A Global Solar Radiation Model for the Design of Solar Energy Systems

RESULTS AND DISCUSSION

Daily Global Solar Radiation
Knowing cloudiness degree K, julian day N and the hour t, the model was computed for two years, the daily global solar radiation was estimated using Eq. 1-6. To evaluate the accuracy of the model three methods are used. The root mean square error is defined as:

(8)

Where,

Fig. 3:	Daily global radiation (W m-2) modeled versus measured for K<0.5

Fig. 4:	Daily global radiation (W m-2), modeled versus measured for 0.5<K<1

The mean bias error is defined as:

(9)

The mean relative percentage error is defined as:

(10)

The sign of the errors is neglected in the Eq. 10.

The daily linear regression coefficients a and the coefficients of determination R² confirmed that the model is more accurate for K<0.5 than for 0.5<K<1 (Fig. 3 and 4).

Table 1:	RMSE, MBE (MJ m-2) and MPE vs cloudiness degree class for daily global radiation

Fig. 5:	Monthly global radiation (MJ m-2), modeled versus measured using the present mode

Fig. 6:	Mean bias errors for monthly global radiation

Table 1 shows RMSE, MBE, MPE versus cloudiness degree class. The analysis of results confirmed that good results can be obtained using this model. For K< 0.5 a negative value of the MBE gives under-estimation in calculated values and vice versa for 0.5<K<1. Finally the model can be employed for estimating daily global solar radiation for the southern region of Tunisian.

Monthly Global Solar Radiation
The monthly global solar radiation at the south of Tunisia (Gabès) was estimated by using Eq. 1-7 for two years.

The high level of agreement between the predicted and measured values of global radiation can be appreciated from the monthly linear regression coefficient a and coefficient of determination R² shown in Fig. 5.

Table 2:	Modeled and measured mean monthly global radiation (MJ m-2)

Table 3:	Annual and seasonal means of measured and estimated global solar radiation (MJ m-2)

Fig. 7:	Mean relative percentage error for monthly global radiation

The MBE and MPE plots in Fig. 6 and 7 show that the performance of the model was excellent during the year but is relatively poor for April and August where the MBE was respectively 78 and 55 and the MPE was 11 and 9%. The model is under-estimated for six months and over estimated for the others.

Seasonal Variation of Global Solar Radiation
Table 2 shows the variation in the average monthly global radiation on horizontal surface. Measurements and computation, both show similar evolutions. The minimum is 353 MJ m^-2 month for January and the maximum is 835 MJ m^-2 month for July. The monthly global radiation was estimated with an accuracy of 4.5%.

Table 3 shows annual and seasonal means in which lower value is 1152 MJ m^-2 in winter and the higher value is 2277 MJ m^-2 in summer. If the mean percent errors are compared, the best results are for Autumn and Summer where the MPE are, respectively 1 and 3.4% while spring and winter were estimated with an accuracy about 4%. Results show that the annual mean of estimated global radiation is 6818 MJ m^-2 with 0.34 of mean percent error.

Table 4:	Monthly linear regression coefficients a, (calculated/measured), coefficients of determination R2, RMSE, MBE (MJ m-2) and MPE for the tow models

Fig. 8:	Mean monthly global insolation (MJ m-2), modeled versus measured using Sivkov model

COMPARISON WITH SIVKOV MODEL

In this part the objective is to compare the results of the newly-developed model with results from other existing model using data obtained in the south of Tunisia. The second model is an empirical formula (11) which was originally proposed by Sivkov (1964a and b) in the form:

(11)

Where, H_m and n_m are the monthly global irradiance (in cal. cm^-2) and the monthly sunshine hours, respectively (Fig. 8), while α is the noon altitude of the sun on the 15th of the month. The Root Mean Square Error (RMSE), Mean Bias Error (MBE) and Mean Percent Error (MPE) are used for this comparison. RMSE, MBE and MPE are shown in Table 4.

Table 4 shows that the low values of RMSE, MBE and MPE are remarkable for the two models but the first one yield the best results. The MBE indicates that the tow models are generally over-estimated. We mention that the Sivkov model was developed on using data from other stations.

CONCLUSIONS

The development of a mathmatical model for prediction of global solar radiation in horizontal surface is presented. The test of this model confirmed that the low values of RMSE, MBE and MPE are remarkable with estimating the daily global solar radiation and that the best results are for k< 0.5, the model produces a low values of RMSE, MBE and MPE with estimating the mean monthly and yearly global solar radiation that have an accuracy of 4.5 and 0.34%, respectively. The model predictions are in good agreement with the experimental data, consequently the developed model will be a useful tool for the design of various solar energy systems.