Modeling Solar Radiation at the Earth’s Surface

# The “k” Indices

The seasonal and diurnal variations of solar irradiance are described by well - established astronomical relationships. At the short-term, the behaviour of solar radiation is mainly ruled by the stochastic parameters: frequency and height of the clouds and their optical properties, atmospheric aerosols, ground albedo, water vapour and atmospheric turbidity (Woyte et al. 2007). As a consequence, the actual solar irradiance can be considered as the sum of two components: deterministic and stochastic. We can isolate the stochastic component by defining the instantaneous clearness index as:

where G is the horizontal global irradiance at ground; I0h, the extraterrestrial hor­izontal solar irradiance; ISC = 1367W/m2, the solar constant; E0, the eccentricity correction factor and 0z, the zenith angle. The sub-indices denote: h, horizontal; n, normal and 0, extraterrestrial.

E0 and 0z depend on astronomical relationships only and can analytically be de­termined for each time instant (Iqbal 1983). The instantaneous clearness index kt accounts for all meteorological, thus stochastic, influences. Therefore, clearness in­dex is the quantity needed to focus on the analysis of fluctuations in solar irradiance. It gives the ratio of the actual energy on the ground to that initially available at the top of the atmosphere accounting, therefore, for the transparency of the atmosphere.

Similarly, we can define the kb and kd indices for the diffuse and direct radiation components, respectively.

is called diffuse fraction (in some literature, diffuse coefficient) and is defined as the ratio of the diffuse irradiance on the ground to the extraterrestrial global horizontal one.

(3.3)

is called direct fraction and is defined as the ratio of the horizontal direct irradiance on the ground to the extraterrestrial global horizontal one.

From the well-known expression, G = In cos 0z + D, it is evident that:

kt = kb + kd. (3.4)

These indices can also be defined for the irradiation by integrating the irradiance values over a given time interval At. The clearness index will be then denoted by kft and defined as the relation between the horizontal global irradiation on the ground and the extraterrestrial global irradiation over the same time interval At:

kAt = fitGdt =

1 /Atl0hdt Ho

The most usual integration periods are the day and the hour, although other peri­ods, as the month, can also be used. When At is less than 5-10 minutes, the clearness index is said to be instantaneous. Therefore, according to the integration period, we deal with monthly (kM), daily (kp), hourly (kj1) or instantaneous (kt) clearness in­dex. Similarly, there can be defined the diffuse index:

kAt _ JAtDdt

d /Atlohdf

and, then, the monthly (kM), daily (kp), hourly (kH) or instantaneous (kd) diffuse fraction. Finally, we can define the direct index:

kA, = /„UosMt_ (3.7)

and, then, the monthly (kM), daily (kp), hourly (kH) or instantaneous (kb) direct fraction.

The ensemble study of the kt, kd and kb indices provides an adequate informa­tion to characterise the actual state of the atmosphere and to know the solar energy availability at a given place.

## Modeling Solar Radiation at the Earth’s Surface

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### Foreword

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