Models for calculating monthly average solar radiation from air temperature in Swaziland

Solar radiation is an important energy source for mankind. Accurate data of solar radiation levels for a particular location is vital for optimum operation of solar energy transducers such as photovoltaic cells and solar thermal collectors. In this work, it is shown a linear relationship exist between monthly average temperatures and solar radiation in Swaziland. The correlation has been utilized to develop two mathematical models for the estimation of solar radiation: one from the measured monthly average temperatures and the other from the square-root of the difference between measured maximum and minimum monthly average temperatures. Both models fit the data well and can be applied to estimate solar radiation in other parts of the region.


INTRODUCTION
Accurate knowledge of solar radiation levels for a particular location is a prerequisite for the determination of the performance of various solar energy transducers such as photovoltaic cells and solar thermal collectors. Solar radiation data is also important in disciplines such as building designs and agricultural processes, for example evapo-transpiration of plants. However, weather stations will, at times do not have data on solar radiation because the instruments for radiation measurement, such as pyranometers and solarimeters, may not be available. As a result, mathematical models have been developed and calibrated to estimate solar radiation in different parts of the world such as in Brazil (Dos Santos et al., 2014), Iran (Saffaripour et al., 2013), India (Bajpai and Singh, 2009), Algeria/Spain (Chegaar et al., 1998), China (Li et al., 2014a, b), Bangladesh (Datta and Datta, 2013), Chile (Meza and Varas, 2000), USA (Allen, 1997) and Nigeria (Umoh et al., 2014). These models estimate solar radiation as a function of meteorological parameters such as temperature, atmospheric pressure, relative humidity, sunshine hours, wind speed, cloud cover, and rainfall.
Minimal empirical models require an input of only one meteorological parameter to estimate solar radiation and they include sunshine-hours based models (Angstrom, 1924;Chegaar et al., 1998;Yang et al., 2006), airtemperature based models (Hargreaves and Samani, 1982;Bristow and Campbell, 1984;Allen, 1997, Dos Santos et al., 2014 and cloud cover based models   (Kostic and Mikulovic, 2017). There are also intermediate models that require the input of two or more meteorological parameters (Okundamiya et al., 2016). In Swaziland, a number of meteorological stations do not measure solar radiation, and the frequently available meteorological records are the daily maximum and minimum temperatures and precipitation. For this reason, this paper evaluates two air-temperature based models for predicting monthly solar radiation in four locations within Swaziland. At one location, measured solar radiation values are available to correctly adjust the empirical coefficients of the models and also to compare with results from other parts of the world.

CLIMATIC CONDITIONS IN SWAZILAND
Swaziland is a small, landlocked country in Southern Africa and is located between South Africa, on the southern, western and northern side and Mozambique on the eastern side. The map of Swaziland is shown in Figure 1. On the western side of the country is the Highveld. The Lubombo plateau lies on the eastern side whilst the Middleveld and Lowveld lie between the Highveld and the Lubombo plateau. The areas of interest in this study, in connection with solar radiation, are Big Bend and Mhlume, both of which lie in the Lowveld, Matsapha (Middleveld) and Mbabane (Highveld). The altitude, latitude and longitude in the various locations are shown in Table 1. The country consists of four seasons, namely, spring (September-October), summer (November-February), autumn (March-May), and winter (June-August).
There are 11 weather stations in Swaziland, but solar radiation levels are recorded in only one, that is in Mhlume. However, daily maximum and minimum temperatures are routinely measured in all the weather stations. In this work, we use the solar radiation data from Mhlume as a reference to develop and calibrate two mathematical models that define solar radiation as a function of temperature. The data is for the year 2004 to 2014. Figure 2 shows that monthly average solar radiation levels at Mhlume are correlated with the monthly average temperature values. Maximum radiation levels were recorded between December and February, during the hottest season, and the lowest solar radiation levels were recorded between May and August, during the coldest season. Generally, the magnitude of the observed solar radiation is proportional to the observed average temperature.

AIR TEMPERATURE BASED MODELS
Given that the most commonly available climatological data in Swaziland is air-temperature, we consider air temperature based models to predict solar radiation at different locations within the country. We first consider that monthly average solar radiation Hav (MJ/m 2 ) is a linear function of the monthly average temperature Tav (°C): , Tmax and Tmin are the average daily maximum and minimum air temperature (°C) for a period of one month, m1 and c1 are empirical constants. The relationship expressed in Equation 1 or Model 1 is motivated by the observation in Figure 2 that the measured solar radiation values are proportional to the observed average temperatures, throughout the year. Previous studies have considered similar linear equations to predict solar radiation, for example, the classical Angstrom model (Angstrom, 1924) assumes that the magnitude of solar radiation is proportional to sunshine hours (Angstrom, 1924;Meza and Varas, 2000;Liu et al., 2012;and Yakubu and Medugu, 2012).
Most temperature-based models in the literature (Meza and Varas, 2000;Liu et al., 2012) assume that solar radiation is a function of the difference between daily maximum and minimum air temperature. This is based on the assumption that the difference generally indicates daily cloudiness. Clear skies corresponds to higher solar radiations levels at the earth surface and cloudy skies corresponds to lower solar radiation levels. In this work, we compare the results of Equation 1 with the classical model of Hargreaves and Samani (1982) that assumes that: where H0 represents extra-terrestrial radiation and m2 is an empirical constant that usually ranges from 0.15 to 0.19, depending on whether the location has an arid or a coastal climate (Allen, 1997). The extra-terrestrial solar radiation, H0 is a function of latitude and can be easily evaluated or obtained in the literature (Duffie and Beckman, 2013). As the solar radiation passes through the earth's atmosphere, it is further modified by processes of scattering and absorption due to the presence of cloud and atmospheric particles. Hence, the solar radiation at the earth's surface is always less than H0.

Evaluation of model parameters
We evaluated the empirical constants for Models 1 and 2 using the measured data of monthly average solar radiation and monthly average temperature at Mhlume during the period 2004 to 2014. These parameters were calculated using the Marquardt-Levenberg algorithm for curve fitting in Gnuplot. The curve-fitting results are also shown graphically in Figure 3 and the best fits are described by m1 = 1.02 ± 0.08, c1 = −4.28 ± 1 and m2 = 0.161 ± 0.002. This means m2 corresponds to the 0.16 that is recommended for this model for locations with arid or semi-arid climates (Hargreaves, 1994;Allen, 1997).

Performance of the models against measured data
The calculated parameters were then utilized to estimate solar radiation levels for each year during the period between 2004 and 2014. The estimated values were then compared with measured values. Figure 4 shows a sample of these results and both models predict values that are consistent with the observed data. The extra-terrestrial solar radiation reaching Swaziland or regions located at the latitude 26.00°S is also given in Figure 4. As expected, the magnitude of the measured or estimated solar radiation is less than the extra-terrestrial solar radiation at the same location since gases and dusts in the atmosphere change the magnitude and spectral composition of the solar radiation that reaches the earth's surface.
In addition, the performance of the models can be quantified by evaluating the root-mean-square-error (RMSE) and the mean-percentage-error (MPE). These are fundamental measures of accuracy in solar energy calculations (Saffaripour et al., 2013;Li et al., 2014, Sonmete et al., 2011Okundamiya et al., 2016). They are, respectively defined as:  where n = 12 is the number of data pairs, is the jth calculated value and is the j-th measured value. Both models give an absolute MPE that is less than five percent, as shown in Table 2 which is considered acceptable in scientific calculations. Model 1 was found to give a lower value of RMSE, an indication of a better performance.
The behavior of the two models is further explored through a scatter plot of the estimated data against the measured data collected for over a decade from 2002 to 2014. Figure 5 shows that both models are consistent with the measured data over this period.

Prediction of solar radiations in other locations
This research utilized the two models with their coefficient calibrated with the measured solar radiation data from Mhlume to estimate monthly average solar radiation levels at Big Bend, Matsapha, and Mbabane. The results are shown in Figure 6 for the year 2012. Both models give consistent results for Mhlume and Matsapha. Model 2 predicts slightly higher radiation levels than Model 1 for Mbabane. The predictions from the two models were then explored over longer periods, using air-temperature data collected between the years 2002 and 2013. The results are given in scatter plot in Figure 7 for the four locations. These results show that both models are consistent in all the locations except in Mbabane, where Model 1 generally predicts lower values than Model 2. This could be due to the fact that the altitude of Mbabane is higher than that of Mhlume by one order of magnitude (Table 1). Given that the value of m 2 is within the recommended range for inland regions (Allen, 1997), Model 1 parameters need to be correctly readjusted for Mbabane.

Conclusions
Air-temperature based models for estimating solar radiation are useful for quantifying solar radiation levels in different locations around the world because they are based on commonly available meteorological data. In this study, a simple linear model that uses average monthly average air temperatures and the classical Hargreaves-Samani model (Equation 2) are shown to accurately predict solar radiation levels at Mhlume in Swaziland with an acceptable mean percentage error. The parameter of the Hargreaves-Samani Model calibrated with measured data from Mhlume is consistent with the value of 0.16 that was recommended by Heagreaves (1994) for inland regions.
For other locations within Swaziland with a slight variation in climatic conditions from Mhlume, the same model parameters can be applied to predict their solar radiation levels. In this regard, we have found that the two models produce consistent results for two locations: Matsapha and Big Bend. However for the third location, Mbabane, which has a wider variation in climatic conditions than Mhlume, the models do not produce consistent results and therefore there is a need to correctly adjust the model parameters for Mbabane or similar regions. Over the past decade, there has been a growing interest in the harnessing of solar energy in Swaziland; mainly using photovoltaic modules to reduce the amount of energy that is imported from neighboring countries. Currently, there is a 100 kW pilot solar farm that is operating in the Lubombo region and there is plan to set up an 850 kW solar plant at a location between Mhlume and Big Bend. Estimation of solar radiation using airtemperature data will therefore assist in the development of the solar energy industry in the country.