CIE standard sky model with reduced number of scaling parameters

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Solar Energy 85 (2011) 553–559 www.elsevier.com/locate/solener

CIE standard sky model with reduced number of scaling parameters M. Kocifaj ⇑ ICA, Slovak Academy of Sciences, 9, Du´bravska´ Road, 845 03 Bratislava, Slovak Republic Received 16 August 2009; received in revised form 23 December 2010; accepted 28 December 2010 Available online 26 January 2011 Communicated by: Associate Editor David Renne

Abstract The calculation of diffuse horizonthal illuminance DV outdoors is generalized and interrelated to parameters that characterize the sky luminance distribution function. The derived analytical formula for DV,rel which is DV normalized using luminance in the zenith can be commonly applicable to all 15 ISO-standardized sky types. This makes the modelling of all 15 skies more uniform than ever before. The presented analytical solution is simplified enough because the set of 60 parameters B1, C1, D1, E1 – B15, C15, D15, E15 scaling the diffuse illuminance is no more necessary in the new solution. In addition, the analytical formulae substituting the empirical ones are valid for all solar altitudes cS (including the situations with sun in zenith). On the contrary, the empirical formulae may become inapplicable for cS > 75° and fail when cS approaches 90°. Compared with detail numerical integration the analytical formula guarantees more rapid calculation of DV,rel. Ó 2010 Elsevier Ltd. All rights reserved. Keywords: Diffuse illuminance; CIE standard general sky; Luminance distribution

1. Introduction Large-scale variability of atmospheric conditions typically results in rapid changes of the scattered light reaching the ground. Cloud coverage may evolve quite dynamically, implying the serious complexities with accurate characterization of most actual luminance distribution. On the one hand, the exact mathematical formulation of sky luminance distribution is almost impossible. But, on the other hand, some quantification of the lighting conditions is essential e.g. to be able to model daylight availability for arbitrarily oriented interiors, or to design the optimum window dimensions. For such modelling purposes it is usually necessary to repeat the entire set of calculations with the aim to simulate natural light in various situations and in any time during a day. Due to high performance requirements, the robust radiative transfer models applicable to any inhomogeneous disperse media (Minin, 1988; Wauben, 1992; Marshak and Davis, 2005) are usually inconvenient for lighting engineer⇑ Tel.: +421 2 59309267; fax: +421 2 54773548.

E-mail address: [email protected] 0038-092X/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.solener.2010.12.024

ing applications. However, rather than searching for appropriate approximations using general theories, the empirical formulae are traditionally preferred by many authors. For instance, Perez et al. (1993) proposed a model that predicts sky luminance distribution based on hourly global and direct irradiances. Igawa and Nakamura (2001) and later Igawa et al. (2004) presented an improved model that simulates all sky conditions using an empirical approach. Here a concept of independent gradation and indicatrix functions (Kittler et al., 1997) is employed to determine sky luminance distribution. At present, Kittler’s model is adopted by CIE (2003) as well as ISO (2004) standards and it is widely used for various sky luminance simulations (Markou et al., 2007; Chirarattananon and Chaiwiwatworakul, 2007; Li and Tang, 2008), and also for evaluation of illuminance on inclined surfaces (Li et al., 2005). Kittler et al. (1998a,b) classified all skies into 15 categories with a set of free parameters. Five parameters a, b, c, d, e characterize sky luminance pattern in relative terms, while another four parameters B, C, D, E are reserved for calculation of both the diffuse illuminance DV and the zenith luminance LZ. Soler and Gopinathan (2002) evaluated the measured zenith lumi-

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Nomenclature a, b parameters of the gradation function c, d, e parameters of the indicatrix function B, C, D, E parameters scaling the diffuse illuminance DV horizontal diffuse sky illuminance DV,rel horizontal diffuse sky illuminance measured in relative units Dexact horizontal diffuse sky illuminance obtained by V ;rel exact numerical integration analytical DV ;rel analytical approximation of DV,rel

K0,int Ki1(y)

Dempirical empirical approximation of DV,rel V ;rel EV extraterrestrial illuminance on a horizontal plane f(v) indicatrix function for sky element with angle v from the sun I1 auxiliary function used in analytical expression for Danalytical V ;rel

Greek symbols a azimuth angle of a sky element c elevation angle of a sky element cS solar altitude (elevation angle of the sun) u(c) luminanace gradation function for sky element with elevation angle c  u average value of the gradation function v angular distance between sky element and sun position

I2 K0

auxiliary function used in analytical expression for Danalytical V ;rel modified Bessel function

nance under cloudless situations and estimated the dependence of LZ on the parameters B, C, D, E. In principle, the diffuse illuminance is not independent of luminance, so the parameters a, b, c, d, e should be sufficient for obtaining DV. For this reason it is necessary to find an integral product of the gradation and indicatrix functions. From mathematical point of view, it is a serious problem to perform a nontrivial analytical integration of sky luminance patterns over the whole upper hemisphere. Instead additional empirical formulae for each sky standard are frequently introduced which are parametrized through B, C, D, E. An implementation of the additional formulae is, however, disadvantageous. It involves extra 60 parameters B1, C1, D1, E1 – B15, C15, D15, E15 into calculation scheme, and consequently, it may lead to a potential incompatibility problem. Namely, there is no reason for coincidence of exactly-integrated and empirically-approximated values of DV if both are obtained using independent formulae (as evident e.g. from Fig. 1 in Darula et al., 2006). Specifically, the diffuse illuminance obtained numerically (by integrating the sky luminance distribution) and empirically (using the parameters B, C, D, E) may differ by definition. The intention of this paper is to present an approximate analytical solution for the diffuse illuminance based on the set of original parameters a, b, c, d, e that are exclusively used to model CIE standard skies (in relative luminance terms). These parameters are the only sufficient to calculate also DV/LZ (using DV,rel) without additional parameters B, C, D, E. However, also the authors of the CIE standard were aware of this fact, but the best-fit formula substituted

integral of the Bessel function K0 Bickeley function that is a supplementary function to K0,int LZ zenith luminance analytical LZ analytical expression for zenith luminance N(cS) normalization factor for analytical expression of DV,rel z zenith angle of the sky element

the absence of an analytical solution. Compared with detail numerical integration, the derived analytical formula can also guarantee the more rapid calculation of diffuse illuminance. Nevertheless, the computational time needed for numerical simulations is not the most important issue. The strength of the presented theoretical solution is a derivation of the analytical formula for diffuse illuminance based on parameters that characterize the luminance pattern. Basically, a minimization of the number of free parameters of any model is always one of the major targets of physical theories which follow the practical applications of the model if an error about 5% is reached. It is true that ISO/CIE standard is also adopted in the relative terms as the arbitrary sky luminance is normalized by zenith luminance via the set of 5 parameters a, b, c, d, e, but absolute luminance values can be computed only after DV/LZ which can be determined from DV,rel derived in this paper. Note that this ratio is characteristic for each sky standard. The presented generalization of the daylight model is therefore a decisive advancement. At present, an enhancement of the sky model by a new sky type would require to find new set of optimized parameters anew, bnew, cnew, dnew, enew and Bnew, Cnew, Dnew, Enew, implying a search over 9-dimensional space. From numerical point of view it means N9 simulations, if N is the number of grid points for each of above listed parameters. If however the later set of Bnew, Cnew, Dnew, Enew can be omitted, the CPU time needed for determination of optimum parameters can be significantly reduced using the theoretical formulae presented in this paper.

M. Kocifaj / Solar Energy 85 (2011) 553–559

2. Theoretical derivations of diffuse illuminance and zenith luminance The relative sky luminance can be calculated as a product of gradation function u and indicatrix function f (Kittler et al., 1998a, 1998b). Kittler et al. defined the twin set of u and f by the following equations uðcÞ ¼ 1 þ a expðb= sin cÞ

ð1Þ

and 

  d f ðvÞ ¼ 1 þ c expðvdÞ  exp p þ e cos2 v; 2

ð2Þ

where a, b, c, d, e are free scaling parameters. The altitude (elevation angle) c characterizes the position of a sky element, while v is the angular distance of the sky element from the sun. Note, that altitude c is a supplementary angle to a zenith angle z, so c = (p/2)  z. Using spherical geometry v ¼ arccosðsin cS cos z þ cos cS sin z cos aÞ;

ð3Þ

where cS is the solar altitude, and a is the azimuth of the sky element measured from the sun-meridian. It is wellknown, that the diffuse illuminance on a horizontal surface is a cosine-weighted integral of sky luminance over the whole upper hemisphere. Therefore, except for a coefficient of proportionality, the relative diffuse illuminance DV,rel can be obtained as Z 2p Z p=2   p exact DV ;rel ¼ u  z sin z cos z f ðvÞdadz: ð4Þ 2 0 0 However, many applications require the absolute values of diffuse illuminance DV. The transition from DV,rel to DV can be made through a scaling factor Q which is the ratio of absolute to relative zenith luminance LZ DV ;rel ; DV ¼ QDV ;rel ¼ p p u 2 f 2  cS

ð5Þ

where the expression for relative zenith luminance (in denominator of Eq. (5)) is evident from Eqs. (1)–(3). The Eq. (5) can be used for any DV,rel independent of whether it is determined in accordance of Eq. (4), or using any other analytical or empirical approximation. The basic idea of this paper is to find a satisfactory accurate analytical approximation of Dexact V ;rel (consult Eq. (4)). In the solution concept, we first rotate the coordinate system by the angle p/2  cS to associate the centre of coordinate system with position of the sun and thus to simplify the inner integration in Eq. (4). In the next step, the gradation function u is substituted by its averaged value Z 2 p=2 ¼ u uðcÞdc ð6Þ p 0 and the second part of the integral (4) is then given as a sum of two terms

ðcS Þ ffi Danalytical V ;rel þ

Z

555

2p u sinðcS Þ N ðcS Þ

Z

cS

0

pcS

f ðvÞ sin vdv )

f ðvÞ sin v sinðp  cS  vÞdv :

ð7Þ

cS

Here N(cS) is a normalization factor which can be determined assuming the limit case of isotropic scattering (i.e. when u = f  1 for any c and any v). Note, that this is just a mathematical manipulation having no relation to the scattering properties of realistic atmosphere. The integral (4) for the isotropically scattering environment can be found in the analytical form Dexact V ;rel ðcS Þju1;f 1 ¼ p

ð8Þ

It is evident from Eqs. (7) and (8) that the following condition must be fulfilled Danalytical ðcS Þju1;f 1 ¼ Dexact V ;rel V ;rel ðcS Þju1;f 1

ð9Þ

to guarantee that Eq. (7) coincides with Eq. (8) (assuming u = f  1). In such a case, the integration of Eq. (7) is straightforward and the normalization factor can be found easily in the form N ðcS Þ ¼ 2



 p  2ðcS þ 1Þ 2 cos cS þ sin cS cos cS :  sin cS 1 þ 2 ð10Þ

The Eq. (7) then reads Danalytical ðcS Þ ffi V ;rel

p u

nR cS 0

f ðvÞsinvdv þ

R pcS cS

f ðvÞsinvsinðp  cS  vÞdv

o

1 þ p2ðc2S þ1Þ coscS þ sincS cos2 cS ð11Þ

 . PlacNow we need to find the analytical expression for u ing the Eq. (1) into Eq. (6), and using some mathematical  can be expressed in the analytical manipulations, the u form as follows Z 2 p=2  ða; bÞ ¼ u f1 þ a expðb= sin cÞgdc p 0 Z 2 1 1 þ a expðbÞ expðbxÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx ¼ p 0 ð1 þ xÞ x2 þ 2x

 2 ð12Þ ¼ 1 þ a 1  K 0;int ðbÞ ; p where K0,int is a tabulated function known as a finite integral of the modified Bessel function K0 and x in Eq. (12) is an arbitrary integration variable. In accordance with MacLeod (1996) the K0,int(y) is the finite integral of K0 Z y K 0;int ðyÞ ¼ K 0 ðtÞdt y P 0; ð13Þ 0

where y is any positive argument. Alternatively, K0,int can be determined from supplementary Bickeley function Ki1(y)

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K 0;int ðyÞ ¼

p  Ki1 ðyÞ; 2

ð14Þ

where the Ki1(y) is formulated in terms of modified Struve functions (Abramowitz and Stegun, 1965; Amos, 1989). Although some useful approximations regarding this function were published (e.g. Luke, 1962) we need only a few values of K0,int, i.e. those for the discrete values of parameters b. Because Eq. (13) is independent of sun position, the set of needed values of K0,int can be calculated numerically and archived for further usage. After Darula et al. (2000), the parameter (b) is smaller than or equal to unity for all 15 sky-types. Then, the following approximation for K0,int is applicable      y2 y4 y y2 y4 1þ þ K 0;int ðyÞ ¼ y 1þ þ  e þ ln 2 36 1600 12 320  2  4 y 3y þ þ ; y 6 1; ð15Þ 12 640 where e = 0.5772 is Euler’s constant. The non-trivial manipulations with the rest integrals result in the final analytical expression for relative diffuse illuminance: ðcS Þ ffi Danalytical V ;rel

p uða; bÞ ½I 1 ðc; d; e; cS Þ þ I 2 ðc; d; e; cS Þ nðcS Þ

ð16Þ

where Z

h p i f ðvÞsinvdv ¼ 1  cexp d ð1  coscS Þ 2 0 e c þ ð1  cos3 cS Þ þ ½1 þ ðd sincS 3 1 þ d2  coscS ÞexpðcS dÞ ð17Þ

I 1 ðc;d;e;cS Þ ¼

cS

and I 2 ðc; d; e; cS Þ ¼

Z

pcS

IðvÞ sin v sinðp  cS  vÞdv c S   c expðcS dÞ 2 ¼ exp½ðp  2cS Þd sin cS þ cos cS d 4 þ d2    2 þ sin cS  cos cS þ ð2 cos 2cS  d sin 2cS Þ sin cS d p i cos cS nh 4 þ e cos 2cS  4c exp d sin 2cS þ 8 h p io2 : ð18Þ þðp  2cS Þ 4 þ e  4c exp d 2

The modified normalization factor n(cS) used in the Eq. (16) is linearly proportional to N(cS), where after Eq. (10) the mapping between n(cS) and N(cS) is through the sinus of solar elevation angle, i.e. N ðcS Þ 2 sin cS p  2ðcS þ 1Þ cos cS þ sin cS cos2 cS ¼1þ 2

nðcS Þ ¼

ð19Þ

3. Validation of the analytical solution Experimental data collected in the framework of daylight measurement program at IDMP stations (IDMP, 2009) are often used for evaluation of the ratio DV/EV. Here EV = 133.8 sin cS is the extraterrestrial illuminance on a horizontal plane. Placing the formula EV = 133.8 sin cS into the Eq. (5) we obtain DV LZ DV ;rel ¼ p p ; EV u 2 f 2  cS 133:8 sin cS

ð20Þ

so, in accordance with Eqs. (5) and (16), the absolute value of zenith luminance is  DV 133:8u p2 Lanalytical ¼ Z uða; bÞ EV p  f p2  cS nðcS Þ sin cS  : ð21Þ I 1 ðc; d; e; cS Þ þ I 2 ðc; d; e; cS Þ The theoretical zenith luminance as well as the diffuse illuminance can be then determined using 5 free parameters a, b, c, d, e, exclusively. In contrast to the former empirical models the new solution (represented by Eq. (21)) does not require the additional four parameters B, C, D, E . To make the evaluation of all discussed approaches possible, we have overtaken the empirical formula for Dempirical V ;rel from (Darula and Kittler, 2005)   u p2 f p2  cS empirical h i 133:8 sin cS : ð22Þ DV ;rel ðcS Þ ¼ B sinC cS þ E sin c S cosD cS The Eqs. 4, 16, and 22 are now in a consistent state and can be compared one to each other. The main intention is to determine the range of validity of both – analytical and empirical solutions. The relative diffuse illuminances analytical Dexact , and Dempirical are calculated numerically V ;rel V ;rel ; DV ;rel and presented in a graphical form (Figs. 1–4) for all 15 sky types. The sky types are characterized in more details in the Table 1. In general, the analytical approximation works well for all sky types. Although some values of Danalytical differ from numerically calculated Dexact V ;rel V ;rel , the common behaviour of the analytical approximation is correct and its overall error D

R

R p=2

p=2 analytical

ðc Þdc

cS ¼0 DV ;rel ðcS ÞdcS  cS ¼0 Dexact S V ;rel S ð23Þ D¼ R p=2 exact D ðc ÞdcS cS ¼0 V ;rel S is smaller than 6%. The smallest discrepancies between analytical and exact calculations are found for the sky types IV–V (Fig. 3 and 4), i.e. for the bright partly cloudy, or clear, or cloudless (polluted) skies. These skies frequently occur in Central Europe and in the Eastern Mediterranean during summer seasons (Bartzokas et al., 2005). The analytical approximation becomes less accurate for overcast skies of the II-type (Fig. 1), but only when the solar altitudes are quite high (Danalytical and Dexact V ;rel V ;rel are almost identical at cS  0°). Similar behaviour is presented in Fig. 3 for

M. Kocifaj / Solar Energy 85 (2011) 553–559

Fig. 1. Relative diffuse illuminance DV,rel which is DV normalized using luminance in the zenith (consult Eq. (5)). The calculations are made for overcast skies I.1–II.2. The solid curve is obtained by exact integration after Eq. (4), dashed curve corresponds to analytical approximation (after Eq. (16)), and dot-and-dashed line is valid for empirical formulae (Eq. (22)).

557

Fig. 3. The same as in Fig. 1, but for partly cloudy skies IV.2–IV.3 and white-blue sky with a clear solar corona (IV.4).

partly cloudy skies. As for the series of sky types III, the best coincidence between analytical and exact calculations corresponds to overcast sky III.1. For brighter circumsolar regions (skies III.2 ? III.4), the values of Danalytical and V ;rel Dexact differ the more the larger is the solar altitude. The V ;rel

empirical formula fits the real values of Dexact V ;rel very good if cS / 75 , but unfortunately, Dempirical is inapplicable for V ;rel very high solar altitudes. This is a limitation of existing empirical formulae for diffuse illuminance and zenith luminance calculations for all 15 sky types. It is evident that the original restriction to cS / 75 of the best fit empirical approximation of DV/LZ (or LZ/DV) modelled by B, C, D, E parameters is quite right, which document all Figs. 1–4 because the error rapidly rises when solar altitude

Fig. 2. The same as in Fig. 1, but for foggy or cloudy sky III.1 and partly cloudy skies III.2–III.4.

Fig. 4. The same as in Fig. 1, but for very clear or cloudless skies V.4– VI.6.

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Table 1 A set of 15 basic types representing Standard Sky Luminance Distributions – SSLD (after Kittler et al. (1998a)). SSLD code

Type of sky

Comments

I.1 I.2

Including the current CIE Standard No direct sunlight sometimes darker or brighter skies

III.4 IV.2 IV.3 IV.4

Overcast with the steep gradation and azimuthal uniformity Overcast with the steep gradation and slight brightening toward sun Overcast moderately graded with azimuthal uniformity Overcast moderately graded and slight brightening toward sun Overcast, foggy or cloudy with overall uniformity Partly cloudy with a uniform gradation and slight brightening toward sun Partly cloudy with a brighter circumsolar effect and uniform gradation Partly cloudy, rather uniform with a clear solar corona Partly cloudy with a shaded sun position Partly cloudy with brighter circumsolar effect White-blue sky with a clear solar corona

V.4 V.5 VI.5

Very clear/non-turbid with a clear solar corona Cloudless polluted with a broader solar corona Cloudless turbid with a broader solar corona

VI.6

White-blue turbid sky with a wide solar corona effect

II.1 II.2 III.1 III.2 III.3

is over 75°. However, many locations in geographical latitudes over ±38.5° N or S where during the whole year the solar altitude cannot reach 75° can use the B, C, D, E approximation (e.g. in about the whole Europe). Of course, in the tropics how Wittkopf (2006) noticed in Singapore, the errors are unavoidable and therefore the integrated DV is used in Darula et al. (2006). Using the analytical formula derived in this paper is more convenient, because the numerical integration is usually time consuming. 4. Conclusions A twin set of empirical formulae is traditionally used to model standard sky luminance patterns and the corresponding diffuse illuminances DV. Although DV is mathematically a well-defined integral of the luminance, both the diffuse illuminance and luminance are frequently expressed using independent formulae. This makes the system incompatible and unnecessarily complex. In the empirical model, five parameters a, b, c, d, e are reserved for characterization the luminance patterns, while another set of 4 parameters B, C, D, E defines the illuminance. It is shown, that the later set of 4 parameters is redundant, implying a possible reduction of model complexity – i.e. the extra 60 parameters (4 for each of 15 sky types) can be eliminated. In the theoretical solution, the parameters scaling the luminance are integrated into the common formula for relative diffuse illuminance DV,rel. This makes the new model more uniform and free of 15 empirical formulae for DV,rel (assuming 15 different sky types). A further improvement deals with accuracy. Although the empirical model is satisfactory accurate for small solar altitudes, it becomes faulty for cS > 75°. The new Eqs. (16)–(18), (and) (21) presented in this paper behave correctly for all cS e h0, 90°i and thus represent a convenient extension to

No No No No

direct direct direct direct

sunlight sunlight sunlight sunlight

sometimes darker or brighter skies exceptionally darker skies sometimes darker or brighter skies exceptionally darker skies

Filtered direct sunlight exceptionally darker skies Filtered or no direct sunlight Filtered or no direct sunlight Filtered direct sunlight Direct sunlight in accordance with luminous turbidity TV in the direction of sun beams Corresponding to the current CIE Clear Standard Corresponding to the current CIE Polluted Standard Direct sunlight in accordance with luminous turbidity TV in the direction of sun beams Direct sunlight in accordance with luminous turbidity TV in the direction of sun beams

the existing ISO-standardized model of sky types. The overall differences between exact and analytical models of DV,rel are smaller than 6%. It needs to be emphasized that the value of 6% represents a weighted error for all realizable solar altitudes cS e h0, 90°i (consult Eq. (23)). Typically, the results obtained by analytical formula (Eq. (16)) and by exact numerical integration (after Eq. (4)) tend to deviate increasingly with growing the solar altitude. Such behaviour is documented in all (Figs. 1–4). The largest discrepancies between analytical and numerical approaches are found at cS > 75°, but they are still much smaller than errors corresponding to empirical formula (see the dot-and-dashed curves in Figs. 1–4). The numerical computations based on both the analytical approximation (Eq. (16)) and the empirical formula (Eq. (22)) are almost equally fast, consuming 0.5s CPU on 1.73 GHz Intel Pentium PC for entire set of 15 skies. The exact numerical integration for the same set of skies lasts 7.5 min, i.e. 30 s per each sky type. Following these facts, a choice of Eq. (4) for elevated solar altitudes and Eq. (16) for the rest cases could be an appropriate compromise between the computational accuracy and computational time. From mathematical point of view, the strength of the presented theoretical solution is a derivation of the analytical formula for diffuse illuminance which is now a function of the parameters of luminance pattern. It represents an important advancement in modeling the daylight conditions as the number of free parameters was essentially eliminated this way. Acknowledgements This paper was supported by the Slovak Research and Development Agency under the Contract No. APVV0264-07.

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