Empirical Model for Estimating Global Solar Radiation on Horizontal Surfaces for Selected Cities in the Six Geopolitical Zones in Nigeria

This study proposes a temperature-based model of monthly mean daily global solar radiation on horizontal surfaces for selected cities, representing the six geopolitical zones in Nigeria. The modelling was based on linear regression theory and was computed using monthly mean daily data set for minimum and maximum ambient temperatures. The results of three statistical indicators: Mean Bias Error (MBE), Root Mean Square Error (RMSE), and t-statistic (TS), performed on the model along with practical comparison of the estimated and observed data, validate the excellent performance accuracy of the proposed model.


Introduction
Global solar radiation is an important parameter necessary for most ecological models and serves as input for different photovoltaic conversion system; hence, it is of economic importance to renewable energy alternative.The solar radiation reaching the earth's surface depends on the climatic condition of the specific site location, and this is essential for accurate prediction and design of a solar energy system [1].
When global solar radiation is used to generate electrical energy for any specific site location, a provision should be made to forecast solar energy which will convert to electrical energy to recover the load demand, that is, the amount of solar energy for that place ought to be known.Technology for measuring global solar radiation is costly and has instrumental hazards [2].Although solar radiation data are available in most meteorological stations, many stations in developing countries (including Nigeria) suffer from a shortage concern of these data.Thus, alternative methods for estimating these data are required [3].One of these methods is the use of empirical models.Accurate modelling depends on the quality and quantity of the measured data used and is a good tool for generating global solar radiation at locations where measured data are not available.
Several models have been proposed to estimate global solar radiation.Page [4] presents a linear regression model used in correlating the global solar radiation data with relative sunshine duration, which is a modified Angstromtype model [5].Badescu [6] studied existing relationships between monthly mean clearness index and the number of bright sunshine hours using the data obtained from Romania, Trabea, and Shaltout [7] studied the correlation between the measurements of global solar radiation and the meteorological parameters using solar radiation, mean daily maximum temperature, mean daily relative humidity, mean daily sea level pressure, mean daily vapour pressure, and hours of bright sunshine data obtained from different parts of Egypt; while Sfetsos and Coonıck [8] used artificial intelligence techniques to forecast hourly global solar radiation.Okogbue and Adedokun [9] estimated the global solar radiation at Ondo, Nigeria, while Ulgen and Hepbasli [10] correlated the ratio of monthly average hourly diffuse solar radiation to monthly average hourly global solar radiation with the monthly average hourly clearness index in form of polynomial relationships for the city of Izmir, Turkey.In another study, Ulgen and Hepbasli [11] correlated solar radiation parameters (global and diffuse solar radiation) with respect to ambient temperature in the fifth order.Several investigations [12][13][14] have demonstrated the predictive ability of the Angstrom-type model, correlating the global solar radiation to relative sunshine duration in a simple linear regression form.Menges et al. [15] reviewed and tested available global solar radiation models to compute the monthly average daily global solar radiation on a horizontal surface using the data obtained from Konya, Turkey.Kaplanis [16] described computationally efficient and intuitively simple model to estimate hourly global solar radiation on a horizontal surface.Falayi et al. [17] based their studies on the correlation between global solar radiation and meteorological parameters using monthly average daily global solar radiations, sunshine duration, temperature, and relative humidity data obtained from Iseyin, Nigeria.
In this study, a solar resource parameter (ambient temperature) has been analyzed to deduce a two-parameter temperature-based linear model for estimating the global solar radiation on horizontal surfaces at different cities in the six geopolitical zones in Nigeria using monthly mean daily minimum and maximum temperatures for Abuja, Benin City, Katsina, Lagos, Nsukka, and Yola.
2.1.Data Analysis.Stochastic analysis was employed on these data to deduce a set of linear models using a sevenyear period (1996)(1997)(1998)(1999)(2000)(2001)(2002) monthly mean daily data set while a three-year period (2003-2005) monthly mean daily data set was used to validate the proposed model.The performance of the proposed model was evaluated using the t-statistic (TS), a statistical indicator proposed by Stone [19], Root Mean Square Error (RMSE), and Mean Bias Error (MBE).These indicators are mainly employed for adjustment of solar radiation data [3,17,20].
The TS was chosen as it allows models to be compared and, at the same time, can induce whether or not a model's estimate is statistically significant at a particular confidence level (in this study, the performance was analyzed based on the 95% confidence level).It takes into account the dispersion of the results, and it can be computed in terms of RMSE and MBE as shown in the appendix.

Process Simulation.
Computer codes were developed in MATLAB for simulating and evaluating the performance of the developed model while stochastic analysis was performed using Microsoft Excel 2007.Table 1 shows the summary of the stochastic analysis performed on clearance index, ambient temperature ratio, and maximum ambient temperature along with the statistical comparison between the observed and estimated values of the monthly mean daily global solar radiation on horizontal surfaces for the study locations.

Results
The results of the analysis suggest that a significant statistical relationship (with coefficient of correlation (CC) of 89.9-97.6%)exists between the clearance index, ambient temperature ratio, and maximum ambient temperature with linear regression model and the ambient temperature ratio is given by where H, H o , R T , T min , and T max are the respective monthly mean daily horizontal global-and extraterrestrial-radiation, minimum to maximum ambient temperature ratio, and minimum-and maximum-ambient temperature, while m 0 , m 1 , and m 2 are empirical constants.Detailed analysis of H o , is given in the appendix.The result of the analysis on ( 1) is presented in Table 1.
Based on the results in Table 1, an empirical model has been proposed for estimating the monthly mean daily global solar radiation on a horizontal surface using minimum and maximum ambient temperature meteorological data.From Table 1, the empirical model investigated in this study is stated as follows.

Model for Abuja:
Model for Benin City: Model for Katsina: Model for Lagos: Model for Nsukka: Model for Yola:  Table 1: Results of stochastic analysis on temperature-based linear regression model of (1) along with the geographical locations for selected cities in Nigeria.

Discussion
The following observations can be made from a study of the results presented in Table 1.The RMSE values, which are a measure of the accuracy of estimation, vary from 0.1632 to 0.2685 (lowest for (4) while (5) gives the highest value).
The MBE values obtained are positive for ( 4) and ( 8) and negative for (3), ( 5), (6), and (7).This shows that the models vary between over and under estimation of H.However, the value of MBE from the model for Lagos (see ( 6)) has the highest underestimation while model for Yola has the highest overestimation.The model for Benin City (see ( 4)) has negligible overestimation.The high values of the coefficient of determination (CD) of the proposed models (0.809-0.952) shows good agreement of estimated to observed values of H (Figure 3).The use of an additional tool, TS, is used to achieve a more reliable statistical result.This is because it is possible to have large RMSE values and at the same time a very small MBE (Table 1).The smaller the value of t, the better the model's performance is.To determine whether a model's estimates are statistically significant, we determined the critical t-value from standard statistical tables at (P − 1) degrees of freedom, that is, t c(α/2) at the α level of significance.For the model's estimates to be judged statistically significant at the (1 − α) confidence level, the calculated t-value must   be between the interval defined by −t c and t c (acceptance region under the reduced normal distribution curve), that is, t-values outside the range of critical t-values show that the equation has no statistical significance.The t-value of the proposed models are within the range of critical tvalue, (t c(0.025) = 1.96).However, a low t-value of the proposed model (0.0295-0.7683) demonstrates the excellent performance of the proposed models.The results on Figure 3 and Table 1 validate the performance of the proposed models for estimating the monthly mean daily global solar radiation on a horizontal surface for different cities in Nigeria.However, to further improve the accuracy (coefficient of determination) of the estimated results additional meteorological parameters should be included in the model.In this study, other parameters are not considered as it is intended to establish simple and accurate means of estimating monthly mean daily global solar radiation data on horizontal surfaces.
As noticed from Figure 3, the cities under study experienced a decrease in the horizontal global solar radiation from March through August (during rainy season) with Benin City having the lowest monthly mean daily horizontal global solar radiation of 3.46 kWh/m 2 /day in July.During the commencement of dry season (September to November) when other cities experienced a rapid increase in their solar radiation, only Katsina maintained a gradual decrease in her global radiation (having her least radiation of 5.37 kWh/m 2 /day in December).Thus, the variation of daily horizontal global solar radiation with month of the year in Katsina differs from others (with identical variation).This is as a result of her geographical location (long.7.60 • E, lat.13.00 • N).The global solar radiation intensity produced by the proposed model can be utilized in design, analysis, and performance estimation of solar energy conversion system, which is gaining significant attention in Nigeria in particular and the world at large.

Conclusion
(2) Solar time (t s ): it is the time that is based on apparent angular motion of the sun across the sky.The time where the sun crosses the meridian of the observer is called solar noon.At solar noon, t s = 12 h.It is given by where t c is the civil time in hours corresponding to the midpoint of each time step, ϑ is the longitude in degrees of the site location, Z e is the time zone in hours to the east of GMT (which is + 1 h for Nigeria), and E is the equation of time in hours.The equation of time is given as (5) Extraterrestrial radiation on a horizontal surface: it is the theoretical possible radiation that would be available if there were no atmosphere.At any point in time (between sunrise and sunset), the extraterrestrial horizontal radiation is Coefficient of correlation (%) CD: Coefficient of determination (%) p: Numbers of observations P: Total numbers of observations.

Figure 1 :
Figure 1: Site map of study locations.

Figure 2
Figure2shows the correlation between the estimated and observed values of the monthly mean global solar radiation while Figure3shows a comparison of the estimated values of the monthly mean daily global solar radiation obtained using (3)-(8) and their corresponding observed values.

( 4 )
82(0.000075 + 0.001868 cos B − 0.032077 sin B −0.014615 cos 2B − 0.04089 sin 2B), B = (d − 1) 360 • 365 , (A.2) where d = Day of the year, starting with 1 for January 1st, that is, 1 ≤ d ≤ 365.(3) Declination angle (δ): it is the angle made by the line joining the centre of the sun and the earth with its projection on the equatorial plane.It is zero at the autumnal and vernal equinoxes, 23.45 • at the summer solstice on June 21, and −23.45 • at the winter solstice on December 21 in the northern hemisphere.Thus, it has a range −23.45 • ≤ δ ≤ 23.45 • .According to Cooper [22] at the intervening periods of the year, solar declination angle can be approximated by a sinusoidal variation as δ = 23.45• sin 360 • 284 + d 365 .(A.3) Hour angle (ω): it is the angular displacement of the sun East or West of the local meridian due to the rotation of the earth on its axis at 15 • per hour, which expresses the time of the day with respect to solar noon as ω = (t S − 12)15 • , ( A . 4 ) while the sunset hour angle is defined as ω s = cos −1 − tan δ tan φ , ( A . 5 ) where ω = hour angle ( • ) and ω s = sunset hour angle ( • ).
is the day of the month and D is the total number of days in the month.The ratio of monthly mean daily global radiation on horizontal surface to monthly mean daily extraterrestrial horizontal radiation is called the clearance index given as obs is the observed H (kWh/m 2 /day), H est is the estimated H (kWh/m 2 /day), and P is the numbers of observations and p corresponds to the twelve months (P = 12) of the year in this study.