Simple mathematical models to link climate-based daylight metrics with daylight factor metrics and daylighting design implications

Abstract

Determination of interior daylight illuminance is the key step in daylighting schemes. Recently, climate-based daylight metrics (CBDMs) which considers the real climatic data for the location has been adopted to evaluate the dynamic daylight performance. However, the usual method to calculate the CBDMs is full scale computer simulations which are quite time demanding and designated skills are required. Architects and building practitioners prefer simple methods to assess daylight performance particularly during initial design process when various building schemes and concepts are being appraised. Daylight factor (DF) is the traditional daylight metric and it has a strong relationship with room parameters which can be simply modified to fit the design criteria. This paper puts forward a series of simple mathematical expressions to correlate the CBDMs with DF metrics (DFMs). The vertical outdoor illuminance at the window center point and the 49 interior points were simulated via the RADIANCE software. The results showed that there are strong correlations between these daylight metrics. The proposed approach would be useful to building professionals conducted in visual comfort, fenestration and daylighting design and evaluation in the preliminary design phase.

1. Introduction

Daylighting is an impressive and sustainable development strategy [1] and design concept is an important issue for the architecture and building research to enhance visual comfort, physical health and green building developments [[2], [3], [4]]. Appropriate daylight-linked lighting controls can substantially save the electric lighting expenditure in commercial buildings [5]. To predict the daylight performance of room spaces and set a scale for architects and building engineers to use when comparing the aspects of various daylighting schemes, there are a number of rules of thumb and design metrics [6]. Traditionally, the daylight factor metrics (DFMs) have been used in evaluating of the interior daylight performance for a long time [7]. Owing to the simplicity, DFMs can be computed by simple equations, figures or several calculation tools [8] which are commonly used among architects and building engineers for conceptual designs and studies [9]. It is adopted by various design manuals [10] for daylighting criteria and widely used in many countries [11]. One of the main strengths is that DFMs are well defined in terms of room variables such as the window area, glazing transmittance and the ratio of window surface to room surface area and the fraction of visible sky [12]. It means that DFMs (under a CIE overcast sky) can represent various room designs which can be conveniently adjusted to fit the daylighting norm. Regression models using input room variables for estimating DFMs are helpful to architects and building engineers at an early design stage [13]. However, DFMs are determined under traditional overcast sky conditions without direct sunlight. In this case, the sky luminance is symmetrical about the zenith and the amount of vertical daylight is identical for all directions. It implies that DFMs are independent of time, non-overcast skies and window orientation [14] which are not the cases especially under clear skies. Large DFMs indicate well daylit space with few artificial lighting demands only, while the visual discomfort such as daylight glare cannot be identified.

Recently, climate-based daylight metrics (CBDMs) containing daylight autonomy (DA) and useful daylight illuminance (UDI) are employed for providing further information on the different daylight conditions throughout the year [15,16]. Such daylighting metrics concern the hours of actual work and real weather conditions for a given place to provide a significant functional account of authentic daylighting environments, which are appropriate for evaluating luminous comfort [17], glare issues [18] and daylight-linked lighting control systems [19,20]. To obtain results that are close to potential actual daylight situations, lighting simulations are the appropriate approach [21]. Nowadays, the advances in computer tools are quite promising, which enable building practitioners to explore daylighting design methodologies more accurately and efficiently. Architects and designers are therefore should evolve towards and familiar with the use of such simulation tools. However, performing full-scale hour-by-hour computer simulation of an entire year is laborious and expensive, as it requires modelling a building geometry over a long simulation period and the use of lengthy input and post-processes. Similarly, parametric analyses for individual building projects are also a laborious undertaking [18,22,23], as the total number of illuminance values that need to be simulated could be close to 12 million [15]. Such sophisticated computational and simulation techniques are difficult for novice students [24] and some building professionals to use [25,26] and typically require expert management [27]. Currently, some studies initiated the link of CBDMs including DA and spatial daylight autonomy (sDA) with DFMs [28,29] but no simple mathematical models are recommended. Lo Verso et al. [22] developed a set of mathematical models to estimate various DA and energy demand for lighting in the early design stages. However, the large sets of input variables in terms of architectural features of building lead to quite complex mathematical expressions and uncertainties. Setting the working period for calculating CBDMs is a major challenge, as although a fixed daytime period may be appropriate for locations near the equator (with small daylength variations throughout a year), it is not appropriate for locations closer to the poles with large daylength differences between summer and winter, which makes it very difficult to set the working period for the latter locations.

Recently, more simple design tools such as simplified geometry maps, regression equations and charts developed through thorough examination using sophisticated simulation tools [[30], [31], [32]] and these are now adopted in the initial planning phase of a project, before overarching design schemes have been finalized. Subsequently, full-scale computer simulation can be conducted to obtain more detailed and accurate findings. This paper attempts to correlate various CBDMs with traditional DFMs (under the CIE Overcast Sky) for buildings facing different directions, where the CBDMs are generated by the simulation software RADIANCE. Ultimately, CBDM can properly be estimated by the room parameters which are simple and fast calculation tools, attaining appropriate architectural and daylighting designs. The performance of daylight-linked lighting controls and visual comfort in terms of daylight glare are evaluated, and the design implications are expounded.

2. Daylight metrics

The daylight metrics being studied here include the DFMs and the latest CBDMs. Generally, there are three forms of DFMs viz. point daylight factor (PDF), average daylight factor (ADF) and vertical daylight factor (VDF). The CBDMs mainly consist of daylight autonomy (DA), continuous daylight autonomy (cDA), maximum daylight autonomy (mDA), spatial daylight autonomy (sDA) and useful daylight illuminance (UDI). This section reviews these daylight metrics covering their main features, merits and limitations.

2.1. Daylight autonomy

DA was originally proposed by the Association Suisse des Electricians in 1989. Later, Reinhart and Walkenhorst redefined the total natural lighting percentage for DA as a dynamic lighting evaluation indicator [33]. DA is expressed as the percentage of daylight hours for a given point in a space above a specified illumination level in a whole year. It takes geographic location and specific weather information into account on an annual basis. Users are free to set the threshold above which DA is calculated. It means that DA could be used to estimate the lighting energy saving under standard daylight-linked on-off lighting controls [34]. However, DA cannot reflect excessive indoor daylight, mDA was proposed to solve this issue [35]. The threshold of mDA is usually ten times the design illuminance of a space. This upper threshold criterion is appropriate to measure the appearance of direct sunlight or other glare conditions. The mDA could provide an indication of the frequency and location for large illuminance contrasts for a given space [36].

Reinhart et al. proposed cDA which is the basic modification of DA [37]. The cDA represents the percentage of occupied hours during the year when the illuminance of a particular point in the room is at or above the target threshold level, with proportional credit given for cases if its daylight contribution meets this level partially. For example, if the value of daylight illuminance is 250 lux at an interior grid point, DA_{500 lux} would give it 0 point whereas cDA_{500 lux} would give it 250/500 = 0.5 point. This daylight metric will help estimate the energy savings by dimming or multistage switching controls. DA and cDA, however, can only evaluate the performance of indoor daylighting environment for a particular point rather than an area. The sDA describes the amount of sufficient daylight received by a given space [38]. Specifically, it describes the percentage of floor area that exceeds a specified illuminance level (e.g., 500 lux) for a standard percentage (e.g., 75%) of the annual occupied hours (e.g., from 8 a.m. to 4 p.m.). sDA_500/75% represents the specified illuminance of 500 lux and the standard percentage of 75%. The Illuminating Engineering Society (IES) guideline recommends two different levels for sDA: the first level is preferred daylight sufficiency, if more than 75% of the analysis area meet the above criteria; the second level is nominally accepted daylight sufficiency, if 55% or more of the analysis area meet the above criteria [27]. The Leadership in Energy and Environmental Design (LEED) awards Credit Points 1, 2 and 3 to the sDA_300/50% for the regularly occupied floor area at least 40%, 55% and 75%, respectively [39]. The sDA can well reflect the indoor daylight uniformity, while it cannot evaluate the visual glare issue. Uribe et al. used sDA to evaluate those louvers with 120 mm spacing and 5%–20% perforations could meet sDA_300/50% between 96% and 100% [40]. DA, mDA, cDA and sDA are not limited to the overcast sky conditions. If the dynamic variations of sky brightness distribution throughout the year is considered, the natural lighting quality of indoor space in a year can be evaluated more accurately.

2.2. Useful daylight illuminance

Nabil and Mardaljevic proposed the UDI in 2005 [41]. For each point in the room, the UDI indicates the percentage of occurrence for a point with its daylight illuminance higher than 100 lux and lower than 2000 lux. Daylight illuminance below the lower threshold (e.g., 100 lux) gives few useful assistances in the perception of the visual environment or performance of visual tasks. Daylight illuminance higher than the maximum threshold (e.g., 2000 lux) may produce visual or thermal discomfort. The UDI scheme reflects the distributions of annual daylight level by determining the occurrence of daylight illuminances within the upper and lower thresholds, indicating whether the indoor daylight is insufficient, desirable, tolerable, and excessive. However, there is some debate about the choice of 2000 lux as an ‘upper threshold’ above which daylight is not required due to potential glare or overheating. For example, LEED V4.1 option 2 adopts the illuminance levels between 300 lux and 3000 lux. Lu et al. [42] used the area-weighted average UDI to prove that the optimized curved facades can improve the daylight efficiency of office building, and Hong et al. used UDI _{300–3000 lux} to evaluate the performance of thermos-chromic glazing window. They reported that UDI _{300–3000lux} for single- and double-glazed windows were 70.9% and 71.9%, respectively [43].

2.3. Daylight factor metrics

DFs can be logically estimated through simple calculation aids [9] without the use of the actual daylight data. PDF is defined as the ratio of daylight illuminance at a point on the indoor working plane to simultaneous outdoor illuminance on a horizontal plane (E_HD) from an unobstructed hemisphere of CIE standard overcast sky. It has been reported that PDF can be adopted in estimating of energy reduction in electric lighting [44]. ADF represents the daylight sufficiency for interior area rather any specific point [45]. Compared with PDF, ADF is closely related to the room parameters and do not require detailed data. It could be used in the sketch design stage [46]. Both PDF and ADF are the summation of sky component, externally reflected component and internally reflected component. VDF consists of light from the sky and daylight reflected from opposite buildings and the ground to the vertical surface. It has been appropriately used to quantify the daylighting performance for compact city regions with vast external obstructions [47], such as Hong Kong [48]. As daylight passes into internal space via window openings, VDF is the sum of the two daylight configuration factors above (C) and below (D) horizon of the daylight flux incident on the mid-height of windowpane.

3. Model and methodology

The data for the correlations can be generated using computer simulation techniques with examples. The model under study is a simple rectangular room of 6 m (m) x 5.4 m (L) x 3 m (H) containing a vertical rectangular window that represents a common case. The room model conforms to the current Standard for design of office building [49]. The reflectance coefficients of ceiling, wall and floor are 0.6, 0.5 and 0.2, respectively. Totally, 49 reference points were adopted to demonstrate the proposed approach. The spacing between the reference points was 0.75 m and the height was 0.8 m above the floor. Fig. 1 presents the layout of the room and the points for simulating the daylight illuminance. The vertical window was considered facing the four cardinal orientations (i.e., N, E, S and W). The window area and the visual transmittance were varied to form different situations for the 49 internal points (Table 1). Unobstructed skies were considered, and such sky conditions are appropriate for rooms at upper floors of high-rise buildings located in compacted areas or houses constructed in low-density cities. Ultimately, the outdoor illuminance transmitted via the center point of the window was also simulated for analysis. The room layout is a simple shoebox model which represents a typical open-plan office, and the findings can be generalisable to a range of applications during early design stage. Such simple ‘shoebox’ model rooms are commonly adopted to demonstrate and evaluate daylighting performance in terms of DFM and CBDM [27,30,31,37,41,46,50].

Fig. 1.

Room layout and reference points.

Table 1.

Window area and visual transmittance for the simulation.

Cases	Window area (m2)	Window transmittance
1	4	0.3
2	6	0.4
3	6	0.6
4	8	0.6
5	9	0.65

Direct sunlight and sky-diffuse illuminance measured from January to December 2004 in Hong Kong were used for the simulation task. The selected daytime for the study were between 8:00 and 16:00 in True Solar Time. The vertical outdoor illuminance at the window center point and the 49 interior points were simulated via the RADIANCE software that was developed by Lawrence Berkeley Laboratory [51]. RADIANCE is a well-established lighting simulation package employing backward ray-tracing technologies to examine high-ranking lighting regimes. It has been used by a number of researchers for predicting outdoor illuminances and solar radiation [52], indoor daylighting and evaluating lighting and daylighting technologies [53,54]. The primary advantage of RADIANCE is only a few limitations on geometry and the many materials may be simulated. In the present study, the RADIANCE version 5.3 running under a Windows 10 workstation was used for the analysis. The simulation accuracy of Radiance software settings are as follows: -ab = 3, -aa = 0.15, -ar = 64, -ad = 512 and -as = 64. The simulation steps and the program used in each calculation step are illustrated in Table 2.

Table 2.

Simulation steps and program.

Step	Command	Explanation
1	Genbox	Create the room model based on the parameters in Table 1 and Fig. 1.
2	Gensky	Define the 15 CIE Standard Sky Types based on Tregenza's model [55].
3	Oconv	Arguments all material and scene files
4	Rtrace	Trace specific rays to find out the illuminance.

4. Results analysis and design implications

The absolute daylight illuminance at all reference points were simulated by RADIANCE. Accordingly, DFMs were calculated by dividing the daylight illuminance to the corresponding E_HD under the CIE Sky 1 [56]. The VDF is calculated at the center point of the window, PDF is the value of 49 points inside the room and ADF is the mean value of the PDF of these 49 points. The DAs and UDI were obtained based on the year-round indoor daylight illuminance simulation. As DFMs are constants under traditional overcast sky condition, their correlations with UDI and DAs should be with respect to the CIE Sky 1. Therefore, DFMs can represent various room designs which are useful for building façade and daylighting design.

4.1. Correlations between dynamic daylight metrics and point daylight factor

The correlations between PDFs (under Sky 1) and DA at the indoor illuminance thresholds of 100, 300 and 500 lux when the window facades facing the four cardinal orientations (i.e., N, E, S and W) under unobstructed skies are shown in Fig. 2(a–f).

Fig. 2.

Correlation between point daylight factor and daylight autonomy for the three indoor illuminance thresholds (a) 100 lux; (b) 300 lux; (c) 500 lux facing north; (d) 500 lux facing east; (e) 500 lux facing south; (f) 500 lux facing west.

The figure can give a convenient way to demonstrate the DA under different PDFs. No substantial orientation effects can be observed for DA₁₀₀ and DA₃₀₀ since daylight illuminance below 500 lux which is mainly sky-diffuse component without direct sunlight for all orientations. Through regression analysis, the model equations correlating DA with PDF were developed and the findings including R² are expressed as Eqs. (1), (2), (3), (4), (5), (6):

$${DA}_{100lux}=101.7-14.54\times {PDF}^{-1.13}{R}^2=0.98$$ (1)

$${DA}_{300lux}=127.2-76.56\times {PDF}^{-0.48}{R}^2=0.92$$ (2)

$$D{A}_{500luxN}=\{\begin{array}{c}0,PDF<0.8\%\\ 120.9-117.8\times PD{F}^{-0.66},PDF\ge 0.8\%R^2=0.96\end{array}$$ (3)

$${DA}_{500luxE}=414.3-387.3\times {PDF}^{-0.08}{R}^2=0.93$$ (4)

$${DA}_{500luxS}=194.3-159.6\times {PDF}^{-0.22}{R}^2=0.95$$ (5)

$${DA}_{500luxW}=247.7-216.2\times {PDF}^{-0.16}{R}^2=0.92$$ (6)

As the indoor illuminance thresholds are usually no more than 500 lux, large DAs can be resulted at relatively low PDFs. These plots also indicate diminishing return for PDF up to 2% or more. As the illuminance threshold increases from 100 to 500 lux, the lowest DA_{100 lux} of 22% reduces to 0% for DA_{500 lux}. It means that at very small DF, 22% of the daytime period for the given point can take a daylight level higher than 100 lux, but it can never be more than 500 lux. Initially, DA_{100 lux} rises rapidly to around 90% with PDF of about 1.2%. Afterwards, DA_{100 lux} increases slightly and becomes the full value of 100% when PDF > 6%. Similar patterns can be found for DA_{300 lux} which reach 90% with the corresponding PDF at 4.2%. With the increase of PDF, DA_{500 lux} is significantly different in four orientations. The DA_{500 lux} values in the south and west are larger than those in the north and east when the PDF is in the range between 1% and 4%. The R²s are more than 0.92, showing that over 92% variations of DA can be explained by the changes in PDF and the correlations are considered very strong. It indicates that the mathematical models can provide a reliable way to estimate the DA from the PDF.

Similarly, the cDA data were determined and correlated with the corresponding PDF under Sky 1. Fig. 3(a–f) presents such correlations between PDF and cDA for the interior illuminance thresholds of (a) 100, (b) 300 and (c) 500 lux.

Fig. 3.

Correlation between point daylight factor and continuous daylight autonomy for the three indoor illuminance thresholds (a) 100 lux; (b) 300 lux; (c) 500 lux facing north; (d) 500 lux facing east; (e) 500 lux facing south; (f) 500 lux facing west.

Since daylight illuminance below the threshold will also be counted as a credit in the cDA which is generally greater than DA for all cases. Using regression analysis, the correlation equations and regression statistics were computed and are depicted in Eqs. (7), (8), (9), (10), (11), (12).

$${cDA}_{100lux}=100.8-5.72\times {PDF}^{-1.23}{R}^2=0.98$$ (7)

$${cDA}_{300lux}=105.9-29.04\times {PDF}^{-0.79}{R}^2=0.96$$ (8)

$${cDA}_{500luxN}=129-73.09\times {PDF}^{-0.42}{R}^2=0.95$$ (9)

$${cDA}_{500luxE}=121.5-61.54\times {PDF}^{-0.46}{R}^2=0.96$$ (10)

$${cDA}_{500luxS}=114.2-49.27\times {PDF}^{-0.55}{R}^2=0.98$$ (11)

$${cDA}_{500luxW}=116.7-53.16\times {PDF}^{-0.52}{R}^2=0.96$$ (12)

The R² is 0.96 or more, indicating a strong relation that correlates cDA to the PDF accurately. The PDFs are 0.6%, 1.9% and 3.4% respectively for cDA_{100 lux}, cDA_{300 lux} and cDA_{500 lux} at around 90%.

Daylighting is an important and useful strategy for reducing the electricity consumption by switching-off the light fittings when the daylight lux levels are appropriate. In a well day-lit space when the available daylight is far larger than the required level, daylighting can provide substantial energy savings when proper daylight-linked lighting controls are adopted. However, such circumstance may cause glare and brings about visual discomfort. DA and cDA are quite appropriate for predicting the electric lighting saving due to standard daylight-linked switching and dimming lighting controls, respectively. Assuming that there are three rows of luminaries parallel to the window façade are separately monitored (neglect the electric lighting) and Points 11, 25 and 39 on Fig. 1 are the locations to detect the illuminance for Rows 1, 2 and 3, respectively. Accordingly, the PDFs, annual daylight illuminance, DA and cDA for the three points representing the 3 rows were determined. If the installed lighting power density was 20 W/m², the annual electric lighting energy consumption (E) for switching or dimming controls can be calculated by Eq. (13):

$$E=\left(1-DAorcDA\right)\times A\times LPD\times h$$ (13)

where A is the room area, m²; LPD is the lighting power density; h is the annual work hours.

Table 3 summaries the percentage of electric lighting consumption under standard on-off and dimming controls for the three rows based on Case 1. As cDA is always larger than DA, dimming control is more efficient than the simple on-off control. The annual electric lighting energy consumption for switching controls can range from 71 to 329 kWh under various numbers of lamp fittings monitored and window orientations. The outcomes can give a convenient alternative to predict the electric lighting savings. It is envisaged that more energy reduction can be achieved due to less of heat dissipation from artificial light and thus a smaller internal cooling load.

Table 3.

Electric lighting energy consumption under different daylighting controls at three zones with the target illuminance = 500 lux.

Daylighting controls (Case 1)		Row 1: sensor location Point 11				Row 2: sensor location Point 25				Row 3: sensor location Point 39
		N	E	S	W	N	E	S	W	N	E	S	W
Standard switching control	E (simulated DA)	108	96	71	76	328	284	262	271	329	327	326	323
	Eqs. (3), (4), (5), (6) (modelled DA)	93	113	83	90	313	238	212	223	329	329	328	329
	Diff.	14	16	11	14	15	46	50	48	0	3	2	7
Dimming control	E (simulated cDA)	39	37	29	29	184	154	132	139	260	240	226	228
	Eqs. (9), (10), (11), (12) (modelled cDA)	43	40	32	33	143	130	113	118	254	234	217	222
	Diff.	−4	−2	−3	−4	41	25	18	21	6	6	9	6

Note: E is annual electric lighting consumption (kWh); Diff. is the difference between modelled and simulated annual electric lighting consumption.

If PDF under CIE Sky 1 is known, DA_{500 lux} and cDA_{500 lux} can be calculated by Eqs. (3), (4), (5), (6), (9), (9), (10), (11), (12). It is easily to get the electric lighting energy usage under different daylight-linked lighting controls. The results of electric lighting consumption are shown in Table 3. The difference at three points is not more than 50 kWh indicating that the proposed model can be used in the initial design of building schemes.

The correlation between mDA and PDF was made and the plots of mDA _{2000 lux} against PDF facing the four cardinal orientations (i.e., N, E, S and W) are presented in Fig. 4(a–d). The mDA indicates a visual glare issue with an intolerable indoor daylighting condition. The graph also gives prominence to the differences that appear with orientation. When PDFs are less than 1.1%, the interior daylight illuminance is less than 2000 lux (mDA _{2000 lux} = 0) for the room facing east, south and west orientations. For north facing room, the PDFs can be lower than 4%. Also, at the mDA _2000lux of 10%, the PDFs are around 5.6%, 2.9%, 2.6% and 2.4% for north, east, south and west orientations, respectively. Regression analysis was carried out for mDA _{2000 lux} and PDF and the regression equations including correlation coefficients for the four principal directions are presented in Eqs. (14), (15), (16), (17):

$${mDA}_N=\{\begin{array}{c}0,PDF\le 4\%\\ 276.6\times {PDF}^{0.14}-341.6,PDF>4\%,R^2=0.99\end{array}$$ (14)

$${mDA}_E=\{\begin{array}{c}0,PDF\le 1.1\%\\ 11.94\times {PDF}^{0.72}-16.24,PDF>1.1\%,R^2=0.96\end{array}$$ (15)

$${mDA}_S=\{\begin{array}{c}0,PDF\le 1.1\%\\ 39.91\times {PDF}^{0.41}-47.23,PDF>1.1\%,R^2=0.98\end{array}$$ (16)

$${mDA}_W=\{\begin{array}{c}0,PDF\le 1.1\%\\ 16.88\times {PDF}^{0.64}-20.87,PDF>1.1\%,R^2=0.96\end{array}$$ (17)

Fig. 4.

Correlation between point daylight factor and maximum daylight autonomy facing four orientations (a) North; (b) East; (c) South; (d) West.

Again, R² is greater than 0.96, showing that the above equations are appropriate to estimate the mDA.

UDI and the corresponding PDF were analyzed and four bell shape curves for north-, east-, south- and west-facing surfaces are depicted in Fig. 5(a–d). At first, the UDI increases with increasing PDF for individual orientation. When PDF is up to around 2%, a peak UDI ranging from 88% to 96% can be viewed for east, south and west orientations. With considerable daylight illuminance of less than 2000 lux, window facing north has a higher UDI (i.e., 99%) than the other vertical directions. As PDF subsequently rises from 2% to 20%, resulting more daylight illuminance over 2000 lux and the UDI drops gently. Large daylight illuminance may cause visual glare. It is interesting to note that PDF just between 2 and 4% would be an appropriate criterion which has been widely adopted by many countries [11].

Fig. 5.

Correlation between point daylight factor and useful daylight illuminance facing four orientations (a) North; (b) East; (c) South; (d) West.

Likewise, regression analysis was conducted to correlate UDI with PDF. It has been found that UDI are better correlated with PDF by quadratic equations. The mathematical expressions and correlation coefficients are determined as Eqs. (18)–(21):

$${UDI}_N=\{\begin{array}{c}101.7-15.19\times {PDF}^{-1.22},PDF\le 4\%,R^2=0.99\\ 409.4-246.1\times {PDF}^{0.16},PDF>4\%,R^2=0.99\end{array}$$ (18)

$${UDI}_E=\{\begin{array}{c}100.6-15.32\times {PDF}^{-1.1},PDF\le 2\%,R^2=0.98\\ 133.3-23.71\times {PDF}^{0.54},PDF>2\%,R^2=0.95\end{array}$$ (19)

$${UDI}_S=\{\begin{array}{c}99.56-13.15\times {PDF}^{-1.1},PDF\le 2\%,R^2=0.98\\ 257.8-140.4\times {PDF}^{0.19},PDF>2\%,R^2=0.98\end{array}$$ (20)

$${UDI}_W=\{\begin{array}{c}95.8-9.49\times {PDF}^{-1.3},PDF\le 2\%,R^2=0.97\\ 146.5-36.03\times {PDF}^{0.45},PDF>2\%,R^2=0.94\end{array}$$ (21)

With the R² of more than 0.94, it indicates that the UDI can be accurately calculated by the corresponding PDF. Such issue can help the design of indoor daylight environment quickly reaching the required standard.

4.2. Correlations between dynamic daylight metrics and vertical daylight factor

The correlations between VDF after transmitted to the room interior under Sky 1 and mDA with 2000 illuminance threshold at the center of vertical window for north, east, south and west are shown in Fig. 6.

Fig. 6.

Correlations between vertical daylight factor and maximum daylight autonomy facing four orientations (t is in the range of 0.2–0.65).

It can be seen that mDA rises with increasing VDF. Containing more amount of sky-diffuse component, the mDA for north- and east-facing surfaces are less than those for south-and west-facing surfaces. When the VDF is 9%, the mDA for north and east directions are around 6% smaller than those for south and west orientations. Through regression analysis, R² values were found of unity indicating perfect correlations and their correlations are expressed in as Eqs. (22)–(25):

$${DA}_{\max N}=99.57-972.3\times {VDF}^{-1.42}{R}^2=1$$ (22)

$${DA}_{\max E}=105.4-422.2\times {VDF}^{-1.01}{R}^2=1$$ (23)

$${DA}_{\max S}=108-204.8\times {VDF}^{-0.76}{R}^2=1$$ (24)

$${DA}_{\max W}=100.8-504.7\times {VDF}^{-1.21}{R}^2=1$$ (25)

The VDF is a constant of around 47% before transmitted into the room space under an unobstructed Sky 1. It would be easily to vary the window transmittance (change the window type) for getting the transmitted VDF to meet the required mDA.

There are simple correlations between PDF and VDF under all sky conditions [57]. The ratio of PDF to VDF can be computed using a number of room parameters and the configuration factor for a vertical rectangle plane to a horizontal point [24]. Such findings are useful to daylight-linked open-loop lighting controls which can get rid of the influence due to the electric lighting. The primary drawback for closed-loop daylight-linked switching control is the sudden and quite often of lamp fittings turning on and off when the sensed daylight plus electric lighting is merely greater than the illuminance set point. The daylight illuminance monitored at the reference point is the recorded vertical daylight illuminance (VDI) times the ratio of PDF to VDF which is well correlated [57]. Further study for open-loop daylight-linked lighting controls based on such technique will be examined in near future.

Daylight glare probability (DGP) is a simplified method for glare evaluation [58]. There are four subjective perception levels for DGP ‘< 35% imperceptible’, ‘35%–40% perceptible’, ‘40%–45% disturbing’ and ‘> 45% intolerable’. Equation (26) presents the mathematical expression for calculating the DGP.

$$DGP<6.22\times {10}^{-5}\times E_V+0.184$$ (26)

where E_V is the vertical illuminance at eye, lux.

Accordingly, the DGPs at the center point of window for the whole year were computed by substituting the transmitted VDI (excluding direct sunlight) into E_V. Fig. 7(a–d) displays the frequency of occurrence (FOC) for the four perception levels against the VDF (under Sky 1) facing various orientations.

Fig. 7.

Correlation between vertical daylight factor and daylight glare probability (t is in the range of 0.2–0.65) facing four orientations (a) North; (b) East; (c) South; (d) West.

For the four orientations, the ‘imperceptible’ drops with VDF but an opposite trend for ‘intolerable’ can be observed. For the other two perceptions, they vary slightly with VDF. Regression analysis was carried out and the mathematical expressions including the R² are summarized in Table 4. The R² values for ‘imperceptible’ and ‘intolerable’ equations are close to unity, representing excellent correlation. The R² for ‘disturbing’ and ‘perceptible’ is in the range of 0.93–1, still denoting a very good correlation. The VDF, therefore, can be used for predicting various daylight glare perceptions. The results also reveal the effects due to orientation. For a given VDF, the FOC of ‘imperceptible’ perception for windows facing north and east is larger than that facing south and west. When VDF is 14%, the FOC of ‘imperceptible’ perception for the glazing facing north, east, south and west are 41%, 38%, 28% and 32%, respectively. To limit the visual glare, the ‘imperceptible’ perception should be the majority. To achieve the FOC of more than 40% for ‘imperceptible’ perception, the VDFs should not be more than 11% for the four orientations. In order to accomplish such small VDF under unobstructed skies, shading devices such as overhang, side-fin and recessed glazed opening with vertical louvres [50,59] may be the appropriate approach and further study will be carried out.

Table 4.

Equations for the relationship between vertical daylight factor and daylight glare probability.

Orientation	DGP	Equation	R2
North	<35%	0.19 × VDF2 – 10.31 × VDF + 150.6	0.99
	35%–40%	– 0.022 × VDF2 + 0.21 × VDF + 18.69	0.93
	40%–45%	0.0078 × VDF3 − 0.52 × VDF2 + 10.74 × VDF − 53.02	0.97
	>45%	−0.32 × VDF2 + 15.77 × VDF − 115.5	0.98
East	<35%	0.13 × VDF2 − 7.14 × VDF + 114.9	1
	35%–40%	0.0014 × VDF2 − 0.5 × VDF + 19.91	0.99
	40%–45%	−0.02 × VDF2 + 0.59 × VDF + 7	0.93
	>45%	−0.23 × VDF2 + 11.86 × VDF − 73.04	0.97
South	<35%	0.11 × VDF2 − 6 × VDF + 92.75	0.99
	35%–40%	0.026 × VDF2 − 1.58 × VDF + 28.08	1
	40%–45%	0.014 × VDF2 − 1.03 × VDF + 22.69	1
	>45%	−0.15 × VDF2 + 8.6 × VDF − 43.52	1
West	<35%	0.14 × VDF2 − 7.46 × VDF + 111.3	0.99
	35%–40%	0.014 × VDF2 − 1.16 × VDF + 26.92	0.99
	40%–45%	−0.014 × VDF2 + 0.13 × VDF + 13.1	0.95
	>45%	−0.14 × VDF2 + 8.5 × VDF − 51.29	1

4.3. Correlation between spatial daylight autonomy and average daylight factor

The correlations between ADF (under Sky 1) and sDA of various occupied space areas (30%, 50%, 70% and 80%) with three illuminance thresholds (100 lux, 300 lux and 500 lux) facing the four cardinal orientations (N, E, S & W) are shown in Fig. 8(a–c). Several important aspects can be noticed. Except north orientation, the sDA_{100 lux 30%} is close to 100%. It means that sDA with low occupied space area and illuminance thresholds is generally independent of the ADF and orientation. When the analysis area and illuminance threshold increase, the sDA can be reduced from 100% to around 10% (sDA_{500 lux 80%}). The sDA for room facing north and east is generally less than that facing south and west. LEED v4.1 regards the sDA_{300 lux 50%} as the scoring criteria for daylight performance. As shown clearly in Fig. 8(b), sDA_{300lux 30%} and sDA_{300lux 50%} are 100% when ADF is more than 4%. It indicates that sDA can only reflect indoor daylighting requirement but cannot evaluate whether indoor daylighting is excessive and may cause visual discomfort. The sDA values can be predicted by corresponding ADFs. For instance, the sDA_{500 lux 30%} with the window wall facing south is 70% when the ADF is 3%. It means that appropriate sDA criteria can be obtained from the corresponding ADF. Based on the data shown in the figure, the range of daylight illuminance for the area concerned can also be specified. The sDA_{100 lux 30%} also implies no daylight illuminance is less than 100 lux when the window facing south. It can be deduced that for 30% of the floor area of the room facing south there is 30% of the annual occupied hours with the daylight illuminance ranging between 100 lux and 500 lux when the ADF is 3%. Such analysis would be important to daylight distribution and uniformity.

Fig. 8.

Correlations between average daylight factor and spatial daylight autonomy with three illuminance thresholds (a) 100 lux; (b) 300 lux; (c) 500 lux.

4.4. Design implications

The CBDMs including DA, cDA, mDA, sDA and UDI are appropriate to evaluate visual performance, indoor daylight distribution, daylight glare and daylight-linked lighting controls. However, the usual method for estimating these CBDMs is to employ sophisticated computer simulation packages containing long-term prevailing weather files. Full-scale computer simulations can be quite complex, costly and time demanding in making the building configurations and running simulations. Sophisticated techniques are required for dealing with pre- and post-processing. Such complicated and advanced simulation software would be appropriately used by a specialist or skillful user [27]. Simple calculation aids formulated via in-depth examination are commonly used by building practitioners particularly in the preliminary design period when various design options and architecture schemes are considered and assessed. Apart from daylight availability, building façade also involves thermal aspect such as the heat gain and loss through the external walls, and the performance of building integrated solar electric and solar thermal. Detailed simulations analysis can be carried out to get subsequent precise findings after all these issues are finalized and the estimated outcomes can be adopted to check with the simulated results. This study demonstrates that such CBDMs are well correlated with DFMs namely PDF, VDF and ADF under the traditional CIE Standard Sky 1. It means that the required criteria in terms of DA, cDA, mDA, sDA and UDI can be easily obtained from the corresponding DFs. The key strength is that the three types of DFMs (i.e., PDF, VDF and ADF) are highly linked with fraction of visible sky and room variables, especially the window area and visual transmittance. There are many simple calculation tools and design aids to determine the DFMs based on these room parameters [10,45] and sky luminance distributions under the traditional CIE Overcast Sky [7]. It could be easily to change the size of these parameters to satisfy the criteria. Previously, a number of studies reported that the CIE Overcast Sky performed the best among many worldwide models adopted in places of various climates under overcast conditions [[60], [61], [62]]. Outdoor illuminance for a particular site is mainly affected by the solar positions and its own sky conditions such that sky-diffuse and direct sunlight datasets are different from one place to another. There may be too many variables for predicting the CBDMs and only the most influencing input factors should be adopted. The CBDMs for a given place may not be accurately predicted from individual input data using a general mathematical model. Using DFMs (i.e., room parameters + fraction of visible sky) as the input data can eliminate such effects. To achieve the LEED [39] ratings system for a 3-point credit, the sDA_{300lux 50%} is 75% for a regularly occupied floor area. Based on Fig. 8b, this would be met by an ADF of 2.7% in a south-facing room in Hong Kong. Using the Longmore model [45], the correct daylight configuration factors above (C) and below (D) horizon (under unobstructed and obstructed CIE Sky 1) and the room parameters can be determined for use in initial daylighting designs. Thus, the corresponding VDF (i.e., C + D) can be computed and then the mDA can be obtained from Fig. 6. Subsequently, the PDFs at various interior grid points will also be found using the sky component table for CIE Sky 1 and a simple equation to determine the internally reflected component [7]. Ultimately, the DA, cDA, mDA and UDI for the corresponding interior points can be predicted based on Fig. 2, Fig. 3, Fig. 4, Fig. 5c, respectively. In addition, the DGP for the four subjective perception levels can be reckoned from Fig. 7. After this initial design scheme has been confirmed, full-scale building simulations will be conducted to get precise outcomes, such as CBDMs. The cases for analysis, however, did not include the use of external shading devices which are appropriate to lower the excessive daylight for rooms located at the top floors of high-rise buildings facing unobstructed skies. Further research studies to compute the amount of daylight reduced by overhang, side-fin and recessed window [50,59] will be conducted soon. To enhance the generalizability and representativeness of the proposed approach, locations with different climates obtained from accessible websites [63] will be examined later.

5. Conclusions and discussions

Correlations between various CBDM including DA, cDA, mDA, sDA and UDI with DFMs under the CIE Overcast Sky (Sky 1) with window wall facing north, east, south and west were established. All the data were obtained via the same software RADIANCE. Generally, all the modelled equations are strongly correlated, indicating that different forms of DA and UDI can be computed from the corresponding DFMs. Accordingly, using the findings in DA and cDA, the lighting energy consumption under standard daylight-linked lighting on-off and dimming controls were computed. Peak UDIs range from 88% to more than 99% when PDFs are between 2% and 4%. Such findings are well agreed with DFMs criteria adopted by many countries. sDA can denote the indoor daylight distribution and uniformity. ADF was employed to correlate with sDA for various occupied room areas at different daylight thresholds. The correlations are useful to obtain proper daylight distributions from a particular room daylighting scheme (i.e., ADF). Further, the frequency of occurrence for the perception levels of DGP were calculated and correlated with VDF. Lower the VDF at the vertical window can allow less ‘distributing’ and ‘intolerable’ perceptions. The present study takes Hong Kong as an example and considers the typical room facing the four cardinal directions under unobstructed skies. Further work will deal with more orientations under unobstructed and obstructed sky conditions and other places with their representative weather files containing sky diffuse and direct sunlight data.

Author contribution statement

Shuyang Li: Conceived and designed the experiments; Performed the experiments; Analyzed and interpreted the data; Wrote the paper.

Danny H W Li: Conceived and designed the experiments; Analyzed and interpreted the data; Wrote the paper.

Wenqiang Chen: Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data.

Siwei Lou, Ernest K. W. Tsang: Contributed reagents, materials, analysis tools or data.

Data availability statement

The authors do not have permission to share data.

Abbreviations

ADF

Average Daylight Factor

CBDMs

Climate Based Daylight Metrics

cDA

Continuous Daylight Autonomy

CIE

International Commission on Illumination

DA

Daylight Autonomy

DF

Daylight Factor

DFMs

Daylight Factor Metrics

DGP

Daylight Glare Probability

FOC

Frequency of Occurrence

IES

Illuminating Engineering Society

LEED

Leadership in Energy and Environmental Design

mDA

Maximum Daylight Autonomy

PDF

Point Daylight Factor

sDA

Spatial Daylight Autonomy

UDI

Useful Daylight Illuminance

VDF

Vertical Daylight Factor

VDI

Vertical Daylight Illuminance