Development of Sky Luminance and Daylight Illuminance Prediction Methods for Lighting Energy Saving in Office Buildings

Accurately predicting indoor illuminance from daylight during the early stages of building design is an important factor in saving energy and the costs associated with lighting. The objective of this study was to predict sky luminance distribution using the Commission Internationale de l’éclairage (CIE) standard sky model, and propose a method that can be used to predict indoor illuminance. Results obtained from the proposed prediction method were compared and verified with simulation values obtained by Desktop Radiance. From the CIE overcast sky, the zenith/horizon ratio was 3:1. From the CIE clear sky, the luminance value was highest around the sun. In contrast, the luminance value was lowest in the opposite direction of the sun when the angle between the sun and sky elements was 90◦. In addition, this study suggested an indoor illuminance prediction method by applying the effects of sky luminance, direct sunlight, and wall reflection elements. When the proposed equation’s calculation results were compared with Desktop Radiance simulation’s value in overcast and clear sky, all statistically analysis (R2, MBE, Cv(RMSE), t-value, p-value) satisfied each standard and showed high correlations. Consequently, it was established that the predicted indoor illuminance obtained from the proposed prediction method was accurate and can be used to predict the level of indoor illuminance. The results further revealed that it is possible to calculate indoor illuminance when installing blinds, by substituting variable values of visible light transmittance (VLT).


Background and Purpose
Daylight is an important factor that influences building lighting energy efficiency and visual satisfaction for the occupants [1,2].Accurate estimation of indoor daylight illuminance is important in saving lighting energy, as indoor daylight illuminance data can be used to predict a building's lighting energy and improve energy efficiency.Especially in a general office building, lighting energy consumption is responsible for 20%-30% of the total building energy consumption [3][4][5][6], and the effective control of daylight can reduce lighting energy consumption by 30%-60% annually [7][8][9].
Sky luminance distribution greatly affects the indoor illuminance making the prediction of sky luminance distribution very important [10][11][12].In other words, we should first predict and evaluate the sky luminance distribution to accurately predict indoor daylight illuminance.As sky luminance can vary considerably, we should have an accurate standard to derive an objective conclusion [13].
indoor visual environments.Chirarattananon and Chaiwiwatworakul [40] measured sky luminance at North Bangkok mainly under clear and intermediate skies.They then evaluated indoor illuminance and visual environments with a detailed analysis of Bangkok's actual sky luminance distribution under overcast conditions.
Igawa et al. [41] proposed models of radiance and sky luminance distributions for all sky conditions from clear to overcast.They classified sky luminance distributions of Tokyo's various sky conditions into indices of clear and cloudless skies.They then analyzed these distributions and compared them against various existing sky model theories.Li and Lou [42] compared non-overcast sky luminance models with data recorded in Hong Kong.They found that sky luminance distributions are affected by various factors, such as the sun's position, turbidity and pollution content in the atmosphere, and the amount, type, and pattern of cloud cover.
Bartzokas et al. [43] studied the sky luminance levels Bratislava, Central Europe and Athens, on the Eastern Mediterranean coast.They defined the prevailing sky luminance distributions for winter and summer.These prevailing sky luminance distributions were intended for use in the design of building openings to increase energy efficiency.
Wittkopf and Soon [44] studied three methods for sky distribution analysis: classification with the relative indicatrix and gradation, along with two independent methods established by Kittler and Tregenza.All three methods were verified to be used for analyzing the extreme daylight conditions in Singapore, which is located at the equator.They found that the analysis of sky distributions is useful for the prediction of daylight patterns and can guide the design of architectural windows and building envelopes.
Numerous studies have been done with respect to indoor daylight illuminance predictions using various methods.Additionally, some studies explain the importance of scale models in artificial and real skies.Navvab [45,46] studied the application of scale models in the prediction of daylight illumination in buildings.In order to measure indoor illuminance, the scale models are used under real sky or simulated sky conditions.In their studies, Navvab reported on the use of scale model in comparative evaluation of model photometry and computer simulation.The scale model provided an increase in the accuracy of the assessment in simplified methods.They explained that scale models are used as an accurate method to evaluate the design performance or to identify a relationship between a proposed design and its elements under real conditions.Navvab et al. [47] also studied the estimated frequencies of CIE luminance distributions and the impact of circumsolar region using TMY weather files, a typical meteorological year (TMY).The TMY can be used to provide hourly results at any location where a TMY or an irradiance time series is available.
Gugliermetti and Bisegna [48] presented simplified algorithms for assessing the indoor daylight illuminance on a prefixed point using external fixed shading devices in various Mediterranean cities.The proposed method in their study was based on splitting the internal illuminance into two components; due to direct and diffuse sun radiations, expressed by the Solar System Luminous Efficacies (SSLEs) and calculated using the package Superlite.SSLEs correlations were tested using experimental measurements on scale models.
In addition, Gugliermetti and Bisegna [49] analyzed the office demand associated with the use of different dynamic window (electrochromic windows) and lighting control systems with the aim of optimizing their usage aspects and characteristics from both visual and energy efficiency point of view.They used the hourly simulation program IENUS (Integrated ENergy Use Simulation).Different electrochromic windows and light management strategies were adopted in their Mediterranean climate studies using TMY weather files.
Tregenza [50] developed a tool for estimating the indoor horizontal illuminance of windows facing large external obstructions under the traditional CIE overcast sky, with a modification of the split-flux formula.Tregenza [51] also formulated simple calculation procedures for determining the indoor daylight illuminance in rooms facing sunlit streets.He described a method for estimating the Energies 2019, 12, 592 4 of 37 mean illuminance on the working plane and on other room surfaces from data about the normal solar illuminance and diffuse horizontal illuminance.
Li et al. [52] suggested a nomograph and a formula for predicting indoor horizontal illuminance that consider indoor reflective components and sky models during initial design stages of a building.They verified the results by comparing them with actual measurements of a scaled-down model.
Li et al. [53] also studied Desktop Radiance program simulation approaches, then compared its results with the indoor daylight illuminance measured in a scale model and in a classroom.In general, the daylight-factor (DF) approach is simple, but it cannot predict dynamic variations in indoor horizontal illuminance that follow changes in sky conditions and the sun's position.With the constant advancement in computer technology, the daylight illuminance can be predicted effectively using Desktop Radiance simulation software.
Vartiainen [31] suggested a formula for predicting daylight illuminance at a particular position indoors.The prediction results were compared against actual measured illuminance, showing that illuminance is more predictable on overcast days with no direct sunlight.However, daylight illuminance is much less predictable with this method if the point is exposed to direct sunlight.
Yoon et al. [54] came up with a simplified equation for evaluating daylight factors in overcast conditions using measured values of vertical illuminance in Seoul, Korea.The results were compared with measurements of a 1/5-scale model room to evaluate the precision of the suggested equation.
Although numerous studies were carried out as shown, methods for predicting sky luminance, such as the sky simulator, artificial sky dome, and sky scanner, were costly.Methods used for predicting indoor daylight illuminance were either using computer software which requires expert knowledge, or using expensive equipment for experimental studies.
Therefore, this study sought to construct an algorithm framework for sky luminance distribution without relying on costly equipment through CIE general sky model theory evaluation; it also utilized the CIE general sky algorithm for indoor illuminance prediction.In addition, it was determined that there is a need for a study that can improve indoor illuminance predictability via a theoretical approach to the prediction formula without relying on costly computer simulation methods that require user expertise.

Selection for Verification Program (Desktop Radiance)
Desktop Radiance [26] was used to check the prediction equation proposed in this study.It was ordered from U.S.A Department of Energy (DOE) and developed at the Lawrence Berkeley National Laboratory (LBNL).This program has been verified through numerous studies, and the program makes it possible to simulate a wide range of visual environments [26][27][28][29][30].In addition, it is possible to calculate illuminance of space using illuminance sensors, then compare its results from those of predicted equation.

Calculation and Verification of Sky Luminance Distribution
Thus, to calculate sky luminance distribution, the CIE standard sky luminance distribution provided by CIE [14] and a theory regarding the CIE general sky standard [20][21][22][23] were examined.The sky luminance distribution of CIE overcast and CIE clear sky was reproduced with 145 sky luminance patches [36] through the algorithm of sky luminance distribution.CIE overcast sky and CIE clear sky luminance distributions were compared with Desktop Radiance simulation results for verification.

Evaluation on daylight illuminance using modified daylight prediction equation
In this study, the DeLight daylight algorithm [31][32][33][34] was analyzed to obtain a modified daylight prediction equation.For a more accurate indoor illuminance prediction, sky luminance, simulation model data, and indoor-reflected components of model were integrated into a modified formula.The modified formula was then used to compare the indoor illuminance prediction values for CIE overcast sky and CIE clear sky with the Desktop Radiance [26][27][28][29][30] simulation results for verification.The daylight factor (DF), a ratio of indoor illuminance to overall sky luminance, was also divided into its sky component (SC) and internally reflected component (IRC) for comparison of CIE overcast sky conditions.For CIE clear sky conditions, the simulation model was unfolded in six directions and evaluated with luminance distribution.The indoor illuminance distribution when external blinds are installed was also evaluated.Figure 1 shows a flow chart of the study process.evaluated with luminance distribution.The indoor illuminance distribution when external blinds are installed was also evaluated.Figure 1 shows a flow chart of the study process.

CIE Standard Overcast Sky
The Northern European countries such as the UK and Scandinavia frequently experience overcast sky.CIE selected the overcast sky as a standard design sky to calculate illuminance, which displays an irregular distribution having a zenith to horizon ratio of 3:1.Equation (1) [15,39,55] expresses the rate of division between luminance of a sky element and zenith luminance for an overcast sky: where Lγ = Luminance of a sky element (kcd/m²), Lz = the zenith luminance (kcd/m²), γ = the elevation angle of a sky element above the horizon and Z = the angular distance between a sky element and the zenith.

CIE Standard Clear Sky
The CIE clear sky model assumes that luminance is highest near the Sun and lowest near the sky perpendicular to the position of the Sun.The elevation angle and azimuth of SC and the Sun are important variables for CIE clear sky's luminance distribution.Kittler's equation is commonly referred to as the luminance equation for a clear sky, which is expressed as shown in Equation (2) [15,56]: where Lγα = luminance in any arbitrary sky element, χ = the angular distance of the sky element from the Sun and Zs = the zenith distance of the Sun.

CIE Standard Overcast Sky
The Northern European countries such as the UK and Scandinavia frequently experience overcast sky.CIE selected the overcast sky as a standard design sky to calculate illuminance, which displays an irregular distribution having a zenith to horizon ratio of 3:1.Equation (1) [15,39,55] expresses the rate of division between luminance of a sky element and zenith luminance for an overcast sky: where L γ = Luminance of a sky element (kcd/m 2 ), L z = the zenith luminance (kcd/m 2 ), γ = the elevation angle of a sky element above the horizon and Z = the angular distance between a sky element and the zenith.

CIE Standard Clear Sky
The CIE clear sky model assumes that luminance is highest near the Sun and lowest near the sky perpendicular to the position of the Sun.The elevation angle and azimuth of SC and the Sun are important variables for CIE clear sky's luminance distribution.Kittler's equation is commonly referred to as the luminance equation for a clear sky, which is expressed as shown in Equation (2) [15,56]: where L γα = luminance in any arbitrary sky element, χ = the angular distance of the sky element from the Sun and Z s = the zenith distance of the Sun.

The CIE General Sky Standard (Kittler's model)
From the ISO 15469/2004 standard [20], CIE general sky is classified into two groups: indicatrix and gradation.CIE general sky is also divided into different classifications and includes sky luminance distribution in Kittler's study [22][23][24][25].The position of the Sun, the arbitrary sky element, and parameters a, b, c, d, and e, which describe atmospheric conditions, are used as input calculation quantities.For window design, glare studies, energy analysis, daylight climate classifications, and other purposes, parameters a, b, c, d and e used in the equations can be selected from Table 1, which lists five standard relative luminance distributions that are based on six groups of a and b values for the gradation function and six groups of c, d, and e values for the indicatrix function.[15,21,[23][24][25]: where L γα = luminance in any arbitrary sky element and L γ = luminance of a sky element (kcd/m 2 ).
Energies 2019, 12, x 6 of 36 luminance distribution in Kittler's study [22][23][24][25].The position of the Sun, the arbitrary sky element, and parameters a, b, c, d, and e, which describe atmospheric conditions, are used as input calculation quantities.For window design, glare studies, energy analysis, daylight climate classifications, and other purposes, parameters a, b, c, d and e used in the equations can be selected from Table 1, which lists five standard relative luminance distributions that are based on six groups of a and b values for the gradation function and six groups of c, d, and e values for the indicatrix function.
where Lrα = luminance in any arbitrary sky element and Lr = luminance of a sky element (kcd/m²).The position of the arbitrary sky element is defined by the zenith angle Z and the azimuth difference Az between the element and the solar meridian, as shown in Figure 2, then its distance from the sun can be found using Equation (4) [15,21,25]: where Az = |α -αs| α and αs are azimuthal angles of the vertical plane of the sky element and Sun position, respectively.The function f(χ) (Equation ( 5)) expresses the scattering indicatrix, which relates the relative luminance of a sky element to its angular distance from the Sun [15,21,25]: where χ = the angular distance of the sky element from the Sun.The position of the arbitrary sky element is defined by the zenith angle Z and the azimuth difference A z between the element and the solar meridian, as shown in Figure 2, then its distance from the sun can be found using Equation (4) [15,21,25]: Energies 2019, 12, 592 7 of 37 where A z = |α − α s | α and α s are azimuthal angles of the vertical plane of the sky element and Sun position, respectively.The function f (χ) (Equation ( 5)) expresses the scattering indicatrix, which relates the relative luminance of a sky element to its angular distance from the Sun [15,21,25]: where χ = the angular distance of the sky element from the Sun.The function f (Z s ) expresses the scattering indicatrix which relates the relative luminance of the zenith distance of the sun.Its value at the zenith is expressed in Equation ( 6) [15,21,25]: where Z s = The zenith distance of the Sun.The luminance gradation function φ relates the luminance of a sky element to its zenith angle, as expressed in Equation (7) [15,21]: where when 0 ≤ Z ≤ π/2, and at the horizon, φ(π/2) = 1.By using the formal study of Kittler [15,[21][22][23][24][25], the zenith luminance and sky luminance, which were based on the data collected from the illuminance sensor, can be calculated.Each coefficient values (A, B, C, D, E, and T v ) were proposed in the prior study.Darula and Kittler [15] explain that one can apply Equation (8) to skies with luminous turbidity factor (T v ) value greater than 12 and Equation ( 9) to clear skies with T v value lower than 12: where E v = extraterrestrial horizontal illuminance, L z = the ratio of zenith luminance to diffuse sky illuminance D v and γ s = elevation angle of the Sun: where A, B, C, D, and E = parameters characterizing a certain sky standard and T v = the luminous turbidity factor.Figure 3 shows the flow of the algorithm explained above for the realization of CIE overcast and clear skies' luminance distribution i.e. one can derive an equation that comprises Equation (3) and the resulting values from Equations ( 4)-( 7) from the Sun and the sky's position (altitude, azimuth).The result can be used to derive the ratio between zenith luminance and sky luminance.Also, the sun's position (altitude, azimuth) can be incorporated into Equations ( 8) and ( 9) to derive the zenith luminance of overcast sky and clear sky.Each sky model's luminance can be obtained by multiplying Equation (3) by the results from Equations ( 8) and (9), respectively.The detailed results are presented in Section 4.
the resulting values from Equations ( 4)-( 7) from the Sun and the sky's position (altitude, azimuth).The result can be used to derive the ratio between zenith luminance and sky luminance.Also, the sun's position (altitude, azimuth) can be incorporated into Equations ( 8) and ( 9) to derive the zenith luminance of overcast sky and clear sky.Each sky model's luminance can be obtained by multiplying Equation (3) by the results from Equations ( 8) and ( 9), respectively.The detailed results are presented in Section 4.

Suggestion of the Daylight Prediction Equation using the DeLight Algorithm
The DeLight algorithm suggests an indoor diffuse illuminance distribution equation, as shown in Equation (10) [31,32] using the position index in Figure 4.
where E p,d = indoor horizontal diffuse illuminance at P point (lx), L = sky luminance (the center of the part of the sky visible to point P through the window) (kcd/m 2 ), z = the distance between the window and P point (m), h p = height between lower edge of the window and P point (m), h w = window height (m), W w = window width (m), X w = distance between the left edge of the window and left wall (m) and X p = distance between the reference point P and left wall (m).The DeLight algorithm suggests an indoor diffuse illuminance distribution equation, as shown in Equation (10) [31,32] using the position index in Figure 4.  Energies 2019, 12, 592 9 of 37 Equation ( 11) [31] considers influence of reflected components from the back wall: where E p,r = indoor horizontal reflected illuminance at P point (lx), τ w = visible light transmittance, L = center of window luminance (kcd/m 2 ), z = the distance between the window and P point (m), h w = window height (m) and W w = window width (m).Equations ( 12) and ( 13) [31,32] are direct component equations, which modify the incidence angle through the window: where E p,b = indoor horizontal beam illuminance at P point (lx), τ w = visible light transmittance of glazing and blinds, θ i = an incidence angle, E b = horizontal beam illuminance measured outdoors (lx), K b = luminous efficacy of the beam radiation (lm/W) and G b = outdoor horizontal beam radiation (W/m 2 ).Visible light transmittance (VLT) of the blind and window can be defined using Equation ( 14) [57] shown below: where τ w (0) = light transmittance of window with the angle of incidence 0 • and θ i = an incidence angle.Equation (15) defines the location of the Sun and azimuth of the window [32]: where θ s = Sun's altitude (rad), Ψ s = Sun's azimuth (rad) and Ψ w = window surface azimuth (rad).Suggested Equation (16) shows the influence of reflected components of all interior walls.The previous equation was limited because it only considered reflected components from the back wall.Therefore, the modified equation includes reflective components from all interior walls: where E p,r = Indoor horizontal reflected illuminance at P point (lx), τ w = visible light transmittance, L = center of window luminance (kcd/m 2 ), z = the distance between the window and P point (m), h w = window height (m), W w = window width (m), R f = average indoor surface reflectance (%) and ρ s = ground surface reflectance (%).
Overall, modified daylight prediction E p considers three factors, the direct component, the sky diffuse component, and the reflected component, which are all employed in Equation (17).When there is an overcast or clear sky without a direct solar component, E p can be calculated except E p,b : Figure 5 shows the flow of the algorithm for predicting the indoor daylight illuminance in CIE overcast and CIE clear sky conditions.The calculation for indoor daylight illuminance of each sky continues from the algorithm flow shown in Table 2 and Figure 5 and uses the equations described throughout this section in conjunction with sky luminance, spatial position data, and the position of the sun.The detailed results are presented in Section 5.

Simulation Analysis Model and Sensors
Figure 6 shows information of the analysis model.This model was used to perform a simulation and apply modified daylight prediction.An analysis model was set as the evaluation space with a model size width of = 3.5 m, depth = 6 m, and ceiling height = 2.7 m, and the window to wall ratio (WWR) of 45%, excluding the frame because it could result in an irregular indoor illuminance value."Window design Guidelines for Building Energy Conservation" published by Korean Ministry of Land, Infrastructure and Transport (MOLIT) [58] suggests the WWR to be 40-50%.Thus, we set the WWR of the analysis model as 45%.Also, the room ratio (room length/width) [59] of the scale model that we are currently making for future experiments study as the analysis model's room ratio.To measure the illuminance of the evaluation space at each measurement point, a total of five (1 × 5) sensor grids were installed at a height of 0.75 m with an interval of 1 m.

Simulation Analysis Model and Sensors
Figure 6 shows information of the analysis model.This model was used to perform a simulation and apply modified daylight prediction.

Simulation Analysis Model and Sensors
Figure 6 shows information of the analysis model.This model was used to perform a simulation and apply modified daylight prediction.[58] suggests the WWR to be 40-50%.Thus, we set the WWR of the analysis model as 45%.Also, the room ratio (room length/width) [59] of the scale model that we are currently making for An analysis model was set as the evaluation space with a model size width of = 3.5 m, depth = 6 m, and ceiling height = 2.7 m, and the window to wall ratio (WWR) of 45%, excluding the frame because it could result in an irregular indoor illuminance value."Window design Guidelines for Building Energy Conservation" published by Korean Ministry of Land, Infrastructure and Transport (MOLIT) [58] suggests the WWR to be 40-50%.Thus, we set the WWR of the analysis model as 45%.Also, the room ratio (room length/width) [59] of the scale model that we are currently making for future experiments study as the analysis model's room ratio.To measure the illuminance of the evaluation space at each measurement point, a total of five (1 × 5) sensor grids were installed at a height of 0.75 m with an interval of 1 m.
Table 3 provides the finishing material's model reflection rate, which included the ceiling (85%), the wall (55%), the floor (20%), and the window (5%), with a window transmittance of 78%.Using these values, the average indoor surface reflection rate was estimated to be 55%.Also, from the input values for the daylight illuminance prediction equation, the location index and penetration ratio of windows for verification points P1 to P5 for daylight illuminance value comparison were shown in Table 4.

Condition of Glazing and Shading Device to Evaluate Daylight Performance
Table 5 presents the set of simulation variables to evaluate daylight performance, by applying a simplified daylight prediction equation.To evaluate the visual environment the change of the sky model and the venetian blind slat angles can be adjusted.The façade orientation of the evaluation model was set to be facing south.Characteristics of the venetian blind [60] were based on a commercially available product, and the blind's material properties and slat angle's setup were presented as shown Figure 7 and Table 6.Venetian blinds were installed on the exterior of the simulation model's windows, so the entire side can be controlled.

Sky Luminance Distribution
Figure 8 shows the sky patch that was used to calculate sky luminance of each location by dividing the sky into 145 elements [36].As indicated previously, the applicable sky models included CIE standard overcast sky and CIE clear sky.To apply the sky luminance L value to the indoor illuminance prediction equation, algorithm framework of sky luminance distribution in Figure 3 were used to create sky luminance distribution (sky luminance/zenith luminance) of CIE overcast sky and clear sky.Table 7 shows luminance distribution on the 21 st of March, June, September, and December a 0 pm with an overcast sky.Sky luminance distribution ratio value was the same for seasona resentative dates.From the standard overcast sky on the 21 st March, June, September, an cember at 2:00 pm, the horizon/luminance ratio was approximately 3:1.When the CIE standar ercast sky's luminance distribution was employed, the ratio of the sky's relative luminanc tribution to zenith luminance ranged from 0.3 to 1, indicating that the zenith luminance wa Table 7 shows luminance distribution on the 21st of March, June, September, and December at 2:00 pm with an overcast sky.Sky luminance distribution ratio value was the same for seasonal representative dates.From the standard overcast sky on the 21st March, June, September, and December at 2:00 pm, the horizon/luminance ratio was approximately 3:1.When the CIE standard overcast sky's luminance distribution was employed, the ratio of the sky's relative luminance distribution to zenith luminance ranged from 0.3 to 1, indicating that the zenith luminance was approximately 3 times brighter than the luminance near the horizon.Also, in overcast sky conditions, the luminance for all azimuths were identical when the elevation of the sky positions were identical.The CIE overcast sky's luminance distribution was at its highest near the zenith (Patch No. 145).Table 8 shows luminance distribution on the 21st of March, June, September, and December at 2:00 pm in clear sky conditions.The sun's altitude was 48.03 • using a standard clear sky on the 21st March at 2:00 pm.The azimuth was 30.09 • , and Patch No.106's luminance value was highest where the sun was located.The opposite side of the sun around Patch No.119 gets low luminance distribution.This luminance value was approximately eight times higher than the ceiling luminance.Also, the distribution on 21st September was similar to that on 21st March.The patch No.105's luminance value was highest where the sun was located.However, on the opposite side of the sun at Patch No.120, the lowest luminance value was shown, and the surrounding distribution also showed the lowest value.The location of the sun's altitude was 67.07 • , and the azimuth was 58.31 • on 21st June at 2:00 pm.From Patch No.136 where the sun was located, a high luminance distribution and ceiling luminance got approximately five times more luminance, whereas the opposite side of the sun around Patch No. 130 got low luminance distribution.The sun's altitude was 25.68 • on 21st December at 2:00 pm., and the luminance distribution was at its highest when the sun was located at Patch No. 58 and at its lowest at Patch No. 131.

Verification of Sky Luminance Distribution
We verified each patch of sky luminance value for the four directions (north, south, east and west) and for the seasonal representative dates (21 st March, 21 st June, 21 st September, 21 st December at 2pm) based on the position of the Sun instead of verifying the entire days' time, because we reviewed and applied CIE general sky theories (in Figure 3).We verified each position of the Sun using the Sun's position of 2 pm where it is tilted towards the west from the southern direction.Figure 9

Verification of Sky Luminance Distribution
We verified each patch of sky luminance value for the four directions (north, south, east and west) and for the seasonal representative dates (21 st March, 21 st June, 21 st September, 21 st December at 2pm) based on the position of the Sun instead of verifying the entire days' time, because we reviewed and applied CIE general sky theories (in Figure 3).We verified each position of the Sun using the Sun's position of 2 pm where it is tilted towards the west from the southern direction.Figure 9

Verification of Sky Luminance Distribution
We verified each patch of sky luminance value for the four directions (north, south, east and west) and for the seasonal representative dates (21 st March, 21 st June, 21 st September, 21 st December at 2pm) based on the position of the Sun instead of verifying the entire days' time, because we reviewed and applied CIE general sky theories (in Figure 3).We verified each position of the Sun using the Sun's position of 2 pm where it is tilted towards the west from the southern direction.Figure 9

Verification of Sky Luminance Distribution
We verified each patch of sky luminance value for the four directions (north, south, east and west) and for the seasonal representative dates (21 st March, 21 st June, 21 st September, 21 st December at 2pm) based on the position of the Sun instead of verifying the entire days' time, because we reviewed and applied CIE general sky theories (in Figure 3).We verified each position of the Sun using the Sun's position of 2 pm where it is tilted towards the west from the southern direction.Figure 9   and applied CIE general sky theories (in Figure 3).We verified each position of the Sun using the Sun's position of 2 pm where it is tilted towards the west from the southern direction.Figure 9 is a diagram depicting the sky element point's altitude and azimuth, which were compared with Desktop Radiance for verification, and the position of the Sun at each seasonal representative date (21st March, 21st June, 21st September, and 21st December) [61].This study calculated the luminance of the four directions (North, South, East and West), and the calculation direction of each sky was -90°, 90°, 0°, and 180°.In other words, the verification of the Desktop Radiance program simulation value and sky luminance was conducted for the 20 positions represented by the four directions (azimuth) and the five sky calculation elevation positions (A, B, C, D, and E).Also, for CIE standard clear sky luminance distribution, the Sun's altitude and azimuth both serve as important variables, altering the distribution.Thus, the changes in the Sun's position (altitude, azimuth) on the seasonal representative dates were also considered.
Table 9 shows interior images of Desktop Radiance, concept diagram, and sky luminance (luminance value at the center of the window) location from each sensor.Window luminance values were different depending on the calculation location because the sensor gets the light through the window from different angles.Sky luminance is the center part of the sky visible to the point sensor through the window.The center of the window luminance was calculated using this concept.Since, sky luminance values are dependent on the sky condition, the center of window luminance in two sky models were calculated using Equations ( 1), (2), and (4).This study calculated the luminance of the four directions (North, South, East and West), and the calculation direction of each sky was −90 • , 90 • , 0 • , and 180 • .In other words, the verification of the Desktop Radiance program simulation value and sky luminance was conducted for the 20 positions represented by the four directions (azimuth) and the five sky calculation elevation positions (A, B, C, D, and E).Also, for CIE standard clear sky luminance distribution, the Sun's altitude and azimuth both serve as important variables, altering the distribution.Thus, the changes in the Sun's position (altitude, azimuth) on the seasonal representative dates were also considered.
Table 9 shows interior images of Desktop Radiance, concept diagram, and sky luminance (luminance value at the center of the window) location from each sensor.Window luminance values were different depending on the calculation location because the sensor gets the light through the window from different angles.Sky luminance is the center part of the sky visible to the point sensor through the window.The center of the window luminance was calculated using this concept.Since, sky luminance values are dependent on the sky condition, the center of window luminance in two sky models were calculated using Equations ( 1), (2), and (4).
It is important to find a location of sky patch to calculate illuminance.It is possible to find an angle looks at the sky from the sensor using a trigonometric function when the height of the window and horizontal distance are known.The location in simulation model is in Incheon, Korea (Latitude: 37.48 • , Longitude: 126.55 • ), and the facade orientation is facing South.The sky location was at position A of the sky patch when looking at the sky from the illuminance sensor at 1 m the altitude was 61.6 • , and since there is a window on the south side, the azimuth was 0 • .When looking at the sky location of spot B-2 m from the illuminance sensor, the altitude was 42.8 • .At 3 m, the sky location of spot C's altitude was 31.7 • .At 4 m, spot D's altitude was 24.8 • , and for 5 m, spot E's altitude was 20.3 • , and since the azimuth's direction is south, the angle was 0 • .(luminance value at the center of the window) location from each sensor.Window luminance values were different depending on the calculation location because the sensor gets the light through the window from different angles.Sky luminance is the center part of the sky visible to the point sensor through the window.The center of the window luminance was calculated using this concept.Since, sky luminance values are dependent on the sky condition, the center of window luminance in two sky models were calculated using Equations ( 1), (2), and (4).For clear sky conditions, sky luminance at spot A was 5197 cd/m 2 , spot B was 6414 cd/m 2 , spot C was 6146 cd/m 2 , spot D was 6027 cd/m 2 , and spot E was 6070 cd/m 2 .On the other hand, a gradually decreasing tendency in sky luminance from spot A to E was observed in overcast sky conditions.Luminance was 2936 cd/m 2 at spot A, 2516 cd/m 2 at spot B, 2185 cd/m 2 at spot C, 1946 cd/m 2 at spot D, and 1814 cd/m 2 at spot E. These luminance values were applied to the modified version of predicted equation, and result of indoor illuminance by different sky models using the modified equation is presented in Section 5.

Verification of Sky Luminance Distribution for Overcast and Clear Sky
Table 10 shows the results of a comparison between sky luminance calculation value (CIE general sky theories in Figure 3) at 5 sky calculation elevation angles for overcast sky (A, B, C, D, and E) and the Desktop Radiance simulation value.
For 21st March, 21st June, 21st September, and 21st December at 2:00 pm of the overcast sky, the luminance values were the same for all four directions (North, South, East and West) when the sky position's elevation was the same.Thus, the sky luminance value for the five positions (A, B, C, D, and E) in the four directions was also the same.When the sky luminance of 21st March results derived from the CIE general sky theories were compared with the Desktop Radiance simulation value, the luminance error rates for positions A, B, C, D, and E in four directions were 1.4%, 1.2%, 1.2%, 1.0%, and 0.5%, respectively.Comparison results of the 21st March, 21st June, 21st September and 21st December at 2:00 pm also showed error rates ranging between 0.5-1.6%, in all four directions.Table 11 shows the results of a comparison between sky luminance at five sky calculation elevation angles for clear sky positions (A, B, C, D, and E) and the simulation value.For 21st March at 2:00 pm, when the window was facing south, the sky luminance ranged from 5269 cd/m 2 to 6509 cd/m 2 due to its close proximity with the sun, and the sky luminance value was relatively high compared to other directions.When the luminance result derived using the CIE sky algorithm was compared with the Desktop Radiance simulation value, the luminance error rates from positions A through E ranged from 0.8% to 1.4%, respectively.When the window was facing west, sky luminance values ranged from 3225 to 3730 cd/m 2 , and the error rates between the calculated value and the Desktop Radiance simulation value ranged from 0.9% to 1.5%, respectively.When the window was facing east, sky luminance values range from 1687 to 2278 cd/m 2 , and the error rates between the calculated value and the simulation value ranged from 0.6% to 1.4%, respectively.When the window was facing north, sky luminance values ranged from 1475 to 2271 cd/m 2 , and the error rates between the calculated value and the simulation value ranged from 0.7% to 1.3%, respectively.Comparison results of the 21st March, 21st June, 21st September and 21st December at 2:00 pm also showed error rates ranging from 0.4% to 1.6%, in each direction.Based on the data given in Tables 10 and 11, Figure 10 shows the Pearson correlation coefficient R 2 [62], MBE [63], RMSE [63], Cv(RMSE) [63] analysis and t-test (t-value, p-value) [64] results of Desktop Radiance simulation and the CIE general sky theories calculation on the two sky types with four seasonal representative dates (21st March, 21st June, 21st September, and 21st December), four directions (North, South, East and West) on five locations (A-E).The luminance distribution was 997-3674 cd/m 2 for an overcast sky and 953-8944 cd/m 2 for a clear sky.For both types of sky models, the luminance value of the simulation and the calculated result of equation had a Pearson correlation coefficient R 2 over 0.99, MBE value less than ±10%, and Cv (RMSE) value less than 30%, showing a high correlation satisfying the American Society of Heating, Refrigerating and Air-Conditioning Engineers (ASHRAE) guidelines [63].The t-test results with a significance level of 0.05 showed a p-value less than 0.05.Therefore, the calculation results from the algorithm framework of CIE sky luminance distribution have a statistical significance with the Desktop Radiance Simulation results.

Verification on Indoor Illuminance Prediction Method (CIE Overcast Sky)
Figure 11 shows the results of a comparison between the Desktop Radiance simulation result and modified equations for overcast sky conditions.Indoor illuminance distribution results for the 21 st of March, June, September, and December at 2:00 pm are presented.The illuminance prediction error rate was 1%-9% from 1 to 5 m when predicting illuminance using the modified equation on 21 st March.At 1 m, the error rate was 5% when comparing result of indoor illuminance using the modified equation and result of the Desktop Radiance simulation.At 2 m, the error rate was 8%.At 3 m, the error rate was 9% when comparing results of indoor illuminance using the modified equation and result of the simulation.Also, at 4 m and 5 m, illuminance prediction error rate of the modified equation ranged from 1% to 3%, prediction of illuminance using modified equation was accurate compare with the Desktop Radiance results.The illuminance prediction error rate was 2%-7% from 1 to 5 m when predicting illuminance using the modified equation on 21 st June.The average error rate of illuminance prediction was 5.1%.The average error rate of illuminance prediction was 5.5% on 21 st September, and the illuminance prediction error rate ranged from 2% to 8% from 1 to 5 m when predicting illuminance using the modified equation on 21 st September.Finally, on 21 st December, the average error rate of illuminance prediction was 6.7%.
While Figure 11 shows detailed results at a specified time (14:00) on 21 st March, 21 st June, 21 st September and 21 st December, Figure 12 shows the results of a comparison between the Desktop Radiance simulation result and a modified equations result in overcast sky for the entire day, excluding the night time, 07:00-18:00 (12 hours).

Verification on Indoor Illuminance Prediction Method (CIE Overcast Sky)
Figure 11 shows the results of a comparison between the Desktop Radiance simulation result and modified equations for overcast sky conditions.Indoor illuminance distribution results for the 21st of March, June, September, and December at 2:00 pm are presented.The illuminance prediction error rate was 1%-9% from 1 to 5 m when predicting illuminance using the modified equation on 21st March.At 1 m, the error rate was 5% when comparing result of indoor illuminance using the modified equation and result of the Desktop Radiance simulation.At 2 m, the error rate was 8%.At 3 m, the error rate was 9% when comparing results of indoor illuminance using the modified equation and result of the simulation.Also, at 4 m and 5 m, illuminance prediction error rate of the modified equation ranged from 1% to 3%, prediction of illuminance using modified equation was accurate compare with the Desktop Radiance results.The illuminance prediction error rate was 2%-7% from 1 to 5 m when predicting illuminance using the modified equation on 21st June.The average error rate of illuminance prediction was 5.1%.The average error rate of illuminance prediction was 5.5% on 21st September, and the illuminance prediction error rate ranged from 2% to 8% from 1 to 5 m when predicting illuminance using the modified equation on 21st September.Finally, on 21st December, the average error rate of illuminance prediction was 6.7%.The R 2 , MBE, RMSE, Cv (RMSE) analysis and t-test (t-value, p-value) was done on the simulation results of each illuminance sensor (1 m-5 m) at 07:00-18:00 (12 hours, all day excluding the night time) and the calculation results of the proposed equation.In case of an overcast sky, the illuminance was distributed between 0-3000 lx for 21 st March and 21 st September, 0-3500 lx for 21 st June and 0-1500 lx for 21 st December.For the overcast sky illuminance, it was the highest at the summer solstice (6.21) and the lowest at the winter solstice (12.21).While Figure 11 shows detailed results at a specified time (14:00) on 21st March, 21st June, 21st September and 21st December, Figure 12 shows the results of a comparison between the Desktop Radiance simulation result and a modified equations result in overcast sky for the entire day, excluding the night time, 07:00-18:00 (12 h).The R 2 , MBE, RMSE, Cv (RMSE) analysis and t-test (t-value, p-value) was done on the simulation results of each illuminance sensor (1 m-5 m) at 07:00-18:00 (12 hours, all day excluding the night time) and the calculation results of the proposed equation.In case of an overcast sky, the illuminance was distributed between 0-3000 lx for 21 st March and 21 st September, 0-3500 lx for 21 st  The R 2 , MBE, RMSE, Cv (RMSE) analysis and t-test (t-value, p-value) was done on the simulation results of each illuminance sensor (1 m-5 m) at 07:00-18:00 (12 h, all day excluding the night time) and the calculation results of the proposed equation.In case of an overcast sky, the illuminance was distributed between 0-3000 lx for 21st March and 21st September, 0-3500 lx for 21st June and 0-1500 lx for 21st December.For the overcast sky illuminance, it was the highest at the summer solstice (6.21) and the lowest at the winter solstice (12.21).
The Desktop Radiance simulation results and the proposed equation calculation results of the four seasonal representative dates showed high correlations, all having a Pearson correlation coefficient R 2 value over 0.99, MBE value less than ±10% and Cv(RMSE) value less than 30%.The t-test results with a significance level of 0.05 showed a p-value of less than 0.05.Therefore the Desktop Radiance simulation result and proposed daylight equation calculation results were shown to have statistical significance.Tables S1-1, S1-2, S1-3, and S1-4 in the supplementary material provide the detailed data for 21st March, 21st June, 21st September, and 21st December at 07:00-18:00 for Figure 12.
Figure 13 shows the comparison between exterior horizontal illuminance and the daylight factor of each seasonal representative dates (overcast sky) using the proposed equation and the Desktop Radiance simulation.Furthermore, the DF values were compared and divided into their SC and IRC.The Desktop Radiance simulation results and the proposed equation calculation results of the four seasonal representative dates showed high correlations, all having a Pearson correlation coefficient R 2 value over 0.99, MBE value less than ± 10% and Cv(RMSE) value less than 30%.The ttest results with a significance level of 0.05 showed a p-value of less than 0.05.Therefore the Desktop Radiance simulation result and proposed daylight equation calculation results were shown to have statistical significance.Tables S1-1, S1-2, S1-3, and S1-4 in the supplementary material provide the detailed data for 21 st March, 21 st June, 21 st September, and 21 st December at 07:00-18:00 for Figure 12.
Figure 13 shows the comparison between exterior horizontal illuminance and the daylight factor of each seasonal representative dates (overcast sky) using the proposed equation and the Desktop Radiance simulation.Furthermore, the DF values were compared and divided into their SC and IRC.The illuminance from natural sources is often determined in terms of the DF, which is the ratio of the indoor illuminance to the outdoor illuminance available on a horizontal plane from the entire overcast sky and is expressed as a percentage.The daylight factor is internationally recognized as the synthetic parameter to relate indoor visual task lighting requirements and daylight availability.From an energy perspective, it is also used in the new daylighting European Standard EN-17037:2018 [65] for evaluating daylight penetration of buildings.Standard sky for DF calculation (overcast sky) has been defined as the most conservative, and from an energy point of view, this is very useful since it represents the condition of peak energy consumption for artificial lighting [65][66][67][68].
Comparing the exterior horizontal illuminance of each seasonal representative dates (overcast sky), the exterior horizontal illuminance of 6.21 (07:00-18:00) was the highest, showing a distribution from 6209 lx to 18,146 lx.The exterior horizontal illuminance of 12.21 (07:00-18:00) was the lowest, showing a distribution from 883lx to 9218 lx.Unlike the case with exterior horizontal illuminance distribution, the four representative dates for an overcast sky had relatively constant DF values between 07:00-18:00, respectively.The calculations from the proposed equation showed that the The illuminance from natural sources is often determined in terms of the DF, which is the ratio of the indoor illuminance to the outdoor illuminance available on a horizontal plane from the entire overcast sky and is expressed as a percentage.The daylight factor is internationally recognized as the synthetic parameter to relate indoor visual task lighting requirements and daylight availability.From an energy perspective, it is also used in the new daylighting European Standard EN-17037:2018 [65] for evaluating daylight penetration of buildings.Standard sky for DF calculation (overcast sky) has been defined as the most conservative, and from an energy point of view, this is very useful since it represents the condition of peak energy consumption for artificial lighting [65][66][67][68].
Comparing the exterior horizontal illuminance of each seasonal representative dates (overcast sky), the exterior horizontal illuminance of 6.21 (07:00-18:00) was the highest, showing a distribution from 6209 lx to 18,146 lx.The exterior horizontal illuminance of 12.21 (07:00-18:00) was the lowest, showing a distribution from 883lx to 9218 lx.Unlike the case with exterior horizontal illuminance distribution, the four representative dates for an overcast sky had relatively constant DF values between 07:00-18:00, respectively.The calculations from the proposed equation showed that the average DF of 3.21, 6.21, 9.21, and 12.21 were 5.9, 6.3, 6.1, and 5.7%, respectively.
After dividing the results into SC and IRC components, the Desktop Radiance simulation value and statistical analysis were done.They showed a high correlation with MBE and Cv(RMSE) values satisfying the ASHRAE standards (MBE less than ±10%, Cv(RMSE) less than 30%) [63].The t-test result with a significant level of 0.05 showed a p-value less than 0.05.Therefore, the DF (SC+IRC) value calculated from the equation proposed in this study have statistical significance with the Desktop Radiance simulation results.We also checked the average DF value compared with the DF of Mangione et al. [66] for the validity of calculated DF value using the proposed prediction method.The calculated DF value was found to be similar to the DF value of Mangione's study [66].Tables S2-1, S2-2, S2-3, and S2-4 in the Supplementary Material provide detailed data for 21st March, 21st June, 21st September, and 21st December at 07:00-18:00 for Figure 13.

Verification on Indoor Illuminance Prediction Method (CIE Clear Sky)
When predicting illuminance for CIE clear sky condition, it is important that solar radiation influxes into the building.For this reason, the simulation model was unfolded into six sides, and indoor illuminance distribution was evaluated.Red represented a relatively high illuminance value, yellow was the median value, and green represented relatively low illuminance.
Figure 14 shows a comparison of the indoor illuminance using the modified equation and the Desktop Radiance simulation on 21st March at 2:00 pm (CIE clear sky).At 2:00 pm on 21st March, the sun was located near Patch No. 106.Therefore, it was possible to see some solar radiation input on the right wall of the six-sided planar figure.The lower depth shows lower luminance.At the illuminance sensor on 21st March at 2:00 pm from 1 to 5 m, the illuminance sensor was not affected by direct sunlight.Thereafter, the equations that consider the influence of diffuse and reflected illuminance were used in the formation of a prediction method.
When comparing Desktop Radiance simulation results at measurement point from 1 to 5 m, the proposed prediction equation's error rate ranged from 2% to 8%, and the overall average error rate was 4.9%.When analyzing the illuminance prediction error rate by distance, the error rate was 8% at 1 m and 6% at 2 m and 3 m.The error rate was 2%-3% at 4 m and 5 m, which is similar to the Desktop Radiance simulation values.The modified equation showed an overall tendency to express higher values than the Desktop Radiance simulation results.As the position of the illuminance sensor changes from 1 m to 5 m, the error rate of illuminance tends to decrease.
Figure 15 shows a comparison graph of indoor illuminance values that were calculated using the proposed prediction equation and the Desktop Radiance simulation on 21st June at 2:00 pm when the sun was located near Patch No. 136.   Figure 15 shows a comparison graph of indoor illuminance values that were calculated using the proposed prediction equation and the Desktop Radiance simulation on 21 st June at 2:00 pm when the sun was located near Patch No. 136.
Since the sun's altitude was high that day at 2:00 pm, which represented a typical summer day (summer solstice), there was no direct sunlight effect at measurement point from 1 to 5 m.When comparing Desktop Radiance simulation result at measurement points of 1-5 m, the error rate ranged from 1% to 6%, and the overall average error rate was 4.3%.When analyzing the illuminance prediction error rate by distance, the error rate was 6% at 1 m, 6% at 2 m, 5% at 3 m, 3% at 4 m, and 1% at 5 m.Since the sun's altitude was high that day at 2:00 pm, which represented a typical summer day (summer solstice), there was no direct sunlight effect at measurement point from 1 to 5 m.When comparing Desktop Radiance simulation result at measurement points of 1-5 m, the error rate ranged from 1% to 6%, and the overall average error rate was 4.3%.When analyzing the illuminance prediction error rate by distance, the error rate was 6% at 1 m, 6% at 2 m, 5% at 3 m, 3% at 4 m, and 1% at 5 m.
Figure 16 shows the indoor illuminance prediction results on 21st September at 2:00 pm when the sun was located near Patch No. 105.Direct solar radiation entered deeply into the building.The solar radiation input on the floor and the right wall of the six-sided planar figure are visible.For this reason, there was a direct sunlight effect at the measurement point at 1 m.Therefore, the influence from the direct solar beam was considered, and Equation ( 12)'s incidence angle was added.When comparing Desktop Radiance simulation result at measurement points of 1-5 m, the overall average error rate was 5.4%.The error rate was 7% at 1 m, 7% at 2 m, 6% at 3 m, 3% at 4 m, and 3% at 5 m.  Figure 17 shows the indoor illuminance prediction results on 21st December at 2:00 pm, which represented a typical winter day (winter solstice), and the sun was located near Patch No. 58.Since the sun's altitude was low, direct solar radiation entered more deeply into the building.The solar radiation input on the floor and the right wall of the six-sided planar figure are visible.For this reason, there was a direct sunlight effect at the measurement point at 1 m.Therefore, the influence from the direct solar beam was considered, and Equation ( 12)'s incidence angle was added.When comparing Desktop Radiance simulation result at the measurement point of 1-5 m, the error rate ranged from 3% to 9%, and overall average error rate was 6.1%.When the modified prediction equation was used for winter solstice conditions, the effect of sky luminance, direct solar radiation, and indoor reflective components were reflected.While Figures 14-17 shows detailed results at a specified time (14:00) on 21st March, 21st June, 21st September and 21st December, Figure 18 shows the result of a comparison between the Desktop Radiance simulation result and modified equations result in clear sky conditions for the entire day, excluding the night time, 07:00-18:00 (12 h).
The R 2 , MBE, RMSE, Cv (RMSE) analysis and t-test (t-value, p-value) was done on the Desktop Radiance simulation results of each illuminance sensor (1 m-5 m) at 07:00-18:00 (12 h, all day excluding the night time) and the calculation results of the proposed equation.
The indoor illuminance of the clear sky was higher than that of the overcast sky, due to the influence of direct solar radiance.The indoor illuminance of a clear sky was distributed between 0-4500 lx for 21st of March and 21st of September, 0-2700 lx for 12.21 and 0-2500 lx for the 21st of June.For clear sky illuminance, it was the highest at the representative dates of the intermediate seasons (21st March & 21st September) and the lowest at the winter solstice (21st December).All four seasonal representative dates showed Desktop Radiance simulation and proposed equation calculation results' Pearson correlation coefficient R 2 equal to or higher than 0.99.The MBE and Cv (RMSE) values were less than ±10%, 30%, respectively.The MBE and Cv (RMSE) value satisfied the ASHRAE standard [63].The t-test result with Significance level set to 0.05 showed that the p-value was also less than the significance level of 0.05.Thus, in comparison to Desktop Radiance simulation value, proposed daylight equation's calculation results possess statistical significance and were considered to be similar.Tables S3-1, S3-2, S3-3, and S3-4 in the Supplementary Material provide detailed data for 21st March, 21st June, 21st September, and 21st December at 07:00-18:00 for Figure 18.The R 2 , MBE, RMSE, Cv (RMSE) analysis and t-test (t-value, p-value) was done on the Desktop Radiance simulation results of each illuminance sensor (1m-5m) at 07:00-18:00 (12 hours, all day excluding the night time) and the calculation results of the proposed equation.
The indoor illuminance of the clear sky was higher than that of the overcast sky, due to the influence of direct solar radiance.The indoor illuminance of a clear sky was distributed between 0-4500 lx for 21 st of March and 21 st of September, 0-2700 lx for 12.21 and 0-2500 lx for the 21 st of June.For clear sky illuminance, it was the highest at the representative dates of the intermediate seasons (21 st March & 21 st September) and the lowest at the winter solstice (21 st December).All four seasonal representative dates showed Desktop Radiance simulation and proposed equation calculation results' Pearson correlation coefficient R 2 equal to or higher than 0.99.The MBE and Cv (RMSE) values were less than ± 10%, 30%, respectively.The MBE and Cv (RMSE) value satisfied the ASHRAE standard [63].The t-test result with Significance level set to 0.05 showed that the p-value was also less than the significance level of 0.05.Thus, in comparison to Desktop Radiance simulation value, proposed daylight equation's calculation results possess statistical significance and were considered to be similar.Tables S3-1, S3-2, S3-3, and S3-4 in the Supplementary Material provide detailed data for 21 st March, 21 st June, 21 st September, and 21 st December at 07:00-18:00 for Figure 18.

Verification on Indoor Illuminance Prediction Method When Installing Venetian Blind (CIE Clear Sky)
Lastly, the possibility of predicting illuminance by applying the modified equation when installing the venetian blind under clear sky conditions was verified.To do this, the blind slat angle was set as a variable and the Desktop Radiance simulation results were compared with the modified equation results.Visible light transmittance of blinds can be calculated using LBNL's Window 7.6 program [69], as shown in Figure 19.It is possible to predict indoor illuminance by substituting the visible light transmittance (VLT) value of glass or blind in the modified equation.

Verification on Indoor Illuminance Prediction Method When Installing Venetian Blind (CIE Clear Sky)
Lastly, the possibility of predicting illuminance by applying the modified equation when installing the venetian blind under clear sky conditions was verified.To do this, the blind slat angle was set as a variable and the Desktop Radiance simulation results were compared with the modified equation results.Visible light transmittance of blinds can be calculated using LBNL's Window 7.6 program [69], as shown in Figure 19.It is possible to predict indoor illuminance by substituting the visible light transmittance (VLT) value of glass or blind in the modified equation.When installing external venetian blinds at 30°, the VLT was 0.519, which was calculated by using the Window 7.6 program [69].When comparing illuminance prediction results of the Desktop Radiance simulation and modified prediction method, the error rates of 21 st March were 9% at 1 m, 8% at 2 m, 10% at 3 m, 8% at 4 and 7% at 5 m.The average illuminance prediction error rate was 8.6%.September, and 21 st December at 2:00 pm (CIE clear sky).
When installing external venetian blinds at 30°, the VLT was 0.519, which was calculated by using the Window 7.6 program [69].When comparing illuminance prediction results of the Desktop Radiance simulation and modified prediction method, the error rates of 21 st March were 9% at 1 m, 8% at 2 m, 10% at 3 m, 8% at 4 and 7% at 5 m.The average illuminance prediction error rate was 8.6%.The error rate was 9% at 1 m, 8% at 2 m, 10% at 3 m, 9% at 4 and 9% at 5 m on 21 st June at 2:00 pm.On the 21 st September and December, installing blinds blocked direct solar radiation to the 1m illuminance sensor and indoor illuminance at 1 m were reduced by 96% and 93%.When comparing illuminance prediction results of the Desktop Radiance simulation and modified prediction method, the average illuminance prediction error rate was 9.7% and 9.4%, respectively.When installing external venetian blinds at 30 • , the VLT was 0.519, which was calculated by using the Window 7.6 program [69].When comparing illuminance prediction results of the Desktop Radiance simulation and modified prediction method, the error rates of 21st March were 9% at 1 m, 8% at 2 m, 10% at 3 m, 8% at 4 and 7% at 5 m.The average illuminance prediction error rate was 8.6%.The error rate was 9% at 1 m, 8% at 2 m, 10% at 3 m, 9% at 4 and 9% at 5 m on 21st June at 2:00 pm.On the 21st September and December, installing blinds blocked direct solar radiation to the 1m illuminance sensor and indoor illuminance at 1 m were reduced by 96% and 93%.When comparing illuminance prediction results of the Desktop Radiance simulation and modified prediction method, the average illuminance prediction error rate was 9.7% and 9.4%, respectively.
Figure 21 shows the results of a comparison between the Desktop Radiance simulation result and modified equations result in clear sky conditions (when installing blind 30 • ) for the entire day, excluding the night time, 07:00-18:00 (12 h).
When exterior venetian blind were installed at 30 • , the Desktop Radiance simulation results and the proposed equation calculation results of the four seasonal representative dates showed high correlations satisfying the ASHRAE standards [63], all having a Pearson correlation coefficient R 2 value over 0.99, MBE value less than ±10% and Cv(RMSE) value less than 30%.The t-test results with significance level 0.05 showed a p-value less than 0.05.Therefore the Desktop Radiance simulation result and proposed daylight equation calculation result were showed to have statistical significance.
Figure 22 is a graph that compares the results of Desktop Radiance and the proposed prediction method when installing external venetian blinds at an angle of 45 • on 21st March, 21st June, 21st September, and 21st December at 2:00 pm (CIE clear sky).When installing external venetian blind at 60˚ on 21 st March, 21 st June, 21 st September, and 21 st December at 2:00 pm, VLT was 0.204, which was calculated using the Window 7.6 program.The VLT value decreased since the direct solar protection rate was higher than that when the slat angle was 45°.The illuminance distribution of slat angle 60° was also lower than that of the case with slat angles 30° and 45°.When comparing illuminance prediction result of the Desktop Radiance simulation and the prediction method, the error rates of 21 st March were 9% at 1 m, 9% at 2 m, 11% at 3 m, 10% at 4 m, and 8% at 5 m.The average illuminance prediction error rate on 21 st June, 21 st September, 21 st December at 2:00 pm were 9.8%, 10.3%, and 10.1%, respectively.
Figure 25 shows the results of a comparison between the Desktop Radiance simulation result and modified equations result in clear sky conditions (when installing blind 60°) for the entire day, excluding the night time, 07:00-18:00 (12 hours).
We analyzed each luminance sensor's (1 m-5 m) Desktop Radiance simulation results and proposed equation calculation results via R 2 , MBE, RMSE, Cv(RMSE) and performed the t-test (tvalue, p-value).When blind was installed at 60°, all statistically analysis (R 2 , MBE, Cv(RMSE), tvalue, p-value) satisfied each standard and showed high correlations, just as when it was installed at 30° or 45°.It was possible to calculate indoor illuminance when installing the blinds, by substituting the variable value of VLT in the prediction method which is comparably accurate for the prediction of illuminance.Tables S6-1, S6-2, S6-3, and S6-4 in the Supplementary Material provide detailed data for 21 st March, 21 st June, 21 st September, and 21 st December at 07:00-18:00 for Figure 25.When installing external venetian blind at 60 • on 21st March, 21st June, 21st September, and 21st December at 2:00 pm, VLT was 0.204, which was calculated using the Window 7.6 program.The VLT value decreased since the direct solar protection rate was higher than that when the slat angle was 45 • .The illuminance distribution of slat angle 60 • was also lower than that of the case with slat angles 30 • and 45 • .When comparing illuminance prediction result of the Desktop Radiance simulation and the prediction method, the error rates of 21st March were 9% at 1 m, 9% at 2 m, 11% at 3 m, 10% at 4 m, and 8% at 5 m.The average illuminance prediction error rate on 21st June, 21st September, 21st December at 2:00 pm were 9.8%, 10.3%, and 10.1%, respectively.
Figure 25 shows the results of a comparison between the Desktop Radiance simulation result and modified equations result in clear sky conditions (when installing blind 60 • ) for the entire day, excluding the night time, 07:00-18:00 (12 h).
We analyzed each luminance sensor's (1 m-5 m) Desktop Radiance simulation results and proposed equation calculation results via R 2 , MBE, RMSE, Cv(RMSE) and performed the t-test (t-value, p-value).When blind was installed at 60 • , all statistically analysis (R 2 , MBE, Cv(RMSE), t-value, p-value) satisfied each standard and showed high correlations, just as when it was installed at 30 • or 45 • .It was possible to calculate indoor illuminance when installing the blinds, by substituting the variable value of VLT in the prediction method which is comparably accurate for the prediction of illuminance.Tables S6-1, S6-2, S6-3, and S6-4 in the Supplementary Material provide detailed data for 21st March, 21st June, 21st September, and 21st December at 07:00-18:00 for Figure 25.
(3) The DeLight algorithm was analyzed and a modified illuminance prediction method was suggested.Illuminance that was calculated by the modified prediction method result was compared with Desktop Radiance simulation result when its CIE overcast sky.The simulation results and the proposed equation calculation results of the four seasonal representative dates showed high correlations, all having the R 2 value over 0.99, MBE value less than ±10% and Cv(RMSE) value less than 30%.The t-test results with a significance level of 0.05 showed a p-value of less than 0.05.Therefore, this study verified that the suggested equation can be utilized in the overcast sky.(4) This study evaluated DF by classifying component into SC and IRC.DF was calculated by the modified prediction method and Desktop Radiance simulation when its CIE overcast sky.As a result DF can be evaluated by SC and IRC since there is no significant difference between the modified prediction method and simulation values.All statistical analysis (R and showed high correlations.It was possible to calculate indoor illuminance when installing the solar shading system, by substituting the variable value of VLT in the prediction method, which is comparably accurate for the prediction of illuminance.(7) By verifying the accuracy of the developed prediction method for calculating indoor illuminance, prediction was deemed possible even when there were changes in the sky conditions, sun location, and blind slat angle.However, it is also necessary to study the factors influencing indoor illumination prediction.The study will proceed considering real experimental case study, the positional direction of the study model, blind control, lighting system control strategy, and influence of surrounding buildings on indoor illumination prediction.

Figure 1 .
Figure 1.Flow diagram of this study.

Figure 1 .
Figure 1.Flow diagram of this study.

Figure 2
Figure 2 presents a diagram of sky relative luminance distribution based on the location of the sun and arbitrary sky elements.The ratio of the luminance L γα in an arbitrary sky element to the zenith luminance L z is expressed as shown in Equation (3) following the current CIE clear sky standard [15,21,23-25]: L γα L z = f (χ)φ(Z) f (Z s )φ(0 • )(3)

Figure 2
Figure 2 presents a diagram of sky relative luminance distribution based on the location of the sun and arbitrary sky elements.The ratio of the luminance Lγα in an arbitrary sky element to the zenith luminance Lz is expressed as shown in Equation (3) following the current CIE clear sky standard [15,21,23-25]:   = ()() ( )(0°)(3)

Figure 2 .
Figure 2. Diagram showing the position of the sun and sky element.

Figure 2 .
Figure 2. Diagram showing the position of the sun and sky element.

Figure 3 .
Figure 3. Algorithm framework of sky luminance distribution.Figure 3. Algorithm framework of sky luminance distribution.

Figure 3 .
Figure 3. Algorithm framework of sky luminance distribution.Figure 3. Algorithm framework of sky luminance distribution.

Figure 4 .where
Figure 4. Room geometry for calculating the indoor horizontal illuminance.

Figure 4 .
Figure 4. Room geometry for calculating the indoor horizontal illuminance.

Figure 5 .
Figure 5. Diagram used in calculating the indoor illuminance in CIE clear and overcast sky.

Figure 5 .
Figure 5. Diagram used in calculating the indoor illuminance in CIE clear and overcast sky.

Energies 2019, 12 , x 10 of 36 Figure 5 .
Figure 5. Diagram used in calculating the indoor illuminance in CIE clear and overcast sky.

Table 4 .
Illuminance sensor geometry and glazing transmittance of analysis model.(z = The horizontal distance between the window and sensor point (m), h p = Height between lower edge of the window and sensor point (m), h w = Window height (m), X p = Distance between sensor point and left wall (m), X w = Distance between the left edge of the window and left wall (m), W w = Window width (m), τ w = Visible light transmittance (%)).

Figure 8
Figure 8 shows the sky patch that was used to calculate sky luminance of each location b iding the sky into 145 elements [36].As indicated previously, the applicable sky models include E standard overcast sky and CIE clear sky.To apply the sky luminance L value to the indoo minance prediction equation, algorithm framework of sky luminance distribution in Figure re used to create sky luminance distribution (sky luminance/zenith luminance) of CIE overcast sk d clear sky.

Figure 8 .
Figure 8. Calculation points for the sky patches.

Figure 8 .
Figure 8. Calculation points for the sky patches.
is a diagram depicting the sky element point's altitude and azimuth, which were compared with Desktop Radiance for verification, and the position of the Sun at each seasonal representative date Energies 2019, 12, x 15 of 36 is a diagram depicting the sky element point's altitude and azimuth, which were compared with Desktop Radiance for verification, and the position of the Sun at each seasonal representative date CIE Standard Clear Sky_9.21 at 2:00 pm (Sun's Location: Altitude 46.53 • , Azimuth 45.39 • ) Energies 2019, 12, x 15 of 36 is a diagram depicting the sky element point's altitude and azimuth, which were compared with Desktop Radiance for verification, and the position of the Sun at each seasonal representative date (21 st March, 21 st June, 21 st September, and 21 st December) [61].Energies 2019, 12, x 15 of 36 is a diagram depicting the sky element point's altitude and azimuth, which were compared with Desktop Radiance for verification, and the position of the Sun at each seasonal representative date (21 st March, 21 st June, 21 st September, and 21 st December) [61].

CIE
Standard Clear Sky_12.21 at 2:00 pm (Sun's Location: Altitude 25.68• , Azimuth 22.31 • )4.2.Verification of Sky Luminance DistributionWe verified each patch of sky luminance value for the four directions (north, south, east and west) and for the seasonal representative dates (21st March, 21st June, 21st September, 21st December at 2pm) based on the position of the Sun instead of verifying the entire days' time, because we reviewed Energies 2019, 12, 592 16 of 37
Concept of sky luminance where looking at the sky from the illuminance sensor (1-5 m), 3.21 at 2:00 pm, Direction: South Sky luminance image of Radiance (The center of the part of the overcast sky visible to the point P through the window) Sky luminance image of radiance (the center of the part of the clear sky visible to the point P through the window)

Energies 2019, 12 , x 20 of 36 Figure 10 .
Figure 10.Comparison of sky luminance between the CIE sky algorithm and Desktop Radiance.(A) Overcast sky (B) Clear sky.

Figure 10 .
Figure 10.Comparison of sky luminance between the CIE sky algorithm and Desktop Radiance.(A) Overcast sky (B) Clear sky.

Figure 14 .
Figure 14.Indoor illuminance distribution of the six-side planar figure and comparison of illuminance between the modified equation and Desktop Radiance (CIE Clear Sky_3.21 14:00).

Figure 14 .
Figure 14.Indoor illuminance distribution of the six-side planar figure and comparison of illuminance between the modified equation and Desktop Radiance (CIE Clear Sky_3.21 14:00).

Figure 15 .
Figure 15.Indoor illuminance distribution of the six-side planar figure and comparison of illuminance between the modified equation and Desktop Radiance (CIE Clear Sky_6.21 14:00).

Figure 16
Figure16shows the indoor illuminance prediction results on 21 st September at 2:00 pm when the sun was located near Patch No. 105.Direct solar radiation entered deeply into the building.The solar radiation input on the floor and the right wall of the six-sided planar figure are visible.For this reason, there was a direct sunlight effect at the measurement point at 1 m.Therefore, the influence from the direct solar beam was considered, and Equation (12)'s incidence angle was added.When comparing Desktop Radiance simulation result at measurement points of 1-5 m, the overall average

Figure 15 .
Figure 15.Indoor illuminance distribution of the six-side planar figure and comparison of illuminance between the modified equation and Desktop Radiance (CIE Clear Sky_6.21 14:00).

Energies 2019, 12 , x 25 of 36 Figure 16 .
Figure 16.Indoor illuminance distribution of the six-side planar figure and comparison of illuminance between the modified equation and Desktop Radiance (CIE Clear Sky_9.21 14:00).

Figure 16 .
Figure 16.Indoor illuminance distribution of the six-side planar figure and comparison of illuminance between the modified equation and Desktop Radiance (CIE Clear Sky_9.21 14:00).

Figure 16 .
Figure 16.Indoor illuminance distribution of the six-side planar figure and comparison of illuminance between the modified equation and Desktop Radiance (CIE Clear Sky_9.21 14:00).

Figure 17 .
Figure 17.Indoor illuminance distribution of the six-side planar figure and comparison of illuminance between the modified equation and Desktop Radiance (CIE Clear Sky_12.2114:00).

Figure 17 .
Figure 17.Indoor illuminance distribution of the six-side planar figure and comparison of illuminance between the modified equation and Desktop Radiance (CIE Clear Sky_12.2114:00).

Figure 20
Figure 20 is a graph that compares the results of Desktop Radiance and the proposed prediction method when installing external venetian blinds at an angle of 30° on 21 st March, 21 st June, 21 st September, and 21 st December at 2:00 pm (CIE clear sky).When installing external venetian blinds at 30°, the VLT was 0.519, which was calculated by using the Window 7.6 program[69].When comparing illuminance prediction results of the Desktop Radiance simulation and modified prediction method, the error rates of 21 st March were 9% at 1 m, 8% at 2 m, 10% at 3 m, 8% at 4 and 7% at 5 m.The average illuminance prediction error rate was 8.6%.

Figure 19 .
Figure 19.Calculation of visible light transmittance of blind using LBNL's Window 7.6 program.

Figure 20
Figure 20 is a graph that compares the results of Desktop Radiance and the proposed prediction method when installing external venetian blinds at an angle of 30 • on 21st March, 21st June, 21st September, and 21st December at 2:00 pm (CIE clear sky).

Figure 21
Figure21shows the results of a comparison between the Desktop Radiance simulation result and modified equations result in clear sky conditions (when installing blind 30°) for the entire day, excluding the night time, 07:00-18:00 (12 hours).

Figure 24
Figure24is a graph that compares the results of Desktop Radiance and the proposed prediction method when installing external venetian blinds at an angle of 60° on 21 st March, 21 st June, 21 st September, and 21 st December at 2:00 pm (CIE clear sky).

Table 3 .
Properties of simulation model.

Table 5 .
Set of simulation variables for indoor illuminance.

Table 6 .
Set of the venetian blind slat angle and dimensions.

Table 6 .
Set of the venetian blind slat angle and dimensions.

Table 7 .
Sky luminance distribution of CIE standard overcast sky.

Table 7 .
Sky luminance distribution of CIE standard overcast sky.

Table 8 .
Sky luminance distribution of CIE standard clear sky.

Table 7 .
Sky luminance distribution of CIE standard overcast sky.

Table 8 .
Sky luminance distribution of CIE standard clear sky.

Table 7 .
Sky luminance distribution of CIE standard overcast sky.

Table 8 .
Sky luminance distribution of CIE standard clear sky.

Table 8 .
Sky luminance distribution of CIE standard clear sky.

Table 8 .
Sky luminance distribution of CIE standard clear sky.

Table 8 .
Sky luminance distribution of CIE standard clear sky.

Table 8 .
Sky luminance distribution of CIE standard clear sky.

Table 8 .
Sky luminance distribution of CIE standard clear sky.

Table 8 .
Sky luminance distribution of CIE standard clear sky.

Table 9 .
Image of Desktop Radiance, concept diagram, and sky luminance from looking at each sensor.

Table 9 .
Image of Desktop Radiance, concept diagram, and sky luminance from looking at each sensor.

Table 9 .
Image of Desktop Radiance, concept diagram, and sky luminance from looking at each sensor.

Table 10 .
Prediction results and error rate of sky luminance distributions in CIE overcast sky.

Table 11 .
Prediction results and error rate of sky luminance distributions in CIE clear sky.

3.21 at 2:00 pm (Sun Alt.48.03 • Azi.30.09
2, MBE, Cv(RMSE), t-test) satisfied each standard and showed high correlations.(5)Predicting illuminance when its CIE clear sky, it is important that there are solar radiation influxes hitting the building.Due to this reason, this study unfolds the simulation model into six sides and has also checked the indoor illuminance distribution.The correlation of Desktop Radiance simulation results and the proposed equation calculation results of the four seasonal representative dates showed a R 2 value over 0.99.The MBE for each seasonal representative date (21st March, 21st June, 21st September, and 21st December) was lower than ±10% at −8.899%, −8.719%, −9.371%, and −9.276%.The Cv(RMSE) values were lower than 30% at 12.631%, 11.771%, 13.167%, and 15.895%, satisfying the ASHRAE standards.The t-test results showed that the p-value were all lower than 0.05.The simulation values and proposed daylight equation's calculated the values that all possessed a statistical significance.(6) Additionally, when the exterior venetian blinds were installed at 30 • , 45 • and 60 • , this study compared the simulation illuminance value of the Radiance program and the calculation value of the proposed equation.All statistical analysis (R 2 , MBE, Cv(RMSE), t-test) satisfied each standard