**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 [12–15], 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.
