Journal of Earth Science & Climatic Change
Open Access

The solar radiation is the most important energy resource to man and indeed it is essential factor for human life. The knowledge of solar radiation distribution at a particular geographical location is vital of solar energy devices; solar energy is the clean, abundant, renewable and sustainable energy resource from the sun which reaches the earth in form of light and heat [1,2]. The developing countries solar radiation data are not easily available for not being able to afford the data equipment and techniques involved [3-5]. The most for renewable and sustainable energy has increased research in in any given location. The solar radiation from the sun is fast becoming an alternative to other conventional sources of energy. The most of variable types of clean and energy basses, solar energy appears to be the most favored option because of its infinite and non-polluting nature [6-13]. The solar radiation is the most ancient source of energy; it is the basic element for almost all fossil and renewable types. Solar energy is freely available and could be easily harnessed to reduce our reliance on hydrocarbonbased energy by both, passive and active designs. Precise solar radiation estimation tools are critical in the design of solar systems [14-16].

The measurement of solar radiation is always a necessary basis for the design of any solar energy conversion device and for a feasibility study of the possible use of solar energy. The measured data are the best but cannot always be available [17-20]. Solar energy is mostly preferred due to being safe, clean, free, limitless and non-polluting. The total solar radiation data in a particular region are essential for energy applications because the photovoltaic panel is directly affected by solar radiation [21-25]. The knowledge of total solar radiation data is essential for the research and basses of the economic viability of systems that use solar energy [26]. The total solar radiation data important for solar energy use are in the form of diurnal variation, monthly mean daily values, frequency distribution of number of constant consecutive days in each certain month, with insulation below above a certain threshold and frequency distribution of monthly mean and annual mean values [27-30]. The measurements of the total solar radiation are very important to design the solar system, but it is not available in the few of countries, which depend on the prediction by using the meteorological parameters in any locality [31-35].

The developed models of correlation used in estimating total solar radiation in Locations of similar latitude, altitude and climatology, the accuracy of such models depends on the quality and quantity of the measured data used. Though less accurate, modeling is a better tool for the estimation of global solar radiation at places where measurements are not available [36-45]. The most common is the Angstrom-type one-parameter equation correlating the total solar radiation to the percentage of bright sunshine hours in a simple linear regression form [46-55].

The models of daily solar radiation con are dividing to models consider the effect of time [56-60], and models’ variation in weather conditions and distribution between hourly total solar radiations of all times which results in a normal distribution [61-68]. The total solar radiation of different local sites is usually globally required [69-75]. The solar radiation data obtained through direct measurements are not available for different places across the world. Thus, the various estimation procedures have been developed to evaluate total solar radiation [76-80].

The initial empirical correlation to estimate of the total solar radiation by Angstrom [81], the Angstrom correlation was adjusted by Prescott [82] and Page [83], and their modifications are being used widely to estimate total solar irradiance [84,85]. The several derivatives for predicting solar radiation over an entire country has been examined in many countries have also modified the correlation [86-100]. These different models exist that correlates with different meteorological [101-115]. Moreover, several empirical models have been used to estimate solar radiation; utilizing available geographical, climatological and meteorological parameters [116-134].

The accuracy of the developed model which is evaluated using statistical errors, a positive MBE value indicates the amount of overestimation in the predicted total solar energy and vice versa. On the other hand, RMSE provides information on the short-term performance of the model and is a measure of the variation of the predicted values around the measured data. RMSE also shows the efficiency of the developed model in predicting future individual values. A large positive RMSE implies a big deviation in the predicted value from the measured value [135-144].

The essence of this research is to development of models for estimating the total solar radiation for the selected location during the present time from 2005 to 2017 and compares the statistical results in the present work. The data in this study were obtained from the Meteorological and Environmental Protection Agency (MEPA) in KSA.

Al-Baha location is situated between (Lat. 200 01' N and Long. 410 28' E) and it is an important city in the south-western region of the Kingdom of Saudi Arabia between the Holy Makkah and Asir region (KSA) shown in Figure 1. The area of study is divided by huge and steep Rocky Mountains into two main sectors, a lowland coastal plain at the west, known as “Tehama”, and a mountainous area with an elevation of 1450 to 2400 m above sea level at the east, known as “Al-Sarat or Al-Sarah” which forms a part of Al-Sarawat Mountains. The selected location in summer is moderate and cold in winter with mean temperatures ranging between 12–23°C. With regard to general weather conditions, the temperature varies from a minimum of 3°C to a maximum of 30 °C and average of 18°C. The surface pressure changes from 750 to 769 mb with a mean value of 759 mb. The relative humidity varies between 5% and 96% with an average value of 51% [144,145].

Figure 1: Map of Al-Baha city is the location studied in the present study.

The factors affecting of the solar radiation absorbed at the Earth’s surface are consider fourth types; Firstly, the effect of the atmosphere in modifying the sun’s radiation before it arrives at earth surface is quite complex. When the sun rays get to within about 40 kilometers at earth surface some of the energy is absorbed in band of zone and some is absorbed and scattered by an upper dust layer which is periodically recharged volcanic eruption or galactic dust clouds. A considerable amount of energy is absorbed and scattered by dry air molecules, water vapor lying close to earth surface and seasonally varying lower layer dust. Secondly the distance between the Earth and Sun, the distance between the earth and sun at aphelion is equal to 152 million kilometer and at perihelia equal to 146.2 million kilometers. Third the incident angle of solar radiation, the earth receives maximum radiation when the radiation is incident at perpendicular to earth surface, when the incident angle of radiation increases the amount of radiation decrease. Also, the amount of radiation decreases with increase of atmosphere thickness which cross it. Fourth the length of day and rotation of earth, the earth rotated about the sun in 365.25 days and rotated by itself in 24 hours and the seasonal variation produce according to the inclined angle of earth axis rotation. The length of the days varies the amount of radiation received per days, then for long day the earth receives more radiation, more than short day these information’s are clear in Figure 2 [1].

Figure 2: The affecting factors of the intensity of solar radiation absorbed at earth’s surface.

The first correlation proposed for estimating the monthly average daily global radiation is based on the model of Angstrom [81]. The original Angstrom-Prescott type regression equation-related monthly average daily radiation to clear day radiation in a given location and average fraction of possible sunshine hours is given by the equation:

G/G_o = a+b (S/S_o) (1)

Where G is the monthly average daily global radiation on a horizontal surface (W/m²), G_o is the monthly average daily extraterrestrial radiation on a horizontal surface (W/m²), S the monthly average daily hours of bright sunshine, so is the monthly average day length, and “a” and “b” values are known as Angstrom constants and they are empirical.

Solar radiation incident outside the earth’s atmosphere is called extraterrestrial solar radiation. On average the extraterrestrial irradiance is 1367 W/m² (solar constant). The extraterrestrial radiation G_o is given as follows [143,146,147]:

G_o = (24/π) × I_SC × E_o × [cos φ cos δ sin ω+ (πω/180) sin φ sin δ] (2)

Where E_o is the correction factor of the Earth's orbit and ω is the sunrise/sunset hour angle given by:

E_o =1+0.033cos (2πdn/365), (3)

ω = cos^-1(-tan φ tan δ) (4)

And φ is latitude and the solar declination angle of the sun (δ) is the angle between a plane perpendicular to a line between the earth and the sun and the earth’s axis, which given in degrees according to Spencer [148] as:

δ = (0.006918-0.399912 cosΓ+0.070257 sinΓ-0.006758 cos2Γ+0.000907 sin2Γ-0.002697 cos3Γ+0.00148 sin 3Γ) (180/π) (5)

Where Γ is the day angle in radiance, it is represented by:

Γ = 2(d_n-1)/365 (6)

Where d_n is day of the year.

Therefore, the monthly mean of daily global radiation G was normalized by dividing with monthly mean of daily extraterrestrial radiation G_o. We can define clearness index (K_T) as the ratio of the observed/measured horizontal terrestrial solar radiation G, to the calculated/predicted horizontal/extraterrestrial solar radiation (G_o). Clearness index (K_T) gives the percentage deflection by the sky of the incoming global solar radiation and therefore indicates both level of availability of solar radiation and changes in atmospheric conditions in a given locality.

K_T = G/G_o (7)

The commonly used solar energy models developed in the past are based on linear and nonlinear models [109]. These models give a correlation between solar energy on a horizontal surface and some meteorological variables such as; shine hour’s s, ambient temperature T, cloud cover c_w, relative humidity R_h, and maximum T_max. and T_min. ambient temperatures. The linear models use simple linear function while the nonlinear models are polynomial function of the third or fourth degree [5,148].

Solar researchers have developed many empirical correlations which determine the relation between solar radiation and various meteorological parameters. As the availability of meteorological parameters, which are used as the input of radiation models is the most important key and output of radiation models (i.e., solar irradiance and solar irradiation). Among the models, some of them are based on ratio of monthly average daily global radiation to the extraterrestrial radiation (G/G_o), non-linear and some are based on empirical coefficients ‘a’ and ‘b’. Proposed models for the present research are listed in Table 1 [1].

Table 1: Proposed models for the present work.

There are numerous works in literature which deal with the assessment and comparison of monthly mean daily solar radiation estimation models. The relative ability of the different models to predict the solar radiation on horizontal and tilted surfaces was tested. The performance of the individual models was determined by utilizing statistical methods. There are numerous works in literature which deal with the assessment and comparison of daily solar radiation estimation models. The most popular statistical parameters are the MBE (mean bias error) and the RMSE (root mean square error). In this study, to evaluate the accuracy of the estimated data, from the models described above, the following statistical estimators were used, MBE, RMSE, MPE (mean percentage error) and the correlation coefficient (R²), to test the linear relationship between predicted and measured values. For higher modeling accuracy, these estimators should be closer to zero, and the correlation coefficient, (R²), should approach to 1. However, these estimated errors provide reasonable criteria to compare models but do not objectively indicate whether the estimates from a model are statistically significant. The t-Test statistic allows models to be compared and at the same time it indicates whether or not a model’s estimate is statistically significant at a particular confidence level. So, the t-Test was carried out on the models to determine the statistical significance of the predicted values [84].

Mean Bias Error (MBE)

To evaluate the accuracy of the prediction data from the models described above, this test provides information on the long-term performance of a model. A low MBE value is desired. A negative value gives the average amount of underestimation in the calculated value. So, one drawback of MBE is that overestimation of an individual observation may cancel underestimation in a separate observation. We can be obtained the values of MBE as follow:

(23)

And the equation of mean percentage error MPE% is expressed by:

(24)

The subscript i refer to the ith value of the daily solar irradiation; n is the number of the daily solar irradiation data. The subscripts “calc.” and “meas.” refer to the calculated and measured daily solar irradiation values, respectively. A percentage error between -10% and +10% is considered acceptable [143].

Root Mean Square Error (RMSE)

The value of RMSE is always positive, representing zero in the ideal case. The normalized RMSE gives information on the short-term performance of the correlations by allowing a term-by-term comparison of the actual deviation between the predicted and measured values. The smaller the value, the better the model’s performance is, and the equation of RMSE as follows [144].

(25)

The t-Test statistic (t)

The tests for mean values, the random variable t with n-1 degrees of freedom may be written here as follows [145].

t = [(n-1) (MBE)²/ (RMSE)² – (MBE)² ]^1/2 (26)

The smaller values of t-statistic the better the performance of modeling.

The correlation coefficient (R²)

In statistics literature, it is the proportion of variability in a data set that is accounted for by a statistical model, where the variability is measured quantitatively as the sum of square deviations. Most often it is defined notationally as:

(27)

This can also be expressed as:

(28)

Herein, Xi and Yi are the measurements and model estimates, respectively. A high value of R² is desirable as this shows a lower unexplained variation. R² is a statistic that gives some information about the goodness-of-fit of a model. In regression, the R² coefficient of determination is a statistical measure of how well the regression line approximates the real data points. An R² of 1.0 indicates that the regression line perfectly fits the data, which is never valid in any solar radiation estimation model.

The relative percentage error (e) of the estimated values of the global solar radiation at each site may be calculated from the following equation:

(29)

The comparison between measured and estimated values of monthly average daily global solar radiation during the period time from 2005 to 2017 in the present study is illustrated in Figure 3. We noticed that a good agreement between measured and estimated values of the global solar radiation for the whole months in the present study. As can be seen from this figure a small deviation between measured and estimated values of global solar radiation during the period time in the current study. The deviation between measured and estimated values does not exceed 6%. It is also noticeable from this figure that the maximum values of global solar radiation in the summer months. While, the lowest values ranging between the values of the global solar radiation in the winter and autumn months and the differences between them are almost equal. We also conclude that the global solar radiation values in the spring months lie between the highest value of summer and the lowest values in the winter and autumn months. The estimated values of the global solar radiation are nearly coinciding with the measured values. It is clear that, the accuracy of model that used in the present work is almost coinciding with measured for average monthly variation in the selected location. It is clear from the previous results that the model that used in the study gives good results compared to the measured results of global solar radiation during the period time in the present research.

Figure 3: Comparison of measured and estimated of monthly average daily global solar radiation from 2005 to 2017 in the present study.

The monthly average values of the extraterrestrial solar radiation (G_o W/m²), measured (G_mes. W/m²), estimated global solar radiation (G_est. W/m²), and the climate parameters; the fraction of sunshine hours (S/S_o), clearness index (kT = G_mes./G_o), relative humidity (RH), the ratio between min. and max. temperature (θ), the differences between max. and min. temperature (ΔT) and the ratio of change in temperature to day length (ΔT/S_o) in the present research during the period time from 2005 to 2017 are listed in Table 2. From this table we notice that the ratio between the measured and estimated global solar radiation with extraterrestrial solar radiation is varies 68-76% and 68-75% respectively. Therefore, the differences between maximum and minimum are varying from 24% to 32%. Moreover, the average differences between measured and estimated solar radiation to the extraterrestrial solar radiation in the present work during the period time not exceed 8%. Also, we noticed that the change between measured and estimated values of solar radiation does not exceed 3%. Also, we noticed that from this table the change between the climate parameters; (KT, RH, θ and ΔT/S_o) for all months during the period time in the present study are small variables.

Table 2: Mean value of input climate parameters for selected site in the present work.

The monthly relative percentage error (e) between the estimated (by each model) and measured values of global solar radiation of the year in the present work during the period time from 2005 to 2017 is summarizing in Table 3. The required variables were substituted into each of the aforementioned models to estimate global solar radiation. The performance of each model in predicting the measured solar radiation was then measured using the above statistics. Then, we first consider the predictability of each model during each month of the year using the relative percentage error. The overall performance using the rest of the statistical indicators will be second step to selecting the best model. A relative percentage error between -10 to 10% is considered acceptable for reasonably predicting global solar radiation data for many applications. The relative percentage error varies by model and by month. Therefore, we briefly discuss the main feature and issues in the performance of the models. Table 3 includes fifteen models, which are used to estimate global solar radiation in the current study during the period time from 2005 to 2017. We can divide these models into five different categories.

Table 3: Monthly relative percentage error (e) of the models in the present study.

The first category, the (e) value ranges between 18.56 and -0.18%. All the models in these categories performed well in the months of February, April, July, August, September and October. In October month the models demonstrated the lowest (e) attained by equation (11). The highest (e) values were observed in January month, when equation (12) had the highest values. As regarding, the second category as performed in equation (9). The lowest (e) value is observed in August month. While the highest value is occurred in January month. In addition to the (e) values in March, July and November months consider small values. The third category exhibited significant differences in performance. Equations (10) and (22) demonstrated extremely variable of (e) in all months, the (e) values range from 15.37 to -0.95% and 15.32 to -2.41 respectively. The lowest value of (e) in equations (10) and (22) are observed in July, August and October months. The fourth category there were major discrepancies and the best and worst performing models were in this category. Equation (14) demonstrated the highest value of (e) of all models considered in this study. The highest (e) was 19.45% in the month of August in equation (14). Equations (14), (15), (20) and (21) reported values of (e) less than 10% through March, July and August months. The last category the (e) value range from 16.32 to -0.65% by equation (16) and (17). The lowest value of (e) is occurring in January month for equation (16) and the highest values observed in equation (17) during May month. We notice from the above discussion that the models are performed differently for different months. Some models performed well in some months, while others performed poorly, and vice versa. In addition, some models demonstrated reasonable predictability, while others produced unreliable estimations.

Summarized values of statistical parameters and regression coefficient in the present study during the period time from 2005 to 2017 is listed in Table 4. From this table we select the model with the most accurate performance in estimating global solar radiation in the current study by assessing overall performances using the overall MBE, RMSE, MPE, R² and t-test. Selected the best model is to reduce the number of equations by allowing MPE values between -10 and 10% to be an acceptable range. Using this criterion, we excluded eleven models, leaving five. Three of these five were from first category Eq (12), (20) and new model, one from first category eq (12) and the last one from fourth category eq (20). Among these models, three of them exhibited on MPE and RMSE less than 2%, and other two models less than 8%. While value of R² for three models Eq 12, 20 and new model are 0.963, 0.959 and 0.972 respectively. The lowest values of t-test observed in models no. (12, 20, and new model), this means that, these models consider the best for using to estimate global solar radiation. Finally, the models of Graica eq 12, Bahel eq 20 and new model was the most suitable relation for estimating the global solar radiation for Al-Baha site, KSA.

Table 4: Statistical parameters and regression constants for different models in selected site.

The comparative study of measured and estimated of global solar radiation for Al-Baha location in the period time from 2005 to 2017 is illustrated in Table 5. From this table it has been noticed that the estimated global solar radiation by using all sixteen models are in close agreement with the measured value of global solar radiation in the present study from 2005 to 2017. Also, it is observed that the values of the global solar radiation estimated by models (G5, G13 and new model) are good with measured data. The values of global solar radiation in other models are highly overestimated or underestimated. The estimated values of global solar radiation by modeling (G5, G13 and new model) and measured data are illustrated in Figure 4. From this figure we observed that good agreement between estimated and measured global solar radiation. The deviation between measured and estimated values is not exceeding 1%.

Figure 4: Comparative of measured and estimated solar radiation in the current study during the period time from 2005 to 2017.

Table 5: Monthly mean measured and estimated values of global solar radiation (W/m²) in the present work during the period time from 2005 to 2017.

The objective of this study is to evaluate various models for the estimation of the monthly average daily global radiation on a horizontal surface from bright sunshine hours, relative humidity and temperatures ratio for Al-Baha, KSA. The collected models were comparative on the basis of the statistical error tests such as mean bias error (MBE), the mean percentage error (MPE), root mean square error (RMSE), correlation coefficient (R²) and t-test in the present study. Good agreement between measured and estimated values of the global solar radiation for the whole months in the present study. The maximum values of global solar radiation in the summer months. While, the lowest values ranging between the values of the global solar radiation in the winter and autumn months and the differences between them are almost equal. The estimated values of the global solar radiation are nearly coinciding with the measured values. The deviation between measured and estimated values does not exceed 6%. The ratio between the measured and estimated global solar radiation with extraterrestrial solar radiation is varies 68-76% and 68-75% respectively. The differences between maximum and minimum are varying from 24% to 32%. Moreover, the average differences between measured and estimated solar radiation to the extraterrestrial solar radiation in the present work during the period time not exceed 8%. Also, we noticed that the change between measured and estimated values of solar radiation does not exceed 3%.

The relative percentage error between -10 to 10% is considered acceptable for reasonably predicting global solar radiation data for many applications. The relative percentage error varies by model and by month. The first category, the (e) value ranges between 18.56 and -0.18%. All the models in these categories performed well in the months of February, April, July, August, September and October. The second category as performed in equation (9). The lowest (e) value is observed in August month. While the highest value is occurred in January month. The third category exhibited significant differences in performance. Equations (10) and (22) demonstrated extremely variable of (e) in all months, the (e) values range from 15.37 to -0.95% and 15.32 to -2.41% respectively. The fourth category there were major discrepancies and the best and worst performing models were in this category. The last category the (e) value range from 16.32 to -0.65% by equation (16) and (17). The models are performed differently for different months. Some models performed well in some months, while others performed poorly, and vice versa. In addition, some models demonstrated reasonable predictability, while others produced unreliable estimations.

The regression coefficients of some collected solar models have been generally presented to estimate the global solar radiation with high accuracy in a given site. The statistical results appear that, models of Graica eq 12, Bahel eq 20 and new model was the most suitable relation for estimating the global solar radiation for Al-Baha site, KSA. The estimated global solar radiation by using all sixteen models is in close agreement with the measured value of global solar radiation in the present study from 2005 to 2017. The values of the global solar radiation estimated by models (G5, G13 and new model) are good with measured data. The values of global solar radiation in other models are highly overestimated or underestimated. Good agreement between estimated and measured global solar radiation. The deviation between measured and estimated values is not exceeding 1%. Finally, according to the climate parameters and statistical results, we observed a new simple linear model:

[(G/G_o) = 0.175 + 0.265 (S/S_o) – 1.84(S/S_o)₂]

Based on modified Angstrom model is recommended to estimate monthly average daily global solar radiation Al-Baha site, KSA, and in other areas with similar climatic conditions where the radiation data is missing or unavailable. The current research will help to advance the state of knowledge of global solar radiation to the point where it has applications.