Accuracy and sensitivity analysis for 54 models of computing hourly diffuse solar irradiation on clear sky

Abstract

Fifty-four broadband models for computation of solar diffuse irradiation on horizontal surface were tested in Romania (South-Eastern Europe). The input data consist of surface meteorological data, column integrated data, and data derived from satellite measurements. The testing procedure is performed in 21 stages intended to provide information about the sensitivity of the models to various sets of input data. There is no model to be ranked “the best” for all sets of input data. However, some of the models performed better than others, in the sense that they were ranked among the best for most of the testing stages. The best models for solar diffuse radiation computation are, on equal footing, ASHRAE 2005 model (ASHRAE 2005) and King model (King and Buckius, Solar Energy 22:297–301, 1979). The second best model is MAC model (Davies, Bound Layer Meteor 9:33–52, 1975). Details about the performance of each model in the 21 testing stages are found in the Electronic Supplementary Material.

Appendix 1

1.1 Clear sky models to computed hourly diffuse solar irradiation

1.1.1 G001: ASHRAE 1972

This is a historical model still widely used by engineers to calculate solar heat gains and cooling loads in buildings, or insolation of simple solar systems. It is based on original empirical work conducted in the 1950s and 1960s (ASHRAE 1972). For more details, see Gueymard (1993, 2012).

1.1.2 G002: ASHRAE 2005

ASHRAE 2005 is similar to ASHRAE 1972, but with some revised coefficients that appeared in ASHRAE (2005).

1.1.3 G003: Badescu

This model is based on MAC (entry G029 in this list) and was proposed by Badescu (2008).

1.1.4 G004: Bashahu

The composite model of Bashahu and Laplaze (1994) is based on the direct irradiance model of King and Buckius (1979); see entry G027 below.

1.1.5 G005: BCLSM

This is the model originally proposed by Barbaro et al. (1979) and later modified in Davies et al. (1988) and Badescu (1987, 1997). The original Barbaro equations are used here, with three modifications: (1) The units in Barbaro's Eq. (10) are converted from cal/(cm² min) to W/m², using 1 cal = 4.184 J; (2) a multiplier of 0.1 that was apparently missing in the second part of Eq. (10) has been added; and (3) Barbaro's Eq. (11) has been corrected for the missing cosZ term involving the zenith angle Z.

1.1.6 G006: Biga

This is the model proposed by Biga and Rosa (1979).

1.1.7 G007: Bird model

This is the broadband transmittances and turbidity model according to Bird and Hulstrom (1980, 1981). For more details, see Gueymard (2003a, 2012).

1.1.8 G008: CEM

This is the broadband model proposed in Atwater and Ball (1978, 1979). See details in Gueymard (2003a).

1.1.9 G009: Chandra

In this model proposed by Chandra (1978), the Linke turbidity coefficient, T _L , was originally a necessary input. This coefficient is obtained here from an average of four linear relationships between Angstrom’s β coefficient and T _L , as proposed in Hinzpeter (1950), Katz et al. (1982), Abdelrahman et al. (1988), and Grenier et al. (1994):

$$ {T_{\text{L}}} = 2.1331 + 19.0204\beta $$

(11)

T _L must be constrained to the range 2–5 to avoid divergence in Chandra’s model. Chandra reported his results as absolute irradiances in cal/(cm² min) and used a solar constant of 1.94 cal/(cm² min) or 1,353 W/m². His results are therefore divided here by 1.94 to obtain transmittances. This empirical model is based on measurements that certainly used the IPS56 radiometric scale; hence, the multiplication by a correction factor of 1.022 (Iqbal 1983) to comply with the current WRR scale (whose announcement by the World Meteorological Organization in 1978 is posterior to the historical data used for the model’s development).

1.1.10 G010: CLS

This is the Cloud Layer Sunshine model developed by Suckling and Hay (1976, 1977). Note that precipitable water must be in centimeters, contrarily to what is indicated in the original papers.

1.1.11 G011: CPCR2

This is the original CPCR2 model by Gueymard (1989) with revised optical masses (Gueymard 1993).

1.1.12 G012: Dogniaux

This combines the models that have been proposed by Dogniaux for direct radiation (Dogniaux 1976) and for diffuse radiation (Dogniaux 1970). The expression for T _L as a function of β is taken here from Dogniaux (1975). See Gueymard (2003a) for details.

1.1.13 G013: DPP

This is the Daneshyar–Paltridge–Proctor model tested in Badescu (1997), Goswami and Klett (1980), and also reviewed in Festa and Ratto (1993). The DPP acronym is from Badescu (1997). The direct irradiance at normal incidence is here from the original paper by Daneshyar (1978) who used the model of Paltridge and Proctor (1976), with corrected unit for Z. (It is actually degrees rather than radians as Badescu or Daneshyar suggested). Coefficients for diffuse irradiance are given by Daneshyar as 0.218 and 0.299 cal/(cm² h) for the USA, as cited from Kreith and Kreider (1975). Daneshyar also suggests 0.123 and 0.181 for Iran. However, their Eq. (5a) suggests that Z is in radians, which is not correct. Goswami and Klett (1980) used the USA values, which are used here, too, after conversion from cal/(cm² h) to W/m². They used the correct unit for Z. Festa and Ratto (1993) mention the Iran values, but with Z in radians rather than degrees, due to the confusion in the original papers. Badescu (1997) also used radians, but with considerably larger values for the coefficients. To avoid further confusion, the modified equations as used here are provided below:

$$ {E_{\text{bn}}} = 950\left[ {1 - \exp \left( { - 0.075h} \right)} \right] $$

(12)

$$ {E_{\text{d}}} = 2.534 + 3.475h $$

(13)

Here E _bn and E _d stand for direct normal irradiance and diffuse irradiance, respectively, both measured in W/m². The sun altitude angle is given by h = 90 − Z where Z is in degrees.

1.1.14 G014: ESRA1

This is the model that was used to develop the European Solar Radiation Atlas, with the broadband transmittances and turbidity computed according to Rigollier et al. (2000) and ESRA (2010). Here, the Linke turbidity coefficient for an air mass of 2 is obtained from the new Page's formula, Eq. (19) of Remund et al. (2003). This revision was suggested by J. Remund (personal communication, 2006). For more details, see Gueymard (2012).

1.1.15 G015: ESRA2

This model is the same as ESRA1, except that T _L is now obtained using formulae proposed in Molineaux et al. (1998) and Ineichen (2006).

1.1.16 G016: ESRA3

This model is similar to ESRA1 or ESRA2, but T _L is obtained from the empirical formula of Dogniaux (1986) as a function of precipitable water and Angstrom’s β coefficient.

1.1.17 G017: ESRA4

This model is again similar to ESRA1, but T _L is here obtained from Eq. (11).

1.1.18 G018: HLJ

This is the broadband model developed by Hottel (1976) for direct irradiance and later modified in De Carli et al. (1986), Jafarpur and Yaghoubi (1989), Aziz (1990), Khalil and Alnajjar (1995), and Togrul et al. (2000), who all added the diffuse transmittance formula of Liu and Jordan (1960), based on Hottel's own recommendation. The Hottel equations corresponding to a visibility of 23 km are used here, which is consistent with the literature cited. For more details, see Gueymard (2012).

1.1.19 G019: Ideriah

This model is proposed in Ideriah (1981) based on the direct irradiance model by King and Buckius (see entry G027 below).

1.1.20 G020: Ineichen

This model comes from Ineichen's parameterization of the SOLIS spectral model (Ineichen 2008). For more details, see Gueymard (2012).

1.1.21 G021. Iqbal A

This is “model A” from Iqbal (1983). It is adapted from the McMaster (MAC) model (Davies 1975). The original formulation for the Rayleigh transmittance is used here, rather than Iqbal's Eq. (7.4.8), which contains a typo. For more details, see Gueymard (1993, 2012).

1.1.22 G022. Iqbal B

This is “model B” from Iqbal (1983). It is adapted from the Hoyt (1978) model. For more details, see Gueymard (1993, 2012).

1.1.23 G023: Iqbal C

This is “model C” from Iqbal (1983). It is adapted from Bird's model (entry G005 above). For more details, see Gueymard (1993, 2003, 2012).

1.1.24 G024: Josefsson

This is the model by Josefsson (W. Josefsson, unpublished manuscript, 1985), as described, used and/or modified in Davies et al. (1988) and Davies and McKay (1989). See Gueymard (1993) for details.

1.1.25 G025: KASM

This is the modified Kasten (1983) model according to Badescu (1997).

1.1.26 G026: Kasten

This is another modification of the original Kasten (1983) model, this time following Davies and McKay (1989)and Krarti et al. (2006). T _L is obtained from Eq. (A1).

1.1.27 G027: King

This model uses the broadband transmittances and turbidity functions according to King and Buckius (1979) and Buckius and King (1978), respectively. The former reference provides and expression that can be used to calculate direct irradiance (Gueymard 2003a). In the latter reference, the diffuse irradiance is to be calculated from its Eq. (36), but two free parameters remain, namely, κ _L and a ₁. For this investigation, the function κ _L has been fitted to the numerical values provided in Buckius and King (1978) for five discrete values of ß, such as

$$ {\kappa_L} = 0.8336 + 0.17257\beta - 0.64595\exp \left( { - 7.4296{\beta^{{1.5}}}} \right). $$

(14)

Similarly, the value of coefficient α ₁ was only described as varying between 0 and 1. The average value α ₁ = 0.5 is assumed here in the absence of more specific indications.

1.1.28 G028: KZHW

This model was proposed by Krarti et al. (2006) as a combination of those of Zhang et al. (2002) model for global radiation and Watanabe et al. (1983) for the direct/diffuse component separation. The empirical coefficients as modified in Krarti et al. (2006) are used here.

1.1.29 G029: MAC

The McMaster model of Davies (1975) was later used and/or modified in Davies et al. (1988) and Davies and Uboegbulam (1979). The formulation (particularly for the Rayleigh and aerosol transmittances) used here is as described in Davies et al. (1988). A corrected Rayleigh transmittance expression is used here since it was misprinted in the latter report. For more details, see Gueymard (1993, 2012).

1.1.30 G030: Machler

This is the model proposed by Machler and Iqbal (1985).

1.1.31 G031: METSTAT

This is a modified version of Bird’s model (entry G007 above) according to Maxwell (1998). For more details, and how to evaluate the Unsworth–Monteith turbidity coefficient it uses, see Gueymard (1993, 2003a, 2012).

1.1.32 G032: MRM4

This model is described by Muneer et al. (1998) and Kambezidis (personal communication, 2002). The numerical coefficients considered here are for the USA and southern Europe, as given in Muneer (2004, p. 73). For more details, and discussion of this model’s shortcomings, see Gueymard (2003b, 2012).

1.1.33 G033: MRM5

More than a mere revision of MRM4 (entry G032), this is actually a completely different algorithm. It is adapted here from Fortran code, version 5, by Kambezidis and Psiloglou (2008). Additional information came from Kambezidis (personal communication, 2007). Ozone, precipitable water, Angstrom’s coefficient ß, and albedo are the input variables here, as discussed in Gueymard (2012).

1.1.34 G034: Nijegorodov

This model (Nijegorodov et al 1997) uses the air mass formula from Nijegorodov and Luhanga (1996) and transmittance expressions from Bird's model (entry G007 above). It is assumed here that the Earth radius is 6,367 km and that the effective atmosphere thickness is 29.7 km.

1.1.35 G035: NRCC

The original model by Belcher and DeGaetano (2005, 2007) was slightly modified according to B. Belcher (personal communication, 2005).

1.1.36 G036: Paltridge

This is the empirical model of Paltridge and Platt (1976).

1.1.37 G037: Perrin

This is the broadband model of Perrin de Brichambaut and Vauge (1982), as discussed in Gueymard (2003a).

1.1.38 G038. PR

A part of this model is described in Psiloglou et al. (2000). However, the aerosol transmittance expression is here from the REST model (Gueymard 2003a) per Psiloglou's request (Psiloglou, personal communication, 2006), with the purpose to improve the model's performance. The new acronym stands for “Psiloglou-REST.”

1.1.39 G039: PSIM

This model is described by Gueymard (2003b), and its performance for direct irradiance predictions was discussed in Gueymard (2003a,b).

1.1.40 G040: REST250

This is version 5.0 of the REST2 model, as described by Gueymard (2008).

1.1.41 G041: Rodgers

This model uses the Unsworth–Monteith turbidity coefficient. It is described by Rodgers et al. (1978). See also Gueymard (2003a).

1.1.42 G042: RSC

Carroll (1985) combined earlier algorithms by Robinson (1966) and Sellers (1965) to derive this model, hence its acronym.

1.1.43 G043: Santamouris

This model is based on atmospheric transmittances from Psiloglou et al. (2000) (entry G029 above), following the advice of M. Santamouris (personal communication, 2002). A pressure correction was added where needed for consistency, as also discussed in Gueymard (2003a).

1.1.44 G044: Schulze

This model was proposed by Schulze (1976).

1.1.45 G045: Sharma

This empirical model was proposed by Sharma and Pal (1965). They used a solar constant of 2 cal/(cm² min), which is replaced here by the more recent value of 1,366.1 W/m² (Gueymard and Kambezidis 2004).

1.1.46 G046: Watt

This model was proposed by Watt (1978), and its performance was studied by Bird and Hulstrom (1981). These authors, however, seem to have misinterpreted some equations, due to Watt’s non-explicit use of the decimal logarithm. This can explain the poor performance results of the Bird and Hulstrom study. The required extinction layer heights are derived here from the original author’s Fig. 4.0.2. A fixed stratospheric turbidity of 0.02 is assumed.

1.1.47 G047. Wesely

This model by Wesely and Lipschutz (1976a, b) uses visibility, which is derived here from Angstrom’s β coefficient using the formula of King and Buckius (1979) in reverse mode.

1.1.48 G048: Yang

The original model of Yang et al. (2001) is used here with the corrections described in Gueymard (2003a), which were eventually included in a later description of the model (Yang et al. 2006).

1.1.49 G049. Zhang

The model proposed by Zhang et al. (2002) and Zhang (2006) described site-specific coefficients empirically derived from radiation measurements in China. For this study, the average of the coefficients tabulated for 24 sites (Zhang 2006) was rather used for improved universality. The model’s Gompertz function that separates the direct and diffuse components appeared to generate unphysical values, which was caused by its coefficient a ₄ being misprinted. The correct value (2.99) is used here (Zhang, personal communication, 2008).

1.1.50 G050: HS

This is a combination between the model by Hourwitz (1945, 1946) to predict direct irradiance and the model by Schulze (1976) to predict diffuse irradiance.

1.1.51 G051: ABCGS

This is a combination between the model by Adnot et al. (1979) to predict direct irradiance and the model by Schulze (1976) to predict diffuse irradiance.

1.1.52 G052: Paulescu and Schlett

In this model by Paulescu and Schlett (2003), the value γ = 0.5 is adopted here to approximate a Rayleigh atmosphere, based on the suggestion by Paulescu (personal communication, 2009).

1.1.53 G053: Janjai

This model is based on publications by Janjai (2010) and Janjai et al. (2011).

1.1.54 G054: REST281

This is version 8.1 of REST2 model, which contains a few corrections compared to the entry G040 above and the original description by Gueymard (2008). For more details, see Gueymard (2012).