A Linear Regression Model for Global Solar Radiation on Horizontal Surfaces at Warri, Nigeria

8/13/2019 A Linear Regression Model for Global Solar Radiation on Horizontal Surfaces at Warri, Nigeria

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Int. Journal of Renewable Energy Development 2 (3) 2013: 121-126P a g e | 121

IJRED ISSN: 2252-4940 , 31 October 2013, All rights reserved

Contents list available at IJRED website

Int. Journal of Renewable Energy Development (IJRED)

Journal homepage: http://ejournal.undip.ac.id/index.php/ijred

A Linear Regression Model for Global Solar Radiation on HorizontalSurfaces at Warri, Nigeria

Michael S. Okundamiya * and Israel E. Okpamen

Department of Electrical and Electronic Engineering, Ambrose Alli University,

P. M. B. 14, Ekpoma-310006, NIGERIA

Article history:

Received August 20, 2013Received in revised form Sept. 18, 2013Accepted September 27, 2013Available online

ABSTRACT : The growing anxiety on the negative effects of fossil fuels on the environment andthe global emission reduction targets call for a more extensive use of renewable energyalternatives. Efficient solar energy utilization is an essential solution to the high atmosphericpollution caused by fossil fuel combustion. Global solar radiation (GSR) data, which are useful forthe design and evaluation of solar energy conversion system, are not measured at the forty-fivemeteorological stations in Nigeria. The dearth of the measured solar radiation data calls foraccurate estimation. This study proposed a temperature-based linear regression, for predicting themonthly average daily GSR on horizontal surfaces, at Warri (latitude 5.02 0N and longitude 7.88 0E)an oil city located in the south-south geopolitical zone, in Nigeria. The proposed model is analyzedbased on five statistical indicators (coefficient of correlation, coefficient of determination, meanbias error, root mean square error, and t-statistic), and compared with the existing sunshine-basedmodel for the same study. The results indicate that the proposed temperature-based linearregression model could replace the existing sunshine-based model for generating global solarradiation data.

Keywords: air temperature, empirical model, global solar radiation, regression analysis, renewable energy,Warri

* Corresponding author:E-mail: [email protected]

1. Introduction

The global warming of the earth poses seriousthreats to the ecosystem. The earths climatic chang e isthe result of increasing concentrations of greenhousegases (GHGs) resulting primarily from fossil fuel

combustion into the atmosphere. Besides the abruptchanges in the earth system, global warming affectshuman societies and ecosystems in a variety of ways(Kamman 2001; IPCC 2007) including adverse impactson water availability, food supply, coastal safety, andhuman health. A typical example of the climate changeimpact is the recent flooding in Nigeria, which claimedlives, displaced citizens and destroyed properties(estimated over one billion naira). In addition, theecological problems associated with the extraction offossil fuel, from the Niger Delta region of Nigeria, havethreatened the existence of her inhabitants (Obi 2009).

The growing anxiety on the impact of fossil fuelscombustion on the environment and the global effort inmitigating their devastating effects, call for a moreextensive use of renewable energy sources. Essentialsolution to the high atmospheric pollution from fossilfuel combustion usually considers renewable energy

resources (Cristobal 2011).Global Solar Radiation (GSR) is the sum of thedirect and the diffuse solar radiation on a surface. Thedirect solar radiation is the solar radiation receivedfrom the sun without having been scattered by theatmosphere. The diffuse solar radiation is the solarradiation received from the sun with its directionchanged due to scattering by the atmosphere. Theknowledge of the availability of GSR data is offundamental importance in order to utilize solar energyeconomically and efficiently. The direct normalirradiation (DNI) is of interest in concentrating solar


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Citation: Okundamiya, M.S. & Okpamen, I.E. (2013) A Linear Regression Model for Global Solar Radiation on Horizontal Surfaces at Warri, Nigeria. Int. Journal ofRenewable Energy Development , 2(3), 121-126

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thermal installations and installations that track theposition of the sun. This is essential for the design andevaluation of solar energy conversion system. GSR dataare not measured at the forty-five meteorologicalstations in Nigeria (NIMET 2012). In the absence ofthese data, one has to rely on available methods ofestimation and also to develop new ones. Theavailability of such data will encourage possible analysisof application and efficient utilization of solar energy, inorder to control GHG emissions.

Several authors (Ulgen & Hepbasli 2002; Falayi &Rabiu 2005; Falayi et al. 2008; Augustine & Nnabuchi2009; Okundamiya & Nzeako 2010) have demonstratedthe significance of the regression theory in theestimation of global solar radiation on a horizontalsurface. Ulgen & Hepbasli (2002) correlated solarradiation parameters (global and diffuse solarradiation) with ambient temperature of the fifth order

for the city of Izmir in Turkey. Falayi & Rabiu (2005)and Augustine & Nnabuchi (2009) applied theAngstrom type model and correlated global solarradiation with relative sunshine duration in a simplelinear regression form. Falayi et al. (2008) based theirstudies on the correlation between global solarradiation and other meteorological parameters such assunshine duration, ambient temperature and relativehumidity, using data collected from Iseyin, Nigeria.Okundamiya & Nzeako (2010) proposed a temperature-based model for estimating the global solar radiation onhorizontal surfaces for Abuja, Benin City, Katsina, Lagos,Nsukka and Yola, representing the six geopolitical zones

in Nigeria. Although Augustine & Nnabuchi (2009) hadpreviously proposed a relationship (based on acorrelation between global solar radiation and sunshinehours), for estimating global solar radiation for the cityof Warri in Nigeria, there is the need for improvementof this method for optimal economic sizing of the solarenergy conversion systems at such location. Thedeveloped model will provide a comprehensivedatabase for the solar energy potential in Warri. Themain objective of this research is to develop a simpleand accurate model, for predicting the monthly averagedaily global solar radiation data on horizontal surfacesin Warri by using linear regression theory. Statisticalanalysis is also performed on the proposed model todetermine its predictive ability for generating globalsolar radiation data.

2. Materials and Methods

2.1 Site Description and Data

Warri is a leading oil city in Delta State, located(latitude 5.02 0N and longitude 7.88 0E) in the south-south geopolitical zone, of Nigeria. The area,surrounded by tropical rain forest and swamp,experiences two distinct seasons: the dry and the rainy

seasons. The dry season lasts from about November toApril and is significantly marked by the coldharmarttan dusty haze from the north east tradewinds. The rainy season spans from May to October,with a brief break in August. The area is characterizedby tropical-equatorial climate with average annualtemperature of about 32.8 0C and average annual rainfallof about 2673.8mm. The natural vegetation is rainforestwith swamp forest in some areas. The forest is rich intimber, palm and fruit trees.

Twenty-two years (July, 1983 June, 2005)monthly average daily datasets of global solar radiationon a horizontal surface and the correspondingmaximum and minimum air temperature are obtainedfrom the archives of the National Aeronautics and SpaceAdministration (NASA) for Warri. NASA derives thesedatasets from a variety of earth-observing satellites andreanalysis research programs, which provide reliable

meteorological resource data over regions wheresurface measurements are scarce or nonexistent, andprovide two unique features: the data are global and, ingeneral, contiguous in time (NASA 2011). Fig. 1 showsthe correlation of the monthly average daily values ofGSR on a horizontal surface and maximum airtemperature.

The existing sunshine-based model (Augustine &Nnabuchi 2009) for predicting average daily globalsolar radiations on the horizontal surface at Warri is:

N n H H o /42.029.0 (1)

where H is the monthly average daily global solarradiations on the horizontal surface, o H is the monthlyaverage daily extraterrestrial radiation on horizontalsurface, n is the monthly average daily bright sunshinehours, and N is the maximum possible monthlyaverage daily sunshine hours.

5 10 15 20 25 30

5

10

15

20

25

30

Monthly mean ma ximum a mbient temperature( o C )

M o n

t h l y m e a n

d a

i l y

G S R o n

h o r i z o n

t a l

s u r f a c e

( k W h / m

2

/ d a y

)

Fig. 1 Correlation of a twenty-two year monthly average daily dataset

of GSR on a horizontal surface and maximum air temperature forWarri, Nigeria.


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Equation 1 was obtained, based on a correlationbetween sunshine hours and global solar radiation,using a seventeen-year (1991 - 2007) monthly averagedaily meteorological data. Detailed analysis of Equation1 is available in the literature (Augustine & Nnabuchi2009).

2.2 Theory

Given the data points of the form ( x p , y p) for p = 1, 2,. . . , P set of points, and that y p depends linearly on x p,then, for accurate prediction of the set points, it isnecessary to obtain the slope m and H-intercept ( mo) ofa line that best fit these dataset points, defined by

po p xmm y (2)

If x p are j independent variables ( x 1p , x 2p , . . ., x jP ), then,Equation 2 becomes

jP j p po p xm xm xmm y ...2211 (3)

and the coefficient of determination defined by

P

p p jP j po y xm xmm R

111

2 ... (4)

where x is the independent variable, j is the number ofindependent variables (which would minimize the sumof squares of the difference between the data and theestimated line) and R is the coefficient of correlation.

This study intends to develop an easy and accuratemethod of predicting the monthly average daily globalsolar radiation data on a horizontal surface. As such, itassumes two independent variables ( j in Equation 4equals 2). If the dependent variable of Equation 4 is theclearance index K and the independent variables are themonthly average daily air temperature ratio, RT andthe monthly average daily maximum air temperature

,maxT then Equation 4 reduces to

p Rpo p T mT mm K max21 (5)

where the subscript p (= 1, 2,. . . 12), refers to themonthly average daily dataset for a typical year.Expressing Equation 5 in matrix form

j P RP

R

P m

m

T T

T T

K

K

.

.

.

1.........

1

.

.

.0

max

1max11

(6a)

i.e.K = AM (6b)

where K is P x 1 column matrix, A is P x 3 matrix and M is ( j + 1) x 1 matrix. To solve for A, it is necessary totransform A to a square matrix (where P j + 1 ).

AT K = AT AM (7)

where AT is the transpose of A. If ( AT A) - 1 exists then

M = (AT A)-1 AT K (8)

The solution of Equation 8 is a matrix of empiricalconstants, M = [m o m 1 m 2], and the proposed model thatbest fits the data is defined by

max21 T mT mm K Ro (9)

Expressing Equation 9 in terms of the monthly averagedaily global solar radiations on the horizontal surface

max21 T mT mm H H Roo pred (10)

where pred H is the predicted monthly average dailyglobal radiation on a horizontal surface (kWh/m 2/day)and o H is the monthly average daily extraterrestrialradiation on a horizontal surface (kWh/m 2/day),

)/( maxmin T T T R is the monthly average daily airtemperature ratio, minT is the monthly average daily

minimum air temperature (oC), maxT is the monthlyaverage daily maximum air temperature ( oC) and m i is

the empirical constants. The available parametersinformed choice of Equation 10. In other words, theinput parameters of Equation 10 are easily obtained.Moreover, these parameters are measured at the forty-five (45) meteorological stations in Nigeria.

2.3 Simulation

Computer codes developed using MATLABprogramming language compute the empiricalconstants of Equation 10 using 15 years (July, 1983

June, 1998) monthly average daily datasets discussedabove. The performance evaluation of the estimation(proposed and existing) models utilized 7 years (July,1998 June, 2005) monthly average daily datasets. Thet-statistic ( t-value) is computed in terms of the meanbias error (MBE) and root mean square error (RMSE)and analyzed at the 95% confidence level. The t-statisticis a known statistical tool (Stone 1993; Okundamiya &Nzeako 2010; 2011) with an exceptionally satisfactoryaccuracy for analyzing solar radiation data. Thecoefficients of determination (R 2) of the estimationmodels are obtained using Microsoft Excel 2007.Detailed analysis of MBE, RMSE, and t-value is given in

the literature (Okundamiya & Nzeako 2010).


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Citation: Okundamiya, M.S. & Okpamen, I.E. (2013) A Linear Regression Model for Global Solar Radiation on Horizontal Surfaces at Warri, Nigeria. Int. Journal ofRenewable Energy Development , 2(3), 121-126

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3. Results Analysis and Discussion

The proposed temperature-based linear regressionmodel for Warri using the 15 years monthly averagedaily dataset of global solar radiation on a horizontalsurface and air temperatures (minimum and themaximum) is

)02995.0084.15001.0( maxT T H H Ro pred (11)

Fig. 2 shows the correlation between the observedand predicted (Equation 11) values of the global solarradiation at Warri using 7 years monthly average dailydata.

Fig.2 Correlation between the observed and the predicted (usingEquation 11) values of global solar radiation at Warri using 7 years

monthly average daily dataset.

3.1 Comparison of Results

It is pertinent to compare the observed with theestimation models (Equations 1 and 11) in order todetermine their performance. The result of thiscomparison (as presented in Table 1 and Figure 3) will

determine the predictive accuracy of the proposedmodels.

Table 1Comparison of predicted (based on the existing and proposed models)values of the monthly average daily global solar radiation on ahorizontal surface at Warri using 7 years monthly average dailydataset

Models

Regression Analysis

(%) Error Terms

R R2MBE

(kWh/m2/day)

RMSE(kWh/m

2/day)

TS*

Present study(Equation 11) 96.2 92.6 0.0056 0.1841 0.1004

Previous study(Equation 1) 89.2 79.5 - 0.1988 0.5093 1.3995

* 96.1)025.0,11( ct

3.2 Analysis and Discussion

A test of MBE and RMSE, which are a measure ofthe accuracy of estimation, gives a long term and shortterm performance of the model studied respectively.Almorox et al. (2005) and Che et al. (2007) recommendthat a zero value for MBE is ideal; however,overestimation of an individual data element will cancelunderestimation in a separate observation.

Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sep. Oct. Nov. Dec.3.5

4.0

4.5

5.0

5.5

6.0

M o n

t h l y a v e r a g e

d a i

l y h o r i z o n

t a l g

l o b a l

s o l a r r a

d i a t o n

( k W h / m

2 / d a y

)

ObservedPresent study (Equation 11)Previous study (Equation1)

Fig. 3 Comparison of observed with the computed (based on the existing and proposed models) values of the global so lar radiation on a horizontalsurface at Warri using 7 years monthly average daily dataset


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In addition, low RMSE values are desirable(Almorox et al. 2005; Che et al. 2007), but few errors inthe sum can cause a significant increase in the indicator.It is possible to have large RMSE values at the same timea low MBE values. Therefore, the use of the MBE and theRMSE statistical indicators is not sufficient for the

evaluation of solar radiation model performance(Okundamiya & Nzeako 2011). This informs the use ofthe t-statistic test indicator.

The t-statistic test is chosen since it allows modelsto be compared, and at the same time can inducewhether a models estimate is statistically significant ata confidence level. It takes into account the dispersionof the results. It can be computed in terms of RMSE andMBE. The smaller the t-value, the better is theperformance of the model. To determine whether theestimates are statistically significant, the critical t-value(t c) at the 95% confidence level and (P - 1) degrees offreedom are obtained from standard statistical table.The following assertions could be deduced from a studyof the results presented in Table 1.

A close to unity value (i.e., 0.926) of thecoefficient of determination (R 2) of the proposedmodel indicates a satisfactory agreement of thepredicted with the observed values of GSR. Thehigher value (0.926 compared to 0.795) of thecoefficient of determination indicates that themapping accuracy of the proposed model inpredicting global solar radiation is over 13%better than the existing model (see Fig. 3).

The MBE value obtained is positive for the

proposed model and negative for the existingmodel. This suggests that the proposed andexisting models vary between overestimationand underestimation of GSR estimatesrespectively. However, the higher magnitude ofthe MBE value from the existing model indicatessignificant (high) underestimation. The proposedmodel has little (insignificant) overestimation.

The RMSE value of the proposed model is lowercompared to the existing model. This indicates abetter accuracy of estimation of the proposedmodel.

For the models estimates to be judged statisticallysignificant at the (1 - ) confidence level, the calculatedt-value must exist between the interval defined by - t c and t c (acceptance region under the reduced normaldistribution curve), i.e., t-values outside the range ofcritical t-values indicate that the model has no statisticalsignificance. The t-values of both models are within therange of critical t-values (t c(11,0.025) = 1.96). However, alower t-value of the proposed model (0.1004 comparedto 1.3995) demonstrates that the predictive efficiencyof the proposed model is better than the existing model.The results presented in Fig. 3 validate the performanceof the proposed temperature-based linear regressionmodel for predicting the daily global solar radiation on

a horizontal surface at Warri, Nigeria. The proposedestimates compared favourably with the observedvalues throughout the year while estimates from theexisting sunshine-based model only comparedfavourably with the observed values during parts of theyear (between April and the August break). However, to

improve on the accuracy of the estimated results,additional climatological parameters such as relativehumidity, relative sunshine duration, solar declination,cloud cover, should be included in the proposed model(Equation 11). This paper ignores other parameters asit intends to create simple and accurate method forestimating the average daily global solar radiation dataon a horizontal surface.

7. Conclusions

This study employs regression analysis and

proposed a temperature-based linear regression modelused to predict the monthly average daily global solarradiation on a horizontal surface at Warri (Nigeria),which is in satisfactory agreement with the observedvalues. The predictive efficiency of the proposedtemperature-based model exceeds the existing modelfor the same study. The results suggest that theproposed temperature-based linear regression modelfor estimating global solar radiation data could replacethe existing sunshine-based model. The estimation ofthe average daily global solar radiation at Warri(necessary for optimal economic sizing of solarphotovoltaic systems), will encourage extensive use ofsolar energy, which is considered as essential solutionto the high atmospheric pollution caused from fossilfuel combustion.

Acknowledgement

This study was partially funded by ETF 2009 AST&DIntervention (Reference no. AAU/REG/ETF.560/475).

References

Almorox, J., Benito, M. & Hontoria, C. (2005) Estimation of monthlyAngstrom-Prescott equation coefficients from measured dailydata in Toledo, Spain, Renewable Energy , 30(6), 931 936.

Augustine, C. & Nnabuchi, M. N. (2009) Correlation between SunshineHours and Global Solar Radiation in Warri, Nigeria, Pacific Journal of Science and Technology , 10(2), 574 579.

Che, H. Z., Shi, G. Y., Zhang, X. Y., Zhao, J. Q., & Li, Y. (2007) Analysis ofsky conditions using 40 year records of solar radiation data inChina, Theoretical and Applied Climatology , 89(1-2), 83 94.

Cristobal, J. R. S. (2011) A multi criteria data envelopment analysismodel to evaluate the efficiency of the Renewable Energytechnologies, Renewable Energy , 36(10), 2742-2746.

Falayi, E. O., Adepitan, J. O. & Rabiu, A. B. (2008) Empirical models forthe correlation of global solar radiation with meteorologicaldata for Iseyin, Nigeria, International Journal of PhysicalSciences , 3(9), 210 216.


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A Linear Regression Model for Global Solar Radiation on Horizontal Surfaces at Warri, Nigeria

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