An analysis of luminous efficacies under the CIE standard skies

ARTICLE IN PRESS

Renewable Energy 33 (2008) 2357–2365 www.elsevier.com/locate/renene

An analysis of luminous efficacies under the CIE standard skies Danny H.W. Li, Tony N.T. Lam, K.L. Cheung, H.L. Tang Building Energy Research Group, Department of Building and Construction, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong SAR, China Received 5 October 2007; accepted 7 February 2008 Available online 25 March 2008

Abstract Designing a building to integrate daylight requires an accurate estimation of the amount of available outdoor illuminance. The common method for predicting daylight has been the derivation of illuminance from the more widely measured solar irradiance using the luminous efficacy approach. Recently, the International Commission on Illumination (CIE) has adopted a range of 15 standard skies which cover the whole probable spectrum of skies in the world. This paper presents the work to model the luminous efficacy of diffuse component under the 15 CIE standard skies. Sky luminance distributions measured between 1999 and 2005 were used for the standard sky classifications. An approach to develop luminous efficacy for inclined surfaces was proposed. The predicted vertical outdoor illuminance data for the four cardinal planes (i.e., N, E, S and W) using the proposed luminous efficacy were evaluated against data measured in 2004. Statistical analysis indicated that the estimated daylight illuminance data give acceptable agreements with measured data for all vertical planes. r 2008 Elsevier Ltd. All rights reserved. Keywords: Sky luminance; Luminous efficacy; Outdoor illuminance; Solar irradiance; Sky conditions

1. Introduction Solar-based energy conversion systems and daylighting schemes are recognized as an important design strategy to generate a sustainable, environmentally friendly and clean energy, reduce the peak electrical and cooling demands and save the total building energy consumption [1,2]. A solar fac- ade is a building component generating electricity and allowing daylight penetrating into the building [3]. Large solar facades receive considerable amount of solar irradiance and daylight illuminance resulting more electricity generated and less electric lighting required, and vice versa [4]. Solar irradiance and daylight illuminance data are crucial to such building envelope designs. Long-term data measurement is regarded as reliable and accurate method of setting up solar irradiance and daylight illuminance databases. The recording practice for solar irradiance is horizontal global and diffuse components but the data often required in building design is the one for vertical Corresponding author. Tel.: +852 27887063; fax: +852 27887612.

E-mail address: [email protected] (D.H.W. Li). 0960-1481/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.renene.2008.02.004

and slope surfaces. In the absence of measurements, the common method for estimating daylight has been the derivation of illuminances from measured horizontal irradiance using the luminous efficacy approach [5–7]. The luminous efficacy models established are mainly for the horizontal surface and very little work has been done for the vertical and inclined planes [8]. Luminous efficacy is not a constant and can vary under various sky conditions [9–11]. In 2003, the International Commission on Illumination (CIE) adopted 15 standard skies [12] covering the whole probable spectrum of skies found in nature. Sky conditions of the same category would have the similar sky distributions and the luminous efficacy would be within certain ranges. Such analyses can help the determination of luminous efficacy for inclined surfaces. Recently, a measuring station has been set up at the City University of Hong Kong to investigate the solar and sky characteristics [13]. The solar irradiance and daylight illuminance data including horizontal (global and diffuse), vertical surfaces for the four cardinal orientations (N, E, S and W), and sky radiance and luminance distributions were systematically recorded. This paper studies the luminous efficacies

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for the 15 CIE standard skies and presents an approach to estimate the luminous efficacy and daylight illuminance on an inclined surface. A comparative assessment of the proposed model is reported. 2. Data measurements A measuring station was established at the City University of Hong Kong in 1991. The instruments were installed on the roof-top in a position relatively free from external obstructions and accessible for general inspection and maintenance. Initially, only measurements of global and diffuse solar irradiance and outdoor illuminance on a horizontal plane were made. In 1996, the measurement was extended to record the vertical global irradiance and illuminance on the four cardinal surfaces facing the north, east, south and west. Totally, six pyranometers and six illuminance sensors were used for the horizontal and vertical solar irradiance and daylight illuminance data measurements. The data were captured simultaneously twice per second and averaged over 10-min intervals. Data collection starts before sunrise and finishes after sunset. All measurements are recorded in terms of true solar-time. This facilitates the computations involving solar altitude for the extraterrestrial irradiance on unit horizontal surface and the subsequent comparison of data for different locations. Details of the solar irradiance and illuminance measurements can be found in some earlier works [14,15]. In 1999, a sky scanner (EKO MS 300LR) was installed at the City University of Hong Kong to record the sky luminance and radiance at 145 points (shown in Fig. 1) of

the sky by scanning the whole sky dome. The scanner was manufactured and calibrated by the EKO Company in Japan. The full view angle of the scanner is 111: that allows each sky patch to be treated as a point source with negligible error [16]. The sky grid pattern shown in Fig. 1 was suggested by Tregenza and Sharples [17] such that the whole sky dome can be considered for subsequent analysis. The important parts of the sky scanner are housed in a weatherproof casing allowing continuous outdoor operation. Output data from the scanner are recorded on a microcomputer located inside the laboratory space on the top floor. To safeguard the sensor, the scanner does not record data greater than 35 kcd/m2 by using an automatic shutter. Each scanning time is about 4 min and measurements are taken every 10 min. It should be pointed out that there are several causes affecting the accuracy of the measured sky distributions. First, the celestial hemisphere was split into 145 circular angular sky patches for the luminance measurement. Such an arrangement, however, led to uncovered regions of the sky. Second, the measured data are based on discrete results rather than continuous analytical functions. Sky luminance between adjoining measurement points may have varied significantly. Third, the scanning time was about 4 min and the measurement interval was 10 min. Substantial variations in sky luminance may have occurred between each record. Also, for ‘out-of-range’ measurements (points close to the solar position under non-overcast skies), an estimation of the sky luminance was made from a simple average of the luminance at nearby points and such conversion could introduce data distortion. To eliminate spurious data and erroneous

N

Patch No. Alt. Azi.

Fig. 1. Measurement points for the sky scanner.

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measurements, some quality-control tests were conducted as follows: 1. Applying a shadow ring correction to the measured diffuse horizontal data according to the method described by LeBaron et al. [18]. 2. Rejecting all diffuse data that were greater than the corresponding global values. 3. Rejecting all global data that were greater than the corresponding extraterrestrial solar component. 4. Rejecting all data with a solar altitude, a, of less than 51. 5. Rejecting all data when horizontal global irradiance was less than 20 W/m2 [19]. 6. Rejecting all data when the direct normal values [i.e., (globaldiffuse)/sin a] exceeded the corresponding extraterrestrial solar component. 7. Rejecting all diffuse data that were greater than half of the corresponding extraterrestrial solar component (because the shadow ring was not properly adjusted). 8. Rejecting all horizontal and vertical data when the computed vertical direct-beam data were greater than the corresponding measured vertical global component. 9. Rejecting all sky-scanned data (i.e., luminance and radiance) when the difference between the corrected horizontal diffuse values and the corresponding integrated diffuse horizontal components from the scanner were greater than 30% [19]. By applying the quality-control tests, horizontal direct solar irradiance and outdoor illuminance data were obtained by subtracting the corrected diffuse components from the corresponding global values. 3. CIE standard sky The set of 15 standard skies includes the CIE clear sky distributions, a uniform luminance distribution, and a close

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approximation to the CIE overcast sky. In general, the standard skies contain five clear, five intermediate and five overcast sky types covering the whole spectrum of skies in the world [20]. The distributions are characterized by continuous mathematical expressions that change smoothly in luminance from the horizon to the zenith and with angular distance from the sun. The standard formula defining the relative luminance distribution on any standard skies can be considered as a combination of gradation function j(Z) and indicatrix function f(w) as follows: L f ðwÞjðZÞ ¼ LZ f ðZS Þjð0 Þ

(1)

where L is the sky luminance in an arbitrary sky element (cd/m2), LZ the sky luminance at the zenith (cd/m2), Z the zenith angle of a sky element (rad), ZS the zenith angle of the sun (rad) and w is the scattering angle, the shortest angular distance between the sun and a sky element (rad). The standard gradations were defined by appropriate a and b variables as jðZÞ 1 þ a expðb=cos ZÞ ¼ jð0 Þ 1 þ a exp b

(2)

The relative scattering indicatrix function can be modeled by an exponential function with adjustable coefficients c, d and e as f ðwÞ 1 þ c½expðdwÞ  expðdp=2Þ þ e cos2 w ¼ f ðZ S Þ 1 þ c½expðdZ S Þ  expðdp=2Þ þ e cos2 ZS

(3)

The exponential term exp(dw) represents the effect of Mie scattering, which decreases rapidly with distance from the sun. The cos2 w term is due to Rayleigh scattering and is zero at 901 to the sun [21]. Both gradation and indicatrix functions are of six types covering the usual range of homogeneous cases from heavy overcast to cloudless skies. The combinations can form a large number of skies but

Table 1 Description of the 15 CIE standard skies No. (code)

1 (I1) 2 (I2) 3 (II1) 4 (II2) 5 (III1) 6 (III2) 7 (III3) 8 (III4) 9 (IV2) 10 (IV3) 11 (IV4) 12 (V4) 13 (V5) 14 (VI5) 15 (VI6)

Type of sky

CIE standard overcast sky, steep luminance gradation towards zenith, azimuthal uniformity Overcast, with steep luminance gradation and slight brightening towards the sun Overcast, moderately graded with azimuthal uniformity Overcast, moderately graded and slight brightening towards the sun Sky of uniform luminance Partly cloudy sky, no gradation towards zenith, slight brightening towards the sun Partly cloudy sky, no gradation towards zenith, brighter circumsolar region Partly cloudy sky, no gradation towards zenith, distinct solar corona Partly cloudy, with the obscured sun Partly cloudy, with brighter circumsolar region White-blue sky with distinct solar corona CIE standard clear sky, low luminance turbidity CIE standard clear sky, polluted atmosphere Cloudless turbid sky with broad solar corona White-blue turbid sky with broad solar corona

For gradation

For indicatrix

a

b

c

d

e

4 4 1.1 1.1 0 0 0 0 1 1 1 1 1 1 1

0.7 0.7 0.8 0.8 1 1 1 1 0.55 0.55 0.55 0.32 0.32 0.15 0.15

0 2 0 2 0 2 5 10 2 5 10 10 16 16 24

1 1.5 1 1.5 1 1.5 2.5 3 1.5 2.5 3 3 3 3 2.8

0 0.15 0 0.15 0 0.15 0.3 0.45 0.15 0.3 0.45 0.45 0.3 0.3 0.15

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only 15 relevant types were selected to form the standard set. Table 1 shows the details of the 15 standard skies [20]. 4. Luminous efficacy Values of luminous efficacy are obtained by simultaneously measuring the illuminance and irradiance on a specified surface, and then computing their ratio. Alternatively, the luminance and radiance of particular sky elements can be measured to calculate the luminous efficacy. It is a convenient quantity in the calculation of daylight availability and lighting energy use in building and, in principle allows for most of the climate- and latitude-related variations. It enables daylight data to be generated from the more widely measured solar irradiance data for places where measured outdoor illuminance data are not obtainable. It has been reported that luminous efficacy of direct solar irradiance (KB) can be expressed as a function of a [22,23]. Luminous efficacy of sky-diffuse irradiance defined as the ratio of simultaneous sky illuminance to irradiance (KD), however, appears to be relatively independent of a [24,25]. Many empirical models for predicting KD were developed under different sky conditions classified by various climatic parameters [26]. As the 15 CIE standard skies cover the whole spectrum of usual skies found in the world, there is a need to model KD under such sky standards. For standard sky classification, sky luminance data were modeled according to the 15 sky standards and compared with the scanned sky luminance readings of the same period. The modeled sky luminance is normalized to the horizontal diffuse illuminance by multiplying all the luminance values with the normalization ratio (NR): NR ¼

SLmea cos y sin y dy df Sl mod cos y sin y dy df

(4)

where Lmea is the measured sky point luminance (cd/m2), lmod is the modeled sky point luminance in relative form, y is the altitude of a sky patch (rad) and f is the azimuth of a sky patch (rad). The relative sky luminance distribution for each sky patch was determined (not individual absolute values) because once the integrated diffuse illuminance or luminance at any given points were obtained, the absolute luminance values of all sky patches can be easily computed. An alternative would be to divide all sky luminance readings by the zenith luminance, but this can cause huge measuring error when the sun is near to the zenith [27]. For low latitude region (e.g., Hong Kong) where high solar altitude dominates, normalized with respect to diffuse horizontal illuminance would, therefore, be more appropriate. Once each modeled sky luminance has multiplied NR, the performance of each standard sky luminance model was assessed by the root-mean-square error (RMSE) which is obtained by subtracting the measured sky patch luminances from the modeled luminances (multiplied the NR) of the 15 CIE standard skies, adding together

the squares of these values, dividing the total by the number of sky patches, then taking the square root. The best-fitting standard sky selected was the one with the lowest RMSE [28]. The 10-min data recorded from January 1999 to December 2005 were gathered for analysis. It is inevitable that there are some periods of missing data for various reasons, including instrumentation malfunction, power failure and sensor calibration. Considerable efforts were made to obtain a continuous record of data and in all about 68,500 sets of 10-min readings were obtained simultaneously from the above measurements. Fig. 2 shows the frequency of occurrence of the best-fitting standard skies in Hong Kong. Large variations can be observed for individual sky types and Fig. 2 illustrates that the overcast and clear skies (i.e., overcast—sky nos. 1–5; clear—sky nos. 11–15) represent about 78% of the Hong Kong sky conditions. The intermediate skies (i.e., sky nos. 6–10) account for the remaining 22%. Sky nos.1 and 3 dominate the subset of overcast skies standing for over 30% of the Hong Kong sky conditions. Sky nos. 6, 7 and 8 are the main sky patterns for the partly cloudy skies. Referring to clear sky type, sky no. 13 has the largest frequency of occurrence. The overall RMSE was found to be 23.7% of the measured mean value. It shows that the prevailing sky conditions in Hong Kong can be described by the 15 CIE standard skies. Accordingly, the sky-diffuse solar irradiance and daylight illuminance data were employed to determine KD under each standard sky. Graphical representation is a simple and direct approach to analyze and interpret meteorological data. The cumulative frequency distribution can indicate the percentage of the working period in which a given luminous efficacy level is exceeded. The cumulative frequency distributions of KD for sky standards 1–5 (overcast conditions) at an interval of 1 lm/W are presented in Fig. 3. It can be observed that the luminous efficacy values range from 71 to 170 lm/W. Generally, sky standards 1 and 4 have the largest and 20 18

Frequency of occurrence (%)

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16 14 12 10 8 6 4 2 0 1

2

3

4

5

6

7 8 9 10 11 12 13 14 15 Sky number

Fig. 2. Frequency of occurrence of the best-fitting 15 standard skies.

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100 Sky no.1 Sky no.2 Sky no.3 Sky no.4 Sky no.5

Cumulative frequency (%)

80

Sky no.11 Sky no.12 Sky no.13 Sky no.14 Sky no.15

90 80 Cumulative frequency (%)

90

70 60 50 40 30

70 60 50 40 30

20

20

10

10 0

0 70

80

90

100

110

120

130

140

150

160

170

180

190

Fig. 3. Cumulative frequency distribution of sky-diffuse luminous efficacy for standard skies 1–5.

100

Sky no.6 Sky no.7 Sky no.8 Sky no.9 Sky no.10

90 80 70 60 50 40 30 20 10 0 70

80

90

100

110

120

130

140

150

160

70

80

90

100

110

120

130

140

150

160

170

180

190

Sky-diffuse luminous efficacy (lm/W)

Sky-diffuse luminous efficacy (lm/W)

Cumulative frequency (%)

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170

180

190

Sky-diffuse luminous efficacy (lm/W)

Fig. 4. Cumulative frequency distribution of sky-diffuse luminous efficacy for standard skies 6–10.

lowest cumulative percentages, respectively. All the five curves exhibit a rapid drop in cumulative percentage when the luminous efficacy lies between 100 and 130 lm/W. The occurrence of these high luminous efficacy values indicates that daylight can provide high illuminance with a relatively small amount of heat generated and most of luminous efficacy values are within this narrow region. Likewise, the cumulative frequency distributions for sky standards 6–10 are plotted in Fig. 4. Akin pattern to that shown in Fig. 3 is observed but with smaller ranges and values for individual curves. The luminous efficacy spreads from 80 to 150 lm/W for the five skies. It is interesting to note that sky standards 7, 9 and 10 have almost identical cumulative percentages from 5% to 75% corresponding to luminous efficacy of between 105 and 130 lm/W. Such findings support that KD is relatively independent of a. Similarly, the cumulative frequency distributions for cloudless skies (sky nos. 11–15) are displayed in Fig. 5. With very little to no cloud amount,

Fig. 5. Cumulative frequency distribution of sky-diffuse luminous efficacy for standard skies 11–15.

all the five skies tend to have large luminous efficacy values. The luminous efficacy can be up to 190 lm/W appearing in skies 13 and 14. Under cloudless skies, the luminous efficacy is affected strongly by a large number of atmospheric elements including aerosols, water vapor, site pressure, ozone and nitrogen dioxide. To reduce the luminous efficacy variations for skies 13 and 14, a complete model should take all these factors into account. However, such parameters are generally not known in everyday applications [29]. For about 95% of the working period, the luminous efficacies for all of them are more than 105 lm/W. It seems that KD is related to the spectral composition of skylight. As mentioned by Littlefair [9] that Rayleigh scattering by air molecules involves removal of the lowest wavelengths of radiation from the direct beam and clear skies exhibit high luminous efficacies. In view of the fact that the luminous efficacy is of a narrow range for a given sky type, a mean value of the luminous efficacy can be specified for each CIE standard sky. Table 2 summarizes the average values with their standard deviations of the experimental sky-diffuse luminous efficacies for the 15 CIE standard skies. The mean values range from 108.6 lm/W with a standard deviation of 9.5 lm/W for sky standard 6 to 137.3 lm/W with a standard deviation of 17.7 lm/W for sky standard 13. As expected, the cloudless skies have large mean and standard deviation values. Since the standard derivations for individual skies are relative small, a constant value of the luminous efficacy in each sky type can be used to predict the outdoor illuminance. The differences of mean KD are within the standard deviations for sky standards 1–10 and an average KD may be used to represent the overcast and partly cloudy skies. 5. Daylight illuminance prediction For daylighting applications, it is desirable to have solar irradiance and daylight illuminance data on inclined surfaces. The daylight illuminance on an inclined plane

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Table 2 Diffuse luminous efficacy KD (lm/W) for 15 CIE standard skies Standard skies

KD S.D.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

115.4 14.6

112.0 12.3

113.9 12.2

108.9 11.1

113.6 11.1

108.6 9.5

111.0 10.4

114.7 12.0

111.1 9.7

111.5 9.5

121.9 13.6

120.3 14.4

137.3 17.7

127.0 19.4

124.8 14.2

with an inclination angle of b (EbG) can be evaluated as the sum of direct beam, sky-diffuse and ground-reflected components. For general applications, b can be from 01 (horizontal) to 901 (vertical) and the following formulae illustrate the approach for a vertical surface (i.e., b ¼ 901). Using luminous efficacy approach, it can be given as E bG ¼ E bB þ E bD þ E bR

(5)

E bG ¼ I bG K bG

(6)

where EbB is the direct-beam illuminance on a plane (lx), EbD the sky-diffuse illuminance on a plane (lx), EbR the ground-reflected illuminance on a plane (lx), IbG the global solar irradiance on a plane (W/m2) and KbG is the global luminous efficacy on a plane (lm/W). The global luminous efficacy on a vertical surface KvG (i.e., b ¼ 901) can be expressed as K vG ¼

E vG E vB þ E vD þ E vR ¼ I vG I vB þ I vD þ I vR

(7)

K vG ¼

I vB E vB I vD E vD I vR E vR þ þ I vG I vB I vG I vD I vG I vR

(8)

where EvG is the global illuminance on a vertical plane (lx), EvB the direct-beam illuminance on a vertical plane (lx), EvD the sky-diffuse illuminance on a vertical plane (lx), EvR the ground-reflected illuminance on a vertical plane (lx), IvG the global irradiance on a vertical plane (W/m2), IvB the direct-beam irradiance on a vertical plane (W/m2), IvD the sky-diffuse irradiance on a vertical plane (W/m2) and IvR is the ground-reflected irradiance on a vertical plane (W/m2). Owing to the strong forward scattering effect of aerosols, the sky-diffuse component tends to be anisotropic. This means that different surfaces would receive different amounts of sky-diffuse components. As solar irradiance and outdoor illuminance have very similar characteristics in nature, methods for predicting solar irradiance on an inclined surface can also be adopted in daylight calculation [30]. In terms of luminous efficacy, the variations would be very small from one surface to another. It is reasonable to assume that the same amount of sky-diffuse luminous efficacy can be considered for both the horizontal and vertical surfaces. By giving the position of the sun and orientation of the plane, it is quite straightforward to calculate the direct-beam component on a vertical surface using solar geometry. For estimating the ground-reflected component, the vertical surface is considered to receive half

of the global component reflected isotropically from the ground. Assuming that the same average ground reflectivity is used for ground-reflected irradiance and illuminance determinations, the amount of ground-reflected luminous efficacy would be equal to the global luminous efficacy on a horizontal plane. Mathematically, IvB, IvR, EvB and EvR can be expressed as follows: I vB ¼

I hB ðcos a cos gÞ sin a

E vB ¼

E hB I hB ðcos a cos gÞ ¼ ðcos a cos gÞK B sin a sin a

(9) (10)

I vR ¼ 0:5rI hG

(11)

E vR ¼ 0:5rI hG K G

(12)

where IhB is the direct-beam irradiance on a horizontal plane (W/m2), IhG the global irradiance on a horizontal plane (W/m2), g the different between solar azimuth and azimuth angle of the normal of the surface (1), r the ground reflectance (dimensionless) and KG is the global luminous efficacy on a horizontal plane (lm/W). Accordingly, vertical global luminous efficacy can be regarded as a combination of the luminous efficacies of global, diffuse and direct on a horizontal surface with different proportions. Rewriting Eq. (8), it becomes K vG ¼

I vB I vD I vR KB þ KD þ KG I vG I vG I vG

(13)

Most researchers proposed polynomial models of different degrees of a for calculating KB [31]. Previous work [24] suggested the following quadratic regression expression to determine KB: K B ¼ 59:15 þ 1:12a  0:0061a2

(14)

Horizontal global component is the sum of direct and diffuse data and the trends in its luminous efficacy can be roughly predicted on this basis. KG can be regarded as a combination of KB and KD that can be written as KG ¼

E hG E hB þ E hD K B I hB þ K D I hD ¼ ¼ I hG I hG I hG

K G ¼ K B ð1  KÞ þ K D K

(15) (16)

where K is diffuse fraction ¼ (IhD/IhG). Using the derived equations, the vertical global luminous efficacy can be determined from the horizontal measurements. As the scanning pattern divided the whole sky dome

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into 145 points, it is possible to determine IvD numerically once the standard sky types were identified [32,33]: P145 Li cos2 y cos f dy df I vD ¼ I hD Pi 145 (17) i Li sin y cos y dy df Accordingly, the IvG and KvG were estimated. To examine the orientation effects, the cumulative frequency distributions of the vertical global efficacy for the north, east, south and west in 2004 were calculated and are presented in Fig. 6. Relatively low luminous efficacy values are obtained when the vertical surfaces receive direct components. With a luminous efficacy less than 110 lm/W, which is due to the presence of a direct component, the north-facing surface has the highest cumulative percentage (mainly diffuse component); while the west-facing plane collects certain amount of direct components in late afternoon giving a smaller percentage value. With luminous efficacy values greater than 110 lm/W, all vertical surfaces indicate a rapid drop in the cumulative percentage. Such a high efficacy 100 North East South West

Cumulative frequency (%)

90 80 70 60 50 40 30 20 10 0 60

70

80

90 100 110 120 130 Global luminous efficacy (lm/W)

140

150

Fig. 6. Cumulative frequency distribution of vertical global luminous efficacy for the four cardinal orientations using predict vertical solar irradiance.

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region corresponds to the diffuse-dominant condition, in which there is little to no direct component. To examine the performance of the proposed luminous efficacy approach, model evaluation with respect to the vertical global daylight illuminance prediction is required. Two widely used statistics, viz. the mean bias error (MBE) and RMSE were used for the assessment. Again, the vertical daylight illuminance for the four principal vertical surfaces (i.e., N, E, S and W) recorded in 2004 was predicted for the comparative study. The global outdoor illuminance on a vertical surface was determined based on the findings in Fig. 6 with the predicted vertical solar irradiance data and evaluated against the measured vertical daylight illuminance data of the same period. The common ground reflectivity of 0.2 (i.e., r ¼ 0.2) [34] was employed for analysis and Table 3 summarizes the MBE and RMSE results for each month as well as the whole year at the four cardinal vertical planes. The statistical analysis shows that in most cases, the vertical daylight illluminance data were over-predicted. The MBE can be up to 34.5% appearing in March for east-facing surface. The computed RMSE values show that the vertical daylight data were estimated more accurately in winter months than those in summer months. The RMSE ranges between 8.1% in December for south-facing plane and 48.4% in March for east-facing plane. A detailed examination of the results found that a is of large values in summer months and the errors were due to direct-beam component (KD) at high a. Also, skies standards of 13 and 14 with relative large KD variations appeared more frequently in summer months. The annual RMSE results are around 38% for the north- and eastfacing surfaces, and about 30% for the south and east vertical planes. It is probably simpler, and possibly more accurate, to model illuminances directly [29]. Likewise, the KvG were estimated using measured IvG. Similar pattern as that shown in Fig. 6 was found. Again, the vertical daylight illuminance data for the four principal orientations were calculated using measured vertical solar irradiance and the MBE and RMSE are

Table 3 Summary of %MBE and %RMSE for global illuminance on vertical surfaces using modeled vertical irradiance data January

February

March

April

May

June

July

August

September

October

November

December

Year

North %MBE %RMSE

9.1 14.8

22.7 33.6

32.6 41.5

32.2 43.0

28.7 39.2

30.4 40.1

25.5 37.5

21.9 35.0

22.4 31.9

11.6 21.4

6.0 23.0

0 17.5

25.6 38.0

East %MBE %RMSE

23.4 33.8

22.4 35.1

34.5 48.4

28.6 42.5

21.4 37.9

22.8 38.8

22.2 38.3

19.2 35.6

22.1 35.4

24.6 37.3

19.9 33.6

13.4 23.9

23.5 38.4

South %MBE %RMSE

8.5 11.4

15.8 23.9

28.4 40.6

30.2 44.0

24.2 37.0

23.3 35.2

19.4 33.0

23.6 39.0

26.0 36.0

17.3 22.0

9.6 16.0

2.3 8.1

21.0 31.8

West %MBE %RMSE

7.2 12.3

11.3 25.2

22.8 32.3

20.5 31.7

19.0 33.0

19.9 33.1

15.6 31.2

17.6 30.1

17.3 30.9

7.9 17.7

1.6 21.2

7.1 18.9

16.1 29.5

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Table 4 Summary of %MBE and %RMSE for global illuminance on vertical surfaces using measured vertical irradiance data January

February

March

April

May

June

July

August

September

October

November

December

Year

North %MBE %RMSE

1.4 7.0

1.8 9.7

4.6 10.6

5.7 12.8

6.8 14.9

4.4 14.4

10.4 18.6

12.5 19.2

8.6 13.5

4.3 8.0

1.6 7.9

1.0 11.0

6.6 15.1

East %MBE %RMSE

6.2 13.5

1.7 13.2

3.0 17.7

2.1 15.8

0.7 10.0

3.1 12.2

1.5 11.5

3.8 12.5

3.9 11.9

0.04 9.8

0.1 12.2

1.1 8.9

0.03 12.7

South %MBE %RMSE

8.1 10.4

2.6 8.5

4.1 11.8

2.8 12.8

9.2 14.2

10.2 15.2

12.7 17.2

9.5 15.3

1.3 9.5

4.6 8.6

1.4 9.9

2.4 6.8

1.9 12.0

West %MBE %RMSE

3.1 9.2

2.0 15.1

1.1 17.8

1.4 16.6

2.4 17.0

1.2 16.7

4.8 16.6

1.9 16.5

2.1 18.2

0.1 14.5

3.5 18.3

8.8 14.8

1.5 16.7

presented in Table 4. With measured vertical solar irradiance data, there are significant improvements in the prediction accuracy, particularly for the east-facing surface. This is because it is a one-step algorithm rather than the two-step routine. The MBE varies from an underestimation of 12.7% in July to an overestimation of 8.1% in January both appearing in south-facing vertical surface. In general, the vertical luminous efficacy models tend to underestimate the vertical global illuminance particularly for north-facing plane which receives mainly diffuse components. It indicates that the proposed KD may underestimate the daylight illuminance collected in 2004. The monthly average RMSE values do not differ a great deal ranging between 7% and 19.2%. In general, the RMSE results in summer months are larger than those in winter months. For the whole year, the RMSE values for north-, east-, south-, and west-facing surfaces are 15.1%, 12.7%, 12% and 16.7%, respectively. With the peak monthly RMSE less than 20%, the predictive ability of the proposed approach is considered acceptable. 6. Conclusions An approach to estimate the diffuse luminous efficacy under the 15 CIE standard skies using measured data from 1999 to 2005 has been conducted. Based on the best-fitting of the standard set determined, the sky types for subtropical Hong Kong were identified. A constant value of the diffuse luminous efficacy (KD) for each standard sky was proposed. It was found that KD ranged from just less than 109 lm/W for sky standard 6 to over 137 lm/W for sky standard 13. Subsequently, luminous efficacy for inclined surfaces were developed and used to predict the vertical luminous efficacy and daylight illuminance for the four cardinal orientations. The estimated vertical outdoor illuminance data were evaluated against measured values in 2004. Based on the identified standard skies, the vertical solar irradiance data were estimated to compute the vertical daylight illuminance. Generally, the vertical outdoor illuminance data were over-predicted with the MBE

up to 34.5%. For the whole year, the RMSE values are around 38% for the north- and east-facing surfaces, and about 30% for the south and west vertical planes. When measured vertical solar irradiance data were used, the predictive ability of the luminous efficacy model performs noticeably better. The estimated results show reasonable good agreements with the measured vertical daylight illuminance data. The annual RMSE was found ranging from 12% to 16.7% and the monthly RMSE less than 20%. With measured solar irradiance data, it would be appropriate to predict the corresponding illuminance values using luminous efficacy approach. When measured solar irradiances are not available, it is probably simpler, and possibly more accurate, to model illuminances directly. Luminous efficacy is a useful parameter in building energy study, especially in the calculation of daylight illuminances for places with only solar irradiance data. The proposed diffuse luminous efficacy was developed under the 15 CIE standard skies which are considered covering the whole probable spectrum of skies in the world. Further research study is required to fit the model for different locations with diverse climates, to select various appropriate climatic indicators for sky classification and to investigate the atmospheric elements under individual sky types to improve the luminous efficacy models. Acknowledgments The work described in this paper was fully supported by a Competitive Earmarked Research Grant from the Research Grants Council of the Hong Kong Special Administrative Region, China [Project no. 9041140 (CityU 116506)]. T.N.T. Lam and H.L. Tang are supported by a City University of Hong Kong studentship. References [1] Li DHW, Cheung GHW, Lam JC. Analysis of the operational performance and efficiency characteristic for photovoltaic system in Hong Kong. Energy Convers Manage 2005;46(7,8):1107–18.

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