Geometric Modeling of Parameters of Variable Natural Light during the Integration of Solar Systems on the Surface of Buildings

Abstract

The design of solar systems on the surfaces of buildings is based on an integrated approach to modeling: the duration of irradiation, zones of effective irradiation, shading zones, and angles of incidence of sunlight on the surface. In this case, the determining factor is the formation of methods for modeling the dynamics of changes in natural light. The conditions for natural exposure of surfaces are determined by the variable exposure to direct sunlight, scattered (diffuse) light, and complex light from the sky and the Sun. Each of these types of light has its own nature of changes presented in the study by means of geometric modeling. The authors proposed to bring together the methods of modeling variables of direct and diffuse sunlight by representing diffuse light in a ray form, that is, light vectors with variable parameters. This made it possible to simulate a rather complex combined effect of direct and diffuse light in solving problems of effective integration of solar systems on the surface of buildings.

1. Introduction

The need for energy is constantly increasing as human civilization develops. The modern energy system is moving toward replacing fossil fuels with renewable energy sources to ensure a sustainable energy supply [1, 2]. Solar energy is a renewable and environmentally friendly source of energy. Therefore, its application is widely used to achieve sustainable energy solutions. The contribution of solar energy to achieving sustainable development is determined by the affordability of meeting energy needs, creating jobs, and protecting the environment. At the same time, solar energy has not yet reached its maximum efficiency in its development. Therefore, research and innovation are important factors in improving the efficiency of renewable solar energy technologies.

Solar energy ranks first among the renewable energy sources introduced into the energy supply of modern buildings [3]. The studies [4, 5] analyze the issues of the effectiveness of the integration of solar systems in the structure. Many studies have been devoted to studying the features of predicting a solar renewable resource when using solar energy in the energy supply of buildings, among them [6, 7, 8, 9].

The studies note the variability of solar energy in space, time, and the complexity of modeling, and various methods are proposed for analysis taking into account the parameters of time and latitude.

The authors of [10] raise the issue of the importance of identifying changes in the composition of solar irradiation for solving problems of integrating photovoltaic technologies.

The amount of energy produced by photovoltaic panels directly depends on the amount of sunlight they receive. Therefore, when modeling the real distribution of solar irradiation on surfaces, it is necessary to take into account the dynamics of changes in light conditions. At the same time, one of the important factors influencing the change in light conditions is the variable influence of direct sunlight, diffuse sunlight, and complex direct and diffuse light.

In the modern practice of real simulation of the irradiation of the surfaces of solar systems, the direction of light is mainly used, simulating the conditions of illumination by direct sunlight, which provides the most effective conditions for obtaining energy from the Sun. In this case, the irradiated part of the surface is considered to be where the rays of direct sunlight fall. The diffuse light of the sky, which also takes part in the irradiation of the surface, is not taken into account. Although there are cases where the diffuse light component is larger compared to direct sunlight. For example, in overwhelming cloudiness, when scattered light predominates, as well as in the zone of own or falling shadow on the surface, where direct sunlight does not fall. Thus, when solving problems of integration of heliosystems, three types of exposure to sunlight should be considered according to the state of the sky:

1. direct variable sunlight with a clear sky (0–3 cloud points),

2. scattered sunlight in an overcast sky (7–10 cloud points),

3. complex light of the sky under the total action of direct sunlight and diffused sunlight of the sky in a cloudy sky (3–7 cloud points).

Each of these types of exposure has its own nature of change and modeling approaches. Direct sunlight is associated with modeling the movement of the Sun. The diffuse light of the sky is characterized by its quantitative indicators and uneven distribution of brightness. Complex light is determined by the total influence of two types of light (the Sun and the sky) in different proportional ratios.

Modeling of the irradiation process when integrating solar systems into a building can be carried out on a geometric basis, including modeling of variable parameters of the irradiation source (direct sunlight, scattered sky light, andcomplex sky light), geometric parameters of the surfaces of the building itself and elements of solar systems. The use of geometric modeling methods for the integration of solar systems on the surface of a building allows solving various problems that arise in the design process. Such tasks include optimal modeling in terms of irradiation of building surface shape parameters, irradiation duration, effective irradiation zones, shading zones, and reflection of light rays from the surface, taking into account the orientation of the building surfaces relative to the cardinal points. The use of geometric modeling tools allows to imagine the process of interaction between the parameters of variable irradiation and the geometric parameters of the shape of the building surface. It is important to determine the means of geometric modeling of light distribution for each type of illumination: direct solar exposure, diffuse (diffuse) sky exposure, and complex direct sunlight and sky exposure.

In this section, the task is to explore and bring together the methods of geometric modeling of direct and diffuse sunlight variables by representing them in a ray form, that is,. light vectors. This will make it possible to geometrically model the complex combined action of direct and diffuse sunlight.

2. Determining the directions of light vectors in dynamics under direct solar irradiation

The process of modeling solar irradiation is a complex system problem in defining many variables in time and space. Among them, primary and derivative ones are distinguished. The primary includes the movement of the Earth around its axis during the day (determines the change in time) and the movement of the Earth around the Sun during the year, which determines the position of the Earth’s axis of rotation relative to the Sun; shape and position in space of a given object and objects shading it (azimuth, shape, and position parameters); position and shape of the solar receiver (azimuth, shape, and position parameters). A change in primary factors in time and space leads to a change in derivative factors that directly determine the flow of solar energy into the solar receiver. Derived factors include the direction of sunlight, which is given by the coordinates of the Sun (azimuth, height), which change with latitude, time of day and day of the year; the intensity of sunlight, which depends on the angle of incidence— the angle between the normal to the surface of the solar receiver and the direction of sunlight and duration of solar exposure. Calculations of the conditions of variable solar irradiation in architecture do not need great accuracy.

Therefore, they can be modeled based on a number of assumptions that simplify the geometric model. A basic model of solar radiation in the form of a diurnal cone of solar rays of variable geometry is proposed based on the following assumptions:

1. The sun’s rays at any time and at any point on the Earth are considered parallel.

2. The shape of the Earth is described by a spherical surface. The surface of the Earth in the vicinity of any point is taken as a plane touching the sphere and perpendicular to the radius of the sphere at the point of contact.

3. The inclination of the Earth’s axis to the plane of the orbit is assumed to be 66.55°.

4. The Earth’s orbit looks like a circle. Therefore, it is possible not to take into account the existing deviations in the speed of movement along the orbit and in time compared to the average time and consider the movement of the Earth along the orbit as uniform.

5. The uniform continuous motion of the Earth in its orbit is replaced by a discrete jump-like motion with a time step equal to a day. The year is taken equal to 365 days. Then during the day, the Earth remains in one point of the orbit. During the transition to the next day, the Earth rotates around the Sun by an angle γ:

   $$\gamma =\frac{360°}{365},$$E1

   and occupies a new daily position.

6. Calculation of time is carried out on the basis of mean solar time, which corresponds to the conditions for the uniform motion of the Earth in a circular orbit.

With such assumptions, many rays during the year break up into many rays that are formed during the movement of the Sun every day. The complex helical guiding cone breaks up into 365 circular guiding cones. In this regard, there is no need to use modeling and calculation of solar irradiation conditions by astronomical data or the results of instrumental observations or graphic materials created on this basis.

The formation of a cone, called the diurnal cone of solar rays, can be represented as a result of the rotation of the direction to the Sun around the axis per day when considering the movement of the Sun in relation to the Earth, which during the day is in one point of the orbit. Due to the parallelism of the sun’s rays on a given day, the diurnal cone will be the same at any point in the vicinity of the Earth and have an axis parallel to the Earth’s axis. The plane of the horizon, drawn through the top of a plane parallel to the Earth, divides the cone into day and night parts. This cone is variable in time since during the year the angle α of intersection of the direction to the Sun with the Earth’s axis changes daily. Let us consider how the cone changes when the Earth moves on the arcs AB, BC, CD, and DA of the orbit g (see Figure 1).

Figure 1.

Change in the diurnal cone of sunlight as the earth moves around the sun.

At point A, the angle α between the generatrix and the axis of the cone is equal to the angle φ = 66.55° since the orthogonal projection of the axis onto the plane of the orbit Ω coincided with the direction to the Sun.

At point B, the direction to the Sun is perpendicular to the axis of the cone. Therefore, during rotation, the sunbeam SB will describe the plane. Consequently, when moving from point A to B, the angle α increases from 66.55° to 90°, and the surface of the diurnal cone straightens and becomes a plane at point B.

At point C, the continuation of the sun’s ray SC makes an angle α = φ with the axis. When the sun ray SC rotates around the axis, a conical surface also appears. In fact, the conical surfaces at points A and C have different cavities (summer and winter) of the same conical surface.

At point D, the sunbeam, as at point B, makes a right angle with the axis. Therefore, when the beam SD rotates, a plane is also formed. With further movement, this plane again turns into a conical surface when the angle α changes to a value of 66.55° at point A.

Thus, for any chosen point in the vicinity of the Earth due to the change in angle α during the year, a family of coaxial diurnal cones of solar rays is formed with a common apex at this point. The area of this family is limited by the summer and winter cavities of the solstice cone, the generatrix of which makes an angle α = φ = 66.55° with the axis. For any intermediate point G1 of the orbit g, the angle α is determined depending on the angle γ of the rotation of the Earth relative to the Sun [11].

Let us consider the features of determining the coordinates of the light vector s under direct solar irradiation, the dynamics of which are related to the geometry of the motion of the Sun. Using the geometric model of the daytime solar cone (Φ), the movement of the sun rays is determined for such initial conditions as the latitude of the area and the day of the year. The desired coordinates of the unit vector s at some point in time are determined by the values of the angles Н^°_s and А^°_s. If to take into account only the daytime part of the cone Φ, that is, the upper one in relation to the horizontal plane P₁, then the value of the applicate (Z) of the vector s will depend only on the magnitude of the angle Н^°_s. The values of the coordinates (X; Y) change in accordance with the position of the projection s1 and its angle А^°_s with the north direction (N).

For further calculations, in addition to the direction of sunlight, it is important to take into account the dynamics of quantitative changes in illumination from the Sun. Thus, the values of the average illumination Е^┴_с of the surface element, which is located perpendicular to the direction of the sun’s rays, change for a given latitude in accordance with the day of the year and the hour of the day. The values of Е^┴_с according to are calculated by the formula:

$${Е}_{с}^{\perp }=\frac{{Е}_{с}^{\circ }}{{\mathrm{\Delta }}^2}\cdot {р}^{М},$$E2

where Δ – the distance from the Sun at a given moment, determined from astronomical tables, Е^°_с – the light solar constant, approximately equal to 135,000 lux, р – air transparency, and M – air mass, determined according to the table, depending on the value of H_s^°.

If to present the value of Е^°_с in graphical form, then it will be displayed as a guide segment attached to the calculated point. To determine the direction of such a segment, it is advisable to use the geometric model of the diurnal cone Φ, which sets the dynamics of changes in the direction of the sun’s rays. Then the lengths of the generators of the diurnal cone Φ will correspond to the values of Е^°_с on the selected scale.

In Figure 2, the change in the hourly directions of the sun’s rays by 50° n.l. for 22.06 is determined by the daytime part of the diurnal cone of solar rays Φ. Variable values of the lengths of the segments on the hourly Φ correspond to the average values of Е^┴_с at a given time of the day.

Figure 2.

The dynamics of changes in the quantitative values of normal illumination Е┴с at a given latitude by 50˚ n.l. for 22.06.

For example, Е^┴_с = 68,200 lux at 12 o’clock, Е^┴_с = 66,600 lux – 10, 14 hours; Е^┴_с = 60,400 lux – 8, and 16 hours with a gradual decrease in their values, which correspond to a zero value during sunrise and sunset (at 4⁰⁰; 20⁰⁰ hours).

For graphical construction of the indicated lengths, let us set aside them in full size on the contour generatrix of the daily cone Ф and transfer them by rotation to the hour generatrix. A curve k is drawn through the upper ends of the guide segments constructed in this way, illustrating the dynamics of changes in the quantitative values of normal illumination Е^┴_с during the day at a given latitude. With its help, it is possible to determine graphically the intermediate hourly values of Е^┴_с. The guide segment, graphically depicting the value of normal solar illumination for given parameters (latitude, day, hour), is recorded with its spatial coordinates. For example, at 4 p.m., the direction of the sunbeam is determined by the unit vector s with coordinates {cosН^°_s sinА^°_s; cosН^°_s cosА^°_s; sinН^°_s}. With this in mind, the coordinates of the guide segment that sets the value of normal solar illumination for the same hour with its length d corresponding to the value of Е^┴_с = 60,400 lux are: {dcos Н^°_ssinА^°_s; dcos Н^°_scosА^°_s; dsinН^°_s}.

According to the well-known formula, the amount of direct solar illumination of a surface element with its different inclination and orientation, given by the direction of its normal n, will be determined:

$$\mathrm{E}={\mathrm{E}}_c^{\perp }cos\theta ,$$E3

where θ – the angle between the vectors s and n, which is calculated by the corresponding formula in analytical geometry, with known values of the coordinates of the vectors.

Thus, the geometric model of the daily cone of solar rays with the constructed curve k — the curve of variable quantitative values of normal illumination Е^┴_с makes it possible to graphically simulate and analytically solve all the parameters that determine the amount of direct solar illumination arriving at a given surface element in dynamics.

3. Determining the directions of light vectors in dynamics under scattered solar irradiation

For geometric modeling of surface irradiation under conditions of scattered sky light, it is necessary to determine the direction of the light flux. As is known, the source of natural light in the light field in this case is the atmosphere, which in geometric modeling is taken as a hemisphere. The atmosphere scatters direct sunlight, respectively, not having a single direction of incidence. That is, the solution of the problems of modeling the irradiation of the sky with scattered light becomes more complicated due to the different directions of the light rays.

To solve this issue, let us assume that when irradiated with scattered sunlight, the light flux has a main direction for each individual point of a given surface. Using the provisions of the theory of the light field [6], let us take the light vector as the predominant direction of the incident light. The light vector characterizes the value and direction of the “pressure” of light on a spherical body (which radius approaches zero) located at a given point on the surface. According to [6], the direction of the light vector coincides with the direction of the solid angle vector, the absolute value of the solid angle vector determines the absolute value of the light vector.

Thus, for modeling spherical illumination at points of a given surface, each of them is conditionally taken as a sphere (radius approaching zero). In Figure 3, the center of the sphere O coincides with the point A under study.

Figure 3.

Geometric modeling of irradiation of point a with scattered sky light.

The rays of diffuse light coming from the points of the sky open for a given point A of the surface k lie inside the solid angle dω. It has a vertex at a given point on the surface A and rests on the contour of a part of the surface of the hemisphere of the sky, which illuminates a given point on the surface A and is determined using the tangent plane τ to it.

Therefore, the solid angle dω is defined as the part of the celestial hemisphere (CH), which is formed by at least two planes intersecting P₁ and τ (Figure 3). The horizontal plane П₁ (XOY), as is well known, is given by the eq. Z = 0 and is perpendicular to the coordinate axis ОZ. The plane τ, with known values of the angle of its inclination ωτ to the horizon plane and the azimuth of the plane А_τ, passes through the origin, its equation:

$$\mathit{Ax}+\mathit{By}+\mathit{Cz}=0,$$E4

where A, B, and C — the coordinates of the vector n to which the plane τ is perpendicular. The coordinate values are determined in accordance with the accepted coordinate system and the specified north direction N.

It should be noted that when solving irradiation problems, it is important to take into account both the “upper” and “lower” sides of the tangent plane τ, which differ from each other in the direction of the vector n and the values of А_τ and ω_τ. So, in Figure 4, when considering the irradiation of the “upper side” of the plane τ, the unit vector n^B has coordinates {−sinω^в_τcosA^в_τ;sinA^в_τsinω^в_τ;sinω^в_τ}. When determining the irradiation of the lower side of the plane τ^н, the value of the azimuth A^н_τ and the angle of inclination ω^н_τ = 180°-ω^в_τ. Are taken into account. The perpendicular τ^н vector n^н has the opposite direction from n^в and coordinates {−sinA^н_τsinω^в_τ;cosA^н_τsinω^в_τ;-sinω^в_τ}.

Figure 4.

Geometric modeling of sky irradiation with scattered light of a given point O of the surface, taking into account the characteristics of the tangent plane τ.

Therefore, the tangent plane τ to a surface element gives a geometric representation of the amount of light coming from the celestial hemisphere to it, dividing the celestial hemisphere into two parts. The direction of the normal specifies the orientation of a given surface element in space and determines the part of the sky that takes part in the irradiation.

According to the above provisions, the coordinates of the vector dω coincide with the coordinates of the axis of the solid angle dω, which is the bisector of the linear angle μ, which measures the dihedral angle between the planes P₁ and τ. So, in Figure 4, when determining the irradiation of the “upper” side of the plane τ, the angle μ^в = (180° - ω^в_τ), and for the “lower” side of the plane τ, the angle μ^н = ω^в_τ. Thus, the coordinates of the vectors dω^в {sinA^вτcosA^в_τ; sin (180°- ω^в_τ)/2}, and dω^н {−sinA^н_τ;, cosA^н_τ; sin ω^в_τ/2}.

The obtained coordinates of the vectors dω^в, and dω^н determine the direction and position of the light vectors in space when a given plane is irradiated with scattered sky light.

According to the provisions of the theory of the light field [6], the length of the light vector ε with a uniform distribution of brightness in the sky changes in accordance with the change in dω, the vector of the solid angle dω, to which it is equal in magnitude. The absolute value of the light vector ε makes it possible to determine the quantitative value of irradiation at a given point on the surface with a scattered sky. Irradiation will change in proportion to the value of cos θ — the angle between the normal n and the light vector ε at a given point:

$$\mathrm{E}=\epsilon cos\theta ,$$E5

In the case, when the irradiated element is perpendicular to the direction of the light vector, that is, the directions of the normal and the light vector coincide, and the angle θ = 0°, there is a position that corresponds to the greatest transfer of light energy. Accordingly, with an increase in the angle between the normal and the light vector at a given point, there is a decrease in exposure. All this makes it possible to analyze the dynamics of irradiation of a surface element according to the change in its real position in space and the parameters of the light source— scattered sky light. When determining the irradiating area of the sky, it is important to take into account the possible shading of a surface element by other objects. In this case, the change in the value of the solid angle dω, and, accordingly, the position of the light vector ε is set by determining the center of gravity of the sky area through which the vector ε passes, or by determining and compiling the vectors ε of simple parts of a given sky area.

4. Determining the directions of light vectors in dynamics when surfaces are irradiated with complex natural light (with the total exposure to direct and diffused sunlight)

Under conditions of complex light, each of the sources, independently of the others, creates an irradiation that can be represented using light vectors. The sum of the light vectors determines the total irradiation at the calculated point, which, in geometric modeling, is represented by the total light vector ε_Σ. Its direction is determined by the light vectors ε_с and ε_н graphically applied to a given point, corresponding to the transfer of direct sunlight ε_с and scattered sky light ε_н to the calculated point for given irradiation conditions. The desired light vector ε_Σ is obtained by the parallelogram rule when the light vectors ε_с and ε_н are replaced by one light vector, the coordinates of which are given by the sum of the corresponding coordinates of the light vectors ε_с and ε_н. The direction of the light vector E_Σ determines the predominant direction of light energy transfer to the calculated point, and its length, according to (5), sets the quantitative value of natural irradiation EΣ entering the calculated point. The use of the above provisions on the geometric modeling of light vectors in direct sunlight and scattered sky light makes it possible to take into account the dynamics of changes in irradiation conditions when determining and geometric modeling the direction of complex light and its quantitative value.

For example, let us determine the total value of exposure to point O, with clear sky at 12:00 22.06 at 50° north latitude. Point O refers to the vertical plane τ (XOZ), the normal n to which coincides with the direction of the OY axis (Figure 5). The direction of the sun’s rays for the given conditions will be determined by the angles δ = 50°, φ = 66.6°, the height of the sun Н^о_s = 180°-(δ + φ) = 63,4°, and the azimuth of the sun А^°_s = 180°. Let us determine the amount of irradiation by direct sunlight E_с arriving at a given surface element. According to (5), at Е^┴_с = 68,200 lux, E_с = Е^┴_сcos(180°-(δ + φ)) = 68200cos63 and 4 εС ° = 30,537 lux. Thus, the transfer of direct sunlight to point O is characterized by the light vector ε_с with coordinates: {0; E_сcosh_o = 13,673; E_сsinh_o = 27,304}.

Figure 5.

Geometric modeling of the total irradiation value of the point O belonging to the vertical plane τ (XOZ) at 12:00 22.06 at 50° north latitude.

Let us consider the definition of the geometric parameters of the light vector ε_н, which characterizes the value and direction of the “pressure“at the point O of scattered sky light with a uniform distribution of brightness L. The light vector ε_н has a direction along the axis of the solid angle dω within which the light flux is transferred.

The length of the light vector corresponds to the amount of scattered light illumination E, determined by the formula:

$${\mathrm{E}}_{\mathrm{{\rm H}}}=\mathit{Ld\omega }cos\theta ,$$E6

where θ — the angle between the normal n and the light vector ε_н.

For the case under consideration (Figure 5), let us assume that with a uniform distribution of the brightness of the clear sky at noon, L = 4000 cd. The solid angle dω formed by two mutually perpendicular planes τ, and P1 is equal to π. Thus, for the given irradiation conditions at the calculated point corresponding to the length of the vector ε_н, it is equal to E_н = cos 45°4000 π =8881 lux. Consequently, the coordinates of the desired light vector ε_н {0; E_нcosθ = 6280; E_нsinθ = 6280}.

The light vector of total exposure εΣ has coordinates {0; 13673 + 6280; 27304 + 6280}, and its length is found from a right triangle whose legs are equal to the coordinates of the light vector ε_Σ:

$${\mathrm{E}}_{\mathrm{\Sigma }}=\sqrt{Y^2+Z^2}=\sqrt{{19953}^2+{33584}^2}=39064,\mathrm{lux}$$E7

For comparison, let us determine the total natural irradiation entering the horizontal plane (XOY) at the calculated point O for the same conditions, that is,. at 12:00 22.06 (Figure 6).

Figure 6.

Geometric modeling of the total irradiation value of the point O belonging to the horizontal plane τ (XOY) at 12:00 22.06 at 50 º north latitude.

The direction of the light vector ε_с, characterizing the irradiation from direct sunlight, will remain unchanged, but the length of the vector will change, which will be equal to Eс = Е^┴_сcos(90°-(180°-(δ + φ))) = 68200cos26,6° = 60,981 lux. In this case, the vector ε_с has coordinates: {0; Escos Нs ° = 27,304; EssinНs° = 54,526}. The amount of scattered light and the direction of the light vector ε_н will also change. Thus, the direction of the light vector ε_н coincides with the OZ axis, the solid angle dω for the given conditions is 2π, and the length of the vector ε_н is E_н = cos0°4000лк2π = 25,120 lux.

The coordinates of the vector ε_н {0;0;25,120}, and the coordinates of the total vector ε_Σ {0;27,304;79,646}. The length ε_Σ, corresponding to the value of the total exposure of the calculated point from direct sunlight and scattered sky light, is determined using the obtained coordinates of the light vector ε_Σ and is equal to:

$$\sqrt{Y^2+Z^2}=\sqrt{{27304}^2+{79646}^2}=84196\mathrm{lux}$$E8

The obtained values of the irradiation parameters from the total action of direct and scattered sunlight are summarized in Table 1 and allow us to draw conclusions regarding the nature of the variability of the complex light irradiating a given point on the surface of the heliosystem.

Parameters	For vertical surface	For vertical surface (in the drop shadow area)	For horizontal surface
Е^┴с (lux)	68200		68200
δ	50º n.l.		50º n.l.
φ	66,6º		66,6º
θ	63,4º		26,6º
Eс	30537		60981
Coordinates εс	{0;13673; 27304}		{0;27304; 54526}
L (cd)	4000	4000	4000
dω	π	π	2π
θ	45	45	0º
Eн (lux)	8881	8881	25120
Coordinates εн	{0; 6280; 6280}	{0; 6280; 6280}	{0;0;25120}
Coordinate εΣ	{0;19953; 33584}	{0; 6280; 6280}	{0;27304;79646}
EΣ (lux)	39064	8881	84196

Table 1.

Variability values of the irradiation parameters on the surfaces by changing their position in space.

Thus, the shift of complex light irradiating a given point on the surface is determined by the uneven weight of the components of direct and diffused sunlight. It should be noted that, in these cases, the change in the parameters of the complex light occurred due to a change in the position of the tangent plane, which determines the position of the calculated point in space. This proves the possibility of modeling the irradiation of planes by changing their position in space for given light conditions.

The use of the above provisions on the geometric modeling of light vectors in direct sunlight and scattered sky light makes it possible to take into account the dynamics of changes in irradiation conditions when determining and geometric modeling the direction of complex light and its quantitative value. The direction of the light vector EΣ determines the predominant direction of light energy transfer to the calculated point, and its length sets the quantitative value of natural irradiation E_Σ entering the calculated point. Determining the length of the vector ε_Σ according to formulas (6, 7) allows you to calculate a rather complex combined effect of direct and diffused light entering the calculation point for simulating its irradiation conditions.

5. Conclusions

The amount of solar radiation reaching the Earth’s surface depends on various atmospheric phenomena (cloudiness, air transparency, uneven illumination, and sky brightness), the position of the Sun both during the day and throughout the year. All these phenomena affect the efficiency of solar panels. Because the amount of energy produced by solar panels is directly dependent on the amount of sunlight they receive. On a cloudless day in direct sunlight, photovoltaic panels receive the maximum amount of light and will produce the maximum amount of energy. When the sun is obscured by clouds, the light level is reduced and the photovoltaic panels will operate at about half their capacity. Thicker cloud cover will significantly reduce the efficiency of photovoltaic panels. Finally, on a very cloudy day, photovoltaic panels will produce the least useful energy. Therefore, the problem of modeling the real distribution of irradiation on surfaces is relevant, taking into account the dynamics of changes in light conditions. Changes are determined by the variable influence of direct sunlight, scattered (diffuse) light, and complex light from the sky and the Sun.

In the course of the study, it was proposed to use a light vector to simulate various types of natural lighting, which allows: based on the image of scattered and direct sunlight by vectors, determine the direction and length of the total vector that simulates the effect of complex lighting in a cloudy sky that best meets the lighting conditions of Ukraine.

The provisions presented in the study on the geometric modeling of natural irradiation of surface points make it possible to further develop them. For example, when modeling the impact of an uneven distribution of the brightness of the sky (clear, overcast, and cloudy) when determining exposure by adjusting the magnitude and direction of the light vector. Taking into account previous studies [12, 13, 14, 15], it is possible to talk about a direct dependence of the change in the brightness of the sky on its state (clear, cloudy, overcast), determined by the degree and nature of the sky cloudiness. For a cloudy and clear sky, geometric modeling can be represented by a conditional pressure on the light vector within the surface or a brightness distribution curve, and for a cloudy sky, by determining the position of the weighted average brightness value within the intermediate surface (curve) between the boundary surfaces (curves) of the brightness distribution of gloomy and clear sky.

On surfaces with self-shadowing or when surfaces are shaded by other objects due to the complication in determining the active area of the sky, it is also convenient to apply irradiation modeling through a light vector and determine the irradiation gradation through the angle between the light vector and the normal at a given point on the surface. Here, the method of stratifying the congruence of normals into a set of surfaces of normals along generating lines or curves can be applied.

Consequently, the geometric modeling of complex natural light makes it possible to more accurately determine its amount and direction when irradiating a given point on the surface. Taking into account the complexity of natural light makes it possible to distinguish the qualitative characteristics of natural irradiation and analyze the conditions for the efficient operation of photovoltaic systems.