Improving the All Sky Model for the luminance and radiance distributions of the sky

Available online at www.sciencedirect.com

ScienceDirect Solar Energy 105 (2014) 354–372 www.elsevier.com/locate/solener

Improving the All Sky Model for the luminance and radiance distributions of the sky Norio Igawa ⇑,1 Graduate School of Human Life Science, Osaka City University, Japan Received 31 January 2013; received in revised form 9 March 2014; accepted 12 March 2014

Communicated by: Associate Editor David Renne

Abstract Energy conservation is a fundamental requirement in buildings today. It is essential to use technology that integrates the thermal environment and the luminous environment to ensure indoor environmental quality. It is important to understand as accurately as possible the distribution of luminance and radiance of the daytime sky. Though the CIE Standard General Sky for sky luminance distributions has been recommended, a method of selecting 15 Sky Types was not shown. In this paper, the author tries to improve the previous All Sky Model based on IDMP measurement data from Osaka with the aim of a high accuracy estimation method for luminance and radiance distributions of the sky. The appearances of the Types of CIE Standard General Sky and the assignment in estimating types are shown based on the measurement data in Osaka. The improved All Sky Model and the validation of this improved model using the measurements in Osaka and Tokyo, Japan was introduced. Ó 2014 Elsevier Ltd. All rights reserved.

Keywords: Luminance distribution; Radiance distribution; Optimized All Sky Model; Measurement

1. Introduction Energy conservation is a fundamental requirement in buildings today. It is essential to use technology that integrates the thermal environment and the luminous environment to ensure indoor environmental quality. Therefore, it is important to understand the real phenomenon that determines the luminance and radiance distributions of sky as accurately as possible. There are many equations for sky radiance distribution, including those of Harrison and Coombes (1988), Harrison

⇑ Address: 2591-1, Higashi-Agenosho, Suo-Oshima, Oshima, Yamaguchi 742-2805, Japan. Tel.: +81 820 77 2567. E-mail address: [email protected] 1 ISES member.

http://dx.doi.org/10.1016/j.solener.2014.03.020 0038-092X/Ó 2014 Elsevier Ltd. All rights reserved.

(1991), Brunger and Hooper (1993), Nakamura et al. (1997a,b,c) and Nagata (1997). UVA, UV-B, and PAR sky distributions were proposed by Grant et al. (1996). Moon and Spencer (1942) proposed a luminance distribution model for the overcast sky which has been recommended as the CIE Standard Overcast Sky (1955). Kittler (1967) proposed a luminance distribution model for the clear sky which has been recommended as the CIE Standard Clear Sky (CIE, 1973). Littlefair (1981) proposed the BRE Average Sky. Nakamura et al. (1985, 1987) proposed the Intermediate Sky. Kittler (1985, 1986) proposed the Homogeneous Sky. Perraudeau (1988) proposed an equation of sky luminance distribution. Perez et al. (1993) proposed the All-weather Model (Aw) as functions of Sky clearness and Sky brightness. Kittler et al. (1997) classified all the sky conditions into 15 categories and proposed numerical equations; afterwards, CIE (2003)

N. Igawa / Solar Energy 105 (2014) 354–372

355

Nomenclature Ce cloud ratio (diffuse fraction) Ces standard cloud ratio Cle cloudless index Eed diffuse irradiance (W/m2) Eeo extraterrestrial normal irradiance (W/m2) Eeg global irradiance (W/m2) Evd diffuse illuminance (klx) f(f) scattering indicatrix function Kc clear sky index Le(cs, c, f) relative sky radiance distribution Lea(cs, c, f) sky radiance distribution (W/m2/sr) Lez(cs) zenith radiance (W/m2/sr) Lv(cs, c, f) relative sky luminance distribution Lva(cs, c, f) sky luminance distribution (kcd/m2) Lvz(cs) zenith luminance (kcd/m2) LzEd inverse of the integration value of relative sky radiance (luminance) distribution m relative optical air mass MBE mean bias error RMSE root mean square error Seeg standard global irradiance (W/m2)

recommended this model as the CIE Standard General Sky (Gs). Igawa et al. (2004) proposed All Sky Model (As) for luminance and radiance distributions of sky. Several researchers (Kittler and Darula, 2006; Wittkopf and Soon, 2007; Markou et al., 2007; Li and Tang, 2008; Kobav et al., 2009) examined frequency of occurrence of the 15 types for the practical use of the CIE Standard General Sky. Hosobuchi et al. (2006) tried to identify the Sky Types using measured solar radiation. However a practical method of assigning the CIE 15 Types was not yet completed. Previous good estimation accuracies for As and Aw were shown in comparisons with measured data by Igawa et al. (2004), but the results for As and Aw are not fully satisfactory. Based on the measurement data in Osaka of 2007, yearly RMSE for luminance distribution of As and Aw were 1.358 (kcd/m2) and 1.369 (kcd/m2) respectively. Yearly RMSE for radiance distribution of As and Aw were 11.59 (W/m2/sr) and 11.36 (W/m2/sr) respectively. There is a problem in the equation of As when the global irradiance is larger than the global irradiance of a clear sky. Estimation accuracy of Aw was not good for a cloudy sky. Therefore, an improvement of the All Sky Model was tried for estimating highly accurate sky radiance and luminance distributions based on the measurement data of IDMP in Osaka. In this paper, the frequencies of occurrence of types of the CIE Standard General Sky were shown based on the measurement data. An improved All Sky Model and its validations were introduced.

Si Siv

sky index of the previous All Sky Model (=Kc + Cle0.5q ) ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 sky index (¼ ð1:0  KcÞ2 þ ð1:0  Cle0:5 Þ )

Greek symbols a azimuth angle of the sky element (rad) as azimuth angle of the sun (rad) c altitude of the sky element (rad) cs solar altitude (rad) f angular distance between the sun and the sky element (rad) u(c) gradation function Symbols of sky models Am All Sky Model divided into 20 kinds at 0.1 intervals of Si As All Sky Model by Igawa et al. (2004) Aw All weather model by Perez et al. (1993) Gs CIE Standard General Sky by CIE (2003) i-As improved All Sky Model proposed in this paper

2. Measurement system and method of coordinating measurement data The solar radiation and daylight measurement station was installed at Osaka City University (34°360 N, 135°300 E) and started measurement on January 1, 2006. The main measurement quantities were shown in Table 1, referring to the guide of IDMP of CIE (1994). General meteorological quantities such as the irradiance, the illuminance, temperature, humidity, and wind are measured in 10 s intervals. The 1 min interval data which is required in the guide of IDMP is arranged as a mean value of the 10 s interval data from 30 s before and after each point. The sky luminance and the radiance distributions are measured in 10 min intervals from sunrise to sunset. Since about four and a half minutes are required for one measurement cycle of the sky luminance and radiance distributions, the 1 min data were matched to the 10 min data averaging from the beginning time of the scanning to 4 min later. The solar altitude corresponding to the 10 min data was calculated based on 2 min after the beginning of each scan cycle. The measurement data in Osaka from January 2007 to December 2008 was used as the basic data for modeling. The measurement data in Osaka from January 2006 to December 2011 was applied for the validation of models concerning the luminance and radiance distributions of sky. The measurement data in Tokyo IDMP station

356

N. Igawa / Solar Energy 105 (2014) 354–372

Table 1 Measurement list. No.

Quantities

Symbols

Sensors and manuf.

Intervals

1

Irradiance

Global

Eeg

1 min from 1-January-2006 to 29-October-2006

2

Diffuse

Eed

MS-802 (EKO) ! CMP11 (Kipp & Zonen)a MS-802 (EKO)

3

Direct normal Vertical north Vertical east Vertical south Vertical west Global

Ees

MS-53 (EKO)

Eegn

9

Diffuse

Evd

MS-802 (EKO) ! CMP11 (Kipp & Zonen)b MS-802 (EKO) ! CMP11 (Kipp & Zonen)b MS-802 (EKO) ! CMP11 (Kipp & Zonen)b MS-802 (EKO) ! CMP11 (Kipp & Zonen)b ML-010S (EKO) ! BAP 30 FCT (LMT)a ML-010S (EKO)

10

Direct normal Vertical north Vertical east Vertical south Vertical west

Evs

ML-010SD (EKO)

Evgn

ML-010S (EKO) ! PHOT02 (Delta OHM)b ML-010S (EKO) ! PHOT02 (Delta OHM)b ML-010S (EKO) ! PHOT02 (Delta OHM)b ML-010S (EKO) ! PHOT02 (Delta OHM)b MS-321LR (EKO) 10 min intervals MS-53 (EKO)

4 5 6 7 8

Illuminance

11 12 13 14

Eege Eegs Eegw Evg

Evge Evgs Evgw

15

Zenith luminance

Lvz

16

Sunshine duration

S

17 18 19 20 21

Air temperature Relative humidity Wind direction Wind speed Atmospheric pressure Amount of rain fall Sky radiance distribution Sky luminance distribution Sky image

Dbt Rh Wd Ws Ap

HMP45A (VAISALA)

Ar Le

MW-010 (0.5 mm/puls) (EKO) MS-321LR (EKO)

22 001-145 201-345 800 a b

STR22 (EKO: sun tracker)

STR22 (EKO: sun tracker)

10 s from 30-October-2006 to present

STR22 (EKO: sun tracker)

SAT-530 (Kaijo Sonic) MY-021 (VAISALA)

10 min

Lv Image

PSV-100 (Prede)

Replaced at the end of 2009. Replaced at the end of 2010.

(35°360 N, 139°300 E) from March 1992 to September 1993 is also applied for the validation of sky models concerning the luminance distribution. The measurement data in Tokyo from January 1993 to September 1993 was applied for the validation of sky models concerning the radiance distribution. When the measurement circumstances meet the following conditions, all data were excluded from the modeling and the validation. The effectively used data was 96.3% of all the measurement sets. (1) When any one datum out of the 145 points for luminance and radiance distributions shows zero or negative in the measurement.

(2) When a shadow covers the sensor temporarily by a part of the surrounding buildings or an unconfirmed object, etc. (3) When the sensor is covered by a flying object or birds. (Transparent fishing lines were arranged above the sensor in parallel at 30 cm intervals at the end of 2010 to prevent the approach of birds.) (4) When a remarkable abnormality is seen in the measurement data. (5) The real open angle of the luminance (radiance) sensor is wider than the11° which is recommended by CIE and also indicated in the catalogue. The data for under 6° altitude of the sky element and the data

N. Igawa / Solar Energy 105 (2014) 354–372

357

Fig. 1. Frequency of occurrence of 15 Types of CIE Standard General Sky in luminance distributions in Osaka (from January 1, 2007 to December 31, 2008).

Fig. 2. Frequency of occurrence of 15 Types of CIE Standard General Sky in radiance distributions in Osaka (from January 1, 2007 to December 31, 2008).

of smaller than 18° of angular distance from the sun could not be measured and thus is excluded from the modeling and validation.

The data coordinated by the above-mentioned methods were assumed as the basic data for this research work.

358

N. Igawa / Solar Energy 105 (2014) 354–372

Table 2 Frequency of occurrence of CIE Sky Type for luminance distribution. Sky Type

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Occurrence area

Freq.

Kc

Cle

0.1–0.3 0.1–0.4 0.1–0.4 0.4–0.5 0.1–0.3 0.4–0.5 0.4–0.7 0.4–1.0 0.1–0.4 0.3–1.0 0.6–0.9 0.9–1.0 0.9 0.9 0.9

0.0 0.0 0.0 0.0 0.0 0.0 0.0–0.3 0.0–0.8 0.0 0.0–0.8 0.3–0.8 0.9–1.0 0.7–0.8 0.9 0.7–0.8

Total a

N 4537 1331 3606 2337 1444 2635 2119 2790 1115 1107 1977 3188 4875 7042 1804 41,907

Max. freq. and area % % 10.8a 3.2b 8.6b 5.6 3.4a 6.3 5.1 6.7 2.7b 2.6 4.7 7.6 11.6c 16.8 4.3c

%

Kc

Cle

29.9 15.6 23.5 23.6 23.7 16.8 9.2 4.6 10.9 5.3 7.4 16.0 20.9 25.7 23.6

0.20 0.30 0.20 0.40 0.20 0.40 0.50 0.90 0.30 1.00 0.80 0.90 0.90 0.90 0.90

0.00 0.00a 0.00b 0.00c 0.00b 0.00c 0.00 0.70 0.00a 0.80 0.50 1.00 0.80d 0.90 0.80d

100.0

Repetition of Kc–Cle areas.

Table 3 Frequency of occurrence of CIE Sky Type for radiance distribution. Sky Type

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Occurrence area Kc

Cle

0.1–0.3 0.2–0.4 0.1–0.4 0.4–0.5 0.1–0.3 0.4–0.5 0.4–0.7 0.4–1.0 0.1–0.4 0.3–1.0 0.7–0.9 0.9–1.0 0.9 0.9 0.9

0.0 0.0 0.0 0.0 0.0 0.0 0.0–0.3 0.0–0.9 0.0 0.0–1.0 0.4–0.9 0.8–1.0 0.8–0.9 0.9 0.7–0.8

Total a

Freq. N 4769 1808 3246 2565 1380 2391 2096 3406 1119 1104 2693 2726 5710 5154 1740 41,907

Max. freq. and area % % 11.4a 4.3 7.7b 6.1 3.3a 5.7 5.0 8.1 2.7b 2.6 6.4 6.5 13.6 12.3 4.2

%

Kc

Cle

29.3 15.7 24.2 22.8 23.9 16.4 8.0 5.1 9.6 5.1 7.4 11.6 19.7 22.6 22.1

0.20 0.40 0.10 0.40 0.10 0.40 0.50 0.90 0.30 0.90 0.90 0.90 0.90 0.90 0.90

0.00 0.00a 0.00b 0.00a 0.00b 0.00a 0.00 0.70 0.00 0.70 0.90c 0.90c 0.80d 0.90c 0.80d

100.0

Repetition of Kc–Cle areas.

Fig. 3. Frequency of occurrence of CIE Sky Type in Osaka for two years from January 1, 2007 to December 31, 2008 about sky luminance distribution.

Fig. 4. Frequency of occurrence of CIE Sky Type in Osaka for two years from January 1, 2007 to December 31, 2008 about sky radiance distribution.

N. Igawa / Solar Energy 105 (2014) 354–372

359

3. Indices to shows sky conditions

Table 4 Constants of CIE Standard General Sky.

In general, the illuminance and the luminance distributions of sky are not measured at meteorological observatories. Therefore, there are lots of proposed methods of estimating the luminance distribution, the radiance distribution, the illuminance, UV, and PAR, etc. based on the irradiance data. The irradiance is useful basic data as the index to show various meteorological sky conditions. At meteorological stations, when the direct and diffuse irradiances are not measured, global irradiance is measured in many cases. In this case, direct and diffuse irradiances are obtained by dividing the global irradiance. Since the global irradiance is the raw data without conversion, it is used as a basis for an index. As indices to show the sky condition, previously Igawa defined Kc and Cle based on the clear sky condition is assumed with a Linke turbidity factor of 2.5. Considering the Linke luminous turbidity factor of 2.45 for CIE Standard Clear Sky by Kittler, the energetic turbidity factor for a clear sky should be lower than 2.45. Here, the Linke energetic turbidity factor for a clear sky was taken as 2.0 for wider correspondence with sky the conditions because they yield the best coincidences for modeling. Since Cle becomes one or more when energetic turbidity factor is less than 2.0, the combination of Kc and Cle can be accepted for worldwide use. This is one of the reasons that the previous All Sky Model should be improved. Kc and Cle are shown as follows.

No.

Symbol

Type

a

b

c

d

e

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Gs1 Gs2 Gs3 Gs4 Gs5 Gs6 Gs7 Gs8 Gs9 Gs10 Gs11 Gs12 Gs13 Gs14 Gs15

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

4.0 4.0 1.1 1.1 0.0 0.0 0.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

0.70 0.70 0.80 0.80 1.00 1.00 1.00 1.00 0.55 0.55 0.55 0.32 0.32 0.15 0.15

0.0 2.0 0.0 2.0 0.0 2.0 5.0 10.0 2.0 5.0 10.0 10.0 16.0 16.0 24.0

1.0 1.5 1.0 1.5 1.0 1.5 2.5 3.0 1.5 2.5 3.0 3.0 3.0 3.0 2.8

0.00 0.15 0.00 0.15 0.00 0.15 0.30 0.45 0.15 0.30 0.45 0.45 0.30 0.30 0.15

Kc ¼ Eeg=Seeg

ð1Þ

Cle ¼ ð1  CeÞ=ð1  CesÞ

ð2Þ

Seeg ¼ 0:84  Eeo=m  expð0:054  mÞ

ð3Þ

Ces ¼ 0:08302 þ 0:5358  expð17:3  cs Þ þ 0:3818  expð3:2899  cs Þ

ð4Þ

where Eeg is the global horizontal irradiance (W/m2), Seeg the standard global irradiance (W/m2), Ce the cloud ratio (=Eed/Eeg), Ces the standard cloud ratio, Eeo the extraterrestrial normal irradiance (W/m2), m the relative optical air mass (Kasten and Young, 1989), Eed the horizontal diffuse irradiance (W/m2) and cs is the solar altitude (rad).

Table 5 Constants of All Sky Model temporarily divide in to 20. No.

Symbol

Si

a

b

c

d

e

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Am1 Am2 Am3 Am4 Am5 Am6 Am7 Am8 Am9 Am10 Am11 Am12 Am13 Am14 Am15 Am16 Am17 Am18 Am19 Am20

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

2.68 2.43 2.14 1.80 1.43 1.05 0.68 0.33 0.03 0.22 0.42 0.58 0.71 0.80 0.86 0.91 0.95 0.97 0.99 1.01

0.89 0.87 0.85 0.83 0.80 0.78 0.75 0.73 0.70 0.67 0.63 0.60 0.57 0.54 0.51 0.47 0.44 0.41 0.38 0.35

0.00 0.02 0.08 0.23 0.49 0.91 1.52 2.35 3.42 4.74 6.29 8.03 9.92 11.84 13.65 15.14 16.04 15.93 14.24 10.02

0.36 0.48 0.63 0.82 1.04 1.28 1.54 1.80 2.04 2.25 2.44 2.59 2.70 2.80 2.86 2.92 2.95 2.98 3.00 3.01

0.00 0.00 0.01 0.01 0.01 0.02 0.03 0.05 0.07 0.10 0.13 0.18 0.22 0.27 0.32 0.36 0.39 0.42 0.44 0.45

is assumed to be the representative Type in the Kc–Cle area. Since the integration value in which the configuration factor and p are multiplied by the measurement value of

4. Frequency of occurrence of the 15 Types of CIE Standard General Sky The frequency of occurrence of the 15 Types of CIE Standard General Sky in the area classified with Kc and Cle is confirmed based on the measurement data in Osaka for two years from 2007 to 2008. Kc is divided into 0.1 (±0.05) intervals within the range from 0 to 1.3. Cle is divided into 0.1 (±0.05) intervals within the range from 0 to 1.2. All the measured data are classified into Kc-Cle areas by using the matrix of Kc and Cle. Root Mean Square Error (RMSE) between the measured skies and 15 Types in the Kc–Cle area were calculated. The Type for which the RMSE is the smallest

Fig. 5. Replace of constant c of previous All Sky Model to c2 when Si is larger than 2.0.

360

N. Igawa / Solar Energy 105 (2014) 354–372

Table 6 Selected models in Kc–Cle areas (solar altitude 35 deg).

Table 7 RMSE of selected model in Kc–Cle areas (solar altitude 35 deg). Kc

Cle 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

0.1

0.2

0.3

0.4

0.5

0.6

0.041 0.073

0.048

0.038 0.149 0.141

0.041 0.094 0.150 0.183

0.053 0.087 0.103 0.138 0.166

0.076 0.085 0.111 0.090 0.116 0.111 0.104

0.085

the sky luminance distribution of 145 points shows a small difference with the horizontal diffuse illuminance, measured values of sky luminance are corrected by multiplying a horizontal diffuse illuminance/integration value. The sky radiance distribution is also corrected similarly. To avoid the influence of the direct solar irradiance, measurements of the luminance and radiance within 18 deg or less angular distance between the sun and the sky element are excluded from the calculation of RMSE and MBE. The Kc–Cle areas where 15 of Gs Type appeared are shown in Figs. 1 and 2 based on the measurements of luminance and radiance distributions of sky acquired for two years in Osaka. As for the frequencies of occurrences of Types in the sky radiance distribution, nearly similar characteristics were obtained though they were not the same as the sky luminance distributions at all. The frequencies of occurrence of the 15 Types in the sky luminance are shown in Table 2 and their frequencies of occurrence in the sky radiance are

0.7

0.080 0.054 0.106 0.119 0.104 0.091 0.099

0.8

0.077 0.097 0.109 0.112 0.084 0.060 0.071 0.036

0.9

1.0

1.1

0.124 0.112 0.122 0.082 0.060 0.059 0.040 0.136

0.147 0.097 0.131 0.107 0.111 0.075 0.119

0.080 0.092 0.181 0.141 0.149 0.196 0.186

1.2

0.135 0.193 0.214 0.231

shown in Table 3. The frequencies of occurrence of CIE Types are shown in Figs. 3 and 4. In the above mentioned examinations, the appearance Kc–Cle areas of each Type are defined where the occurrence frequency is the highest or more than 50% of the highest value. Here, the appearance circumstances of the sky luminance distributions are considered. The CIE Standard

Table 8 Constants for improved All Sky Model.

A B C D E F G H

a

b

c

d

e

1.0193 0.0955 0.0823 0.4530 0.1294 0.2876 0.3169 6.4046

0.3646 0.8806 1.6503 0.3319 0.6525 0.2681 0.5434 12.3328

3.3246 1.8413 0.8436 0.3009 8.3642 0.8183 0.5424 9.1901

3.8472 2.1573 0.5050 0.6257 61.0275 3.2725 1.2096 31.1039

0.6370 0.5995 1.0259 1.3334 0.0022 1.0765 0.7066 0.5187

Note: if b > 0, b = 0; if c < 0, c = 0; if e < 0, e = 0.

N. Igawa / Solar Energy 105 (2014) 354–372

361

Table 9 Constants for LzEd. k

j

i 4

3

2

5

6 5 4 3 2 1 0

5.6146 17.9921 20.0121 12.0503 8.2042 2.2514 0.4774

29.4046 93.4316 103.1918 55.2228 28.2605 7.3074 1.2853

47.2024 142.8905 142.9116 58.2657 23.5534 5.7338 0.8565

43.8510 130.9200 130.0067 49.5379 13.0987 2.4593 0.2806

8.2509 17.7456 3.1167 14.3877 9.0805 2.3038 0.1641

0.9358 2.6364 3.7005 3.5037 2.2572 1.2745 0.7447

4

6 5 4 3 2 1 0

17.2129 63.0588 86.5230 64.5195 36.9118 8.3944 1.6652

85.8973 298.9370 382.9478 250.6187 122.2518 26.3761 4.5943

129.4606 420.7243 477.7507 249.3821 103.4001 19.1065 3.1165

125.4744 391.1156 419.8383 189.4251 56.5677 8.7967 1.4959

16.6675 25.7323 28.0500 70.2059 38.5437 9.4755 0.5221

1.7011 8.4401 10.4232 1.0365 4.9664 3.6080 1.9573

3

6 5 4 3 2 1 0

21.5603 88.8005 140.5464 115.2602 58.4325 12.5318 1.7622

98.3234 376.6700 549.7882 408.1553 188.1080 38.1286 5.0850

133.2000 473.6141 617.7442 389.1329 158.1039 26.3229 2.9477

134.7364 443.8715 524.2791 279.5759 90.2370 14.5404 2.1838

5.7213 15.9462 92.1837 121.5988 60.4685 13.3797 0.5745

7.9890 31.5361 41.4865 18.9449 0.8295 2.5300 1.2611

2

6 5 4 3 2 1 0

16.1603 68.1074 110.3658 88.4298 39.1455 8.5411 0.5530

62.0261 249.5476 384.7705 291.6143 122.2380 25.5973 1.8213

68.6303 263.2480 376.5734 255.1865 95.2499 17.1831 0.3930

66.7874 233.4506 301.1853 180.4192 60.1343 11.9369 1.0051

9.3995 51.2836 105.3289 100.9524 43.8912 7.4727 0.2158

8.0240 30.4587 41.6451 24.4274 5.8629 0.8271 0.0791

1

6 5 4 3 2 1 0

5.6538 22.4881 34.5496 26.0768 10.1609 1.4801 0.0550

18.5946 72.5977 109.0127 80.1132 30.7499 4.7414 0.2373

15.3888 58.6626 83.4590 55.9029 19.0722 1.9300 0.0316

15.0642 54.7188 75.1759 49.8447 17.7449 2.6996 0.0642

6.8261 28.0338 45.1168 34.7254 11.9372 1.2676 0.0032

2.4525 9.9369 15.8059 12.6379 5.3456 1.0207 0.0227

0

6 5 4 3 2 1 0

0.8791 2.7495 3.0179 1.1932 0.0024 0.0089 0.0018

3.2070 10.1893 11.6684 5.4566 0.7879 0.1344 0.0124

2.8856 8.5197 8.6199 3.0029 0.0560 0.1890 0.0062

3.0796 10.6148 14.0185 8.7173 2.4222 0.1446 0.0134

0.2823 1.0694 1.3755 0.5736 0.1517 0.1348 0.0078

0.1061 0.2046 1.7036 2.7262 1.4338 0.1598 0.4086

5

Clear Sky by Kittler (CIE, 1973) is assumed as a typical clear sky appears in the areas where Kc is from 0.9 to 1.0 and Cle is from 0.9 to 1.0. The appearance of CIE Standard Clear Sky by Gusev (CIE, 1973) is assumed to be a slightly turbid clear sky appears in the areas where Kc is from 0.7 to 0.8 and Cle is 0.9. In Osaka, the frequency of occurrence of the blue sky (Type12) is not so high but the frequency of occurrence of Type 14 is the highest, that is, the typical perfect clear sky does not appear so often. Although the frequency of occurrence frequency of a Type depends on the meteorological characteristics of the measurement site, the sky patterns of luminance or radi-

1

0

Fig. 6. Comparison of LzEd between the integration of sky vault and the calculation by Eq. (13).

362

N. Igawa / Solar Energy 105 (2014) 354–372

Table 10 Yearly RMSE and MBE for luminance distributions of sky. Year

RMSE (kcd/m2)

N

MBE (kcd/m2)

As

Aw

i-As

As

Aw

i-As

10,807 9697

1.028 1.109

1.352 1.159

1.041 1.080

0.137 0.148

0.117 0.156

0.131 0.141

Osaka 2006 2007 2008 2009 2010 2011

20,724 21,138 20,769 13,844 17,308 19,041

1.284 1.358 1.323 1.301 1.353 1.381

1.333 1.369 1.362 1.354 1.376 1.350

1.237 1.291 1.268 1.261 1.284 1.288

0.099 0.138 0.100 0.089 0.204 0.174

0.087 0.118 0.072 0.058 0.163 0.144

0.061 0.096 0.059 0.050 0.167 0.135

Total

133,328

1.293

1.342

1.239

0.135

0.112

0.101

Tokyo 1992 1993

Fig. 9. Yearly RMSE of radiance distributions in Tokyo and Osaka.

Table 11 Yearly RMSE and MBE for radiance distributions of sky. Year

Tokyo 1993 Osaka 2006 2007 2008 2009 2010 2011

RMSE (W/m2/sr)

N

MBE (W/m2/sr)

As

Aw

i-As

As

Aw

i-As

7865

11.95

12.80

11.81

1.67

1.62

1.60

20,724 21,138 20,769 13,844 17,308 19,041

10.93 11.59 11.31 11.04 11.39 11.75

10.99 11.36 11.22 11.16 11.32 11.36

10.35 10.84 10.63 10.55 10.69 10.94

0.98 1.38 0.99 0.97 1.55 1.37

0.86 1.20 0.74 0.70 1.21 1.12

0.69 1.06 0.67 0.67 1.28 1.08

Fig. 10. Yearly MBE of radiance distributions in Tokyo and Osaka.

ance distributions in the Kc–Cle area are affected only by Kc and Cle and have no site dependency. Therefore, the Kc–Cle areas where the Type appears became considerably clear. However, the Type estimated by the measurement falls Total 120,689 11.39 11.34 10.75 1.24 1.02 0.95 on over somewhat wide area, because there are no extreme differences in the sky patterns between adjoining areas. The sky pattern in the Type is necessary to consider the influence of the clouds and the atmosphere, since the sky vault does not vary homogeneously. Also when clouds shine brilliantly, the diffuse irradiance increases and the global irradiance increases to be larger than that of the clear sky, and Kc becomes larger than one. On the other hand, near the clear sky condition, if clouds lie scattered and the direct solar beam is interrupted by a cloud, the Kc–Cle area is recognized as a near overcast sky or overcast sky. In these occasions, the indices become unstable because the direct solar radiation varies rapidly and becomes difficult to specify the Type clearly by only momentary direct, diffuse and global irradiances. Therefore the selection method of the Type of CIE General sky is not Fig. 7. Yearly RMSE of luminance distributions in Tokyo and Osaka. yet proposed. Considering the CIE General Sky, it will be better to propose the overall equations which show average skies in small Kc–Cle areas for the practical use of the sky model. It will then be possible to express all sky conditions and their distributions based on the sets of average sky of small areas.

Fig. 8. Yearly MBE of luminance distributions in Tokyo and Osaka.

N. Igawa / Solar Energy 105 (2014) 354–372

363

Kc 0.01

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95 1.00

1.05

1.10

1.15

1.20

1.25

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

Cle

0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10

Fig. 11. Tentatively classified sky conditions into 5 categories.

Table 12 RMSE and MBE of luminance distributions in Osaka for sky conditions. Sky conditions

Overcast I-Overcast Intermediate I-Clear Clear Total

N

RMSE (kcd/m2)

MBE (kcd/m2)

As

Aw

i-As

As

Aw

i-As

20,710 20,032 28,590 20,005 23,487

0.679 1.560 1.860 1.381 1.039

0.701 1.589 1.888 1.457 1.005

0.663 1.507 1.767 1.286 0.992

0.072 0.000 0.105 0.138 0.461

0.015 0.025 0.020 0.196 0.367

0.073 0.088 0.130 0.090 0.359

112,824

1.334

1.357

1.272

0.134

0.108

0.095

Table 13 RMSE and MBE of radiance distributions in Osaka for sky conditions. Sky conditions

Overcast I-Overcast Intermediate I-Clear Clear Total

N

RMSE (W/m2/sr)

MBE (W/m2/sr)

As

Aw

i-As

As

Aw

i-As

20,710 20,032 28,590 20,005 23,487

5.64 13.47 16.13 11.59 8.54

5.82 13.60 16.17 11.32 7.93

5.51 12.90 15.37 10.39 7.85

0.49 0.34 0.89 1.27 3.79

0.07 0.53 0.17 1.68 3.10

0.51 0.39 1.09 0.90 3.05

112,824

11.35

11.24

10.67

1.21

0.98

0.91

5. Comparison between average sky luminance distributions and previous sky models in Kc–Cle areas To obtain detailed, basic information on the sky luminance distributions, the average sky luminance distributions of the Kc–Cle areas obtained by actual measurement are compared with the CIE Standard General Sky, All Sky Model and the All Weather Model. Measurements from January 1, 2007 to December 31, 2008 are used as the basic data.

The average sky luminances of the areas where Kc and Cle are divided into 0.1 intervals respectively are obtained, and are compared with the following four kinds of models. (1) 15 Types of CIE Standard General Sky. (Gs1–Gs15: Table 4). (2) 20 kinds of All Sky Model at 0.1 intervals of Si. (Am1–Am20: Table 5). (3) All Sky Model calculated based on the time series irradiance data (As).

364

N. Igawa / Solar Energy 105 (2014) 354–372

ing Mean Bias Error (MBE) and RMSE. The model for which RMSE is the smallest is selected as the best fit model in the area. MBE and RMSE are calculated by the following equations respectively. MBE ¼

N X ðEi  M i Þ=N

ð5Þ

i¼1

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi XN 2 RMSE ¼ ðEi  M i Þ =N i¼1

Fig. 12. RMSE of luminance distributions for sky conditions in Osaka.

ð6Þ

where Ei is the estimated value, Mi the measured value, and N the number of data points. The best corresponding sky model to the average sky luminance distribution was shown in Table 6 when the solar altitude is 35 deg (±2.5 deg) and their RMSEs were shown in Table 7. Gs and Am correspond better than estimations of As and Aw. Am corresponds well in the cloudy sky, and Gs corresponds well in the clear sky. No model corresponds well in the areas of Kc > 1 and 0.2 6 Cle 6 0.5. Since the current sky models are not fully perfect, a better sky model should be proposed. 6. Improvement of the All Sky Model

Fig. 13. MB of luminance distributions for sky conditions in Osaka.

Fig. 14. RMSE of radiance distributions for sky conditions in Osaka.

(4) All Weather Model calculated based on the time series irradiance data (Aw).

It is thought that the estimation accuracy of the All Sky Model is excellent for the overcast sky from the examinations. If the estimation accuracy of the Kc–Cle areas of which Kc > 1 and 0.2 6 Cle 6 0.5 is improved, that is As can be used as a practical model. Following the basic equation of As, the equations to obtain the coefficients are improved. First of all, Kc and Cle which are assumed to be the sky index are defined as shown in Section 3. Next, the solar altitude band is divided into 5 deg (±2.5 deg) based on the measurements in 2007 and 2008. Kc and Cle are subdivided in the divided solar altitude band at 0.05 intervals, and the average sky luminance distributions of the Kc–Cle areas are obtained. Coefficients (a–e) of each Kc–Cle area are obtained using regression analysis as a function of the altitude of a sky element and the angular distance between the sun and a sky element based on the average sky luminance distributions.

In the calculation of previous All Sky Model, As for indices, Si (=Kc + Cle0.5) are calculated based on the previous equations of Kc and Cle. Since the coefficient c for diffusion indicatrix function (see Eq. (10)) of previous All Sky Model indicates a negative value when Si is larger than 2.1, tentatively, the value of c is replaced with 10 when Si is larger than 2.0 as shown in Fig. 5. This is also one of the reasons that the previous All Sky Model should be improved. These four kinds of models are compared with the average sky luminance distribution in the Kc–Cle area calculatFig. 15. MBE of radiance distribution for sky conditions in Osaka.

N. Igawa / Solar Energy 105 (2014) 354–372

365

Table 14 RMSE and MBE of the sky luminance distributions in Osaka for sky regions. Entire

Overcast sky As Aw i-As Mean Luminance (kcd/m2) Number of data Number of sets Intermediate Overcast As Aw i-As Mean Luminance (kcd/m2) Number of data Number of sets Intermediate sky As Aw i-As Mean Luminance (kcd/m2) Number of data Number of sets Intermediate Clear Sky As Aw i-As Mean Luminance (kcd/m2) Number of data Number of sets Clear sky As Aw i-As Mean Luminance (kcd/m2) Number of data Number of sets Whole sky As Aw i-As Mean Luminance (kcd/m2) Number of data Number of sets

Zenith

South

North

East and west

MBE (kcd/m2)

RMSE (kcd/m2)

MBE (kcd/m2)

RMSE (kcd/m2)

MBE (kcd/m2)

RMSE (kcd/m2)

MBE (kcd/m2)

RMSE (kcd/m2)

MBE (kcd/m2)

RMSE (kcd/m2)

0.072 0.015 0.073 3.688

0.679 0.701 0.663

0.152 0.182 0.126 4.366

0.655 0.688 0.638

0.290 0.108 0.226 3.846

0.855 0.903 0.828

0.018 0.087 0.046 3.434

0.626 0.634 0.613

0.096 0.037 0.101 3.489

0.530 0.548 0.522

2,249,299 20,710

0.000 0.025 0.088 7.756

381,793

1.560 1.589 1.507

2,170,715 20,032 0.105 0.020 0.130 8.893

1.860 1.888 1.767

1.381 1.457 1.286

1.773 1.737 1.673

0.285 0.506 0.243 6.689

1.039 1.005 0.992

0.715 0.568 0.594 4.480

1.244 1.271 1.131

12,217,291 112,824

0.440 0.175 0.365

0.616 0.499 0.565 15.413

1.011 0.701 0.159 15.113

1.051 0.895 0.931

1.066 0.019 0.615 10.406

2.947 2.962 2.779

2,025,325

0.332 0.025 0.177

1.220 1.218 1.189

0.211 0.131 0.089 5.892

2.269 2.238 1.984

0.208 0.633 0.252 4.270

1.381 1.461 1.350

0.138 0.605 0.331 3.037

0.944 1.239 1.032

2,294,286

0.011 0.182 0.062

0.107 0.190 0.044 7.663

1.241 1.292 1.182

0.117 0.231 0.085 6.116

0.921 1.032 0.906

920,230

0.652 0.942 0.739

563,688

2.069 2.032 1.926 11.190

1.062 1.094 1.049

1,315,140

480,120

1.656 1.382 1.542

0.052 0.084 0.131 7.176

921,472

696,160

474,452

1.284 1.254 1.201 6.821

0.307 0.042 0.116 6.440

952,660

480,768

404,227

413,060

1.334 1.357 1.272 6.695

2.356 2.428 2.236

585,593

360,516

2,531,602 23,487 0.134 0.108 0.095

0.497 0.024 0.372 9.095

0.968 0.004 0.457 9.690

497,040

412,208

513,689

2,165,093 20,005 0.461 0.367 0.359 5.150

1.549 1.552 1.497

356,267

3,100,582 28,590

0.138 0.196 0.090 7.412

0.490 0.033 0.456 9.022

417,806

0.273 0.318 0.177 4.309

0.729 0.765 0.706

1,080,402

0.984 1.119 1.002 4.661

2,717,776

0.001 0.081 0.031

0.915 0.964 0.889 5.838

5,189,904

366

N. Igawa / Solar Energy 105 (2014) 354–372

Table 15 RMSE and MBE of the sky radiance distribution in Osaka for sky regions. Entire

Zenith

South

North

East and west

MBE (W/m2/sr)

RMSE (W/m2/sr)

MBE (W/m2/sr)

RMSE (W/m2/sr)

MBE (W/m2/sr)

RMSE (W/m2/sr)

MBE (W/m2/sr)

RMSE (W/m2/sr)

MBE (W/m2/sr)

RMSE (W/m2/sr)

Overcast sky As Aw i-As Mean Radiance (W/m2/sr) Number of data Number of sets

0.49 0.07 0.51 28.62 2,249,299 20,710

5.64 5.82 5.51

1.13 1.48 0.90 33.98 381,793

5.42 5.68 5.29

2.36 0.85 1.79 30.07 417,806

7.19 7.56 6.96

0.04 0.59 0.21 26.51 497,040

5.16 5.21 5.04

0.63 0.39 0.67 27.01 952,660

4.38 4.55 4.31

Intermediate Overcast As Aw i-As Mean Radiance (W/m2/sr) Number of data Number of sets

0.34 0.53 0.39 63.13 2,170,715 20,032

13.47 13.60 12.90

3.59 0.62 3.28 74.16 356,267

13.36 13.34 12.85

8.05 0.15 3.76 79.6 412,208

20.39 20.75 19.29

3.23 0.33 0.28 51.91 480,768

10.52 10.34 10.04

1.04 1.27 0.48 58.09 921,472

9.19 9.45 8.95

Intermediate sky As Aw i-As Mean Radiance (W/m2/sr) Number of data Number of sets

0.89 0.17 1.09 72.58 3,100,582 28,590

16.13 16.17 15.37

2.72 1.15 1.75 75.92 513,689

15.37 14.99 14.60

5.25 4.51 4.89 125.13 585,593

25.50 25.11 24.05

1.63 1.11 0.70 48.09 686,160

12.08 12.62 11.80

0.45 1.23 0.04 62.21 1,315,140

10.88 11.22 10.41

Intermediate Clear Sky As Aw i-As Mean Radiance (W/m2/sr) Number of data Number of sets

1.27 1.68 0.90 54.65 2,165,093 20,005

11.59 11.32 10.39

1.26 2.99 1.04 50.71 360,516

10.37 10.20 9.38

10.09 2.54 3.72 108.99 404,227

19.15 16.79 15.81

2.30 3.85 1.07 32.39 480,120

7.97 9.76 8.50

0.66 1.86 0.45 45.12 920,230

7.61 8.17 7.24

Clear sky As Aw i-As Mean Radiance (W/m2/sr) Number of data Number of sets

3.79 3.10 3.05 36.36 2,531,602 23,487

8.54 7.93 7.85

4.86 3.86 4.02 32.42 413,030

8.00 6.83 7.10

9.21 1.74 5.98 72.72 474,452

13.83 10.70 12.10

0.88 4.17 2.24 21.85 563,688

5.32 7.41 5.99

2.56 2.84 1.85 30.25 1,080,402

6.11 6.22 5.67

Whole sky As Aw i-As Mean Radiance (W/m2/sr) Number of data Number of sets

1.21 0.98 0.91 52.12 12,217,291 112,824

11.35 11.24 10.67

2.77 0.66 2.21 54.35 2,025,295

10.77 10.44 10.09

3.17 1.18 2.15 85.96 2,294,286

17.67 16.64 16.12

0.05 1.22 0.39 36.56 2,707,776

8.40 9.26 8.45

0.37 0.91 0.11 45.33 5,189,904

7.81 8.10 7.48

The equations of the improved All Sky Model (i-As) for luminance and radiance distributions of sky were shown respectively by Eqs. (7) and (8).

Leaðcs ; c; fÞ ¼ Lezðcs Þ  Leðcs ; c; fÞ ¼ Lezðcs Þ

uðcÞ  f ðfÞ uðp=2Þ  f ðp=2  cs Þ

ð8Þ

uðcÞ ¼ 1 þ a  expðb= sin cÞ

ð9Þ

f ðfÞ ¼ 1 þ cfexpðd  1Þ  expðd  p=2Þg þ e  cos2 f 2

Lvaðcs ; c; fÞ ¼ Lvzðcs Þ  Lvðcs ; c; fÞ ¼ Lvzðcs Þ

uðcÞ  f ðfÞ uðp=2Þ  f ðp=2  cs Þ

ð7Þ

ð10Þ

where Lva(cs, c, f) is the sky luminance (kcd/m ), Lvz(cs) the zenith luminance (kcd/m2), Lv(cs, c, f) the relative sky luminance, Lea(cs, c, f) the sky radiance (W/m2/sr), Lez(cs) the zenith radiance (W/m2/sr), Le(cs, c, f) the relative sky radiance, u(c) the gradation function, f(f) the diffusion

N. Igawa / Solar Energy 105 (2014) 354–372

367

practical use of sky radiance and luminance distributions. Therefore, the zenith luminance and the zenith radiance have to be shown. When relative sky luminance distribution (Lv) is obtained from the global irradiance and the diffuse irradiance, the zenith luminance can be shown using Lvzðcs ; Kc; CleÞ ¼ R p=2 R 2p c¼0

Fig. 16. RMSE of luminance distributions for sky regions in Osaka.

a¼0

Evd Lvðcs ; c; fÞ  cos c  dc  da

ð12Þ

The integration calculation of the denominator of Eq. (10) requires much effort. When LzEd was shown by the reciprocal of the integration value as in Eqs. (13) and (14), it may be useful to shorten the calculation time. Lvzðcs ; Kc; CleÞ ¼ Evd  LzEd

ð13Þ

Lezðcs ; Kc; CleÞ ¼ Eed  LzEd

ð14Þ

When LzEd of each Kc–Cle area of each solar altitude band is calculated beforehand, LzEd can be expressed as a function of the solar altitude, Kc, and Cle as in LzEd ¼

5 X ½AðkÞ  Kck  k¼0

Fig. 17. RMSE of radiance distribution for sky regions in Osaka.

AðkÞ ¼

4 X j¼0

½Bðj; kÞ  Clej ;

Bðj; kÞ ¼

5 X ½Cði; j; kÞ  cis  i¼0

ð15Þ The constants to obtain LzEd were shown in Table 9. The comparison of LzEd between the integration of sky vault and the simple calculations by Eq. (15) is shown in Fig. 6. A simple calculation of LzEd is almost equal with the integration value of the relative sky distribution, although sufficient accuracy is practicably obtained. 7. Validation of luminance and radiance distributions of sky

Fig. 18. Typical varying of illuminances on a fine day in Osaka (May 3, 2006).

indicatrix function, cs the solar altitude (rad), c the altitude of a sky element (rad), and f the angular distance between the sun and a sky element. The coefficients a–e were calculated using x of Eq. (11). The constants A–H were shown in Table 8.     Gkc GCle x ¼ A þ B  exp  þ E  exp  2 2   GKc þ GCle 2 þ H  exp  GKc ¼ fðKc  CÞ=Dg ; 2 GCle ¼ fðCle  F Þ=Gg

2

ð11Þ

The relative sky luminance and radiance distributions are obtained based on the irradiance data. The absolute values of sky radiance and luminance are required for the

The Improved All Sky Model (i-As) was compared with All Weather Model (Aw) by Perez et al. (1993) and the previous All Sky Model (As) by Igawa et al. (2004). The data used for the comparison was the measurement data in Osaka (January 2006–December 2011) and Tokyo (March 1992–September 1993) acquired based on the IDMP of CIE. The calculated luminance and radiance values are compared with measured sky luminance and radiance distributions. Generally, the zenith luminance and radiance are not available from ordinary weather stations. Therefore, relative distribution of the sky is estimated from the irradiance. The zenith luminance and radiance are calculated from relative luminance and radiance distributions and the diffuse illuminance and irradiance. The absolute values of luminance and radiance distributions of sky are obtained using the zenith luminance and radiance, and multiplying the relative luminance and radiance distributions.

368

N. Igawa / Solar Energy 105 (2014) 354–372

Fig. 19. Comparisons of sky luminance distributions between the measurements and the estimations by i-As on a fine day in Osaka (May 3, 2006) (dark line: measurement, thin line: estimation).

N. Igawa / Solar Energy 105 (2014) 354–372

(4) Intermediate Overcast Sky 0.9 6 Siv < 1.15. (5) Overcast Sky (Overcast): Siv P 1.15.

Fig. 20. Varying of illuminances on a unstable day in Osaka (May 6, 2006).

7.1. Yearly comparison of models The yearly mean values of RMSE and MBE concerning the sky luminance for eight years and radiance for seven years in Tokyo and Osaka were shown in Tables 10 and 11 and Figs. 7–10. The tendencies of the estimations of luminance and radiance are similar. For luminance distributions, MBEs and RMSEs of 3 models lie between 0.101 and 0.135 (kcd/m2) and between 1.239 and 1.342 (kcd/m2) respectively. For radiance distributions, MBEs and RMSEs of 3 models lie between 0.95 and 1.24 (W/ m2/sr) and between 10.75 to 11.39 (W/m2/sr). All the models seem to satisfy the estimation accuracy. The differences of RMSE between As and Aw are not large through all years. For luminance distribution, RMSE of 1.239 (kcd/ m2) for i-AS is better than 1.293 (kcd/m2) for As and 1.342 (kcd/m2) for Aw. For radiance distribution, RMSE of 10.75 (W/m2/sr) for i-AS is better than 11.39 (W/m2/ sr) for As and 11.34 (W/m2/sr) for Aw. Improved i-As is the best, though all models compared here are highly accurate models. 7.2. Comparison according to sky conditions The estimation accuracies of the sky models according to the sky conditions are compared. The sky conditions are tentatively classified into five categories; (1) Clear Sky, (2) Intermediate Clear Sky, (3) Intermediate sky, (4) Intermediate Overcast Sky, and (5) Overcast Sky. The sky condition is classified as follows. Sky index (Siv) is defined by Kc and Cle as shown in qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ð16Þ Siv ¼ ð1:0  KcÞ þ ð1:0  Cle0:5 Þ The sky conditions are classified by the position of the Kc– Cle area, where the area with Kc = 1 and Cle = 1 is representative of a clear sky. The sky conditions are defined as follows. (1) Clear Sky (Clear): Siv < 0.15. (2) Intermediate Clear Sky (I-Clear): 0.15 6 Siv < 0.3. (3) Intermediate Sky (Intermediate): 0.3 6 Siv < 0.9.

369

(I-Overcast):

The classified sky using the Kc–Cle area was shown in Fig. 11. The areas where Kc is larger than one are influenced by the state of the clouds where the diffuse component increases and the global irradiance grows compared with the clear sky. When this occurs, Cle decreases because of the decrease in direct component. The estimation accuracies of the sky luminance and radiance distributions in Osaka according to the sky conditions for 6 years from 2006 to 2011 were shown in Tables 12 and 13. RMSE and MBE of sky luminance distributions are shown in Figs. 12 and 13. Those of the sky radiance distributions are shown in Figs. 14 and 15. The estimations of luminance and radiance distributions show almost similar characteristics. The estimating accuracy of the Intermediate Sky which the movement of the clouds is fast and unstable is the lowest and the Intermediate Overcast Sky in which the clouds are also unstable follows. The estimating accuracy of Overcast Sky, in which the sun is covered and the sky is comparatively steady and the absolute values of diffuse illuminance and irradiance are small compared with other sky conditions, is the highest. The models picked up here estimate considerably highly accurate sky luminance and radiance distributions and are practicable. Among the selected sky models, RMSE and MBE show that i-As has the best estimation accuracy compared with other models in all sky conditions. 7.3. Estimation according to sky regions The sky vault is temporally divided into 4 regions: (a) the zenith region: the sky element of which altitude is higher than 60° above the horizon, (b) the south region: the region in the direction of the sun (azimuth <45° and altitude <60°), (c) the north region: the region opposite to the sun region (azimuth >135° and altitude <60°), (d) the east–west region: the region on both sides of the sun (45° < azimuth < 135° and altitude <60°), and the estimating accuracy of the sky element of each region is compared. The basic data of sky distribution is rotated by interpolating the measurement data of the same sky altitude, so that the basic direction of dataset always matches with the solar meridian. Note that the south region, the north region and the east–west region described here are not shown in real directions. The south was shown as a direction from the sun. The RMSE and MBE of the sky luminance distribution in all sky conditions in Osaka according to the sky regions are shown in Table 14 and those of the sky radiance distribution are shown in Table 15. RMSE of sky luminance and radiance distributions for all years according to sky regions are shown in Figs. 16 and 17. As for the sky luminance distribution, RMSE of South facing the sun tends to be larger

370

N. Igawa / Solar Energy 105 (2014) 354–372

Fig. 21. Comparisons of sky luminance distributions between the measurements and the estimations by i-As on an unstable day in Osaka (May 6, 2006) (dark line: measurement, thin line: estimation).

N. Igawa / Solar Energy 105 (2014) 354–372

than that of other directions, followed by Zenith. In the measurement of the sky element near the sun, it is difficult to acquire data with high reliability because of the influence of the direct sun beam. In the comparison of the sky elements according to the sky regions, the sky radiance distributions show the similar characteristic with the sky luminance. In all cases, RMSEs of i-As are the smallest, that is, i-As can be used to estimate all sky regions. 7.4. Measurements and estimations on typical days The typical varying of illuminances on a fine day (May 3, 2006) was shown in Fig. 18, and the comparisons of sky luminance distributions between the measurements and estimations by i-As every hour was shown in Fig. 19. The estimations fit the measurements very well, excluding the sky elements near the sun where measurements are affected by the direct beam of the sun. The reproducibility of the estimating values of the sky luminance distributions near the clear sky is high. The varying of illuminance on an unstable day (May 6, 2006) was shown in Fig. 20, and the comparisons of sky luminance distributions between the measurements and estimations were shown in Fig. 21. When the illuminance varies rapidly at 12:00 and 14:00, etc., the calculation values based on the indices become different from the measurements because the irradiances and illuminances varied and the estimating errors are caused by the varying of the values of the indices as well as a relatively long measurement period of the sky scanner. Overall estimations reproduce the real phenomenon well. 8. Conclusions In this research work, an improved All Sky Model was proposed and confirmed based on the IDMP measurement data in Osaka (2006–2011) and Tokyo (1992–1993). In the long term measurements, a large amount of the basic observation data was obtained. These data were useful for this research work and also will be useful for the further advanced examinations. The sky luminance and radiance distributions were measured using the sky scanner and the difficulties of measurement originating from the measurement method and the measurement instruments were brought into relief. It is necessary to note that the error margin depends on not only the estimation error by the models but also the quality of the measurement data. The nature of the measurement data is of paramount importance. The improved All Sky Model (i-As) proposed here is regarded as the final model based on the current measuring instruments. In the yearly comparison of luminance distribution of three models based on two years of Tokyo data and six years of Osaka data, MBEs and RMSEs of 3 models lie between 0.101 and 0.135 (kcd/m2) and between 1.239 to 1.342 (kcd/m2) respectively. For radiance distributions based on one year of Tokyo data and six years of Osaka

371

data, MBEs and RMSEs of 3 models lie between 0.95 and 1.24 (W/m2/sr) and between 10.75 to 11.39 (W/m2/ sr) respectively. These values show that three models seem to satisfy the estimation accuracy. For luminance distribution, RMSE of 1.239 (kcd/m2) for i-AS is better than 1.293 (kcd/m2) for As and 1.342 (kcd/m2) for Aw. For radiance distribution, RMSE of 10.75 (W/m2/sr) for i-AS is better than 11.39 (W/m2/sr) for As and 11.34 (W/m2/sr) for Aw. Among them the lowest values of MBE and RMSE were shown by i-As, that is, i-As is the best, though all models compared here are highly accurate models. In the comparison with the estimation values through 6 years in Osaka data according to sky conditions, sky regions, it was confirmed that previous All Sky Model (As) and All Weather Model (Aw) were both considerably accurate and practicable models. In addition, the improved All Sky Model (i-As) proposed here was confirmed to have the highest estimating accuracy in all conditions, and will be able to reproduce luminance and radiance distributions of the sky more realistically. Acknowledgements This research work was supported by the fund of Japan Society for the Promotion of Science (B) 17360285 and (B) 21360279. References Brunger, A.P., Hooper, F.C., 1993. Anisotropic sky radiance model based on narrow field of view measurements of shortwave radiance. Solar Energy 51 (1), 53–64. CIE, 1955. Natural Daylight, Official Recommendation. Compte Rendu, CIE 13th Session, Committee E-3.2, vol. II, part 3-2, II–IV, pp. 35–37. CIE, 1973. Standardization of Luminance Distribution on Clear Skies. Pub. CIE No. 22 (TC-4.2). CIE, 1994. Guide to Recommended Practice of Daylight Measurement. Pub. CIE 108, ISBN 3 900 734 50 X. CIE, 2003. Spatial Distribution of Daylight – CIE Standard General Sky. Standard CIE S 011/E:2003. Grant, R.H., Gao, W., Heisler, G.M., 1996. Photosynthetically active radiation: sky radiance distributions under clear and overcast conditions. Agric. For. Meteorol. 82, 267–292. Harrison, A.W., Coombes, C.A., 1988. An opaque cloud cover model of sky short wavelength radiance. Solar Energy 41, 387–392. Harrison, A.W., 1991. Directional sky luminance versus cloud cover and solar position. Solar Energy 46 (1), 3–19. Hosobuchi, H., Yoshida, H., Uetani, Y., 2006. Study on identifying the sky types recommended as CIE Standard General Sky by measured flux of solar radiation. J. Environ. Eng., Archit. Inst. Jpn. 609, 31–38 (in Japanese). Igawa, N., Koga, Y., Matsuzawa, T., Nakamura, H., 2004. Models of sky radiance distribution and sky luminance distribution. Solar Energy 77 (2), 137–157, 2004.06. Kasten, F., Young, A.T., 1989. Revised optical air mass tables and approximation formula. Appl. Opt. 28 (22), 4735–4738. Kittler, R., 1967. Standardization of outdoor conditions for the calculation of daylight factor with clear skies. In: Proc. the CIE International Conference on Sunlight in Buildings, Bouwcentrum International, Rotterdam, pp. 273–285.

372

N. Igawa / Solar Energy 105 (2014) 354–372

Kittler, R., 1985. Luminance distribution characteristics of homogeneous skies: a measurement and prediction strategy. Light. Res. Technol. 17 (4), 183–188. Kittler, R., 1986. Luminance models of homogeneous skies for design and energy performance predictions. In: Proc. 1, 1986 International Daylighting Conference, Long Beach, pp. 18–22. Kittler, R., Perez, R., Darula, S., 1997. A new generation of sky standards. In: Proc. Lux Europa ’97, pp. 359–373. Kittler, R., Darula, S., 2006. The method of aperture meridians: a simple calculation tool for applying the ISO/CIE Standard General Sky. Light. Res. Technol. 38, 109–119. Kobav, M.B., Bizjak, G., Dumortier, D., 2009. Complete analysis of the luminance measurements gained with a sky scanner. In: Proc. of 11th European Lighting Conference Lux Europa 2009, Istanbul, September 9–11, pp. 273–278. Li, D.H.W., Tang, H.L., 2008. Standard skies classification in Hong Kong. J. Atmos. Sol.-Terrest. Phys. 70, 1222–1230. Littlefair, P.J., 1981. The luminance distribution of an AVERAGE SKY. Light. Res. Technol. 13 (4), 192–198. Markou, M.T. et al., 2007. A new statistical methodology for classification of sky luminance distributions based on scan data. Atmos. Res. 86, 261–277. Moon, P., Spencer, D.E., 1942. Illumination from a non-uniform sky. Trans. Illum. Eng. Soc. 37, 707–726.

Nagata, T., 1997. Radiance distribution on stable overcast skies. J. Light Vis. 21 (1), 6–9. Nakamura, H., Oki, M., Hayashi, Y., 1985. Luminance distribution of intermediate sky. J. Light Vis. Environ. 9 (1), 6–13. Nakamura, H., Oki, M., Iwata, T., 1987. Mathematical description of the Intermediate Sky. In: Proc. 21st CIE Session, Venice 1, pp. 230–231. Nakamura, H. et al., 1997a. A study on sky radiance distribution: part 1. Relative sky radiance distribution by solar altitude. J. Illum. Eng. Inst. Jpn. 81 (2), 26–36 (in Japanese). Nakamura, H. et al., 1997b. A study on sky radiance distribution: part 2. Equation for the relative yearly average mean sky radiance distribution. J. Illum. Eng. Inst. Jpn. 81 (2), 110–120 (in Japanese). Nakamura, H. et al., 1997c. A study on sky radiance distribution: part 3. Equation for the zenith radiance and the absolute yearly a mean sky radiance distribution. J. Illum. Eng. Inst. Jpn. 81 (5), 12–20 (in Japanese). Perez, R., Seals, R., Michalsky, J., 1993. All-weather model for sky luminance distribution – preliminary configuration and validation. Solar Energy 50 (3), 235–245. Perraudeau, M., 1988. Luminance models. In: National Lighting Conference 1988, Cambridge, pp. 291–292. Wittkopf, S.K., Soon, L.K., 2007. Analysing sky luminance scans and predicting frequent sky patterns in Singapore. Light. Res. Technol. 39 (1), 31–51.