Empirical correlations for the estimation of global solar radiation using meteorological data in WA, Ghana

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A. Uba Department f Mechanical Engineering, Wa Plytechnic, Wa, Ghana ABSTRACT Mnthly average daily glbal lar radiatin data are eential in the deign and tudy f lar energy cnventin device.

1 Available nline at Advance in Applied Science Reearch, 2013, 4(4):63-71 ISSN: CODEN (USA): AASRFC Empirical crrelatin fr the etimatin f glbal lar radiatin uing meterlgical data in WA, Ghana Emmanuel. A. Sarah and Felix. A. Uba Department f Mechanical Engineering, Wa Plytechnic, Wa, Ghana ABSTRACT Mnthly average daily glbal lar radiatin data are eential in the deign and tudy f lar energy cnventin device. In thi tudy, multiple linear regrein mdel were develped t etimate the mnthly average daily glbal lar radiatin uing even parameter during a perid f tw year frm 2010 t 2011 fr Wa Plytechnic weather tatin. The parameter ued were the extraterretrial radiatin, mean ambient temperature, mean il temperature, relative humidity, declinatin,rati f the difference between the maximum and minimum mnthly mean ambient temperature t the minimum mnthly mean ambient temperautre and rati f unhine duratin. Selected mdel were cmpared n the bai f the tatitical errr tet; mean bia errr (MBE), mean percent errr (MPE), rt mean quare errr (RMSE) and the t-tet. Baed n the tatitical reult, the crrelatin equatin that culd be emplyed fr the purpe f etimating glbal lar radiatin f lcatin that have the ame climate, latitude and altitude a Wa Plytechnic weather tatin i given a H = RH n N in δ The preent wrk will help t advance the tate f knwledge f glbal lar radiatin t the pint where it ha applicatin in the etimatin f mnthly average daily glbal lar radiatin. Keywrd: Daily glbal lar radiatin, empirical mdel, multiple linear regrein, t-tatitic. INTRODUCTION Infrmatin f lar radiatin at a given lcatin i eential fr many applicatin, uch a in the deign f prject, meterlgical frecating, lar heating, drying and architectural deign. Hence, lar radiatin data are required by lar engineer, architect and agriculturit. Furthermre, mnthly mean data are needed fr the etimatin f lngterm lar ytem perfrmance. In the applicatin f lar energy lited abve, the mt imprtant parameter that are ften needed are the average lar irradiatin and it cmpnent. Unfrtunately, few meterlgical tatin, epecially in many develping cuntrie, meaure accurately and cntinuuly thee data. One way f lving thi i uing apprpriate crrelatin which are empirically etablihed, that can be ued t etimate glbal lar radiatin frm mre readily available meterlgical parameter uch a unhine hur, daily temperature and relative humidity. Empirical mdelling i an eential and ecnmical tl fr the etimatin f glbal lar radiatin. The accuracy f uch mdel depend n the quality f the meaured data ued. Thugh le accurate, mdelling i a better tl fr the etimatin f glbal lar radiatin at place where meaurement are nt available [20]. Several empirical mdel have been develped t calculate lar radiatin uing variu parameter. [7] develped the earliet mdel ued fr etimating glbal radiatin, in which the unhine hur data and clear ky radiatin data, were ued. Many 63

2 Emmanuel. A. Sarah and Felix. A. Uba Adv. Appl. Sci. Re., 2013, 4(4):63-71 reearcher have ued Angtrm mdel t develp empirical crrelatin [1,17, 20, 21, 22, 24, 25]. In additin, ther empirical mdel have been develped t calculate lar radiatin frm ther parameter uch a relative humidity, ambient temperature, il temperature, number f rainy day and evapratin [2, 11, 13, 14, 15, 20, 25]. In thi tudy, the firt aim wa t determine mnthly variatin f mean glbal lar radiatin, extraterretrial radiatin, mean ambient temperature, relative humidity, mean unhine hur and mean il temperature in Wa Plytechnic.The ecnd aim wa t t develp equatin that crrelate mnthly average daily glbal radiatin and the ther variable t elect the mt apprpriate mdel fr Wa Plytechnic in nrthern Ghana. We then reviewed all the equatin and retained me f them. Thi wa fllwed by tatitical cmparin methd uch a MBE, RMSE and t-tatitic, f the retained equatin t elect the bet. ESTIMATION METHODS Mnthly mean daily extraterretrial radiatin, daylength,angle f declinatin and unet hur angle, uing the average day f the mnth, were calculated frm equatin (1) (4), repectively [12] n πωs H O = G c cφ cδ in ω + in φ in δ c π (1) 2 c 1 N = ( tan φ tan δ ) 15 (2) n δ = in (3) ( ) 1 = c tan tan (4) ω φ δ METHODOLOGY Extraterretrial radiatin n a hrizntal urface in mnthly perid were calculated numerically uing declinatin angle, latitude and unet hur angle uing the etimatin methd. Available mnthly mean daily glbal lar radiatin, unhine duratin hur, relative humidity, il temperature and ambient temperature were taken frm the ite f the weather tatin f Wa Plytechnic, fr the perid 2010 and The graphical lcatin f Wa Plytechnic weather tatin i latitude 10.01º N, with an altitude f 322 m abve ea level.mnthly average ver the tw year perid f the data, prceed in preparatin fr the crrelatin, are preented in Table 1. Table 1: Glbal Slar Radiatin and relevant meterlgical data fr Wa Plytechnic Mnth (MJ/m 2 ) (MJ/m 2 ) (ºC) (%) / (ºC) (ºC) Jun Jul Nv The change f the glbal lar radiatin and extraterretrial radiatin in mnthly perid thrughut the tw year perid were invetigated and hwn in Fig.1. 64

3 Emmanuel. A. Sarah and Felix. A. Uba Adv. Appl. Sci. Re., 2013, 4(4): H H Slar Radiatin (MJ/m 2 ) Jun Jul Nv Jun Jul Nv Figure 1: The change f the mean glbal lar radiatin and extraterretrial radiatin fr the tw year The mnthly change f the ther meterlgical parameter uch a il temperature, ambient temperature, relative humidity and unhine hur were een in Fig Ta T RH 80 Ta, T, RH(%) Jun Jul Nv Jun Jul Nv Figure 2: The mean mnthly change f ambient temperature, il temperature and relative humidity 65

4 Emmanuel. A. Sarah and Felix. A. Uba Adv. Appl. Sci. Re., 2013, 4(4):63-71 We then develped equatin t etimate the mnthly mean glbal lar radiatin by applying multiple linear regrein t variu parameter uch a H 0, n / N,in δ, θ, T a, T and RH (%). The value f H were etimated uing thee equatin. Starting with ne parameter, the equatin in linear regrein tk the frm; Y = β + β x (5) 0 1 where a and b are regrein cefficient and i the crrelated parameter. Multiple linear regrein equatin fr etimating with even parameter i a fllw; Y = β + β x + β x + β x + β x + β x + β x + β x (6) CORRELATIONS The etup f the weather tatin at Wa Plytechnic meaure the wind peed, lar radiatin, il temperature, relative humidity, unhine hur and ambient temperature. In rder nt t verlk any parameter r grup f parameter, multiple linear regrein analyi f the even driving variable were emplyed t develp equatin t etimate the mean mnthly glbal lar radiatin. The variu linear regrein analye are a fllw in Table 1. Table 2: Multiple Linear Regrein Mdel Equatin Number f Variable Number f Mdel Equatin One variable crrelatin 7 Tw variable crrelatin 21 Three variable crrelatin 35 Fur variable crrelatin 35 Five variable crrelatin 21 Six variable crrelatin 7 Seven variable crrelatin 1 A ttal f 127 mdel equatin were frmed and analyed fr the bet mdel equatin that can predict the mnthly average lar radiatin (in MJ/m 2 ) with the leat errr.seventeen mdel equatin were then chen fr cmparin uing % errr analyi. The value frm equatin were cmpared with the crrepnding meterlgical value. The eventeen linear regrein mdel equatin are a fllw: H = RH n N in δ ( R = 0.999, σ = 0.054) (7) H = n N in δ T a ( R = 0.100, σ = 0.054) (8) H = RH n N in δ T ( R = 0.999, σ = 0.057) (9) H = RH n N in δ T ( R = 0.999, σ = 0.057) (10) H = RH n N in δ T a H ( R = 1.000, σ = 0.053) (11) H = RH n N in δ T a T ( R = 0.100, σ = 0.058) (12) H = RH n N in δ T a θ( R = 1.000, σ = 0.054) (13) H = RH n N in δ T θ ( R = 1.000, σ = 0.054) (14) H = RH n N in δ T H ( R = 1.000, σ = 0.053) (15) H = n N inδ 0.045T a T θ ( R = 0.999, σ = 0.085) (16) H = RH n N in δ T θ ( R = 1.000, σ = 0.054) (17) 66

5 Emmanuel. A. Sarah and Felix. A. Uba Adv. Appl. Sci. Re., 2013, 4(4):63-71 H = RH n N in δ T a H ( R = 1.000, σ = 0.058) (18) H = RH n N in δ T a θ( R = 1.000, σ = 0.059) (19) H = RH n N inδ H θ ( R = 1.000, σ = 0.057) (20) H = n N inδ 0.046T T H 0.412θ ( R = 0.999, σ = 0.093) a H RH n N in δ H θ ( R = 1.000, σ = 0.057) (22) H = RH n N inδ H θ ( R = 1.000, σ = 0.054) (23) METHODS OF COMPARISON The perfrmance f the eventeen mdel wa evaluated n the bai f the tatitical errr tet, (i.e. the mean percentage errr (MPE), rt mean quare errr (RMSE) and mean bia errr (MBE)) and al teted whether they are tatitically ignificant uing t-tatitic. % Errr Thi i meant t tet fr the range within which the errr f the mdel equatin can be quantified, expreed in percentage frm. A lwer percentage errr interval, appraching zer i deirable and a negative value f, hw that the reult i mall than the actual value and vice vera. A deirable value f huld be between 5% t+5%. Percentage errr and percentage errr interval are defined a; H i, errr ( H i, cal H i, mea ) ( H i, mea ) = 100 (24) ( ) H errr,int erval = H i, errr,max H i, errr,min (25) The Mean Bia Errr 1 (,, ) 1 n i cal i mea (26) MBE = H H n Thi tet prvide infrmatin n lng-term perfrmance. A lw MBEvalue i deired. A negative value give the average amunt f under-etimatin in the calculated value and vice vera. S, ne drawback f thi tet i that veretimatin f an individual bervatin will cancel under-etimatin in a eparate bervatin. The Mean Percentage Errr n 1 H i, cal H i, mea MPE(%) = 100 n 1 H (27) i, mea A percentage errr between 10% and +10% i cnidered acceptable. (21) 67

6 Emmanuel. A. Sarah and Felix. A. Uba Adv. Appl. Sci. Re., 2013, 4(4):63-71 Table 3: Cmparin between meaured and etimated value f the crrelatin equatin M MODELS H (MJ/m 2 Eqn. Eqn. Eqn. Eqn. Eqn. Eqn. Eqn.1 Eqn. Eqn. Eqn. Eqn. Eqn. Eqn. Eqn. Eqn. Eqn. Eqn. ) Jun Jul Nv TOTAL

7 Emmanuel. A. Sarah and Felix. A. Uba Adv. Appl. Sci. Re., 2013, 4(4):63-71 The Rt Mean Square Errr The value f RMSE i alway pitive, repreenting zer in the ideal cae. The nrmalized rt mean quare errr give infrmatin n the hrt term perfrmance f the crrelatin by allwing a term by term cmparin f the actual deviatin between the predicted and meaured value. The maller the value, the better i the mdel perfrmance. 1/2 1 n 2 ( i, cal i, mea ) n (28) 1 RMSE = H H t-tatitic The randm variable t with n 1 degree f freedm may be written here a fllw: 2 ( n 1)( MBE) ( RMSE) ( MBE) 1/ 2 t = 2 2 The maller the value f t the better i the mdel perfrmance in appraching the real value. T determine whether a mdel etimate are tatitically ignificant, ne imply ha t determine, frm tandard tatitical table, the critical t value, i.e. t α/2 at α level f ignificance and (n 1) degree f freedm. Fr the mdel etimate t be judged tatitically ignificant at the (1 α) cnfidence level, the calculated t value mut be le than the critical value. The t-tatitic ued in additin t the MBE and RMSE give mre reliable and explanatry reult [25]. RESULTS AND DISCUSSION A ttal f 127 mdel equatin were frmed and % errr analyi were perfmed n them. Seventeen mdel equatin were then extracted baed n the %errr interval f -1% t +1%. The applicability f the prped crrelatin in predicting i teted by etimating value fr Wa Plytechnic lcatin ued in the analyi. Etimated value f alng with meaured data, are hwn in Table 3. The reulting equatin were then taken thrugh tatitical errr tet. The reult are hwn in Table 4. Table 4: Errr value (in MJ/m 2 ) fr the develped crrelatin mdel (29) ERRO R Eqn. 7 Eqn. 8 Eqn. 9 Eqn. 10 Eqn. 11 Eqn. 12 Eqn.13 Eqn. 14 MODELS Eqn. 15 Eqn. 16 Eqn. 17 Eqn. 18 Eqn. 19 Eqn. 20 Eqn. 21 Eqn. 22 Eqn. 23 MBE MPE RMSE t-stat The t-tatitic value btained were all le than the critical t fr each equatin implying that the equatin were all tatitically ignificant. Baed n MBE tet, equatin (7), (13), (16), (19) and (21) qualified. The reult hw that Equatin (7) i the bet while Equatin (13) i the wrt. With repect t MPE tet, all equatin qualified. Equatin (7) i the bet while Equatin (11) i the wrt. Fr the RMSE tet, Equatin (7) i the bet while Equatin (11) i the wrt. Since the tet n RMSE cnvey infrmatin n the hrt-term perfrmance f different equatin, it enable a term-by-term cmparin f the actual variatin between the etimated and meaured value and ince Equatin (7) pae all tet, ur mdel i then that equatin. Figure 3 hw plt f the meaured and Equatin (7). 69

8 Emmanuel. A. Sarah and Felix. A. Uba Adv. Appl. Sci. Re., 2013, 4(4): Mean Glbal Slar Radiatin (MJ/m2) H mea. Eqn Jun Jul Mnth Nv Figure 3: Cmparin f meaured and etimated(eqn. 7) f mean mnthly glbal lar radiatin CONCLUSION The bjective f thi tudy wa t evaluate variu mdel fr the etimatin f mnthly average daily glbal radiatin n a hrizntal urface frm unhine hur, relative humidity, ambient temperature and il temperature and t elect the mt apprpriate mdel fr Wa Plytechnic weather tatin. One hundred and twenty even equatin in different cmbinatin were btained. Seventeen mdel having the bet %errr were cmpared uing the tatitical rutine. Even thugh up t even variable crrrelatin have been develped, it wa berved that the bet crrelatin invlved nly three variable namely Relative Humidity, declinatin and mnthly percent pible unhine. The MBE, MPE, RMSE and %errr value f the elected equatin (Eqn. 7) are 0.01, 0, and t 0.272, which are within the range f acceptable value. Hence, the multiple linear regrein mdel that culd be emplyed fr the purpe f etimating glbal lar radiatin f lcatin that have the ame climate, latitude and altitude a Wa Plytechnic tatin i the crrelatin equatin with the leat value f RMSE a H = RH n N in δ Acknwledgment The authr wuld like t thank the taff f Agricultural Engineering Department, Wa Plytechnic, fr their iniative fr etting up the weather tatin fr academic wrk. NOMENCLATURE # $ # % multiple linear regrein cefficient & &$ crrelated parameter in regrein equatin MBE mean bia errr MPE mean percent errr RMSE rt mean quare errr t-tat t-tatitic ' ( lar cntant [1367 W/m 2 ] 70

9 Emmanuel. A. Sarah and Felix. A. Uba Adv. Appl. Sci. Re., 2013, 4(4):63-71 mnthly mean daily glbal radiatin [MJ/m 2 ] $ mnthly mean daily extraterretrial radiatin [MJ/m 2 ] )* mnthly average daily hur f bright unhine [hur] mnthly average f maximum pible daily hur f bright unhine [hur] +* mnthly mean ambient temperature, ºC +* mnthly mean ambient temperature, ºC, - determinatin cefficient, relative humidity, %. unet hur angle (in degree) declinatin angle / rati f the difference between the maximum and minimum mnthly mean ambient temperature t the minimum mnthly mean ambient temperautre latitude, the angular lcatin nrth r uth f the equatr, nrth pitive 1 tandard deviatin REFERENCES [1] Akinglu B, Ecevit A, Energy,1990, 15, [2] Akpabi L, Ud S, Etuk S, Turkih. Jurnal Phyic, 2004, 26, [3] Akpabi L, Ukpng U, Etuk S, Turkih. Jurnal Phyic, 2003,6, [4] Alnaer W, Renewable Energy, 1993, 3, [5] Alrx J, Benit M, Hntria C, Renewable Energy, 2005, 30, [6] Ampratwum D, Drvl A,Applied Energy, 1999, 63, [7] Angtrm A, Slar and terretrial radiatin meterlgical ciety, 1924, 50, [8] Bakirci K, Energy, 2009, 34, [9] Burari FW, Abdul Azeez MA, Medugu DW, Advance in Applied Science Reearch, 2010, 1 (1), [10] Bulut H, Buyukalaca O, Applied Energy,2007, 84, [11] Chw T, Chan A, Fng F, Lin Z, Slar Energy, 2006, 80, [12] Duffie JA, Beckman WA,Slar Engineering f Thermal Prcee, Jhn Willey Publicatin, New Yrk, 1994, pp [13] Elagib N, Manell M, Energy Cnverin and Management, 2000, 41, [14] El-Metwally, Jurnal f Atmpheric and Slar-Terretrial Phyic, 2005, 67, [15] Ertekin C, Yalbiz O, Energy Cnverin and Management,2000, 41, [16] Ituen, En E, Een, Nikan U, Nwkl, Samuel C, Ud, Ema, G, Advance in Applied Science Reearch, 2012, 3 (4), [17] Jin Z, Yezheng W, Gang Y, Energy Cnverin and Management, 2005, 46, [18] Lin W, Ga W, Energy Cnverin and Management,1999, 40, [19] Medugu DW, Yakubu D, Advance in Applied Science Reearch, 2011, 2 (2), [20] Menge H, Ertekin C, Smmet M, Energy Cnverin and Management, 2006, 47, [21] Muneer T, Yune S, Applied Energy, 2006, 83, [22] Paulecu M, Fara L, Tulcan E, Atmpheric Reearch, 2006,79, [23] Rehman S, Halawani T, Renewable Energy, 1997, 12, [24] Sheiker K, Energy Cnverin and Management, 2006, 47, [25] Tgrul I, Onat E, Energy Cnventin and Management, 1999, 40, [26] Yrukglu M, Celik A, Energy Cnverin and Management, 2006, 47,

MODULE 5 Lecture No: 5 Extraterrestrial Radiation

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