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Evaluation of overcast-sky luminance models against measured Hong Kong data
Applied Energy 70 (2001) 321–331 www.elsevier.com/locate/apenergy
Evaluation of overcast-sky luminance models against measured Hong Kong data Danny H.W. Li*, Chris C.S. Lau, Joseph C. Lam Building Energy Research Group, Department of Building and Construction, City University of Hong Kong, Kowloon, Hong Kong, China Received 12 September 2001; received in revised form 3 October 2001; accepted 13 October 2001
Abstract Sky luminance distribution is one of the most important quantities for predicting indoor daylight illuminance levels. Overcast-sky types are essential because they are used in more general sky models and appear quite frequent in some places. This paper presents the work on the evaluation of six worldwide overcast-sky models against two-year (1999–2000) measured Hong Kong sky luminance data. Overcast-sky conditions were identified using cloud cover (CLD) and a subsequent interpretation the overcast skies into thin and heavy overcast types was conducted in conjunction with the clearness index (Kt). A statistical analysis of the models has indicated that the International Commission on Illumination (CIE) standard overcast sky model performed the best, in particular for the heavy overcast-sky condition. The Building Research Establishment (BRE) quasi-overcast-sky model showed a good agreement with the thin overcast distributions which may include a circumstance component and the sky luminance patterns being orientation dependent. # 2001 Published by Elsevier Science Ltd. All rights reserved.
1. Introduction Daylighting has long been recognised as an important means for energy conservation in buildings. Effective daylighting will reduce not only the need for electric lighting but also for air-conditioning, due to less heat dissipation from artificial light fittings [1]. The more daylight is made available in a building, the less artificial lighting is required. The first step towards evaluating the visual performance and * Corresponding author. Tel.: +852-278-87063; fax: +852-278-87612. E-mail address: [email protected] (D.H.W. Li). 0306-2619/01/$ - see front matter # 2001 Published by Elsevier Science Ltd. All rights reserved. PII: S0306-2619(01)00044-7
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Nomenclature b C, D CLD Id Kt Lmea Lpred Lz L N Z s
Sky luminance distribution index (dimensionless) Empirical coefficients (dimensionless) Cloud cover (Oktas) Horizontal diffuse irradiance (W/m2) Clearness index (dimensionless) Measured sky point luminance (kcd/m2) Predicted sky point luminance (kcd/m2) Sky luminance at the zenith (kcd/m2) Sky luminance at elevation angle (kcd/m2) Number of readings (dimensionless) Solar zenith angle (radians) Solar altitude (radians) Azimuth angle of the sky patch (radians) Azimuth angle of the Sun (radians) Elevation angle of a sky patch (radians) Scattering angle between the sun and the sky patch (radians)
energy efficiency due to daylighting requires an accurate estimation of the amount of daylight available outside the building. This includes not just the total amount of light coming from the sky, but also the distribution of luminance over the sky vault since the actual daylight illuminance of a room is mainly contributed by the luminance pattern of the part of the sky in the direction of view of the window [2]. This is of particular interest when considering a point in a deep plan side-lit room, which will only receive daylight from a certain part of the sky [3]. Traditionally, the daylighting performance of a building is always evaluated in terms of daylight factor, which by definition is the ratio of the internal illumination to the illumination simultaneously available outdoors under overcast-sky conditions [4]. The overcast sky is often considered to provide the worst design condition for daylighting. Consequently, the diversity of daylight will be improved under other sky types. For example, if sunlight is present, the contribution of reflected sunlight would increase the interior lighting levels. For places where overcast days occur more frequently, such as temperate regions and areas during rainy seasons, investigating the overcast-sky luminance is particularly important. Moreover, the overcast sky may be used as a component within more general sky luminance models which treat the sky as a linear combination of overcast and cloudless. Therefore, studying the characteristics of the overcast-sky luminance distribution has been a subject of great interest [5]. Many overcast-sky luminance distribution models have been developed to describe the sky patterns [6]. The empirical Moon-and-Spencer [7] equation for the luminance distribution of an overcast sky was adopted by the International Commission on Illumination (CIE) in 1955 as the standard for the overcast-sky luminance. Recently, Enarun and Littlefair [8] have reported that the
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Fig. 1. Measurement points for the sky scanner.
CIE overcast-sky performed the best among all worldwide models adopted in southern England under fully overcast skies. However, it has been pointed out in the literature that the overcast-sky type is not unique. There are two patterns of overcast sky, namely thin and heavy covered with bright and dark clouds, respectively [9]. The thin overcast-sky may include a circumsolar component and the sky luminance pattern is no longer orientation independent. The CIE overcast-sky model is applicable only to a heavy overcast sky. In 1999, a sky scanner was installed at the City University of Hong Kong to record the sky luminance distribution. As the measured sky luminance data are available, there is a need to examine closely the characteristics of overcast-sky conditions, particularly the locations in subtropical regions. This paper evaluates various overcast-sky luminance models against the data measured in subtropical Hong Kong. Characteristics of the findings and their implications are discussed.
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2. Sky luminance measurement The sky-luminance distribution is measured by means of a sky scanner (EKO MS 300LR) which was manufactured and calibrated by the EKO Company of Japan. The scanner is located on a roof top in a position relatively free from any external obstructions, and readily accessible for inspection, general cleaning and maintenance. It measures the luminance at 145 points (shown in Fig. 1) of the sky by scanning the sky dome. Data collection starts before sunrise and finishes after sunset. All measurements are referred to true solar time (TST), which is defined by the position of the true Sun (i.e. the Sun reaches its highest elevation precisely at solar noon). The important parts of the sky scanner are housed in a weatherproof casing, so allowing continuous outdoor operation. Output data from the scanner were recorded on a microcomputer placed inside the laboratory space on the top floor. A Visual Basic computer program was used to capture and transmit the measured data. To safeguard the sensor, the scanner does not record luminance data greater than 35 kcd/m2 by using an automatic shutter. Each scanning time is about 4 min and measurements are taken at 10 min intervals.
3. Overcast sky database Hourly sky-luminance data, recorded during the 24-month period from January 1999 to December 2000, were gathered for the study. To eliminate spurious data and
Fig. 2. Overcast-sky data set.
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inaccurate measurements resulting from low solar altitude, sky luminance data at an elevation of 6 or less and the whole sky luminance data-set recorded at a solar altitude angle of less than 5 were excluded from the analysis. It is inevitable that there will be some short periods of missing data for various reasons, including instrumentation malfunction and power failure. Considerable effort was made to obtain a continuous record of data and, in all, about 8000 sets of hourly readings were made from the sky scanner. It is perhaps logical and intuitive using cloud cover to denote the state of the sky as the whole sky is being considered. The basic criterion to describe overcast skies is that the sky is totally covered with clouds. It means that sky conditions correspond to a cloud cover (CLD) of 8 Oktas (i.e. skies totally covered by cloud) being defined as overcast. In total, around 970 sets of hourly overcast-sky luminance data were selected. Fig. 2 presents the distribution of the data set according to solar altitude. It can be seen that the data are quite evenly distributed between the solar altitudes 5 and 75 , and less data are found at solar altitudes of more than 75 . As Hong Kong is located at a low latitude (22.3 ), the peak solar altitude is up to 90 (i.e. the zenith) [10]. The distribution shown in Fig. 2 is quite different from places located at high latitudes [8].
4. Overcast-sky luminance models In total, six overcast-sky luminance models (CIE [7], Muneer and Angus [11], Perez [12], Harrison [13], Perraudeau [14] and Littlefair [15]) were used for the evaluation. In general, these models can be classified into two categories, namely orientation independent and orientation dependent. Brief descriptions of the models are given as follows: 4.1. Orientation independent models The general luminance distribution formula for the overcast sky is expressed as: 1 þ bsin ð1Þ 1þb For the overcast-sky, Moon and Spencer [7] proposed b=2, and this was adopted by CIE in 1955 as the standard for the overcast-sky luminance distribution, i.e. L ¼ Lz
L ¼ Lz ð1 þ 2sinÞ=3
ð2Þ
Using data supplied from the Building Research Establishment (BRE) daylight monitoring station at Garston, UK, Muneer and Augus [11] proposed different b values for overcast skies. They put b=1.32 and b=0 for shaded windows and Sunfacing windows, respectively. Compared with the CIE standard model, larger sky luminance values resulted for shaded windows. With b=0, it means that a uniform sky luminance pattern is assumed for Sun-facing surfaces under overcast skies.
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Instead of the sine function in the general sky luminance distribution formula, Perez et al. [12] have suggested an exponential alternative as: L ¼ Lz
1 þ CexpðD=sinÞ 1 þ CexpD
ð3Þ
Kittler et al. [16] proposed C=4 and D=0.7 for densely overcast skies. All three models listed above assume that the luminance distribution of the sky is symmetrical about the zenith and changes with the elevation above the horizon. This means that under an overcast sky, the sky luminance distribution computed is independent of the solar azimuth and the vertical outdoor illuminance would be the same for all orientations except that the Muneer and Angus model changes along the azimuth axis in the form of a step function (i.e. shaded and Sun-facing surfaces). 4.2. Orientation dependent The orientation-dependent models Sun with respect to the observed sky sky luminance is proportional to the between the Sun and the sky patch (
consider the effects due to the position of the element as being important. Apart from , the solar zenith angle (Z) and the scattering angle ), where :
cos ¼ cosZsin þ sinZcoscosðs Þ
ð4Þ
Generally speaking, the peak sky-luminance would appear near to the solar position, and decreases rapidly with the distance from the Sun. Based on the measurements of sky-luminance at 121 equally-spaced points over the sky domes, Harrison [13] proposed the overcast-sky model as follows: L ¼ 0:4 þ 0:21Z þ 0:27sin þ 1:45expð2:41 Þ
ð5Þ
Another orientation dependent model was developed by Perraudeau [14]. The formulation of the model is a product of three functions as: L ¼ Id 32:33 þ 13:16expð3 Þ þ 3:24cos2 1:180:23sin0:6 ð0:76 þ 0:13cosZ þ 0:2sinZÞ
ð6Þ
The final overcast-sky luminance model (i.e. quasi-overcast) for evaluation was established by Littlefair [15] using BRE data. The luminance distribution is given as: L ¼ ð1 þ sinÞ½40 þ 4078 þ 1350expð2 Þ
ð7Þ
As indicated in Eq. (7), the quasi-overcast has a modest circumstance component and its zenith luminance is around twice the horizon luminance.
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5. Data analysis The sky-luminance data were determined using the six models and compared with the measured readings for the same period. The modelled sky luminance distribution is normalised to the horizon diffuse illuminance by multiplying all the luminance values by the normalisation ratio (NR) as follows [8]: NR ¼
Lmea cossindd Lpred cossindd
ð8Þ
We are interested in the relative sky-luminance distribution (i.e. not absolute values) because once the horizon diffuse illuminance is determined, absolute luminance values of all sky patches can be easily computed. The performance of the each sky-luminance models was assessed by the following two widely used statistics: mean-bias error (MBE) and root-mean-square error (RMSE) as given by 1 Lpred Lmea MBE ¼ ð9Þ N Lmea ( " #)1=2 1 Lpred Lmea 2 RMSE ¼ N Lmea
ð10Þ
The MBE provides information on the long-term performances of the modelled regression equations. The RMSE gives information on the short-term performances, and indicates the scattering of data around the modelled regression equations. The predicted sky-luminance data for the six models were determined. Taking account of the symmetry about the solar position, the MBE and RMSE values were achieved by dividing the sky vault into six zones namely, by the Sun low and high regions, by side low and high regions and opposite Sun low and high regions [8]. Table 1 summarises the evaluation results. The statistical results show that all models predict the sky-luminance values better for the high zone than for low region. Low RMSE results are found when the sky patches are far away from the Sun (i.e. opposite sun-high and at side high regions) and large RMSE values appear in the by sun-low sector. The CIE overcast-sky model shows the best agreement with the measured Hong Kong data. The exponential model of Perez et al., with the modelling coefficients given by Kittler et al. [16], shows the next best agreement. Both models exhibit similar results and the RMSE for any zone does not exceed 25%. The quasi-overcast sky, which relies on solar position, has the next best performance. The RMSE is just over 25% for the whole sky. The models of Muneer and Angus and Harrison and Perraudeau perform less satisfactorily against the subtropical Hong Kong data. In the ‘by sun low sector’, these three models always tend to over-predict the sky luminance, causing the MBE and RMSE values to be the highest among all sectors. A peak RMSE of 65.5% occurs in the by sun-low
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Table 1 Results of model evaluation for overcast sky conditions By Opposite At Whole By Opposite At sky sun-low sun-low sidea low sun-high sun-high sidea high
Model
Error (%)
Moon and Spencer [7] (CIE)
MBE 1.6 RMSE 20.1
2 23.2
1 23.3
0.2 23.9
1.6 17.1
4.4 17.2
3.4 17.5
Perez et al. [12] Kittler et al. [16]
MBE 0.5 RMSE 20.8
8.4 24.1
4.3 23.6
5.2 24.3
3.9 17.7
6.7 18.2
5.7 18.4
Muneer and Angus [11] MBE 7.1 RMSE 30.9
35.1 49.9
0.2 28
29.5 46.5
2.9 18.2
5.6 17.5
1.3 18.2
Littlefair [15] (quasi-overcast sky)
MBE 3.8 RMSE 25.2
19.1 42.3
0.6 23.1
3.3 24.9
6.3 20.7
4.4 16.2
2.3 16.4
Harrison [13]
MBE 5.2 RMSE 35.8
39 65.5
3.9 24.5
0.8 26.2
16.1 33.8
14.5 20.9
9 18.4
Perraudeau [14]
MBE 7.9 RMSE 36.3
39.8 57.4
26.1 43.9
25.2 43.4
4.2 22
12.7 21.9
11.3 22
a According to Littlefair’s division of the sky-luminance algorithm studies [8], the sky elements located between 65 and 115 from the Sun’s position are classified as ‘At side’.
sector for the Harrison model. The findings are quite similar to the work reported by Enarun and Littlefair [8] using data measured in southern England under fullyovercast skies. 5.1. Subsequent analysis The above analyses indicate that, in general, orientation-independent models, which are functions of the altitude of the sky patch only, out-performed other complex rivals. However, the results are limited to the overcast sky-luminance data obtained when the sky is fully covered by the cloud (CLD=8 Oktas). This skyluminance database may include a mixture of thin and heavy overcast-sky types. It seems that only using the meteorological CLD, cannot identify these two overcastsky patterns. It has been suggested that of clearness index (Kt), which is defined as the ratio of horizontal global solar-irradiance to the extraterrestrial solar-irradiance, of less than 0.15 is a good criterion for representing a heavy overcast-sky [17]. In terms of sky-clearness classification, Kt is a widely used index because it depends only on global solar irradiance (i.e. one measured parameter). The global solar irradiance is recorded by many meteorological stations around the world and the measurement is quite straightforward. A low Kt means a low global solar-irradiance, which usually represents a cloudy sky with a high portion of diffuse irradiation. A large Kt means a high global solar-irradiance which is dominated by the direct component. Since CLD is the basic criterion for describing overcast-skies, CLD=8 Oktas was used in conjunction with Kt for further sky classification as follows:
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1. Heavy overcast sky (CLD=8 Oktas and Kt40.15); 2. Thin overcast sky (CLD=8 Oktas and Kt > 0.15). The same overcast luminance database was used for the analysis. Around 750 and 220 sets of sky-luminance data were further categorised into heavy and thin overcast skies, respectively. The majority is heavy overcast sky data, and it is not surprising that the CIE overcast sky model gave the best results in the previous analyses. The performance of sky-luminance models, hence, depends on the selection of the database. It is argued that a thin overcast sky may consist of a circumsolar component and the sky luminance distributions should be better interpreted by orientation dependent models rather than independent models. The CIE overcast sky model, however, is applicable only for a heavy overcast sky, when the complete sky canopy is covered with uniform dark clouds. In the previous section, it has been shown that the CIE overcast model and the quasi-overcast model are in good agreement with the experimental data. Therefore, these two models were selected in the subsequent analyses. Again, the performance of each model was assessed statistically in terms of MBE and RMSE. In order to examine the predictive abilities of the two models in Table 2 Results for the CIE standard overcast-sky and quasi-overcast-sky models under heavy and thin overcast skies Clearness index (Kt)
Global irradiance (W/m2)
Number of cases
Kt40.15
0–20 20–40 40–60 60–80 80–100 100–120 120–140 140–160 160–180 180–200
163 136 125 105 76 55 46 24 19 1
Kt >0.15
50–100 100–150 150–200 200–250 250–300 300–350 350–400 400–450 450–500 550–600 600–650
15 30 55 51 26 18 6 11 1 1 1
CIE standard overcast-sky
Quasi-overcast-sky
MBE (%)
RMSE (%)
MBE (%)
RMSE (%)
3.3 1.4 0.6 1 0.8 1.1 1.5 0.7 3.1 1.7
25.9 20.7 16.1 16.3 16.7 19.6 19.2 15.7 28.5 9.7
7.2 4.3 3 3 2.6 2.5 2.8 1.7 3.8 1
38.4 26.4 21.1 19.5 19.2 21.3 20.7 16.4 28.4 9.9
2.3 1.8 1.5 1.6 2.6 2.6 1.5 3 4.4 6.1 1.9
21.3 20.2 18.7 17.9 18.1 20.5 21.8 20.7 27.4 22.6 19.3
3.6 3.4 2.4 2.4 3.4 3 1.8 3.5 4.2 6.4 3.9
23.9 22.9 19 17.8 17.6 18.4 19.8 20.8 24.1 22 18.2
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detail, evaluations were carried out at intervals of 20 and 50 W/m2 of global irradiance, respectively, for heavy and thin overcast skies. Table 2 presents the assessment results. It can be seen that the two models tend to over-estimate the skyluminance data. For the heavy overcast sky type (i.e. Kt40.15), no global irradiance data exceed 200 W/m2 and the majority is less than 100 W/m2. The CIE overcast model out-performs the quasi-overcast model particularly for low global irradiance. At global irradiances of more than 160 W/m2, the CIE overcast model is on a par with the quasi-overcast model. The RMSE values range from 9.7 to 25.9% for the CIE overcast model and from 9.9 to 38.4% for the quasi-overcast model. Different statistical results are found for the thin overcast sky data. The global irradiance can be up to 650 W/m2 but only 18% of the data are greater than 300 W/ m2. The performance of the CIE overcast model is better than the quasi-overcast model at global irradiances less than 200 W/m2. At other global irradiance values, the quasi-overcast model is more effective. In terms of RMSE, the predictive abilities of the two models are better for thin-overcast than heavy-overcast skies. The RMSE values are less than 28 and 25% for the CIE overcast model and quasi-overcast models, respectively.
6. Conclusions An evaluation of the predicted sky-luminance distribution based on six overcast sky models (CIE [7], Perez et al. [12], Muneer and Angus [11], Littlefair [15], Harrison [13] and Perraudeau [14]) against two years measured sky luminance data in Hong Kong has been conducted. Using CLD to identify the sky conditions, around 970 hourly sets of sky luminance data were interpreted as overcast skies. It has been found that the CIE overcast and quasi-overcast models have the lowest RMSE results for individual sectors. The CIE overcast model, which changes with the altitude of the sky patch, shows the best agreement with the measured sky luminance data among all other complicated equations. The RMSE values range from 17.1% in the sun-high region to 23.9% in the side-low region. The quasi-overcast model performs better than all other orientation-dependent rivals that are functions of both altitude and azimuth of the sky element. The RMSE for the whole sky is just above 25%. The characteristics of overcast skies were also discussed. To identify thin and heavy overcast skies, the same overcast database was subsequently categorised using the clearness index (Kt). Again, the CIE overcast and quasi-overcastsky models were adopted for further model assessment. It has been found that the CIE overcast sky model is more effective under heavy overcast-sky patterns, particularly when the global irradiance data are low. The quasi-overcast-sky model, however, out-performs the CIE overcast-sky model under thin overcast-sky distributions with high global irradiance values. The findings also support that CLD in conjunction with Kt is a good hybrid index to identify thin and heavy overcast-sky conditions. The work presented can be considered a preliminary study on worldwide overcast-sky luminance models. In view of the fact that all these models were developed in Europe and North America, where the low solar-altitudes dominate,
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further research is required to establish an empirical overcast-sky luminance model for places in subtropical regions of low latitude.
Acknowledgements The work described in this paper was supported by a grant from the City University of Hong Kong (Project No. 7100193). C.C.S. Lau. is supported by a City University of Hong Kong Studentship. The authors would like to thank E.K.W. Tsang for his help with data collection.
References [1] Li DHW, Lam JC. Measurements of solar radiation and illuminance on vertical surfaces and daylighting implications. Renewable Energy 2000;20(4):389–404. [2] Rutten AJF. Sky luminance research imperative for adequate control of temporary supplementary artificial lighting installations. Building and Environment 1994;29(1):105–11. [3] Tregenza PR, Waters IM. Daylight coefficients. Lighting Research and Technology 1983;15(2):65– 71. [4] Hopkinson RG, Petherbridge P, Longmore J. Daylighting. London, UK: Heinemann, 1966. [5] Fritz S. Illuminance and luminance under overcast skies. Journal of the Optical Society of America 1955;45(10):820–5. [6] Kittler R. Universal modeling of daylight climates for design purposes. Architectural Science Review 1999;42(2):75–8. [7] Moon P, Spencer DE. Illumination from a non-uniform sky. Illuminating Engineering 1942;37(10): 707–26. [8] Enarun D, Littlefair P. Luminance models for overcast skies: assessment using measured data. Lighting Research and Technology 1995;27(1):53–8. [9] Muneer T. Evaluation of the CIE overcast sky model against Japanese data. Energy and Buildings 1998;27(2):175–7. [10] Lam JC, Li DHW. Study of solar radiation data for Hong Kong. Energy Conversion and Management 1996;37(3):343–51. [11] Muneer, T. and Angus, R.C. Daylight illuminance factors based on real sky distributions. In: Proceedings of the CIBSE National Lighting Conference, Cambridge, UK, 1994, 260–6. [12] Perez R, Seals R, Michalsky J. All-weather model for sky luminance distribution—preliminary configuration and validation. Solar Energy 1993;50(3):235–45. [13] Harrison AW. Directional sky luminance versus cloud cover and solar position. Solar Energy 1991; 46(1):13–19. [14] Perraudeau M. Luminance models. In: Proceedings of the CIBSE National Lighting Conference, Cambridge, UK, 1988. [15] Littlefair J. The luminance distributions of clear and quasi-clear skies. In: Proceedings of the CIBSE National Lighting Conference, Cambridge, UK, 1994, 267–83. [16] Kittler R, Darula S, Perez R. A new generation of sky standards. In: Proceedings of the Lux Europa Conference, Amsterdam, 1997, 359–73. [17] Li DHW, Lam JC. An analysis of climatic parameters and sky condition classification. Building and Environment 2001;36(4):435–45.
Evaluation of overcast-sky luminance models against measured Hong Kong data Danny H.W. Li*, Chris C.S. Lau, Joseph C. Lam Building Energy Research Group, Department of Building and Construction, City University of Hong Kong, Kowloon, Hong Kong, China Received 12 September 2001; received in revised form 3 October 2001; accepted 13 October 2001
Abstract Sky luminance distribution is one of the most important quantities for predicting indoor daylight illuminance levels. Overcast-sky types are essential because they are used in more general sky models and appear quite frequent in some places. This paper presents the work on the evaluation of six worldwide overcast-sky models against two-year (1999–2000) measured Hong Kong sky luminance data. Overcast-sky conditions were identified using cloud cover (CLD) and a subsequent interpretation the overcast skies into thin and heavy overcast types was conducted in conjunction with the clearness index (Kt). A statistical analysis of the models has indicated that the International Commission on Illumination (CIE) standard overcast sky model performed the best, in particular for the heavy overcast-sky condition. The Building Research Establishment (BRE) quasi-overcast-sky model showed a good agreement with the thin overcast distributions which may include a circumstance component and the sky luminance patterns being orientation dependent. # 2001 Published by Elsevier Science Ltd. All rights reserved.
1. Introduction Daylighting has long been recognised as an important means for energy conservation in buildings. Effective daylighting will reduce not only the need for electric lighting but also for air-conditioning, due to less heat dissipation from artificial light fittings [1]. The more daylight is made available in a building, the less artificial lighting is required. The first step towards evaluating the visual performance and * Corresponding author. Tel.: +852-278-87063; fax: +852-278-87612. E-mail address: [email protected] (D.H.W. Li). 0306-2619/01/$ - see front matter # 2001 Published by Elsevier Science Ltd. All rights reserved. PII: S0306-2619(01)00044-7
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Nomenclature b C, D CLD Id Kt Lmea Lpred Lz L N Z s
Sky luminance distribution index (dimensionless) Empirical coefficients (dimensionless) Cloud cover (Oktas) Horizontal diffuse irradiance (W/m2) Clearness index (dimensionless) Measured sky point luminance (kcd/m2) Predicted sky point luminance (kcd/m2) Sky luminance at the zenith (kcd/m2) Sky luminance at elevation angle (kcd/m2) Number of readings (dimensionless) Solar zenith angle (radians) Solar altitude (radians) Azimuth angle of the sky patch (radians) Azimuth angle of the Sun (radians) Elevation angle of a sky patch (radians) Scattering angle between the sun and the sky patch (radians)
energy efficiency due to daylighting requires an accurate estimation of the amount of daylight available outside the building. This includes not just the total amount of light coming from the sky, but also the distribution of luminance over the sky vault since the actual daylight illuminance of a room is mainly contributed by the luminance pattern of the part of the sky in the direction of view of the window [2]. This is of particular interest when considering a point in a deep plan side-lit room, which will only receive daylight from a certain part of the sky [3]. Traditionally, the daylighting performance of a building is always evaluated in terms of daylight factor, which by definition is the ratio of the internal illumination to the illumination simultaneously available outdoors under overcast-sky conditions [4]. The overcast sky is often considered to provide the worst design condition for daylighting. Consequently, the diversity of daylight will be improved under other sky types. For example, if sunlight is present, the contribution of reflected sunlight would increase the interior lighting levels. For places where overcast days occur more frequently, such as temperate regions and areas during rainy seasons, investigating the overcast-sky luminance is particularly important. Moreover, the overcast sky may be used as a component within more general sky luminance models which treat the sky as a linear combination of overcast and cloudless. Therefore, studying the characteristics of the overcast-sky luminance distribution has been a subject of great interest [5]. Many overcast-sky luminance distribution models have been developed to describe the sky patterns [6]. The empirical Moon-and-Spencer [7] equation for the luminance distribution of an overcast sky was adopted by the International Commission on Illumination (CIE) in 1955 as the standard for the overcast-sky luminance. Recently, Enarun and Littlefair [8] have reported that the
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Fig. 1. Measurement points for the sky scanner.
CIE overcast-sky performed the best among all worldwide models adopted in southern England under fully overcast skies. However, it has been pointed out in the literature that the overcast-sky type is not unique. There are two patterns of overcast sky, namely thin and heavy covered with bright and dark clouds, respectively [9]. The thin overcast-sky may include a circumsolar component and the sky luminance pattern is no longer orientation independent. The CIE overcast-sky model is applicable only to a heavy overcast sky. In 1999, a sky scanner was installed at the City University of Hong Kong to record the sky luminance distribution. As the measured sky luminance data are available, there is a need to examine closely the characteristics of overcast-sky conditions, particularly the locations in subtropical regions. This paper evaluates various overcast-sky luminance models against the data measured in subtropical Hong Kong. Characteristics of the findings and their implications are discussed.
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2. Sky luminance measurement The sky-luminance distribution is measured by means of a sky scanner (EKO MS 300LR) which was manufactured and calibrated by the EKO Company of Japan. The scanner is located on a roof top in a position relatively free from any external obstructions, and readily accessible for inspection, general cleaning and maintenance. It measures the luminance at 145 points (shown in Fig. 1) of the sky by scanning the sky dome. Data collection starts before sunrise and finishes after sunset. All measurements are referred to true solar time (TST), which is defined by the position of the true Sun (i.e. the Sun reaches its highest elevation precisely at solar noon). The important parts of the sky scanner are housed in a weatherproof casing, so allowing continuous outdoor operation. Output data from the scanner were recorded on a microcomputer placed inside the laboratory space on the top floor. A Visual Basic computer program was used to capture and transmit the measured data. To safeguard the sensor, the scanner does not record luminance data greater than 35 kcd/m2 by using an automatic shutter. Each scanning time is about 4 min and measurements are taken at 10 min intervals.
3. Overcast sky database Hourly sky-luminance data, recorded during the 24-month period from January 1999 to December 2000, were gathered for the study. To eliminate spurious data and
Fig. 2. Overcast-sky data set.
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inaccurate measurements resulting from low solar altitude, sky luminance data at an elevation of 6 or less and the whole sky luminance data-set recorded at a solar altitude angle of less than 5 were excluded from the analysis. It is inevitable that there will be some short periods of missing data for various reasons, including instrumentation malfunction and power failure. Considerable effort was made to obtain a continuous record of data and, in all, about 8000 sets of hourly readings were made from the sky scanner. It is perhaps logical and intuitive using cloud cover to denote the state of the sky as the whole sky is being considered. The basic criterion to describe overcast skies is that the sky is totally covered with clouds. It means that sky conditions correspond to a cloud cover (CLD) of 8 Oktas (i.e. skies totally covered by cloud) being defined as overcast. In total, around 970 sets of hourly overcast-sky luminance data were selected. Fig. 2 presents the distribution of the data set according to solar altitude. It can be seen that the data are quite evenly distributed between the solar altitudes 5 and 75 , and less data are found at solar altitudes of more than 75 . As Hong Kong is located at a low latitude (22.3 ), the peak solar altitude is up to 90 (i.e. the zenith) [10]. The distribution shown in Fig. 2 is quite different from places located at high latitudes [8].
4. Overcast-sky luminance models In total, six overcast-sky luminance models (CIE [7], Muneer and Angus [11], Perez [12], Harrison [13], Perraudeau [14] and Littlefair [15]) were used for the evaluation. In general, these models can be classified into two categories, namely orientation independent and orientation dependent. Brief descriptions of the models are given as follows: 4.1. Orientation independent models The general luminance distribution formula for the overcast sky is expressed as: 1 þ bsin ð1Þ 1þb For the overcast-sky, Moon and Spencer [7] proposed b=2, and this was adopted by CIE in 1955 as the standard for the overcast-sky luminance distribution, i.e. L ¼ Lz
L ¼ Lz ð1 þ 2sinÞ=3
ð2Þ
Using data supplied from the Building Research Establishment (BRE) daylight monitoring station at Garston, UK, Muneer and Augus [11] proposed different b values for overcast skies. They put b=1.32 and b=0 for shaded windows and Sunfacing windows, respectively. Compared with the CIE standard model, larger sky luminance values resulted for shaded windows. With b=0, it means that a uniform sky luminance pattern is assumed for Sun-facing surfaces under overcast skies.
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Instead of the sine function in the general sky luminance distribution formula, Perez et al. [12] have suggested an exponential alternative as: L ¼ Lz
1 þ CexpðD=sinÞ 1 þ CexpD
ð3Þ
Kittler et al. [16] proposed C=4 and D=0.7 for densely overcast skies. All three models listed above assume that the luminance distribution of the sky is symmetrical about the zenith and changes with the elevation above the horizon. This means that under an overcast sky, the sky luminance distribution computed is independent of the solar azimuth and the vertical outdoor illuminance would be the same for all orientations except that the Muneer and Angus model changes along the azimuth axis in the form of a step function (i.e. shaded and Sun-facing surfaces). 4.2. Orientation dependent The orientation-dependent models Sun with respect to the observed sky sky luminance is proportional to the between the Sun and the sky patch (
consider the effects due to the position of the element as being important. Apart from , the solar zenith angle (Z) and the scattering angle ), where :
cos ¼ cosZsin þ sinZcoscosðs Þ
ð4Þ
Generally speaking, the peak sky-luminance would appear near to the solar position, and decreases rapidly with the distance from the Sun. Based on the measurements of sky-luminance at 121 equally-spaced points over the sky domes, Harrison [13] proposed the overcast-sky model as follows: L ¼ 0:4 þ 0:21Z þ 0:27sin þ 1:45expð2:41 Þ
ð5Þ
Another orientation dependent model was developed by Perraudeau [14]. The formulation of the model is a product of three functions as: L ¼ Id 32:33 þ 13:16expð3 Þ þ 3:24cos2 1:180:23sin0:6 ð0:76 þ 0:13cosZ þ 0:2sinZÞ
ð6Þ
The final overcast-sky luminance model (i.e. quasi-overcast) for evaluation was established by Littlefair [15] using BRE data. The luminance distribution is given as: L ¼ ð1 þ sinÞ½40 þ 4078 þ 1350expð2 Þ
ð7Þ
As indicated in Eq. (7), the quasi-overcast has a modest circumstance component and its zenith luminance is around twice the horizon luminance.
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5. Data analysis The sky-luminance data were determined using the six models and compared with the measured readings for the same period. The modelled sky luminance distribution is normalised to the horizon diffuse illuminance by multiplying all the luminance values by the normalisation ratio (NR) as follows [8]: NR ¼
Lmea cossindd Lpred cossindd
ð8Þ
We are interested in the relative sky-luminance distribution (i.e. not absolute values) because once the horizon diffuse illuminance is determined, absolute luminance values of all sky patches can be easily computed. The performance of the each sky-luminance models was assessed by the following two widely used statistics: mean-bias error (MBE) and root-mean-square error (RMSE) as given by 1 Lpred Lmea MBE ¼ ð9Þ N Lmea ( " #)1=2 1 Lpred Lmea 2 RMSE ¼ N Lmea
ð10Þ
The MBE provides information on the long-term performances of the modelled regression equations. The RMSE gives information on the short-term performances, and indicates the scattering of data around the modelled regression equations. The predicted sky-luminance data for the six models were determined. Taking account of the symmetry about the solar position, the MBE and RMSE values were achieved by dividing the sky vault into six zones namely, by the Sun low and high regions, by side low and high regions and opposite Sun low and high regions [8]. Table 1 summarises the evaluation results. The statistical results show that all models predict the sky-luminance values better for the high zone than for low region. Low RMSE results are found when the sky patches are far away from the Sun (i.e. opposite sun-high and at side high regions) and large RMSE values appear in the by sun-low sector. The CIE overcast-sky model shows the best agreement with the measured Hong Kong data. The exponential model of Perez et al., with the modelling coefficients given by Kittler et al. [16], shows the next best agreement. Both models exhibit similar results and the RMSE for any zone does not exceed 25%. The quasi-overcast sky, which relies on solar position, has the next best performance. The RMSE is just over 25% for the whole sky. The models of Muneer and Angus and Harrison and Perraudeau perform less satisfactorily against the subtropical Hong Kong data. In the ‘by sun low sector’, these three models always tend to over-predict the sky luminance, causing the MBE and RMSE values to be the highest among all sectors. A peak RMSE of 65.5% occurs in the by sun-low
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Table 1 Results of model evaluation for overcast sky conditions By Opposite At Whole By Opposite At sky sun-low sun-low sidea low sun-high sun-high sidea high
Model
Error (%)
Moon and Spencer [7] (CIE)
MBE 1.6 RMSE 20.1
2 23.2
1 23.3
0.2 23.9
1.6 17.1
4.4 17.2
3.4 17.5
Perez et al. [12] Kittler et al. [16]
MBE 0.5 RMSE 20.8
8.4 24.1
4.3 23.6
5.2 24.3
3.9 17.7
6.7 18.2
5.7 18.4
Muneer and Angus [11] MBE 7.1 RMSE 30.9
35.1 49.9
0.2 28
29.5 46.5
2.9 18.2
5.6 17.5
1.3 18.2
Littlefair [15] (quasi-overcast sky)
MBE 3.8 RMSE 25.2
19.1 42.3
0.6 23.1
3.3 24.9
6.3 20.7
4.4 16.2
2.3 16.4
Harrison [13]
MBE 5.2 RMSE 35.8
39 65.5
3.9 24.5
0.8 26.2
16.1 33.8
14.5 20.9
9 18.4
Perraudeau [14]
MBE 7.9 RMSE 36.3
39.8 57.4
26.1 43.9
25.2 43.4
4.2 22
12.7 21.9
11.3 22
a According to Littlefair’s division of the sky-luminance algorithm studies [8], the sky elements located between 65 and 115 from the Sun’s position are classified as ‘At side’.
sector for the Harrison model. The findings are quite similar to the work reported by Enarun and Littlefair [8] using data measured in southern England under fullyovercast skies. 5.1. Subsequent analysis The above analyses indicate that, in general, orientation-independent models, which are functions of the altitude of the sky patch only, out-performed other complex rivals. However, the results are limited to the overcast sky-luminance data obtained when the sky is fully covered by the cloud (CLD=8 Oktas). This skyluminance database may include a mixture of thin and heavy overcast-sky types. It seems that only using the meteorological CLD, cannot identify these two overcastsky patterns. It has been suggested that of clearness index (Kt), which is defined as the ratio of horizontal global solar-irradiance to the extraterrestrial solar-irradiance, of less than 0.15 is a good criterion for representing a heavy overcast-sky [17]. In terms of sky-clearness classification, Kt is a widely used index because it depends only on global solar irradiance (i.e. one measured parameter). The global solar irradiance is recorded by many meteorological stations around the world and the measurement is quite straightforward. A low Kt means a low global solar-irradiance, which usually represents a cloudy sky with a high portion of diffuse irradiation. A large Kt means a high global solar-irradiance which is dominated by the direct component. Since CLD is the basic criterion for describing overcast-skies, CLD=8 Oktas was used in conjunction with Kt for further sky classification as follows:
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1. Heavy overcast sky (CLD=8 Oktas and Kt40.15); 2. Thin overcast sky (CLD=8 Oktas and Kt > 0.15). The same overcast luminance database was used for the analysis. Around 750 and 220 sets of sky-luminance data were further categorised into heavy and thin overcast skies, respectively. The majority is heavy overcast sky data, and it is not surprising that the CIE overcast sky model gave the best results in the previous analyses. The performance of sky-luminance models, hence, depends on the selection of the database. It is argued that a thin overcast sky may consist of a circumsolar component and the sky luminance distributions should be better interpreted by orientation dependent models rather than independent models. The CIE overcast sky model, however, is applicable only for a heavy overcast sky, when the complete sky canopy is covered with uniform dark clouds. In the previous section, it has been shown that the CIE overcast model and the quasi-overcast model are in good agreement with the experimental data. Therefore, these two models were selected in the subsequent analyses. Again, the performance of each model was assessed statistically in terms of MBE and RMSE. In order to examine the predictive abilities of the two models in Table 2 Results for the CIE standard overcast-sky and quasi-overcast-sky models under heavy and thin overcast skies Clearness index (Kt)
Global irradiance (W/m2)
Number of cases
Kt40.15
0–20 20–40 40–60 60–80 80–100 100–120 120–140 140–160 160–180 180–200
163 136 125 105 76 55 46 24 19 1
Kt >0.15
50–100 100–150 150–200 200–250 250–300 300–350 350–400 400–450 450–500 550–600 600–650
15 30 55 51 26 18 6 11 1 1 1
CIE standard overcast-sky
Quasi-overcast-sky
MBE (%)
RMSE (%)
MBE (%)
RMSE (%)
3.3 1.4 0.6 1 0.8 1.1 1.5 0.7 3.1 1.7
25.9 20.7 16.1 16.3 16.7 19.6 19.2 15.7 28.5 9.7
7.2 4.3 3 3 2.6 2.5 2.8 1.7 3.8 1
38.4 26.4 21.1 19.5 19.2 21.3 20.7 16.4 28.4 9.9
2.3 1.8 1.5 1.6 2.6 2.6 1.5 3 4.4 6.1 1.9
21.3 20.2 18.7 17.9 18.1 20.5 21.8 20.7 27.4 22.6 19.3
3.6 3.4 2.4 2.4 3.4 3 1.8 3.5 4.2 6.4 3.9
23.9 22.9 19 17.8 17.6 18.4 19.8 20.8 24.1 22 18.2
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detail, evaluations were carried out at intervals of 20 and 50 W/m2 of global irradiance, respectively, for heavy and thin overcast skies. Table 2 presents the assessment results. It can be seen that the two models tend to over-estimate the skyluminance data. For the heavy overcast sky type (i.e. Kt40.15), no global irradiance data exceed 200 W/m2 and the majority is less than 100 W/m2. The CIE overcast model out-performs the quasi-overcast model particularly for low global irradiance. At global irradiances of more than 160 W/m2, the CIE overcast model is on a par with the quasi-overcast model. The RMSE values range from 9.7 to 25.9% for the CIE overcast model and from 9.9 to 38.4% for the quasi-overcast model. Different statistical results are found for the thin overcast sky data. The global irradiance can be up to 650 W/m2 but only 18% of the data are greater than 300 W/ m2. The performance of the CIE overcast model is better than the quasi-overcast model at global irradiances less than 200 W/m2. At other global irradiance values, the quasi-overcast model is more effective. In terms of RMSE, the predictive abilities of the two models are better for thin-overcast than heavy-overcast skies. The RMSE values are less than 28 and 25% for the CIE overcast model and quasi-overcast models, respectively.
6. Conclusions An evaluation of the predicted sky-luminance distribution based on six overcast sky models (CIE [7], Perez et al. [12], Muneer and Angus [11], Littlefair [15], Harrison [13] and Perraudeau [14]) against two years measured sky luminance data in Hong Kong has been conducted. Using CLD to identify the sky conditions, around 970 hourly sets of sky luminance data were interpreted as overcast skies. It has been found that the CIE overcast and quasi-overcast models have the lowest RMSE results for individual sectors. The CIE overcast model, which changes with the altitude of the sky patch, shows the best agreement with the measured sky luminance data among all other complicated equations. The RMSE values range from 17.1% in the sun-high region to 23.9% in the side-low region. The quasi-overcast model performs better than all other orientation-dependent rivals that are functions of both altitude and azimuth of the sky element. The RMSE for the whole sky is just above 25%. The characteristics of overcast skies were also discussed. To identify thin and heavy overcast skies, the same overcast database was subsequently categorised using the clearness index (Kt). Again, the CIE overcast and quasi-overcastsky models were adopted for further model assessment. It has been found that the CIE overcast sky model is more effective under heavy overcast-sky patterns, particularly when the global irradiance data are low. The quasi-overcast-sky model, however, out-performs the CIE overcast-sky model under thin overcast-sky distributions with high global irradiance values. The findings also support that CLD in conjunction with Kt is a good hybrid index to identify thin and heavy overcast-sky conditions. The work presented can be considered a preliminary study on worldwide overcast-sky luminance models. In view of the fact that all these models were developed in Europe and North America, where the low solar-altitudes dominate,
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further research is required to establish an empirical overcast-sky luminance model for places in subtropical regions of low latitude.
Acknowledgements The work described in this paper was supported by a grant from the City University of Hong Kong (Project No. 7100193). C.C.S. Lau. is supported by a City University of Hong Kong Studentship. The authors would like to thank E.K.W. Tsang for his help with data collection.
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