Mathematically Integrable Solar Irradiance Model

An approach in two steps for including air temperature on the input parameters list of global solar irradiation models follows. First, an empirical solar irradiance model with air temperature besides cloudiness is employed. Subsequently, daily global solar irradiation is computed by integration of the irradiance model between sunrise and sunset.

Fig. 7.1 Daily global solar irradiation computed with Eqs. (7.3) (bold line) and (7.4) (a band delimited by two curves corresponding to At = 10°C, lower one, and At = 20°C, upper one) in respect to cloudiness

We start with searching for an appropriate correlation between global solar irradi – ance at the ground G(N, t, h) and outside atmosphere Ge, with air temperature t and solar elevation angle h as parameters: G(N, t,h) = Gef (N, t, sinh). An acceptable response yields from the following function:

f(N, Tr • sin h) = c1(N) • c2(N) – Nc3(N) • [Tr sin h]c4(N) (7.5)

where Tr = 1 +1/273 with temperature t in degree Celsius.

The coefficients ci, i = 1,2,3,4 are the subject of a fitting process that is run­ning for different classes of cloudiness. The result is a discrete set of coefficients which could be carried on in a secondary fitting process to approximate it with con­tinuous functions. Thus, in addition to the determination coefficient, the monotone behavior of the discrete coefficients with respect to cloudiness is used as a selection criterion in the first fitting process. Turning f(N, t, h) into a continuous function is a requirement for solar irradiation computation by mathematically integration over time. As an example, Fig. 7.2 shows the coefficients cj, c2, c3 fitted with data coming also from Timisoara (Paulescu and Fara 2005) and the corresponding approximation functions f(N), i = 1,2,3 (c4(N) = f (N) = 1.16):

f (N) = e—0.1341+2.44181-4.66676N2+3.83066N3

f2(N) = 0.7988 + 0.27829N073642 f3(N) = (1.46112 – 0.91168N)-1

It is remarkable that the coefficients are along regular curves. Therefore, for all range of cloudiness the correlation can be expressed as:

Fig. 7.2 Discrete coeffi­cients from Eq. (7.5) and the continuous approximation functions, Eqs. (7.6 a, b,c) in respect to cloudiness

Introduction of a discontinuity near to N = 1 is an ordinary practice in solar en­ergy estimation to improve the model accuracy in most cloudy or overcast situation. Even if these corrections are present, such models cannot estimate global solar ir- radiance at a high level of accuracy in predominantly clouded conditions. However the results are acceptable when the solar irradiation is computed. It can be done by integrating between sunrise and sunset of irradiance:

r ®0

Hj = C G(N, a, Ta(j, a)) da (7.8)

-®0

where cloudiness is replaced with its daily average. For Ta(j, a) a suitable model based on daily air temperature extreme is described as:

Equation (7.9a) empirical adjust the Eq.(7.9b) to a local meteo-climate (a = 0.99 and b = -0.41, at Timisoara). t(j, a), in °C, is the estimated air temperature in the Julian day j at hour angle a. ao(j) is the sunset hour angle and am is the hour angle at which the maximum air temperature is reached. In this model we assume am to be the same in every day. C in Eq. (7.8) is accordingly to the unit of H: for C = 12/п, H is inWh/m2.

The quality of the air temperature estimation using the sine-cosine Eqs. (7.9 a, b) can be assessed from Fig. 7.3. It is a scatter plot of estimated versus observed air temperature at the station of Timisoara. Figure 7.3a shows the scattering of instan­taneous values at 9.00, 12.00 and 15.00 local standard time in the year 2000 while Fig. 7.3b shows the daily mean air temperature in the years 1998-2000. It can be

seen that, when a mathematical integration is performed, the prediction accuracy increase. It is due to the fact that the model performance is high in the middle of the day when maximum of solar energy is collected.

There are a variety of methods with various degrees of complexity developed to approximate diurnal temperature from its maxima and minima. We emphasize here an empirical model from (Cesaraccio et al. 2000) as being acceptably accurate for estimating the hourly mean of air temperature. It is useful when Eq (7.8) is applied for the computation of hourly solar radiation.

All these models demonstrate that the daily global solar irradiation can be related to the corresponding extraterrestrial value using at input only daily minimum and maximum air temperature besides daily mean of cloudiness. But, from common ob­servations, daily extremes of air temperature encapsulate information about weather condition. Consequently the solar energy estimation can be made straightforward by eliminating the cloudiness from the input.

Updated: August 3, 2015 — 6:49 pm