On the Interaction between the Depth and Elevation of External Shading Devices in Tropical Daylit Classrooms with Symmetrical Bilateral Openings

Abstract

External shading devices are an important design feature in tropical buildings, particularly for climate mitigation. However, the interaction between the depth and elevation of the shading devices and their impact on indoor daylight performance is not fully understood, especially for the case of tropical buildings with bilateral openings. This study therefore aims to evaluate the design possibilities of external shading devices with various depth and elevation in terms of daylight performance for the case of tropical school classrooms with bilateral openings in an Indonesian city. A computational simulation method using Radiance is utilized to perform annual daylight metrics calculations. Geometry, material, and simulation settings are prepared using the Ladybug tool under Grasshopper for four building orientations, namely 0°, 45°, 90°, and 135°. Sensitivity and uncertainty analyses are conducted for all design combinations. The results show that the interaction between a shading device’s depth and elevation is unique, depending on the building orientation and the availability of direct sunlight. In general, shading elevation is more influential, compared to shading depth, on the observed daylight metrics and the combined objective functions at all orientations.

1. Introduction

External horizontal shading is historically rooted in tropical architecture, such as on vernacular buildings that belong to many ethnic groups in the tropical regions. Many vernacular buildings are equipped with roofs that protrude continuously to shade the building as can be found, for example, in the traditional houses of Aceh [1] and Toraja [2] in Indonesia. In some cases, verandas are also designed to provide shade, thus avoiding excessive solar radiation and heat throughout the year, as in Javanese traditional houses [3]. Despite the vernacular wisdom in integrating horizontal shading devices, daylight potential in modern tropical architecture is often not considered optimally.

Horizontal external shading devices are still widely available on modern building facades in the tropics. However, in many public buildings, such as government-funded schools, shading devices are often not intentionally designed to obtain the most benefit from daylight. In Indonesia, for example, school classrooms are typically designed according to government regulations [4] which are focused on the physical appearance, instead of indoor environmental performance, of the buildings.

Around 175,000 (64%) government-funded schools across Indonesia are elementary level [5]. Early studies investigating daylight performance in elementary school classrooms have been conducted by several researchers in Indonesia, using either static [6,7] or dynamic or climate-based metrics [8]. Those studies concluded that there is not enough daylight in most of the school classroom. In addition, several recent studies have evaluated more than 250 elementary schools in terms of annual diffuse and direct daylight metrics. The studies showed that approximately 14% of the classrooms suffered from an insufficient amount of indoor daylight illuminance [9], and approximately 43% received over-excessive amounts of annual direct sunlight [10].

It has been understood that, on the one hand, insufficient amounts of daylight may affect hormonal patterns of the building occupants which in turn may cause drowsiness [11,12,13,14]. On the other hand, excessive sunlight also contributes to the increased risk of overheating and glare [15,16,17]. Meanwhile, daylight comes in abundance in tropical climate regions. As previously mentioned, horizontal external shading is an essential feature for mitigating the problem. Furthermore, around 70% of building energy use [18] can be associated with the façade design [18,19]. It is therefore important for the building designers or architects to carefully design the external shading features.

Several attempts have been made in the literature to optimize indoor daylight performance in tropical buildings by using horizontal external shading devices [16,20,21,22,23,24]. These studies have evaluated horizontal shading devices with multiple configurations of input parameters, such as depth and rotation of the devices, in rooms with a unilateral (single-sided) opening type. However, in Indonesia, the use of bilateral (double-sided) opening types is well-known, particularly in government-funded school classrooms which are typically designed as a row in single-loaded buildings. This is mainly to maximize the potential of having cross-ventilation inside the rooms, which in turn can reduce the need for mechanical cooling. From a daylight perspective, if properly designed, such an opening type can also be beneficial. Current investigations in typical Indonesian school classrooms found that shading depth and elevation play an essential role in determining the availability of direct and diffuse daylight in the rooms [9,10]. However, interaction of shading depth and its elevation has not been discussed in detail.

Therefore, this study aims to evaluate the design possibilities of various depths and elevations of horizontal shading device to understand the interaction of those parameters in determining daylight performance in tropical school classrooms with bilateral openings. To achieve the goal, this study implements dynamic daylight performance criteria, combined with sensitivity and uncertainty analyses, to observe the interaction of the design parameters throughout the year.

2. Methods

2.1. Verification

This study employs the computational simulation method using the Rhinoceros (RH) interface to rationally investigate the annual dynamic daylight performance with Radiance as the daylight simulation engine, which has been validated for various lighting scenes [25,26,27,28,29]. Note that Radiance has undergone long-term as well as short-term validations [16], which is considered representative for achieving the objective.

Nevertheless, verification is conducted in this case to determine the appropriate Radiance simulation parameter by comparing the analytical values of the sky component (SC) with those from simulation with high, medium, and low settings (

Table 1. Radiance parameter settings.

Low	Medium	High
-aa 0.25–ab 2–ad 512–ar 16–as 128–lw 0.05	-aa 0.2–ab 3–ad 2048–ar 64–as 2048–lw 0.01	-aa 0.1–ab 6–ad 4096–ar 128–as 4096–lw 0.005

). The SC is selected as a criterion for verification due to its relatively low uncertainty, since it only considers the effective daylight opening with no internal or external reflections and without glass transmission. Thus, the Radiance setting verification can be effectively conducted. In this study, the standard CIE overcast sky model is assumed. Theoretical SC calculation under the standard CIE overcast sky has been discussed in detail elsewhere [30].

A hypothetical object, which is CIE test case 5.13 [31], is introduced for this purpose, together with the modified version that includes another opening on the opposite wall to create bilateral openings. The room dimension is 4.0 m × 4.0 m with a height of 3.0 m. Shading depth of 2.0 m at 3.0 m elevation from the floor is introduced. The opening dimension is 1.0 m × 2.0 m at 1.0 m elevation from the floor. Additionally, eight calculation points are considered at intervals of 0.5 m between points and 0.25 m from the facade wall. All calculation points, labelled as A to H in Figure 1, lie on the room’s axis.

Next, the simulated SC values on each calculation point for the two scenarios are compared with the analytical values. Their absolute difference (AD) is defined in Equation (1).

AD = |SC_analytical − SC_simulation|

(1)

2.2. Digital Modeling

The hypothetical classroom size is modelled based on Indonesian regulations for elementary school classrooms, which is 7 m × 8 m [4] with a height of 3.5 m (Figure 2). The window-to-wall ratio (WWR) in this study is 30% [32], distributed symmetrically on each side of the classroom to represent the bilateral opening type. Furthermore, the external shading device is dynamically modelled with respect to its elevation and depth. All modelling processes are performed using the Grasshopper (GH) algorithm [33,34]. In addition, the rest of the fixed parameters are shown in

Table 2. Input variables.

Fixed Design Parameter	Value and Attribute
WWR (%)	30%: distributed windows
Window elevation [m]	1.5 m
Window height [m]	1.2 m
Window distance [m]	1.5 m
Radiance Material
Wall reflectance (ρw) [-]	0.5
Floor reflectance (ρf) [-]	0.2
Ceiling reflectance (ρc) [-]	0.8
Ground reflectance (ρg) [-]	0.2
Shading reflectance (ρshd) [-]	0.3
Glass transmittance (τ) [-]	0.7
Dynamic Design Parameter
Shading depth (d) [m]	1.0–2.5 m: 0.1 m step size
Shading elevation (z) [m]	2.7–3.5 m: 0.1 m step size

.

After the digital model was completed, the geometry was given Radiance material through Honeybee [+] (HB [+]) components [35,36,37]. Furthermore, the interior surface material properties are also tabulated in Table 2.

2.3. Simulation Settings

There are 195 horizontal illuminance sensor points in the room, with the outer ones being 0.5 m from the nearest wall [38], as shown in Figure 3a. The north direction is shown by the green (positive y) axis. All sensor points are located at a height of 0.75 m from the floor, as required by the Indonesian standard on daylighting [39], with an upward vector direction [0 0 1]. To assess visual comfort, a sensor point is placed at the center of the room 1.2 m from the floor. The view direction is represented by the view vector towards the front part of the classroom, namely [0 1 0] for 0° orientation; [−1 1 0] for 45° orientation; [–1 0 0] for 90° orientation; and [–1 –1 0] for 135° orientation, as depicted in Figure 3b.

Furthermore, this study assumes occupancy throughout the year from January to December and from 08:00 to 17:00 (10 h/day), i.e., 3650 h per year. The city of Lhokseumawe, Indonesia (5°10′0″ N, 97°8′0″ E, 2~24 m above sea level) is assumed to be the location of the classrooms. The weather file for Lhokseumawe is implemented (EPW file converted to WEA file for daylight data) to create a sky matrix with 145 patches [34]. Furthermore, high simulation settings are used for normal scenes (ab = 6), black scenes (ab = 1), and black analemma (ab = 0). The typical Radiance material properties for the interior surfaces are shown in Table 2. The black scene is where all the material is set to black or no reflection to calculate annual sunlight exposure (ASE), and the black analemma is where the original sun patch is replaced with the original sun position. As shown by previous studies, the high setting is applied to avoid significant differences in simulation results [29].

Next, a simulation folder outside RH and GH is generated to perform parallel simulations using the Radiance console. Parallel simulations are carried out by running 8 scenarios at once out of 144 scenarios per orientation. Using a quad-core machine (Intel (R) Core (TM) i5 1.60 GHz) with 8 logical processors equipped with 12 GB RAM, simulations for eight scenarios can be completed in about 7 min. Finally, annual illuminance is divided into two categories, the total annual illuminance value and the direct illuminance value for direct sunlight metric calculations. The daylight metrics are computed using an Excel spreadsheet.

For assessing visual comfort, a separate folder is created following the workflow of the daylight metrics simulation. For the point-in-time visual comfort metric, rpict function of Radiance is employed to generate high dynamic range (HDR) images. Next, the point-in-time glare metric is computed using Ladybug Tools (LBT) components (HB Glare Postprocess component to access Evalglare of Radiance). Finally, a false-color image is created using an HB False Color component.

2.4. Daylight and Visual Comfort Metrics

This study employs several annual daylight metrics such as UDI_250–750lx [40], UDI_100–3000lx [41,42], sDA_300/50%, and ASE_1000,250 [43]. Daylight glare probability (DGP) is employed as a visual comfort indicator [44]. All daylight metrics are computed based on horizontal illuminance, while DGP is based on vertical illuminance and luminance contrast. The UDI metrics are defined in Equations (2) and (3) as follows:

$${\mathrm{UDI}}_{250-750\mathrm{lx}}=\frac{t_{250\mathrm{lx}\le E<750\mathrm{lx}}}{T}\times 100\%$$

(2)

$${\mathrm{UDI}}_{100-3000\mathrm{lx}}=\frac{t_{100\mathrm{lx}\le E<3000\mathrm{lx}}}{T}\times 100\%$$

(3)

where t is duration in which workplane daylight illuminance satisfies the designated range, and T is the total observation time in the entire year. Results from the UDI_250–750lx and UDI_100–3000lx calculations are spatially averaged to yield the metrics aUDI_250–750lx and aUDI_100–3000lx, respectively.

The sDA_300/50% is defined as follows:

$${\mathrm{sDA}}_{300/50\%}=\frac{A_{\mathrm{DA}300\ge 50\%}}{A_{\mathrm{total}}}\times 100\%$$

(4)

where A_DA300≥50% is the floor area with daylight autonomy (DA) 300 lux ≥ 50%, and A_total is the total floor area of the room. According to the IES, a minimum value of 55% is recommended for sDA_300/50% in order to gain credits for daylighting [40].

The ASE_1000,250 is defined as follows:

$${\mathrm{ASE}}_{1000,250}=\frac{A_{s_E{}_{1000\mathrm{lx}\ge 250\mathrm{h}}}}{A_{\mathrm{total}}}\times 100\%$$

(5)

where $A_{s_E{}_{1000\mathrm{lx}\ge 250\mathrm{h}}}$ is the floor area with direct sunlight illuminance of 1000 lux for at least 250 h in a year.

Meanwhile, the DGP is defined as follows:

$$\mathrm{DGP}=5.87 \times {10}^{-5} E_v+9.8 \times {10}^{-2} \log (1+{{\displaystyle \sum }}_i\frac{L_{s,i }^2{\omega }_{s,i}}{E_v^{1.87}{P}_i^2})+0.16$$

(6)

where E_v is the vertical illuminance on the observer’s eye, L_s,i is the source luminance of the i-th source, ${\mathsf{\omega }}_{\mathrm{s},\mathrm{i}}$ is the solid angle of the i-th source, and P_i is the Guth position index of the i-th source. DGP is observed on critical days of the year, which are the solstices, on 21 December and 21 June, and the equinox on 20 March. Since, in the tropics, glare potentially happens in the morning and afternoon, DGP is observed at 09:00 and 15:00 h on those three days.

Furthermore, the daylight metrics are combined into an objective function Y. In addition, this metric is utilized as the basis to define the most sensitive daylight metric at every orientation.

$$Y={\mathrm{aUDI}}_{250-750\mathrm{lx}}+{\mathrm{aUDI}}_{100-3000\mathrm{lx}}+{\mathrm{sDA}}_{300/50\%}-({\mathrm{ASE}}_{1000,250})$$

(7)

The combined metrics shall inform a more rational judging criterion by involving various daylight metrics that may have different sensitivity [45]. Design combinations with the highest and lowest Y values are later evaluated in terms of DGP at the observer’s viewpoint in Figure 2. In this way, the improvement from the lowest to the highest objective Y values are observable with respect to daylight discomfort glare.

The DGP criteria suggested by reference [46] are employed in this study. A previous study has also suggested that subjects in the tropics prefer lower DGP values, compared to the originally proposed values [44]. The criteria are shown in

Table 3. DGP criteria employed in this study, adopted from [46].

Range	DGP < 0.22	0.22 ≤ DGP ≤ 0.23	0.24 ≤ DGP ≤ 0.25
Category	Imperceptible	Imperceptible–Perceptible	Disturbing–Intolerable

.

2.5. Sensitivity and Uncertainty Analyses

The obtained simulation data are further processed using sensitivity and uncertainty analyses. First, sensitivity analysis is conducted by constructing a multilinear regression model with normalized input and output variables. Sensitivity is measured with the standardized regression coefficient (SRC), whose values typically range from –1 to +1. For each daylight metric, the SRC of each input variable is computed. The closer the SRC to ±1, the more sensitive the output variable to the given input. In this case, the input variables are the standardized depth (d’) and elevation (z’) of the external shading devices. Standardization is also applied to the output variables. For instance, aUDI_250-750lx is standardized to become aUDI_250-750lx′ and so forth.

Next, uncertainty analysis is employed by using the coefficient of variation (CV), which is defined as the ratio between the standard deviation and mean values of a given dataset. In this study, a CV value above 0.1 is considered as having high uncertainty.

Finally, each daylight metric is assessed in terms of its linear relation to the objective function Y by observing their coefficient of determination (R²) at each of the building orientations (0°, 45°, 90°, 135°). This step is necessary to understand whether the observed daylight metric is actually influential on the defined objective function. An R² ≥ 0.95 indicates a strong linear relation, meaning that the input variable alone can explain the vast majority (≥95%) of the linear changes in Y.

3. Results

3.1. Verification

From CIE test case 5.13 (Figure 1a), the comparison for SC values at every calculation point, according to analysis and simulation for all Radiance settings (low, medium, and high), is shown in Figure 4a. Meanwhile, absolute difference (AD) is depicted in Figure 4b. Figure 4a shows that high and medium settings consistently follow the pattern of the analytical values. The average AD values for low, medium, and high settings are 0.19%, 0.15%, and 0.14%, respectively, as also depicted in Figure 4b.

For the bilateral case (Figure 1b), analytical calculation shows the SC value is peaked at the central calculation point. This is expected since the openings are located on the opposite side of the walls. Thus, simulation settings that closely follow the analytical trend are considered legitimate. Figure 5a shows that the settings that closely follow the analytical profile are the medium and high settings, with an average AD value of 0.47%. Meanwhile, the low setting has the highest AD value (0.54%). The overall profile AD values in the bilateral test case are shown in Figure 5b. By considering the two test cases, the high Radiance setting is thus considered for further simulation in this study, as also suggested in the literature which recommended ab ≥ 6 to achieve accurate and reliable simulation results in Radiance [29].

3.2. Daylight Availability

The simulation result shows that sDA_300/50% has the highest value of 100% for all scenarios and orientations, meaning that the target illuminance of 300 lux can be achieved by daylight alone at any time in all considered cases. At all orientations, aUDI_250–750lx shows relatively linear trends as the shading depth (d) increases at each elevation (z), as illustrated in Figure 6a,e and Figure 7a,e. Meanwhile, non-linear trends are observed in other daylight metrics (aUDI_100–3000lx and ASE_1000,250) and in the objective function Y, except at an orientation of 90° (Figure 7d). For aUDI_250–750lx, at all orientations, the linearity gradient decreases as shading elevations heighten.

At an orientation of 0°, where the openings face east and west, most of the scenarios have met sDA_300/50% > 55% and aUDI_100–3000lx > 80% (Figure 6b); however, the majority of them still do not comply with ASE_1000,250 < 10% (Figure 6c). This suggests that the sensor points receive relatively high illuminance values, which most likely exceed 750 lux, as indicated by the lowest aUDI_250–750lx values in all scenarios compared to those at other orientations. Furthermore, to avoid overexposure to sunlight at this orientation, a design scenario with a combination of a longer d with a lower z is preferred. This is evident as shown in Figure 6c.

At the orientations of 45° (Figure 6e–h) and 135° (Figure 7e–h), almost all scenarios have met the target values for daylight metrics (sDA_300/50% > 55% and aUDI_100–3000lx > 80%), except for aUDI_250–750lx. Only three scenarios at the orientations of 45° and 135° are below the aUDI_100–3000lx target, as shown in Figure 6f and Figure 7f (d = 1.0 m, z = 3.4 m; d = 1.0 m, z = 3.5 m; and d = 1.1 m, z = 3.5 m). Additionally, most of the scenarios do not exceed the recommended criterion for ASE_1000,250 (< 10%). Scenarios that do not satisfy the target are the combination d < 1.8 m, z = 3.5 m; d < 1.6 m, z = 3.4 m; d < 1.5 m, z = 3.3 m; d < 1.4 m, z = 3.2 m; d < 1.3 m, z = 3.1 m; and d < 1.1 m, z = 3.0 m (Figure 6g and Figure 7g).

At an orientation of 90°, where the windows face north and south, in the combination with the shortest d and the highest z, where the presence of shading devices are not visible from inside the classroom, the resulting daylight metrics are above the minimum target (sDA_300/50% > 55% and UDI_100–3000lx > 80% in Figure 7b) but do not exceed the maximum ASE_1000,250 of 10% (Figure 7c). Meanwhile, the aUDI_250–750lx at 90° has the highest value compared to those at the remaining orientations.

In addition, the maximum objective Y, which is considered as the optimum design scenario, is discovered at z = 2.7 m for d = 1.0 m until 2.5 m at each of the building orientations (0°, 45°, 90°, and 135°). Except for the orientation of 0°, the increment of Y values from d = 1.0 m until d = 2.5 m is relatively linear (Figure 6h and Figure 7d,h). Even though the orientation of 0° has a non-linear trend (Figure 6d), it is suggested that longer shading depth still corresponds to greater Y values, as is the case with the other orientations.

3.3. Visual Comfort

This study evaluates the minimum and maximum objective Y values in all orientations in terms of DGP. The minimum objective Y value always occurs at d = 1.0 m with z = 3.5 m. Meanwhile, the maximum objective Y occurs at d = 2.5 with z = 2.7 m in all orientations. Those conditions are defined as the worst- and the best-case scenarios in this study.

At the orientation of 0° (

Table 4. DGP values at orientations of 0° and 45° at 09:00 and 15:00 h on 20 March, 21 June, and 21 December *.

Dates	Hours	Orientation 0°		Orientation 45°
		Min Y	Max Y	Min Y	Max Y
20 March	09:00	DGP: 0.24	DGP: 0.23	DGP: 0.23	DGP: 0.22
	15:00	DGP: 0.24	DGP: 0.23	DGP: 0.24	DGP: 0.23
21 June	09:00	DGP: 0.31	DGP: 0.28	DGP: 0.28	DGP: 0.25
	15:00	DGP: 0.28	DGP: 0.26	DGP: 0.26	DGP: 0.24
21 December	09:00	DGP: 0.23	DGP: 0.23	DGP: 0.23	DGP: 0.22
	15:00	DGP: 0.26	DGP: 0.25	DGP: 0.29	DGP: 0.26

* Note: Color scale unit is in cd/m².

), almost all DGP values for the minimum objective Y fall under the disturbing–intolerable glare category, except on 21 December 09:00 h (DGP: 0.23). With maximum objective Y values, the DGP values only slightly improve for some evaluation hours. The improvement is shown with the shift from the disturbing–intolerable category to the imperceptible–perceptible one, which occurs on 20 March at 09:00 and 15:00 h and on 21 December at 09:00 h. The rest remain in the same category, even though there is slight reduction in the DGP values.

At the angled orientations of 45° and 135° (Table 4 and

Table 5. DGP values at orientations of 90° and 135° at 09:00 and 15:00 h on 20 March, 21 June, and 21 December *.

Dates	Hours	Orientation 90°		Orientation 135°
		Min Y	Max Y	Min Y	Max Y
20 March	09:00	DGP: 0.23	DGP: 0.22	DGP: 0.23	DGP: 0.22
	15:00	DGP: 0.24	DGP: 0.23	DGP: 0.24	DGP: 0.23
21 June	09:00	DGP: 0.25	DGP: 0.23	DGP: 0.25	DGP: 0.23
	15:00	DGP: 0.27	DGP: 0.25	DGP: 0.28	DGP: 0.25
21 December	09:00	DGP: 0.23	DGP: 0.22	DGP: 0.23	DGP: 0.22
	15:00	DGP: 0.28	DGP: 0.26	DGP: 0.26	DGP: 0.24

* Note: Color scale unit is in cd/m².

), most of the scenarios associated with the minimum objective Y values fall under the disturbing–intolerable category, except on 20 March and 21 December in the morning. Meanwhile, for the maximum objective Y values, at the orientation of 45°, the scenarios on 20 March at 09:00 and 15:00 h and 21 December at 09:00 h have been improved to yield DGP values within the imperceptible–perceptible category. The rest remain in the disturbing–intolerable category. At the orientation of 135°, more design scenarios have been improved to the imperceptible–perceptible category. Only on 21 June and 21 December int the afternoon did the resulting DGP remain in the disturbing–intolerable category.

At the orientation of 90° (Table 5), for the minimum objective Y the situation is similar to the orientations of 45° and 135°. Meanwhile, for the maximum objective Y, DGP in most of the design scenarios falls under the imperceptible–perceptible category. Only on 21 June and 21 December in the afternoon did the DGP values fall under the disturbing–intolerable category.

3.4. Sensitivity and Uncertainty Analyses

Figure 8 displays the SRC values for d’ and z’ with respect to the observed daylight metrics and the objective function Y at all orientations. It is found that shading depth (d’) positively influences all daylight metrics and the objective function Y with SRC > 0.50. Meanwhile, shading depth negatively influences ASE_1000,250, which is expected since deeper shading correlates to a lesser amount of direct sunlight penetration. The opposite phenomenon occurs for shading elevation (z’) because the higher the shading device, the less effective it is in reducing direct sunlight. Shading elevation is found to be more influential on Y and aUDI_250–750lx at all orientations.

Moreover, shading depth is slightly more influential on aUDI_100–3000lx in a positive direction (SRC ≥ 0.60). However, shading elevation is more influential than shading depth on ASE_1000,250, except at the orientation of 90° where ASE_1000,250 is relatively unchanged regardless of input variables (R² = 0.04, SRC for d’ = –0.15, SRC for z’ = 0.14). Meanwhile, aUDI_250–750, aUDI_100–3000, and Y have R² = 0.88, 0.62, and 0.96, respectively, where the SRC values are depicted in Figure 8. The low R² of ASE_1000,250 at the orientation of 90° is because the openings in that case face north and south, where the annual direct sunlight contribution is at a minimum. Meanwhile, for inter-orientation comparison, the R² for ASE_1000,250 at orientations 0°, 45°, and 135° is 0.96, 0.57, and 0.60, respectively, suggesting that direct sunlight contribution in those orientations is still noticeable.

In terms of uncertainty,

Table 6. Coefficient of variation (CV) values for each daylight metric at each orientation.

Metric Orientation	0°	45°	90°	135°
aUDI250–750lx	0.77	0.71	0.59	0.65
aUDI100–3000lx	0.09	0.06	0.03	0.06
ASE1000,250	0.58	1.89	11.19	1.83
sDA300/50%	0.00	0.00	0.00	0.00
Y	0.23	0.10	0.04	0.10

shows that the sDA_300/50% results are 100% in all scenarios at all orientations. The metric thus has the lowest uncertainty since the value from this metric is unchanged (CV = 0.00) in any case. aUDI_100–3000lx is the metric with the next lowest uncertainty (CV ≤ 0.09) at all orientations. Furthermore, the objective function Y has low uncertainty (CV ≤ 0.10) in most orientations, except at 0° (CV = 0.23). This situation can also be observed from Figure 6d, which suggests that Y has widely diverged at all elevations. The most extreme uncertainty occurs for ASE_1000,250 at the orientation of 90°. This is because, at d = 1.0 m with z = 3.4 m and z = 3.5 m, non-zero values are obtained, whereas the remaining scenarios return zero values (Figure 7c). Consequently, the standard deviation becomes large, which in turn yields a high CV value at the orientation of 90° (CV = 11.19).

This finding suggests that aUDI_250–750lx and ASE_1000,250 are sensitive metrics with regard to shading depth and elevation. Since the risk of direct sunlight exposure in the tropics is consistently high, design considerations of horizontal external shading devices in tropical school classrooms should be performed carefully, particularly in relation to shading geometry.

For the maximum Y values at z = 2.7 m for d = 1.0 m until 2.5 m, the CV values are relatively low (<0.10) for orientations 45°, 90°, and 135°. This means that Y is insensitive to changes of input variables, which in this case are shading depth and elevation. However, at the orientation of 0°, the CV value for the optimum design scenario is high (>0.10). The CV values for optimum design scenarios are tabulated in

Table 7. CV values for the maximum Y values at every orientation.

Metric Orientation	0°	45°	90°	135°
Y	0.11	0.04	0.03	0.03

.

To further observe the influence of each daylight metric on the objective function Y, the relationship of each metric is depicted in Figure 6a–d, with linear regression being applied to each model. At the orientation of 0°, ASE_1000,250 has R² = 1.00, meaning that it is perfectly linear with respect to Y (Figure 9a). This is reasonable since the bilateral openings in the classrooms face east and west, so the contribution of direct sunlight is expected to be large enough to influence Y. It can also be inferred that when choosing for the depth and elevation of external shading devices at an orientation of 0°, the primary consideration is to reduce the risk of having direct sunlight, i.e., minimizing ASE_1000,250. As shown in Figure 6c, scenarios with d > 2.0 m and z = 2.7 m or 2.8 m are the most recommended solution to reduce excessive sunlight exposure in tropical classrooms with bilateral openings.

At orientations of 45° and 135°, the relationships between Y and daylight metrics are somewhat identical to each other. aUDI_100–3000lx is found as the most linearly related metric to y (Figure 6b,d, R² = 0.98), whereas Y is also highly sensitive to the change of aUDI_100–3000lx, as shown by the steep gradients which are also reported in Table 6. This particular metric plays a crucial role in defining overall daylight performance at these orientations.

Meanwhile, at the orientation of 90°, aUDI_250–750lx is the most linearly related metric to Y (Figure 9c). Overall, the changes in Y values are smaller compared to the cases of other orientations, as previously indicated by the low uncertainty (CV = 0.04) in Table 6. Notice that, as previously explained, ASE_1000,250 at this orientation has the lowest R² because the metric values are always zero, except at d = 1.0 m, z = 3.4 m (ASE_1000,250 = 0.5%), and d = 1.0 m, z = 3.5 m (ASE_1000,250 = 6.7%). This indicates that, in general, for north–south bilateral openings the risk of having excessive sunlight in the classroom is very small, so long as the shading depth is not smaller than 1.0 m and the elevation is not higher than 3.5 m.

In addition, based on Figure 9, one can obtain the linear regression model Y = Ax + B, where x is the relevant daylight metric in each relationship. The A and B coefficients are reported in

Table 8. The A and B coefficients of linear regression models between Y and each daylight metric at each orientation.

Orientation	aUDI250–750lx		aUDI100–3000lx		ASE1000,250
	A	B	A	B	A	B
0°	7.66	112.8	4.26	–228.0	–1.45	213.5
45°	2.78	173.9	3.71	–151.4	–1.66	205.0
90°	1.30	194.3	2.26	–12.4	–4.51	208.3
135°	2.81	173.7	3.61	–143.0	–1.69	205.7

, where a greater A coefficient indicates a steeper gradient.

It is observed that for aUDI_250–750lx and aUDI_100–3000lx, the steepest gradient (7.66 and 4.26) occurs at the orientation of 0°, as also evident from Figure 9. Meanwhile, the steepest gradient for ASE_1000,250 is −4.51, found at the orientation of 90°, where the bilateral openings face north and south. However, as depicted in Figure 9, R² > 0.95 is only found for aUDI_100–3000lx at orientations of 45° and 135°, and for ASE_1000,250 at the orientation of 0°. It shows that aUDI_100–3000lx is more influential, compared to ASE_1000,250, in defining the objective function Y. In other words, Y is more sensitive to the change of aUDI_100–3000lx at the described orientations. Moreover, as suggested in Figure 9, the relation between Y and aUDI_100–3000lx at orientations of 45° and 135° is identical, and Table 8 confirms that their gradients only differ by 0.1.

4. Discussion

This study focuses on the interaction between shading depth and elevation of external shading devices in tropical school classrooms, which is yet to be discussed in detail in the literature. Ingenuity from tropical vernacular architecture has nonetheless indicated the importance of horizontal shading devices [2] to mitigate the risk of sunlight exposure and overheating in indoor spaces. Previous studies have suggested that geometrical parameters, particularly the depth and elevation of the shading devices, are critical in determining optimum daylight performance in existing school classrooms in Indonesia [9,10]. However, the interaction of shading depth and elevation in determining daylight and direct sunlight metrics is not fully understood. This study has demonstrated, through annual daylight simulation, that choosing the appropriate shading geometry is indeed crucial for enhancing daylight performance in modern tropical classrooms with bilateral openings.

This study has also revealed that the interaction between shading depth and its elevation is unique, depending on building orientations and the metrics being evaluated. At the orientation of 0° where the bilateral openings face east and west, extending the shading depth beyond 1.7 m (Figure 6b) at any elevation does not have significant impact on aUDI_100–3000lx, since all design scenarios have satisfied the minimum target of 80%. At this orientation, higher shading elevations yield insignificant impact on aUDI_250–750lx (Figure 6a), even when the shading depth is extended farther. Furthermore, at lower elevations (2.7 m and 2.8 m), extending shading depth beyond 2.0 m is unnecessary, as it also yields no impact on ASE_1000,250 (Figure 6c). However, when all metrics are combined into the objective function Y, a longer shading depth (d > 2.0 m) is still recommended at all elevations (Figure 6d). This is also indicated by the high CV value (0.23) at the orientation of 0°, where alteration in the input parameters shall significantly affect the objective function Y.

At orientations of 45° and 135°, all design scenarios have satisfied aUDI_100–3000lx > 80%, except the scenarios with the shortest depth and the highest elevation (Figure 3f and Figure 4f). To satisfy the target for aUDI_100–3000lx, it is not recommended to have a shading depth greater than 1.2 m because the daylight illuminance would most of the time be within the range of 100~3000 lux. Meanwhile, with respect to ASE_1000,250 (Figure 3g and Figure 4g), even at the highest shading elevation it is not recommended to have a shading depth greater than 1.7 m, since the metric is already lower than 10%. With respect to the objective function Y, as shading depth becomes greater, the Y values tends to converge to a certain final value. This also suggests that a shading depth larger than 2.0 m is not recommended.

At the orientation of 90°, all design scenarios have satisfied aUDI_100–3000lx > 80% and ASE_1000,250 < 10% (Figure 7b,c). At any elevation, the objective function Y values converge to a final value (Figure 7d), which is indicated by low uncertainty (CV = 0.04). Since the bilateral openings in this case face north and south, the risk of excessive direct sunlight is low, as shown in Figure 7c. The aUDI_250–750lx metric has the highest value, indicated by steep gradients at each elevation (Figure 7a). This finding shows the presence of more diffuse daylight illuminance within the range of 250 to 750 lux.

Furthermore, this study also discovers that shading depth greater than 2.0 m is not recommended as it has already satisfied the daylight metrics criteria with shading depth < 2.0 m, except at the orientation of 0° where the bilateral openings face east and west. This knowledge is essential in the design process toward more effective and cost-efficient shading devices. Next, shading elevation is found to have greater influence on daylight metrics compared to shading depth (Figure 8). This finding is significant since the majority of past studies did not include shading elevation as one of the noteworthy design features for tropical daylighting [15,16,20,22]. Finally, the contribution of each performance criterion is somewhat different in defining the objective function Y in each orientation. Attention should be paid carefully to the most influential metric at each orientation, as suggested in Figure 9.

Lastly, even in the case where the design scenario has met the daylight metrics requirement, the resulting visual comfort is not necessarily acceptable. This study has shown that there may be an increase of DGP at the observer point. However, even with the maximum Y value, most of the design scenarios in this study still yield DGP values that fall under the imperceptible–perceptible category. Several design scenarios also fall under the disturbing–intolerable category, for example at the orientation of 0° in the morning and afternoon on 21 June and in the afternoon on 21 December. Consequently, additional design features, such as curtains or blinds [47], should be installed on the internal side of the opening.

5. Conclusions

This study has investigated the interaction between the depth and elevation of horizontal external shading devices in tropical school classrooms that are equipped with bilateral openings with respect to indoor daylight performance. It has been revealed that the device’s elevation influences all of the observed daylight metrics. In addition, shading depth greater than 2.0 m is somewhat ineffective, as all design scenarios with fixed input parameters as shown have satisfied most of the performance criteria. Among the daylight metrics, sDA_300/50% has the lowest uncertainty (CV < 0.1), followed by the spatial average of UDI_100–3000lx and the combined objective function Y. Other metrics have higher uncertainty, so careful attention is required when choosing the depth and elevation of horizontal external shading devices, particularly when the observed metric has high uncertainty. The most influential daylight metric for each building orientation is also unique.

While most design scenarios in this study have met daylight performance criteria, visual comfort criteria may not be satisfied at all times. Even with the highest objective Y value, high DGP values are still present at some critical times of the year, so that a combination of curtain or blind installation is recommended. Overall, this study has contributed to the knowledge enhancement of designing passive strategies for daylighting in tropical buildings with bilateral openings.

The findings of this study are crucial for passive design practice, particularly in the tropical region. The outcomes can be implemented by architects or building designers to create an effective design solution considering shading device depth and their elevation for daylight-friendly school classrooms. As has been discussed, the optimum design solution is different for each orientation and daylight metric. Therefore, careful attention is required in order to conform with daylight and visual comfort criteria.

Furthermore, the design solution provided in this study is limited to daylight and visual comfort performance metrics. The case is also limited to classrooms with symmetrical, bilateral openings located in Lhokseumawe, which is a coastal tropical city. In the future, asymmetrical, bilateral opening typology should also be considered and optimized due to its potential complexity. In addition, tropical cities at higher altitude could also be considered due to the possibility of having greater solar radiation.