Variability of Diffuse Daylight Due to the Diversity of Cloud Arrays

: Illuminance modeling that allows us to mimic or even replicate the dynamics of daylight changes is increasingly becoming a challenge for the development of more accurate prediction systems of natural light availability in building interiors or variations of insolation at arbitrarily oriented façades. We demonstrate that illuminance amplitude due to random cloud arrangement can vary over a wide range even when other atmospheric parameters remain unchanged. It follows from our systematic numerical modeling that diffuse illuminance predictions can be signiﬁcantly improved by incorporating cloud coverage, mean cloud size, and cloud base altitude into daylight models. We show that any model of homogeneous luminance patterns would fail in modeling the illuminance amplitudes we can expect on horizontal and vertical planes under partial cloud coverage with individual clouds distributed randomly. However, these situations occur with high frequency in most of regions worldwide, thus the modeling results we obtained here are of high relevance to daylight modeling and solar energy systems as well.


Introduction
Conventional energy resources such as fossil fuel are dwindling drastically as population grows, and, this goes hand in hand with constantly increasing needs for alternative energies. Therefore, it is not surprising that the importance of renewable energy supplies rises noticeably in all aspects of human life. In particular, solar energy is becoming an attractive candidate to meet future energy demands because of its abundance and global availability in the form of direct sunbeams or diffuse daylight [1]. In addition to solar energy storing systems, direct and scattered sunlight are found to be very effective tools for energy saving in buildings [2]. Daylight delivery deeper into building interiors strongly depends on window orientation and sun path through the sky. Unshielded direct sunbeams are known to be a major component of total luminous flux crossing the interface between outer and inner window interface. As the sunlight incidence angle approaches 90 degrees, the rapid crossover from direct to diffuse light dominance can be observed. Diffuse light is the only source of natural light that enters windows oriented sideward of the position of sun. Not surprisingly, the accurate modeling of sky luminance and illuminance is a key element in daylight research nowadays [3,4].
Angular distribution of diffuse light changes significantly with aerosol microphysics, especially the size of particles suspended in air [5]. Due to their large variability, the aerosol particles can intensify scattered light by several orders of magnitude depending on the angle contained by the direction to the sun and direction to a sky element. Therefore, empirical sky luminance models independent of aerosol properties usually fail to fulfil accuracy criteria imposed on advanced daylight predictions. Along with aerosols, clouds are the largest source of uncertainty in modeling the diffuse light since the clouds are of random shapes and distributed arbitrarily in the sky [6]. There are not many simple ways Appl. Sci. 2021, 11, 9190 2 of 8 to treat the irregularity of a sky luminance pattern having its origin in cloud-reflected or cloud-masked light. Although the clouds can traverse sky or part of it, time-averaging over luminance distributions is not a reasonable approach because the mean luminance distribution obtained this way satisfies no one singular momentary sky state. Therefore, a use of average luminance distribution can result in unrealistic predictions of daylight availability for a few or many situations during a day. Averaged luminance distribution also suffers from loss of information on daily dynamics of luminance amplitudes. The latter can be important, e.g., for clouds accumulating in a sky quadrant opposite to the azimuthal position of the sun [7]. These clouds appear exceedingly bright compared to clear sky surrounding which luminance is reduced because of weak backscattering from realistically shaped aerosols.
In this paper we show the range of diffuse illuminances on the horizontal surface and vertically oriented façades we can expect from broken cloud arrays in different situations. Contrasting to homogeneous luminance models, the real illuminance on the horizontal or a vertical plane exhibits a number of anomalous features under randomly arranged cloud fields. Numerical experiments are conducted to demonstrate illuminance amplitudes for a set of parameters, specifically cloud coverage, statistically averaged size of clouds, and cloud base altitude. The simulations are introduced in Section 2, while the results obtained are interpreted in Section 3.

Sky Luminance Simulations
The sky luminance patterns and corresponding illuminances on a few selected surfaces were modeled by a computational tool UniSky [8]. This tool allows us to simulate arrays of regularly or randomly distributed spherical clouds with specified dimensions and altitudes, as well as various aerosol conditions [7].
Since we were primarily interested in the influence of clouds, we assumed fixed aerosol characteristics in computations. In particular, the single scattering albedo of aerosol particles was considered to be 0.93 and their asymmetry parameter was 0.64. These could be representative values for aerosol particles in an urban environment [9]. To simulate a moderately polluted environment, we set the aerosol optical thickness of the atmosphere to 0.3 (at 550 nm).
The UniSky simulator solves the radiative transfer problem in the atmosphere up to the second order of scattering, therefore also the Earth's albedo has to be specified. The value 0.15 was chosen because it appears to be typical for urban regions (see [10] and references there). However, this parameter is not important for our purposes.
The main focus was on clouds configuration in the simulations. We concentrated on low-layer clouds (the cloud base up to 2 km above the ground) because their diversity can lead to greater diffuse light fluctuation than higher-layer clouds. Cumulus clouds, forming a low-level broken cloud array, obviously have the geometrical thickness of several hundreds of meters [11,12]. Therefore, we modelled them by the spherical clouds with a radius of 0.25 and 0.5 km. The cloud base altitude was considered to be 0.5 km and 1 km for the both cloud sizes. For each cloud type (specified by its radius and altitude), various random distributions of clouds in the sky were simulated at three different cloud fractions: 0.1, 0.3, and 0.5. The cloud fraction is defined here as the ratio of a horizontal cloud area to the total area of a selected zone [7].
Cumulus clouds are optically dense, with an average extinction coefficient equal to 30 km −1 or greater [13,14]. This provides an optical thickness of at least 15 for the smaller clouds, although the UniSky simulator allows for the maximum value of 10. A cloud albedo of such optically thick clouds can be approximately calculated (see, e.g., [14]) and for the thickness of 15 we obtain the cloud albedo equal to 0.64. We held this value fixed in the simulations.
We calculated a diffuse illuminance on the horizontal plane as well as four vertical surfaces to demonstrate the possible fluctuations of diffuse light. In particular, the vertical planes oriented exactly to the north, east, south and west. These illuminances could illustrate how different distributions of clouds in the sky affect the sky luminance pattern.

Results
We will present three classes of results in this section: • Diffuse light variability due to the cloud size and altitude; • Fluctuations due to the different distributions of the same clouds; • Comparison of two specific cases with the same direct illuminance and very similar diffuse horizontal illuminance.

Different Cloud Size and Altitude
As mentioned in Section 2, we considered two different cloud radii R = 0.25 km and 0.5 km, and two cloud base altitudes h C = 0.5 km and 1 km. The diffuse illuminances calculated for each combination of these parameters at one specific cloud distribution and the fixed value of the cloud fraction C F = 0.3 are shown in Figure 1. The illuminances on the horizontal plane and the three vertical planes are depicted as a function of the solar zenith angle.
Appl. Sci. 2021, 11, x FOR PEER REVIEW 3 of 9 We calculated a diffuse illuminance on the horizontal plane as well as four vertical surfaces to demonstrate the possible fluctuations of diffuse light. In particular, the vertical planes oriented exactly to the north, east, south and west. These illuminances could illustrate how different distributions of clouds in the sky affect the sky luminance pattern.

Results
We will present three classes of results in this section: • Diffuse light variability due to the cloud size and altitude; • Fluctuations due to the different distributions of the same clouds; • Comparison of two specific cases with the same direct illuminance and very similar diffuse horizontal illuminance.

Different Cloud Size and Altitude
As mentioned in Section 2, we considered two different cloud radii = 0.25 km and 0.5 km, and two cloud base altitudes ℎ = 0.5 km and 1 km. The diffuse illuminances calculated for each combination of these parameters at one specific cloud distribution and the fixed value of the cloud fraction = 0.3 are shown in Figure 1. The illuminances on the horizontal plane and the three vertical planes are depicted as a function of the solar zenith angle. One can see that lower clouds can cause noticeably greater diffuse horizontal illuminance (and, for example, north wall illuminance) than clouds of the same size but placed higher. Even more significant differences can be seen when clouds of different radii but with the same base altitude are compared-the smaller clouds with h C = 0.5 km give greater illuminances than the bigger ones. However, these conclusions are not universal. A different distribution of clouds in the sky can result in completely different behavior. Figure 1 only demonstrates how different cloud parameters can have a noticeable effect on the magnitude of diffuse illuminance on various surfaces.
The dependence of the illuminance on the solar zenith angle follows approximately the trend observed under clear sky conditions. This is except for the case of the south wall, in which the clouds simply shade the bright vicinity of the sun at the middle elevations.

Different Cloud Distributions
Now, let us take one specific type of clouds and examine how their different distribution affects illuminance values. We specifically simulated 10 various random distributions of clouds with the radius of 0.25 km and the base altitude of 0. One can see that lower clouds can cause noticeably greater diffuse horizontal illuminance (and, for example, north wall illuminance) than clouds of the same size but placed higher. Even more significant differences can be seen when clouds of different radii but with the same base altitude are compared-the smaller clouds with ℎ = 0.5 km give greater illuminances than the bigger ones. However, these conclusions are not universal. A different distribution of clouds in the sky can result in completely different behavior. Figure 1 only demonstrates how different cloud parameters can have a noticeable effect on the magnitude of diffuse illuminance on various surfaces.
The dependence of the illuminance on the solar zenith angle follows approximately the trend observed under clear sky conditions. This is except for the case of the south wall, in which the clouds simply shade the bright vicinity of the sun at the middle elevations.

Different Cloud Distributions
Now, let us take one specific type of clouds and examine how their different distribution affects illuminance values. We specifically simulated 10 various random distribu-  (c) (d) Figure 2. Diffuse illuminance on the horizontal plane (a) and vertical planes oriented to the south (b), east (c), and north (d) as a function of the solar zenith angle calculated for ten random distributions of identical clouds with the cloud fraction equal to 0.1.  It is evident that a variation of cloud layout in the sky (while preserving their fraction) can significantly alter the illuminance magnitude. Individual values can differ by more than 10 klx at low cloud fractions; but also at higher fractions the illuminance variation is not negligible. To demonstrate quantitatively this variability, we calculated relative standard deviations of the illuminances within the examined sample. The results for the horizontal illuminances are presented in Table 1.

Comparison of Two Specific Skies
During the simulations, two sun and cloud configurations were found which produced the same direct horizontal illuminance and nearly the same diffuse horizontal illuminance. However, the corresponding sky luminance patterns differed due to the different cloud types and their distribution. These luminance patterns are shown in Figure 4.

Comparison of Two Specific Skies
During the simulations, two sun and cloud configurations were found which produced the same direct horizontal illuminance and nearly the same diffuse horizontal illuminance. However, the corresponding sky luminance patterns differed due to the different cloud types and their distribution. These luminance patterns are shown in Figure  4.  The clouds have a radius of 0.5 km and the base 0.5 km above the ground in the first case (case 1). In the second case (case 2), it was 0.25 km and 1 km. The cloud fraction was 0.3 and the zenith angle of the sun was 40° in both cases, though the random distribution of clouds in the sky vault was diverse. The sun is not covered by clouds, so the direct horizontal illuminance was equal to 80.917 klx. The calculated diffuse horizontal illuminances were 28.059 klx for case 1 and 28.19 klx for case 2.
Nearly identical illuminances indicate nearly identical irradiance. According to the standard empirical models [15][16][17], the sky luminance distribution in these two cases should be very similar, not regarding the specific cloud arrays. One such sky luminance map calculated by the Perez all-weather model [16] is shown in Figure 5. The clouds have a radius of 0.5 km and the base 0.5 km above the ground in the first case (case 1). In the second case (case 2), it was 0.25 km and 1 km. The cloud fraction was 0.3 and the zenith angle of the sun was 40 • in both cases, though the random distribution of clouds in the sky vault was diverse. The sun is not covered by clouds, so the direct horizontal illuminance was equal to 80.917 klx. The calculated diffuse horizontal illuminances were 28.059 klx for case 1 and 28.19 klx for case 2.
Nearly identical illuminances indicate nearly identical irradiance. According to the standard empirical models [15][16][17], the sky luminance distribution in these two cases should be very similar, not regarding the specific cloud arrays. One such sky luminance map calculated by the Perez all-weather model [16] is shown in Figure 5. The almost identical luminance patterns predicted by the empirical model for these two cases result in very close illuminances on the same vertical walls. To demonstrate this fact, we estimated diffuse illuminances on the four vertical surfaces under consideration using the Perez all-weather model [16] and compared them to the values obtained  The almost identical luminance patterns predicted by the empirical model for these two cases result in very close illuminances on the same vertical walls. To demonstrate this fact, we estimated diffuse illuminances on the four vertical surfaces under consideration using the Perez all-weather model [16] and compared them to the values obtained by the UniSky simulator. The results are shown in Figure 6. Apart from the fact that the Perez model underestimates the illuminances when compared to the UniSky model, it is clear that it is incapable of capturing the differences between the two cases. The almost identical luminance patterns predicted by the empirical model for these two cases result in very close illuminances on the same vertical walls. To demonstrate this fact, we estimated diffuse illuminances on the four vertical surfaces under consideration using the Perez all-weather model [16] and compared them to the values obtained by the UniSky simulator. The results are shown in Figure 6. Apart from the fact that the Perez model underestimates the illuminances when compared to the UniSky model, it is clear that it is incapable of capturing the differences between the two cases.

Discussion and Conclusions
The presented results clearly show that diverse cloud configurations cause considerable variability in diffuse sky light. A different altitude and size of the low-level clouds affect the diffuse illuminances at the ground in various ways depending on their distribution in the sky and cloud fraction. For example, the horizontal illuminance or illuminance on a vertical wall can differ by several klx for clouds of the same size but 500 m apart in altitude. However, there is no universal rule how to predict such results. They are determined by particular circumstances.
Considering clouds with the same parameters, a variation in their distribution in the sky at the same cloud fraction changes the sky luminance pattern significantly, resulting in noticeably different diffuse illuminances on the horizontal plane and façades. The differences between individual values can be by more than 10 klx. In principle, this could

Discussion and Conclusions
The presented results clearly show that diverse cloud configurations cause considerable variability in diffuse sky light. A different altitude and size of the low-level clouds affect the diffuse illuminances at the ground in various ways depending on their distribution in the sky and cloud fraction. For example, the horizontal illuminance or illuminance on a vertical wall can differ by several klx for clouds of the same size but 500 m apart in altitude. However, there is no universal rule how to predict such results. They are determined by particular circumstances.
Considering clouds with the same parameters, a variation in their distribution in the sky at the same cloud fraction changes the sky luminance pattern significantly, resulting in noticeably different diffuse illuminances on the horizontal plane and façades. The differences between individual values can be by more than 10 klx. In principle, this could lead to ambiguous classification of the standard sky type [18,19], although we still have the same cloud cover (only a different cloud layout).
However, the opposite situation is also possible. Many empirical models of sky radiance/luminance distribution [15][16][17] use various diffuse and direct (or global) irradiance ratios to classify the sky quality and to determine corresponding coefficients for calculation of the relative sky radiance/luminance. Therefore, if some two cloud configurations result in the same direct and diffuse horizontal irradiance, these models are not able to distinguish them, although the actual luminance distributions may be different.
All the preceding statements imply that if we want more realistic predictions of diffuse light availability, we need to use more realistic models that account for various actual sky conditions. We should simulate different cloud configurations to gain a sense of how the luminance might be distributed in the sky. The empirical models only provide averaged sky types that do not correspond to actual conditions, which can result in noticeable deviations in predicted diffuse light amount.
Mass calculations using more realistic models simulating actual sky conditions could serve as an alternative or supplement to climate-based modeling, which obviously uses the averaged sky luminance models applied to some typical illuminance/irradiance conditions at given time during a year. Knowing the typical properties of a cloud coverage (cloud fraction, cloud altitudes . . . ) and properties of atmospheric aerosol during a year, one could be able to obtain an estimation of possible diffuse light variations due to random cloud configurations in a particular period. Data Availability Statement: The numerical solver used to model diffuse illuminance on the horizontal surface and vertical planes is publicly available on http://unisky.sav.sk/?lang=en&page= aplikacia&subpage=sim (accessed on 2 August 2021). The computational data are available from the corresponding author upon reasonable request.