Correlations for estimating solar radiation using sunshine hours and temperature measurement in Osogbo, Osun State, Nigeria

Abstract

In this study, the global solar radiation on horizontal surface in Osogbo, Osun state, Nigeria was analyzed using 11-year data (1997–2007). Correlations using linear and quadratic expressions were developed to relate the global solar radiation on horizontal surface based on relative sunshine hours and temperature measurements for evaluating the monthly average daily global solar radiation. The calculated monthly clearness index values indicate that the prevailing weather condition in Osogbo is heavily overcast. All the developed quadratic correlations gave better correlation coefficients (0.834, 0.872 and 0.823 respectively) than the linear models. However, the Hargreaves and Samani related based quadratic model gave the best among the three developed quadratic expressions and is therefore suggested for estimating the monthly global radiation in this site and its surroundings.

Keywords

global solar radiation / Hargreaves model / sunshine hour / ambient temperatures / Osogbo

Introduction

One of the most viable options needed to reduce the energy crisis in Nigeria is the utilization of abundant solar energy falling on her surface, due primarily to her geographical location which is close to the equator. The knowledge of solar energy radiation is very vital for the optimal design and performance of any solar energy conversion systems (SECs) - for heating and electricity, architectural design, greenhouse structures and selection of cooling systems. According to Falayi et al. [1], solar radiation reaching the earth’s surface depends on the climatic conditions of a particular location and this is thus very vital toward the prediction and design of SECs. Augustine and Nnabuchi [2] mentioned that the best solar radiation information is that obtained from experimental measurements of the direct and diffuse components of the solar insolation at the particular location. However, due to lack of availability of solar radiation actonometers and other associated instrumentation as well as the high cost of maintenance and calibrations of these instruments, alternative option is to correlate solar radiation with other meteorological parameters for locations with similar geographical characteristics.

Research works on solar applications/resources have been conducted in limited locations in Nigeria [1-20]. However, none has been related to Osogbo or the nearest geographical vicinity in south-west region of Nigeria where enormous solar resources exist. The aim of this study is therefore to develop empirical equations based on sunshine hours and ambient temperatures that will be suitable for correlating monthly daily global solar radiation on a horizontal surface for locations having the similar latitude and topography with Osogbo, Osun State, Nigeria. The work will provide information necessary in green house crop management and in the prediction of factors that are useful for controlling the environment especially the domestic air-conditioner and dehumidifier.

Methodology

Site description and solar parameters

The data on sunshine hours, global solar radiation and minimum and maximum temperature used in this study were obtained from Nigerian Meteorological Agency (NIMET), Oshodi, Lagos. The geographical coordinates of the meteorological station in Osogbo are latitude of 7.77°N, longitude of 4.57°E and altitude of 288 m. The respective data collected for the location were the monthly averages for 11 years (1997-2007). The global solar radiation data were captured using Gun-Bellani distillate (measured in milliliter) and were converted into MJ/(m²·d) using a conversion factor of 1.1364 H_GB (where H_GB is the Gunn Bellanni reading) proposed by Sambo [3]. It should be mentioned here that, another conversion factor proposed by Folayan as adopted in Augustine and Nnabuchi [2] exist as (1.35±0.176) × H_GB. However, the Folayan conversion factor may provide an over-estimation of the values when compared with exact values unlike the factor of 1.1364 by Sambo [3], which gives a good approximation with the experimental data.

The Gunn-Bellani distillate solar radiation integrator provides a time integrated assessment of radiation falling on a black body by measuring the volume of the liquid distilled in a receiving graduated tube. It is always being calibrated against standard solar radiation recorders like the pyranometer, pyrheliometer to ascertain the high level of accuracy associated with the respective actinometer. Its low cost, simple observation procedure and absence of any replaceable mechanical or electronic part favors its use for weather data gathering across a number of experimental sites in a developing country like Nigeria [21]. The daily sunshine hour was measured using the Campbell Stokes sunshine recorder which consists of a glass sphere mounted concentrically in a section of a spherical bowl, the diameter of which is such that the sun’s rays are focused sharply on a card held in the grooves in the bowl. The length of the burn trace left on the card represents the sunshine duration (in hour). To obtain valuable results, both the spherical part and the sphere should be made with great precision, and the mounting should be designed so that the sphere can be accurately centered on it. The achievable measurement uncertainty in this instrument is the larger of 0.1 h or 2% of the reading [22,23]. The maximum and minimum thermometers were used for recording the lowest and highest temperature of the air for each day.

Empirical correlations between solar radiation and meteorological data

The major models considered for solar correlations are derived from the Angstrom-Page equation as expressed in Eq. (1) [2,4-6].

(1){\displaystyle \frac{H_m}{H_{\mathrm{0}}}}=a+b{\displaystyle \frac{\overline{n}}{\overline{N}}},

where a and b are regression constants, \overline{n}is the monthly mean daily bright sunshine hours, \overline{N} is the maximum possible monthly mean daily sunshine (MJ/(m²·d)), H₀ is the monthly mean extraterrestrial solar radiation on horizontal surface (MJ/(m²·d)), and H_m is the measured monthly mean daily global solar radiation on a horizontal surface. The clearness index (K_T=H_m/H_{\mathrm{0}}) is the ratio of the monthly averaged daily global solar radiation to the monthly averaged daily extraterrestrial solar radiation, while \overline{n}/\overline{N} is the ratio of the monthly averaged daily of sunshine to the monthly averaged daylight hour [2,5]. The expressions for H₀ and \overline{N} are given in Refs [5-7]. As

(2)H_{\mathrm{0}}={\displaystyle \frac{\mathrm{24}}{\pi }}{I}_{\mathrm{sc}}\left(\mathrm{1}+\mathrm{0.033}\cos {\displaystyle \frac{\mathrm{360}n}{\mathrm{365}}}\right)\left(\cos \varphi \cos \delta \sin {\omega }_s+{\displaystyle \frac{\pi }{\mathrm{180}}}{\omega }_s\sin \varphi \sin \delta \right),

(3)\overline{N}={\displaystyle \frac{\mathrm{2}}{15}}{\omega }_s,

where

(4)\delta =\mathrm{23.45}\sin \left({\displaystyle \frac{\mathrm{360}(\mathrm{284}+n)}{\mathrm{365}}}\right),

(5){\omega }_s=\arccos (-\tan \varphi \tan \delta ),

where I_sc is the solar constant with a mean value of 4.9212 MJ/(m²·d), \varphi is the latitude of the site, \deltais the solar declination, {\omega }_sis the hour angle, and n is the day number ranging from n = 1 on 1st January to n = 365 on 31st December. Equation (1) is a linear form of the Angstrom-Page relation. Some researchers have further expressed the clearness index and fraction of sunshine duration in exponential, logarithmic, power and quadratic forms [1,2,4-10].

For comparison with the Angstrom model, temperature based models were also developed for Osogbo in line with the Hargreaves and Samani model together with average temperature models in Eqs. (7) and (8) as expressed in Refs [11,12].

(6){\displaystyle \frac{H_m}{H_{\mathrm{0}}}}=a+b\mathrm{\Delta }{T}^{\mathrm{0.5}},

(7){\displaystyle \frac{H_m}{H_{\mathrm{0}}}}=a+b{T}_a,

where \mathrm{\Delta }T and T_a are the difference between the mean monthly maximum and minimum temperatures and the monthly averaged temperature respectively. The temperature based models could be very useful in a situation when the measurement of sunshine hours is not available. More so as in Chineke et al. [24], the daily temperature difference is found to be highly correlated with the clearness or cloudiness index. In addition, the quadratic form of the above-mentioned models (Eqs. (1), (6) and (7)) are presented in this study.

Methods of comparison of correlations

The calculated values of global solar radiation generated from the simulation were compared with the measured values for the location under study. The performance of the above models were investigated using the methods of stochastic analysis to calculate the root mean square error (RMSE), mean bias error (MBE), mean percentage error (MPE), mean absolute percentage error (MAPE) and mean absolute bias error (MABE) using Eqs. (8)-(12).

(8)\mathrm{RMSE}=\sqrt{{\displaystyle \frac{{\displaystyle \sum {(H_{\mathrm{pred}}-H_{\mathrm{obs}})}^{\mathrm{2}}}}{n}},}

(9)\mathrm{MPE}={\displaystyle \frac{{\displaystyle \sum \left({\displaystyle \frac{H_{\mathrm{obs}}-H_{\mathrm{pred}}}{H_{\mathrm{obs}}}}\times \mathrm{100}\right)}}{n}},

(10)\mathrm{MBE}={\displaystyle \frac{{\displaystyle \sum (H_{\mathrm{pred}}-H_{\mathrm{obs}})}}{n}},

(11)\mathrm{MAPE}=\mathrm{ABS}\left[{\displaystyle \frac{{\displaystyle \sum \left({\displaystyle \frac{H_{\mathrm{obs}}-H_{\mathrm{pred}}}{H_{\mathrm{obs}}}}\times \mathrm{100}\right)}}{n}}\right],

(12)\mathrm{MABE}=\mathrm{ABS}\left[{\displaystyle \frac{{\displaystyle \sum (H_{\mathrm{pred}}-H_{\mathrm{obs}})}}{n}}\right],

H_pred and H_obs are the estimated and measured values of global solar radiation while ABS represents the absolute value of the expression. The low values of RMSE, MBE, MPE, MABE and MAPE are desirable. It is also possible to have large RMSE values at the same time a small MBE or vice versa. The positive MBE and MPE shows over-estimation while the negative MBE and MPE indicates under-estimation of the values; for long-term performance of the examined regression equations, MPE is preferred. Error measures with absolute values (such as MABE and MAPE) provide valuable information because they do not have the problem of errors of opposite signs cancelling themselves out; a low mean error can at times be misleading but this is avoided with absolute error measures which provide the average total magnitude of the error. On the other hand, absolute error measures can also assume a symmetric loss function through the total magnitude of error without the true bias or direction of the error. Hence, when using the mean absolute values, it is also useful to compute a measure of bias.

Results and Discussion

Global solar radiation and clearness index

The measured values of global solar radiation on a horizontal surface H_m extraterrestrial solar radiation on a horizontal surface H₀, clearness index K_T, measured values of monthly mean daily sunshine hours S_o together with the sunshine fraction S/S_o using Eqs. (1) to (8) are shown in Table 1. Seasonal and monthly variations are observed in both the measured global solar radiation and sunshine hours. The values of these parameters are generally lower in the raining season (April to October) when compared with the dry season (November to March). The maximum values of the monthly mean daily sunshine hours and the monthly mean daily global solar radiation on a horizontal surface in this location are 6.827 h and 15.569 MJ/(m²·d), respectively in November; while the minimum values occur in August as 2.818 h and 9.845 MJ/(m²·d) for the monthly mean daily sunshine hours and global solar radiation on a horizontal surface respectively. These months coincide with the dry and wet periods respectively at the site. The low values of sunshine hours and global solar radiation observed during the raining season resulted from the prevailing cloudy conditions during this period. It should be mentioned that the observed trend in global solar radiation in this study is similar to that of other locations in south-west Nigeria [1,25]. The general variation in trend noticed across the months may also be caused by the sun’s elevation, variations in cloud cover, angle of inclination on which the intensity of solar radiation depends [25,26].

The clearness index, which is defined as the fraction of solar radiation at the top of the atmosphere that reaches a location on earth’s surface varies between 0.271 in August and 0.438 in November, with an annual average of 0.365. It should be noted that the clearness index indicates the level of availability of solar radiation and weather condition at a particular location on the earth’s surface. Using the weather condition classification proposed by Refs [22,27], which are ① heavily overcast weather (K_T≤0.4), ② partly overcast weather (0.4≤K_T≤0.6), and ③ clear weather (K_T≥0.7); the prevailing weather condition in Osogbo can be classified as mainly heavily overcast except from November to February when it is slightly partly overcast. The relative low clearness index (and hence, the weather condition) observed in this study is a common trend in most locations at the southern part of Nigeria. This is because of the prevailing high moisture content and heavy cloud during the raining season, and dust and forest fire smoke (from preparation for farm land for planting season) during the dry season.

Sunshine hours based models

The scattered plots showing the variation of the clearness index with the relative sunshine hour are presented in Fig. 1, where a linear curve (Fig. 1(a)) and a quadratic curve (Fig. 1(b)) which fits to these data points are also presented. For the linear model, the coefficient of determination of 0.807 exists between the clearness index and relative to the sunshine hour while for the quadratic model, the coefficient of determination is 0.834. As can be seen from Fig. 1(a), the values of a and b are determined as 0.1943 and 0.3986, respectively. The sum of the empirical constants (a and b), which is a measure of atmospheric transmissibility and thickness of the cloud across the region is then determined as 0.593. It should be mentioned that a is a measure of the global solar radiation received at the ground as a fraction of the extraterrestrial radiation through an overcast sky while b represents the sensitivity of normalized global radiation to normalized sunshine duration (direct radiation). The summary of derived coefficients for the sunshine hours based models and other models proposed for Osogbo in this study are given in Table 2.

The comparison between the values of a, b and (a + b) determined for Osogbo and selected locations in Southern Nigeria are demonstrated in Table 3. The value of empirical constant a can be observed to be similar across all the sites while there is a slight variation in the value of empirical constant b across the listed locations in Table 3. The observed variation may be caused by the fact that values of the constants (a and b) depend on the latitude, relative sunshine hours, elevation above sea level, relative humidity, maximum air temperature and other meteorological parameters of the selected site.

The comparison between the measured and predicted global solar radiation using linear expression (see Fig. 1(a)) and quadratic expression (see Fig. 1(b)) is illustrated in Fig. 2. It is observed clearly from Fig. 2 that both the linear and quadratic models predict the trend of the global solar radiation in this site quite well. However, the monthly averaged daily global solar radiation is over predicted by both models in the month of January and slightly under predicted in the months of February and March.

Temperature based models

Two types of temperature based correlation models are determined for Osogbo, which are the correlation equation which relates the clearness index to the square root of the difference between the monthly averaged daily maximum and minimum temperatures (Hargreaves and Samani related type model), and the correlation equation which relates the clearness index with the monthly daily averaged temperature model.

The scattered plots showing the variation of the clearness index with the square root of the difference between the monthly averaged daily maximum and minimum temperatures are displayed in Fig. 3, where two linear curves (Fig. 1(a)) and a quadratic curve (Fig. 1(b)) which fit to these data points are also presented. A linear curve which fits with the intercept of zero was applied to this plot to produce the Hargreaves type correlation. In this case, the correlation coefficient of 0.433 exists between the clearness index and the square root of the difference between the monthly averaged daily maximum and minimum temperatures (Fig. 3(a)). The derived coefficient is found to be 0.1143 for this site. This value is significantly different from 0.16 that is assumed to be applicable for tropical climate like Nigeria [28]. The second linear model with intercept component indicate a better correlation between the clearness index and the square root of the difference between the monthly averaged daily maximum and minimum temperatures (see Fig. 3(a)). In this linear type model, the correlation coefficient is 0.509. However, for the quadratic model the correlation coefficient of 0.872 exists between the clearness index and the square root of the difference between the monthly averaged daily maximum and minimum temperatures. This indicates that the quadratic model is better than the two linear models and therefore, is suggested to be used for Osogbo and neighboring sites. The comparison between the measured and predicted global solar radiation using linear expressions (see Fig. 3(a)) and quadratic expression (see Fig. 3(b)) is exhibited in Fig. 4. It is seen clearly from Fig. 4 that all models predict the trend of the global solar radiation in this site quite well. However, the quadratic model predict the monthly averaged daily global solar radiation better than the linear models which over-predicted and under-predicted the solar radiation over a wide range of months.

The scattered plots showing the variation of the clearness index with the monthly daily averaged temperature are demonstrated in Fig. 5, where again, a linear (Fig. 5(a)) and a quadratic curve (Fig. 5(b)) which fit to these data points are also presented. The correlation coefficient that exists between the clearness index and the monthly daily averaged temperature are 0.668 and 0.828 for the linear and the quadratic model, respectively. This indicates that based on the monthly daily average temperature, 66.8% of the clearness index can be accounted using the linear model whereas 82.8% of the clearness index can be accounted using the quadratic model. The comparison between the measured and predicted global solar radiation using linear expressions (see Fig. 5(a)) and the quadratic expression (see Fig. 5(b)) is depicted in Fig. 6. It is observed clearly from Fig. 6 that the quadratic model predicts the monthly averaged daily global solar radiation better than the linear model.

Error analysis

For each of the models, performance indicators (namely MPE, MAPE, MBE, MABE and RMSE) were adopted to compare their respective predictive efficiencies with the observed values (see Table 4). Since low values are desirable of the indicators for better performance, the Hargreaves and Samani based quadratic model recording values of 0.21, 3.43, 0.00, 0.47 and 0.62 for the MPE, MAPE, MBE, MABE and RMSE indicators respectively, gave the strongest fittings between the predicted and the observed values, having recorded the lowest among the seven developed models. This is followed by the Angstrom quadratic and linear dependent models respectively; under-estimation of the predicted to the observed values is further noticed with these two models for the MBE.

In addition, the low and positive values obtained for the MBE and MPE is an indication of good performance on the short and long-terms of the examined regression equations respectively together with an average amount of overestimation in the calculated values. The Hargreaves and Samani based quadratic model gives the best prediction for simulating global solar radiation for Osogbo and near locations with similar geographic characteristics followed by the Angstrom based quadratic model.

Conclusions

Six empirical models (linear and quadratic) are developed, based on sunshine duration and temperature parameters for the estimation of global solar radiation for Osogbo. The following conclusions were made:

1) The maximum monthly mean daily sunshine hours and daily global solar radiation on a horizontal surface are 6.827 h and 15.569 MJ/(m²·d) respectively in November. The minimum values occur in August as 2.818 h and 9.845 MJ/(m²·d) for the monthly mean daily sunshine hours and global solar radiation on a horizontal surface respectively. The mean annual sunshine hours in this location is found to be 5.1 h while the mean annual daily global solar radiation is estimated as 13.110 MJ/(m²·d).

2) In the absence of the sunshine hour data, it is shown that temperature based models (especially quadratic expression type) can be used to predict the global solar radiation within a reasonable level of accuracy in Osogbo and other similar locations.

3) Among the developed expressions and based on statistical error analysis, the temperature based quadratic model shows the best agreement with the measured data and therefore, is suitable for the estimation of global solar radiation for Osogbo and sites with similar latitudes and altitudes, especially in the absence of sunshine hours measurement.

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