Modeling Solar Radiation at the Earth’s Surface

Berlynd Model for Determining Global Irradiation on Clear Days

In 1956 Berlynd proposed a model, which is a function of astronomical parameters, albedo and meteorological parameters, that is reported in Kondratyev (1969) and is given as

Gc = Gsccos 0z/[1 + fsecOz], (4.8)

where the coefficient f is a function of albedo, atmosphere optical thickness in the zenith direction and parameters that characterize the diffuse portion of the global irradiation. The value for f is determined by adjusting the calculated values for Gc to the values measured on a clear day at the site under consideration. It should be emphasized that the Berlynd model is the only one of the models discussed that also takes into consideration meteorological parameters, i. e., the magnitude of the coefficient f.

The authors have recently reported (Ianetz et al. 2007) upon the application of Berlynd and the above listed models to the solar global irradiation being moni­tored at three meteorological stations located in the semi-arid Negev region of Is­rael, viz., Beer Sheva, Sde Boker and Arad. The following discussion will be limited to the results for Beer Sheva, which are similar to those corresponding to the other two sites.

The global irradiation is measured utilizing an Eppley PSP pyranometer and a Campbell Scientific Instruments datalogger monitors and stores the data at 10 minute intervals (i. e., the meters are scanned at 10 s intervals and average values at 10 minute intervals are calculated and stored). The database utilized to test the clear sky correlations consisted of measurements made from January 1991 through December 2004. In addition, a much larger database for Beer Sheva, January 1982 through December 2004, was utilized to perform an in-depth analysis of this site. The Beer Sheva meteorological station is part of the national network and the in­strument calibration constant is checked periodically. Only those days for which all hourly values were recorded were included in the analysis. The validity of the individual measured hourly values was checked in accordance with WMO recom­mendations (WMO 1983). Those values that did not comply with the WMO recom­mendations were considered erroneous and rejected (i. e., the corresponding daily value was rejected).

The Iqbal filter was applied to the database to determine monthly average clear sky global irradiation. The threshold value for a clear sky was set as Kt > 0.7 but it was observed that there were a number of months that had either a very small number or zero days that met this criterion. Consequently, the threshold was lowered to Kt > 0.67 for January, February and October and further reduced to Kt > 0.65 for November and December.

It was observed from a preliminary screening of the simple clear sky correlations, viz., Eqs. (4.1-4.6), that Eqs. (4.2, 4.5 and 4.6) significantly underestimated the clear sky global irradiation as determined by applying the Iqbal filter to the site databases. The same result was observed when Eq. (4.7) was applied to these sites. Consequently, only those models corresponding to Eqs. (4.1, 4.3 and 4.4), referred to as the H, BD and ABCG models, respectively, were tested.

The Berlynd model was also applied to the database. There exist tabulated monthly values of Berlynd coefficient f corresponding to moderate latitudes, cf., Kondratyev (1969), but a set of monthly values for Beer Sheva was determined as was done previously by Kuusk (1978) for his site in Estonia. As mentioned above, the f values were determined by adjusting the calculated values for Gc to those val­ues measured on a clear day at the site, i. e., Beer Sheva. It was observed that the f values exhibit a maximum in June (0.320) and a minimum in January (0.170), cf., Table 4.3.

An inter-comparison between the monthly average clear day solar global irra­diation as defined by the Iqbal filter and that determined by the application of the four models to the Beer Sheva database is reported in Table 4.4. The deviation (%) of the monthly average clear day solar global irradiation is on the average smallest for the case of the Berlynd model. Its maximum deviation is 5.06% (October) and its average monthly deviation is 1.95%. It should be emphasized that the Berlynd model is fitted to the site conditions via the coefficient f, whereas the other models are regression equations expressed as a function of the zenith angle.

The solar global irradiation database consisting of the years 1982 to 2004 (23 years) was utilized to perform an in-depth analysis for Beer Sheva. The monthly average daily extraterrestrial solar irradiation, the absolute monthly maximum mea­sured daily global irradiation (i. e., a single value for each month), the monthly aver­age of the maximum daily global irradiation measured each year (i. e., average of 23 values for each month), monthly average clear day global irradiation using the Iqbal filter, monthly average measured daily global irradiation, the monthly average of the minimum daily global irradiation measured each year (i. e., average of 23 values for each month) and the absolute monthly minimum measured daily global irradi­ation (i. e., a single value for each month) during the years 1982 to 2004 is shown in Fig. 4.1. It should be noted that both the maximum and minimum solar global

Table 4.3 Monthly ‘f’ values for Berlynd model

Month

J

F

M

A

M

J

J

A

S

O

N

D

Beer

0.170

0.190

0.230

0.275

0.310

0.320

0.312

0.300

0.280

0.210

0.190

0.176

Sheva

Table 4.4 Inter-comparison of the clear day solar global irradiation as determined by the Iqbal filter and the models (kWh/m2)

Month (days)

Iqbal

Berlynd

B-D

A-B-C-G

H

January (54a)

4.06

4.02c

3.81d

3.43

3.87

February (104a)

4.87

5.02

4.70

4.30

4.87

March (62)

4.34

4.30

5.98

5.62

4.28

April (96)

7.39

7.34

7.09

4.76

7.53

May (147)

8.02

7.93

7.87

7.53

8.34

June (278)

8.24

8.18

8.18

7.84

8.68

July (198)

8.09

8.10

8.04

7.70

8.54

August (91)

7.52

7.69

7.44

7.13

7.89

September (41)

4.62

4.51

4.41

4.08

4.75

October (60a)

5.34

5.61

5.13

4.73

5.34

November (32b)

4.14

4.27

4.10

3.70

4.21

December (28b)

3.56

3.70

3.58

3.19

3.60

aKT > 0.67. bKT > 0.65.

c Bold type - model result with smallest deviation from Iqbal. d Italic type - model result with > 5% deviation from Iqbal.

irradiation values for a particular month for each year are independent and can be applied to determine the standard error of the average. The standard error of the average has been found previously (Kudish et al. 2005) to be less than the inherent measurement error of the instrument.

Fig. 4.1 Analysis of the solar global irradiation for Beer Sheva (1984-2004)

It is observed from Fig. 4.1, that the magnitudes of the monthly average clear day global irradiation values based upon the Iqbal filter are very close to the correspond­ing average of the maximum daily global irradiation measured each year. Also, it is observed that the magnitudes of the monthly average daily global irradiation values approach that of the corresponding Iqbal filter values during the summer months. This testifies to the prevalence of clear sky conditions during the summer months in Beer Sheva, cf., Ianetz et al. (2000).

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