equations, the dependencies of the heating temperature of SMs on solar irradiance and on temperature and the emissivity factor were calculated under various operating conditions. The presented curves show that the lowest heating temperature of SMs is observed at Тa=30 • C and ε=0.8 and at a decrease in the heating temperature from 111 to 38 • C. The highest heating temperature of SMs is observed at Тa=30 • C and ε = 0.3 and at a decrease in the heating temperature from 164 to 60 • C.
Thus, to reduce the heating temperature of SMs, it is necessary to increase the emissivity factor. The efficiency of SMs increases significantly due to the use of heat pipe systems.