**2. Methodology**

In order to perform an accurate daylight analysis, in Section 2.1, state-of-the-art metrics are presented and commented to show their peculiarities and provide a rationale for their use. Then, in Section 2.2, the building and the experimental campaign are described; finally, the development of a model that is used for running daylight simulations is described in Section 2.3. The simulations were run for both the current configuration of the building and for a series of proposed interventions that are aimed to improve daylight fruition.

### *2.1. Static and Dynamic Metrics: Which Ones do Fit the Purpose?*

As introduced in Section 1 and reported in ref. [9], a number of static and dynamic metrics have been employed in previous studies concerning daylight in schools. In this research, we chose to use both traditional 'static' and more recent 'dynamic' metrics to highlight the differences in terms of applicability and derived outcomes. Furthermore, standard metrics, such as the average Daylight Factor (*aDF*) and the Uniformity Ratio (*UR*) have been considered to inform about daylight distribution under the worst-case condition of an overcast sky. These static metrics are also prescribed by Italian regulations, which set the achievement of minimum thresholds for compliance purposes. The mathematical formulation for these metrics is reported in Equations (1)–(3):

$$DF = \frac{E\_i}{E\_o} \tag{1}$$

$$aDF = \frac{T \cdot \mathcal{W} \cdot \theta \cdot M}{A \cdot (1 - R^2)} \tag{2}$$

$$LIR = \frac{E\_{\rm min}}{E\_{\rm avg}} \tag{3}$$

The first equation represents the classical definition of punctual Daylight Factor, expressed as the percent ratio of the indoor illuminance value *Ei* to the outdoor horizontal illuminance value *Eo* under a standard CIE overcast sky luminance distribution. The second equation expresses the spatial average *DF*, according to the simplified formulation proposed by Crisp and Littlefair [3] for a room of total enclosing surface area *A* (opaque and transparent), showing an average reflectance *R* and equipped with windows of effective transmittance *T* and area *W*. In Equation 2, *θ* is the angle subtended by the visible sky from the centre of a window in a vertical plane and *M* is the maintenance factor.

The Uniformity Ratio (*UR*) is instead defined as the ratio of the indoor minimum illuminance value *Emin* to the average value *Eavg*, under the same overcast sky conditions that are used for the *DF* calculations.

The CBDM metrics adopted are the Useful Daylight Illuminance (*UDI*), the spatial Daylight Autonomy (*sDA*), and the Annual Sunlight Exposure (*ASE*), all of which are defined with reference to an ideal horizontal grid of sensor points. The rationale behind their selection is that all of them are complimentary to each other and help in depicting both the temporal and spatial distribution of daylight in a comprehensive way. In fact, the *UDI* metrics informs about the percent of occupied hours in which the illuminance value in a specific point of the room falls within a specific illuminance bin; in many cases, the *UDI* is also determined in relation to the spatial mean illuminance value. In this study, three different bins are considered for the illuminance: lower than 100 lux (insufficient daylight), between 100 and 2000 lux (appropriate daylight) and higher than 2000 lux (potential glare), as suggested in the original formulation of Nabil and Mardaljevic [5]. It is worth mentioning that a later study suggested increasing the upper threshold to 3000 lux [14]. On the other hand, the *sDA* metrics informs on the percentage of space achieving a target illuminance value for a specified amount of time in a year. According to the definitions that are given in the IES-LM-83-12 Standard [4], the target illuminance value for *sDA* is 300 lux and the amount of time above which this value should be retained is 50% of the occupied hours.

Finally, the *ASE* metrics expresses the percentage of space for which direct sunlight only (i.e., without considering any internal and exterior reflections and blinds operation) exceeds a threshold value for a fixed number of hours. The illuminance threshold is set to 1000 lux, while the number of annualhoursexceedingthisvalueshouldbelowerthan250,otherwisediscomfortglaremayoccur.

 The mathematical formulation adopted for these metrics in this paper is reported in Equations (4) and (5):

$$LIDI = \frac{\sum\_{i=1}^{n} t\_i x\_i}{\sum\_{i=1}^{n} t\_i} \quad \text{with } x\_i = \begin{cases} 1 \text{ if } E\_{\text{avg}} \text{ is within the bin} \\ 0 \text{ if } E\_{\text{avg}} \text{ is outside the bin} \end{cases} \tag{4}$$

$$sDA = \frac{\sum\_{i=1}^{n} \sum\_{j=1}^{p} x\_{i,j}}{\sum\_{j=1}^{p} p\_j \cdot \sum\_{i=1}^{n} t\_i} \quad \text{with } x\_{i,j} = \begin{cases} 1 \text{ if } E\_{i,j} \ge E\_{threshold} \\\ 0 \text{ if } E\_{i,j} < E\_{threshold} \end{cases} \tag{5}$$

In Equation (4), the spatial average of the illuminance values is considered; here, *ti* is the *i*-th occupied hour and *xi* is a binary function that, according to the mean illuminance recorded in the *i*-th occupied hour, is *xi* = 1 if this falls within the bin and *xi* = 0 otherwise. In Equation 5, *pj* is the generic *j*-th sensor node on the horizontal calculation grid, while the binary function becomes *xij* to account for a double summation over both the temporal and spatial domains being this metrics defined as a spatial average. The expression for the *ASE* metrics is the same as for the *sDA* metrics, with differences in the calculation method (i.e., only the direct sunlight contribution is considered for illuminance appraisal) and thresholds (see Table 1).

Finally, the risk of glare occurrence can be studied for some points of view by calculating the Daylight Glare Probability (*DGP*) index, which, according to Carlucci et al. and Pierson et al. [15,16], is one of the most robust approaches for glare assessment. In fact, it is able to account for both the saturation effect when the amount of light reaching the eye is too large, and the contrast effect when the contrast between the visual task and the field of view is too strong. These two aspects are considered in the vertical illuminance at eye level *Ev* and in the logarithmic term of Equation (6):

$$DGP = 5.87 \cdot 10^{-5} E\_{\upsilon} + 0.0918 \cdot \log \left[ 1 + \sum\_{i=1}^{n} \left( \frac{L\_{s,i}^{2} \cdot \omega\_{s,i}}{E\_{\upsilon}^{1.87} \cdot P\_{i}^{2}} \right) \right] + 0.16\tag{6}$$

Here, *Ls* is the luminance of the *i*-th glare source, *ωs* is the solid angle through which the glare source is seen, and *Pi* is the position index. It is important to note that, unlike the other metrics, the *DGP* reports the probability that a person is disturbed by glare in a scale of 20% to 80% [17], rather than a percent of time or space achieving a target value (see Table 1).

Because of current computational limitations, most of the available daylight simulation tools make use of a simplified version of the *DGP*, defined by Wienold in [18,19] and known as *DGPs*:

$$DGPs = 6.22 \cdot 10^{-8} E\_{\upsilon} + 0.184\tag{7}$$


**Table 1.** Thresholds and target values of the metrics employed \*.

\* Except for *aDF* and *UR*, all the metrics have been derived for office spaces.

Here, the influence of the single glare sources is neglected, an assumption that proved to provide reliable results in the absence of direct sunlight transmission and peak reflections through the façade in the observer's direction [19]. Within these validity constraints, four different bands are suggested for categorization purposes: imperceptible glare (*DGPs* < 0.35), perceptible glare (0.35 ≤ *DGPs* < 0.4), disturbing glare (0.4 ≤ *DGPs* < 0.45), and intolerable glare (*DGPs* ≥ 0.45) [18]. In this way, numerical outcomes can directly be used as for the other metrics, since they do not refer any more to a probability distribution. For all these reasons, *DGPs* is used as a glare metrics in this study.

### *2.2. Case Study Building and the Experimental Campaign*

The case study building is an elementary school located in Agira (Italy), a town that is characterized by the warm and sunny climate conditions of a typical Mediterranean island like Sicily. The building shows an E shape with the main façade and the two lateral wings hosting about 30 classrooms on two floors (rooms' height of 4.50 m, total floor area of around 4600 m2), while the central wing is separated from the lateral ones by means of two courtyards and hosts the gym (see Figure 1). A long corridor running along the north side gives access to the different rooms, all of which are side lit by two or more double-glazed PVC windows that are equipped with external plastic roller shades. This configuration is different for the gym, which has windows placed on all three external walls and has no shading provision.

**Figure 1.** Guglielmo Marconi elementary school in Agira. (**a**) External view of the main facade; and, (**b**) Aerial view.

For the sake of assessing the daylight availability inside the school, three rooms with different functions, orientations, and features have been surveyed: a typical classroom and a computer classroom facing south, and the gym with openings to the north, west, and east oriented facades (see Figure 2).

**Figure 2.** Axonometric views and pictures of the study rooms.

Unfortunately, by the time that the measurement campaign was carried out, the rooms on the west and east wings were not accessible. However, most of them are used for office purposes and not for teaching and learning activities.

Concerning the features of the selected rooms, the standard and computer classrooms have the same size (8.8 × 6.1 × 4.5 m3), three openable windows of 1.6 m<sup>2</sup> size each exposed due to south, placed at 1.2 m from the floor, and equipped with a 0.9 m<sup>2</sup> wide clerestory. The interior finishing layer of the walls and the ceiling is of cement plaster painted with a white colour, except for a light blue belt at 2 m from the floor, while the floor itself is made up of granite tiles. In order to keep a high level of detail for modelling, the existing furniture has been considered as well. The standard classroom is furnished with a traditional blackboard and an interactive whiteboard on one of the shortest sides of the room, a closet on the opposite side and a number of desks to accommodate the pupils and the teacher (see Figure 2). The computer classroom instead presents a series of desks with 25 LCD screens and cases.

The gym shows an open plan layout (18.5 × 9.8 × 4.5 m3), the walls and the ceiling have the same interior finishing layer of the standard and computer classrooms painted white but with a light pink coloured belt of 2 m height from the floor, which is finished with an anti-slip green coloured PVC layer. Ten windows (four on each of the long sides, two on the north oriented wall) that are placed at 2 m from the floor and equipped with a clerestory (for a resulting window area of 4.8 m2) bring daylight inside.

The visible reflectance of all indoor opaque surfaces has been calculated through Equation (8), under the hypothesis that the opaque finishing layers behave as a Lambertian diffuser (i.e., equal reflectance for all directions). The luminance *L* to be used in Equation (8) is the mean value from three spot luminance measurements, which are gathered through a MINOLTA LS 100 luminance meter (measurement range 0.01–50 kcdm−2, accuracy ± 0.2%). Similarly, the average of three illuminance measurements, collected through a MINOLTA T-10A lux meter (measurement range 0.01–300 klux, accuracy ± 3%), allows for assessing the illuminance value *E* in Equation (8).

The visible transmittance of the glazing is calculated at the center of the glass as the ratio of the recorded indoor vertical illuminance close to the windows (measured with the lux meter facing the glazing) to the outdoor vertical illuminance measured just outside the window with the lux meter facing outdoor, see Equation (9). The indoor and outdoor illuminance measurements have been taken simultaneously using tripods on 26 February under partly overcast sky conditions. The resulting values, which are used as an input to the software, are listed in Table 2.

$$
\rho = \frac{\pi L}{E} \tag{8}
$$

$$
\tau = \frac{E\_{indoor}}{E\_{outdoor}} \tag{9}
$$


**Table 2.** Optical properties of the surfaces in the selected rooms.

\* The transmittance is different despite the use of the same glazing type because of dirt deposition.

### *2.3. Daylighting Simulations and Calibration of the Model*

The appraisal of daylight conditions has been conducted by means of dynamic simulations in DIVA [22], a tool that makes use of the well-validated backward ray-tracing method that was provided by Radiance [23]. Given the similarities between the standard classroom and the computer room, the calibration of the model has been carried out only for the standard classroom and the gym, by fine-tuning the more relevant radiance parameters, namely ambient bounces –ab, ambient accuracy–aa, ambient sample–as, ambient divisions–ad, and ambient resolution–ar. The reader can ge<sup>t</sup> a thorough understanding of these and other Radiance parameters by consulting [24].

The rooms are supposed to be occupied all year long from 8AM to 1PM on Monday to Friday, except from half of June to half of September (summer break) and for some weeks during the winter break, for a total of 1122 hours per year.

The grid of sensor points has been placed on an ideal 'work plane' identified at desks height in the standard and computer classrooms and at 1 m from the floor in the gym. In both cases, the grid has an offset of 0.5 m from the walls, as suggested by the Italian legislation [20], which also prescribes the use of a calculation grid whose maximum spacing *p* is defined as a function of the maximum room size *d* through Equation (10):

$$p = 0.2 \cdot 5^{\log d} \tag{10}$$

This computation results in a square grid of 0.8 m spacing in the case of the standard and computer classrooms, and of 1.8 m spacing in the gym.

For the sake of brevity, the results of the model calibration are presented in Figure 3 for the standard classroom only in the form of gradient contour lines, where the measured illuminance values ((a) panel) are compared to those that are resulting from the simulations ((b) panel). From this picture, it is easy to appreciate how well the simulated illuminance values follow the measured values in terms of both magnitude (illuminance differences are always lower than 12%) and spatial distribution. In both cases, the half of the classroom that is farther from the windows keeps below 500 lux, while in the other half, the illuminance values frequently keep between 500 lux and 1000 lux. The main differences are detected close to the windows, where the effects of direct sunlight is more evident in the measurements. As an example, the measurements pointed out a spot of direct sunlight close to the middle window that is not detected by the simulations. The main reason for this discrepancy is the approximation of climate conditions due to the lack of weather time series for the town of Agira (LAT. 37◦39 N, LON. 14◦31 E, altitude of 824 m above the sea level). Indeed, the closest weather dataset available, provided in the Typical Meteorological Year format (TMY, [25]) and used in this research, is that pertaining to the city of Catania (LAT. 37◦30 N, LON. 15◦05 E, altitude of 7 m above the sea level).

**Figure 3.** Comparison between measured (**a**) and simulated (**b**) illuminance levels for the standard classroom on 26 February at 11:30 a.m.

Notwithstanding this limitation, similar results are obtained for the gym (a maximum punctual difference of around 23% is reached for the points furthest away from the windows) and the model can be considered to be calibrated. The resulting Radiance parameters are listed in Table 3. Here, it is worth to discuss how the coarser values that are chosen for the gym derive mainly from its geometry and the absence of any furniture to be modelled. More in detail, the number of ambient bounces -*ab* (that is to say the maximum number of bounces of the diffuse indirect component) have been set to two instead of four because the vast amount of glazed surfaces, as well as their orientation and the room size, make the diffuse component lower than the direct one. For the same reasons, the ambient resolution value -*ar* has been set to 128 (that expresses the maximum density of ambient values used for interpolation). These parameters, together with the use of ambient super-samples -*as* and ambient divisions *–ad* values of 256 and 512, respectively, (that are mainly used for reducing noise effects when rendering), help keeping the simulation time reasonable without significantly affecting the simulation outcomes.


