Modeling Solar Radiation at the Earth’s Surface

Daily Distributions of Global Radiation

The most important, probably, study on the daily global irradiation distributions, cited in most works, is that by Liu and Jordan (1960). Based on this study, several works have been carried out using different methodologies as well as different types of equations to model the solar radiation variability.

For instance, the daily global irradiation scaled to the mean daily global ir­radiation over a month (Liu and Jordan 1960), and the daily global irradiation scaled to the clear-sky daily global irradiation (Bois et al 1977; Exell 1981), have been studied. Nevertheless, most of the studies deal with the daily global clear­ness index, as those by Whillier (1956), Liu and Jordan (1960), Bendt et al. (1981),

Hollands and Huget (1983), Olseth and Skarveit (1984), Saunier et al. (1987), Graham et al. (1988), Gordon and Reddy (1988), Feuillard and Abillon (1989), Rbnnelid and Karlsson (1997) or Babu and Satyamurty (2001).

The earlier work by Liu and Jordan (1960) examines the daily clearness index distribution for certain monthly mean values of the clearness index, kM. They used 5 years of daily data from 27 locations in the United States with latitudes from 19° to 55° North. The authors pointed out that the curves of CDFs of kp do not significantly change with the month and location, but they rather depend on the monthly average, kM, of daily values, for each considered month. The cumulative distributions^ = F(kp |kM), were generated for the monthly averages clearness index values, kM = (0.3|0.1|0.7). The authors made the hypothesis of universal validity of the CDF curves although they did not provide any fit function for the distributions (Fig. 3.1).

Bendt et al. (1981) studied the different frequency distributions from which purely randomsequences of daily clearness index, kp, can be generated, with the restriction of kM to be bound to a specified value. They proposed the following expression for the density function:

_ Y eYx

f(x |x ) = —-! —, (3.12)

eYxmax — eYxmin

being x = kp and x = kM, with xmin < x < xmax and where xmin = 0.05 and xmax conveniently selected for each month. The parameter у can be calculated from the following equation:

I x ■ 1 e^^min x _____ 1 e^max

_ = (Xmin y) e ^Xmax r)e. (313)

eYXmin — erxmax

The corresponding CDF is expressed as:

__ erxmin — erx

F (x lx ) =------------------- . (3.14)

V ' erxmin — erxmax

Figure 3.2 shows a plot of Eq. (3.14) for different values of x. The authors also found that the distributions depend on the season (Fig. 3.3).

Because of the difficulty in obtaining the parameter у from Eq. (3.13), Suehrcke and McCormick (1987) proposed the following simplified expression:

У = a ,tg fnx — (xmax — xmin) /2 , (3.15)

xmax - xmin

where xmin = 0.05 and A = 15.51 — 20.63xmax + 9.0xmax.

Based on this distribution, Reddy et al. (1985) suggested that the maximum value of the Bendt’s distribution can be yielded from the linear expression:

xmax = 0.362 + 0.597x.

Fig. 3.2 The CDFs based on Bend’s model. The curves are similar to that of Liu and Jordan (1960), but exhibit a different behaviour around the unity value of fractional time. Adapted from Bendt etal. (1981)

Hollands and Huget (1983) proposed the use of a modified gamma PDF such as:

f(x |X ) = C(Xmax ~ x) eXx, (3.17)

xmax

with x = kp , x = kM, and 0 < x < xmax. The parameters C and X depend on xmax and x and C is yielded by:

The relation between xmax and X is given by:

- = [ (X + xmax) (1 - eXxmax)+ 2xmaxeXxmax] eXxmax - 1 - Xxmax

The CDF is then:

Figure 3.4 shows the gamma functions based onEq. (3.17). Notice the unimodal character of the distributions. Figure 3.5, obtained based on Eq. (3.20), shows the

CDFs for different monthly mean values of kM. It can be observed the similarity with those generated from the equations suggested by Bendt.

Olseth and Skarveit (1984) used another type of normalisation based on an index. This index, ф, depends on the maximum and minimum irradiation values of the considered location on the Earth, defined as:

, x — xmin

ф = - - , (3.21)

xmax xmin

with partitions ф = (0.0|0.1|1.0) and being x = kf3. Then, they fitted the curves using two modified gamma distributions in such a way that the bimodality of the frequency distributions was captured. That is,

f (ф |ф) = rnG (ф, Х) + (1 - rn)G (ф, Хг), (3.22)

being:

Х1 = -6.0 + 21.3 ф, (3.23)

Х2 = 3.7 + 35є-53<ф . (3.24)

These authors introduced for the first time a bimodal distribution with a clear - sky mode at high ф values and an overcast mode at low ф values. By means of this new methodology, they achieved to accurately reproduce the distributions for high latitudes. The results were compared with those predicted by the Hollands - Huget model showing significant differences and yielding a better fitting to the data. Figure 3.6 shows the observed differences between the fitting of the Olseth-Skarveit model and that of the Hollands-Huget model.

Most of these research studies on solar radiation have been carried out with the main aim of predicting the long-term average energy delivered to solar collectors. The daily distributions have been studied by Hansen (1999) for 10 locations in the United States, with the aim to be used in biological applications. This author describes three alternative models for the distributions and emphasises the strong non-normality of them. Wang et al. (2002) also analysed the behaviour of these distributions and the fluctuations introduced by the topography, with the aim of using the results for terrestrial ecosystem studies. All the distributions showed to be asymmetrical. The asymmetry shown by the annual irradiation distributions at high latitudes was studied by Rbnnelid (2000). The analysis conducted by Ibanez et al. (2003) for 50 locations in the USA aimed to test the modality of the daily clear­ness index distributions. This study concluded that 60% of the distributions showed a bimodal behaviour. Given the predominance of the bimodal shape of the probabil­ity density distributions, Ibanez et al. (2002) proposed a bi-exponential probability density function. This function used the mean monthly clearness index and the mean monthly solar altitude at noon to fit the observed behaviour of the daily clearness indices.

Tiba et al. (2006) analysed the CDFs for 23 sites located in the Southern hemi­sphere. These authors concluded that the Liu and Jordan CDFs do not have a uni­versal character as previously stated by other authors, such as Saunier et al. (1987).

~2.a

1.0

Fig. 3.6 Differences between the fitting of the Olseth-Skarveit model and that of the Hollands - Huget model. The Olseth-Skarveit PDF is bimodal. Adapted from Olseth and Skarveit (1984)

Gueymard (1999) presented two new models to predict the monthly-average hourly global irradiation distributions from its daily counterpart; whereas Mefti et al. (2003) used the monthly mean sunshine duration to estimate the probability density func­tions of hourly clearness index for inclined surfaces in Algeria.