Energy Conversion and Management 50 (2009) 1227–1235

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Energy Conversion and Management

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Statistical comparison of monthly mean hourly and daily diffuse and globalsolar irradiation models and a Simulink program development forvarious Algerian climates

M. Koussa *, A. Malek, M. HaddadiRenewable Energy Development Center, Postal Box 62, Observatory Road Bouzaréah, Algiers, Algeria

a r t i c l e i n f o

Article history:Received 30 June 2008Accepted 27 January 2009Available online 12 March 2009

Keywords:IrradiationDiffuseMeteorologicalBright sunshine hoursGlobal

0196-8904/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.enconman.2009.01.035

* Corresponding author. Tel.: +213 021 90 15 03; faE-mail addresses: mustaphakoussa@cder.dz (M

(A. Malek), mourad.haddadi@enp.edu.dz (M. Haddadi

a b s t r a c t

This work has two main objectives. The first objective is to carry out a statistical comparison of specificmodels for estimating monthly mean daily and hourly global and diffuse irradiations on a horizontal sur-face from the monthly average per day of the main meteorological data and to recommend the mostaccurate models to be used for estimating solar radiation in sites belonging to the Algerian climate areasrepresented by each of the three sites presented in this work. The second objective is to propose a Sim-ulink program where the methodology allows for reconstituting daily and hourly monthly mean horizon-tal solar radiation data. From the 10 global radiation models proposed in several journal papers,Rietveld’s, Garg et al.’s and McCulloch’s relations are retained, respectively, for estimating the monthlymean daily global ratio for Adrar, Bouzareah and Ghardaia sites and Page and from the nine retained dif-fuse relations, Page’s, Liu et al.’s and Erbs et al.’s relations are used, respectively, for the same sites forcalculating the monthly averages per day of diffuse irradiation. On the other hand, the models of Jainand Liu et al. are, respectively, recommended for the reconstitution of the monthly averages per hourof the global and diffuse irradiations. However, the obtained results are used to develop a Simulinkprogram which can be used in the estimation of long-term performance of flat plate solar radiationsystems.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Data of global and diffuse solar radiation at a location are of sig-nificant use in the design and in the long-term evaluation of the so-lar energy conversion systems performances. Furthermore,monthly mean daily data are needed for the estimation of long-term solar systems performances, but for more precision andaccording to the considered system application, generally, scien-tists use the hourly data. Thus, for this, the required solar radiationcomponent is the total solar irradiation incident on an inclined sur-face according to the slope of the considered sensor. If this compo-nent was not available, then, the horizontal global and diffuseradiation data are the best source of information for its calculation.However, measurements of the main meteorological parametersare generally available, but horizontal diffuse and global solarradiations are only measured at few locations due to the cost ofthe equipment required, and due to the care needed for regular

ll rights reserved.

x: +213 021 90 16 54.. Koussa), amalek@cder.dz).

adjustment to allow for the seasonal progression path of the sunacross the sky, especially for the diffuse component measurement.This is the main case in Algeria where few of the 35 existing mete-orological offices record the solar radiation data.

However, this paper has two objectives: the first consists in car-rying out statistical comparison of specific models for estimatingthe horizontal monthly mean daily and hourly of global and diffuseirradiations and retaining the most accurate models. The secondone consists in developing a simple program in Simulink in whichthe retained models are used to reconstitute the monthly meanhourly and daily diffuse and global irradiations from the monthlyaverage per day of a minimum main meteorological data.

Many approaches have been proposed in several journals pa-pers and retained in our work to estimate monthly mean daily dif-fuse and global irradiations as function of the main meteorologicaldata. On the other hand, some approaches have been used to ob-tain estimations of the monthly mean hourly global and diffuseirradiations from monthly mean daily data. Furthermore in ourpaper, we first performed a literature review of all the existingmodels and we made a description of each retained model. Thiswas followed by a statistical comparison of the monthly meandaily and hourly retained models to the measured data obtainedfrom three Algerian sites.

1228 M. Koussa et al. / Energy Conversion and Management 50 (2009) 1227–1235

As results, for the three considered Algerian sites and for thelocations included in the same climatic zones, we propose, respec-tively, Rietveld’s [1], Glover’s and McCulloch’s [2] and Garg’s andGarg’s [3] relations for reconstituting monthly mean total solarenergy for Adrar, Bouzareah and Ghardaia, and Liu’s and Jordan’s[4], Erbs et al.’s [5] and Page’s [6] models for reconstituting themonthly mean daily diffuse solar energy for Bouzareah, Ghardaiaand Adar. Furthermore, Jain’s [7] and Liu’s and Jordan’s [4] modelsare retained to reconstitute, respectively, the monthly mean hourlyglobal and diffuse irradiations. In this way, a Simulink program hasbeen developed to calculate the monthly average per day and perhour of the horizontal global and diffuse irradiations from onlythe main meteorological data.

2. Models description

2.1. Monthly mean daily models

2.1.1. Total solar energy models2.1.1.1. Global irradiation bright sunshine hours relations.2.1.1.1.1. Rietveld’s model [1]. Rietveld [1] examined several linearcorrelations developed in different locations in the world and relat-ing the clearness index KT to the sunshine duration ratio (DI/SSo),was used, and noted that the a and b coefficients are, respectively,related linearly and hyperbolically to the appropriate mean valuesof DI/SSo such that

a ¼ 0:10þ 0:24DISSo

� �ð1Þ

b ¼ 0:38þ 0:08SSo

DI

� �ð2Þ

and by substituting these two relations in the Angstrom equationform he proposed the following equation:

Gh

Gho¼ 0:18þ 0:62

DISSo

� �ð3Þ

This formulation is believed to be applicable anywhere in the world.2.1.1.1.2. Glover’s and McCulloch’s [2] relation. In their model,attempting to improve on the total irradiation–sunshine correla-tion, Glover and McCulloch [2] have included the latitude effectsand presented the following formulation:

Gh

Gho¼ 0:29 cosðuÞ þ 0:52

DISSo

� �ð4Þ

where u is the latitude of the considered site, and this relation is va-lid for u 6 60�.2.1.1.1.3. Hay’s [8] model. The coefficients a and b of the linear cor-relation form relating the global irradiation to the sunshine durationare affected by the optical properties of the atmosphere constitu-ents, ground reflectivity and average air mass. Garg and Garg [3]developed the second type of equation relating the global irradiationto the sunshine duration, where he took into account the consideredmeteorological factors and proposed the following expression:

Gh

Gho¼

0:1572þ 0:5566 DISS0o

h i1� q qa

DISS0o

� �þ qc 1� DI

SS0o

� �� � ð5Þ

Table 1Values of relative humidity factors wij as function of the month number and humidity sub

i/j 1 2 3 4 5 6

1 1.28 1.38 1.54 1.77 2.05 2.302 1.46 1.77 2.05 2.15 2.05 2.053 1.60 1.81 2.00 2.17 2.25 2.26

where q, qa, and qc are, respectively, the ground, cloudless sky andcloud albedo. SS0o is the modified day length which excludes thefraction of day where the solar zenith angle is greater than 85� be-cause the Campbell–Stokes sunshine recorder used by most meteo-rological stations does not function when the solar elevation is lessthan 5�. The proposed modified day length can be obtained from

SS0o ¼2

15cos�1 cosð85Þ � sinðuÞ sinðdÞ

cosðuÞ cosðdÞ

� �ð6Þ

2.1.1.2. Global irradiation, relative humidity, sunshine hours andtemperature relations.2.1.1.2.1. Reddy’s [9] model. Reddy [9] proposed a relation thattakes into consideration some theoretical parameters such as lati-tude u, mean sun declination for the considered month, and themaximum possible bright sunshine hours in addition to the mainmeteorological parameters such as the measured bright sunshinehours, relative humidity and the number of rainy days in a month.The formula is as follows:

Gh ¼ K 1þ 0:8DISSo

� �� �ð1� 0:2tÞffiffiffiffiffiffiffi

HRp ð7Þ

where

K ¼ 100ðkSSo þ wij cosð/ÞÞ ð8Þ

k is the latitude factor given by the following relation:

k ¼ 1ð1þ 0:1/Þ ð9Þ

u is the latitude of the place in degrees, SSo is the theoretical daylength in hours, DI is the daily bright sunshine hours, and t is de-fined by the following ratio:

t ¼ number of rainy days in the monthtotal number of days in the month

ð10Þ

HR is the mean relative humidity per day in the month, and wij

is the seasonal factor (i = 1, 2 and 3) for inland stations, coastal sta-tions and hilly stations, respectively, while (j = 1,2, . . .,12) standsfor January–December. The wij values are listed out in Table 1.2.1.1.2.2. Relation of Sayigh [10]. The model of Sayigh [10] wasdeveloped by using the data relating to several Indian sites whoselatitude lies between 8� North and 28.6� North. Thus, Sayigh’s [10]model is expressed by the following relation:

GH ¼ KN exp uDISSo

� �� HR

15

� �� 1

Tmax

� �� �ð11Þ

where

N ¼ 1:7� 0:458u ð12ÞK, DI, SSo, and HR were previously defined, and Tmax is the monthlymaximum dry temperature. In Eq. (12), the latitude u of the consid-ered site is expressed in radians. Furthermore, Sayigh [10] subdi-vides the entire range of relative humidity into three subranges:HR 6 65%, HR P 70% and 65% < HR < 70%, and corresponding toeach subrange, a graph has been drawn between the humidity fac-tor wij and each month of the year.2.1.1.2.3. Relations of Swartman and Ogunlade[11]. Swartman andOgunlade [11] have proposed two empirical equations. These

ranges 1, 2, 3.

7 8 9 10 11 12

2.48 2.41 2.36 1.73 1.38 1.172.10 2.17 2.14 1.96 1.60 1.432.24 2.20 2.10 1.92 1.74 1.60

M. Koussa et al. / Energy Conversion and Management 50 (2009) 1227–1235 1229

formulas predict global radiation on a horizontal surface frombright sunshine hours and relative humidity only. The two rela-tions are given as follows:

Gh1 ¼ 490DI12

� �0:357ðHRÞ�0:262 ð13Þ

Gh2 ¼ 460 exp 0:607DI12� HR

� �� �ð14Þ

2.1.1.2.4. Relation of Garg and Garg [3]. Garg and Garg [3] showedthat the daily global irradiation data averaged over each monthfor 14 Indian measurement stations can be correlated with sun-shine duration and atmospheric water vapor content. They foundthat a least squares fit of data gives the following relationship:

Gh ¼ Gh00:414� 0:4

DISSo

� �� 0:0055Ha

� �ð15Þ

where DI is the sunshine duration, SSo is the day length, and Ha isthe atmospheric water vapor content per unit volume of dry airwhich may be computed from the following expression:

Ha ¼ HR 4:7923þ 0:3647Ta þ 0:0055T2a þ 0:0003T3

a

� �ð16Þ

Ta is the dry air temperature in degree Celsius.2.1.1.2.5. Relation of Hussain [12]. Following Garg’s and Garg’s [3]model, Hussain [12] proposed an empirical expression. Thus, thisexpression makes it possible to estimate the total solar energy asfunction of monthly average per day of the absolute humidityand the monthly average per day of bright sunshine duration. Thisrelation is given as follows:

Gh ¼ Gh0 0:394þ 0:364DISSo

� �� 0:0035Ha

� �ð17Þ

DI, Gho, SSo and Ha were previously described.2.1.1.2.6. Relation of Sambo [13]. Based on the main meteorologicalparameters such as sunshine duration DI, minimum and maximumdaily dry temperatures Tmax and Tmin, relative humidity HR, whereall are monthly averaged values and the monthly mean per dayof the theoretical values of day duration SSo, Sambo [13] developeda correlation given as follows:

Gh

Gho

� �¼ 0:62� 0:294

DISSoþ 0:178

DISSo� HR� Tmin

Tmax

� �

þ 0:491DISSo

Tmin

Tmax

� �ð18Þ

2.1.2. Reconstitution of the monthly mean daily diffuse solar energy2.1.2.1. Diffuse radiation and sunshine hours relations.2.1.2.1.1. Relations of Iqbal [14]. Iqbal [14] proposed two correla-tions where the first one is in a linear form which requires mea-sured global irradiation and the sunshine hours data as input. Inhis second proposed correlation, Iqbal [14] removed the globalirradiation requirement and expressed a correlation which re-quires only sunshine duration. The two proposed relations are gi-ven in the following:

Dh

Gh¼ 0:791� 0:635

DISSo

ð19Þ

Dh

Go¼ 0:163þ 0:478

DISSo� 0:655

DISSo

� �2

ð20Þ

2.1.2.1.2. Hay’s [15] relation. In his work, Hay [15] considered theincreasing magnitude of the global and diffuse solar radiationsafter multiple reflections between the earth and the cloud cover.The relationships between the radiation before and after multiplereflections are given below.

For a given global irradiation Gh, the multiple reflections phe-nomena increase the received global irradiation, which is quanti-fied by G0h and is obtained from the following equation:

G0h ¼ Gh 1� q qaDISS0oþ qc 1� DI

SS0o

� �� �� �ð21Þ

and the diffuse irradiation obtained after the multiple reflectionphenomena is calculated by using the correlation in following:

D0hG0h¼ 0:9702þ 1:6688K 0T � 21:303ðK 0TÞ

2 þ 51:288ðK 0TÞ3

� 50:08ðKoTÞ

4 þ 17:551ðKoTÞ

5 ð22Þ

where K 0T ¼G0hGho

, and finally Dh can be calculated from the followingrelation:

Dh ¼ D0h þ Ghq qaDISS0oþ qc 1� DI

SS0o

� �� �ð23Þ

2.1.2.2. Diffuse radiation, sunshine hours and atmospheric water con-tent relation.2.1.2.2.1. Relation of Hussain [12]. In his same paper, Hussain [12]proposed an expression relating the diffuse irradiation to the abso-lute humidity and the bright sunshine hours, this relation is asfollows:

Dh ¼ Gh00:306� 0:165

DISSo

� �� 0:0025Ha

� �ð24Þ

2.1.2.3. Diffuse radiation and clearness index relations.2.1.2.3.1. Model of Liu and Jordan [4]. Liu and Jordan [4] firstreported the correlation between the fraction of monthly meandiffuse to monthly mean total solar energy KD as function of themonthly mean clearness index KT and gave

KD ¼ 1:39� 4:027KT þ 5:531K2T � 3:108K3

T ð25Þ

For 0.30 < KT 6 0.702.1.2.3.2. Page’s model [6]. Page [6] using data from 10 widelyspread sites in 40� North to 40� South latitude belt proposed thisgeneral expression:

KD ¼ 1:0� 1:13KT ð26Þ

2.1.2.3.3. Collares-Pereira’s and Rabl’s model [16]. Collares-Pereiraand Rabl [16] used measured data from five United States loca-tions, and to test for seasonal variations they grouped the data intobins according to their sunset hours angles xs. And the leastsquares fit yields the correlation

KD ¼ 0:775þ 0:347 xs �p2

� �� 0:505þ 0:261 xs �

p2

� �h i� cos 2ðKT � 0:9Þð

�ð27Þ

According to the yearly variation of the sunset angle hour, thedata are grouped as follows:

xs � p2 � 0:15 for winter season

p2 � 0:15 � xs � p

2 � 0:15 for spring and fall season

xs � p2 � 0:15 for Summer season

1230 M. Koussa et al. / Energy Conversion and Management 50 (2009) 1227–1235

2.1.2.3.4. Relation of Erbs et al. [5]. Erbs et al. [5] used a data set forfour United State stations and developed two third–order seasonalpolynomial correlations for monthly diffuse fraction as function ofclearness index and gave

KD ¼ 1:391� 3:560KT þ 4:189K2T � 2:137K3

T ð28-aÞ

for xs 6 1.4208 and 0.30 6 KT 6 0.80 and

KD ¼ 1:311� 3:022KT þ 3:427K2T � 1:821K3

T ð28-bÞ

for xs > 1.4208 and 0.30 6 KT 6 0.80, where xs is a sunset hour an-gle in radians.2.1.2.3.5. Iqbal’s relation [17]. Iqbal [17] used measured datarelative to several Canadian sites and developed the followingrelation:

KD ¼ 0:958� 0:982KT for 0:30 6 KT 6 0:60 ð29Þ

Table 2Geographical co-ordinates of the considered sites.

Location Latitude in degrees Longitude in degrees Height in meters

Bouzaréah 36.8N 3.08E 345Adrar 27.82N 0.18W 263.9Ghardaia 32.4N 3.80E 468.4

2.2. The hourly models

2.2.1. Global irradiation model2.2.1.1. Model of Jain [7]. In his work, Jain [7] used 11 years monthlyaverage values of the ratio hourly to daily of global rt as well as thediffuse irradiations rd on a horizontal plan data relating to severalCanadian sites and had shown that the distribution of these ratiosfollows a Gaussian law form. However, he established the follow-ing relation for global irradiation:

rt ¼1ffiffiffiffiffiffi

2pp

rexp �ðts � 12Þ2

2r2

" #ð30Þ

where ts is the true solar time in hours, and r is defined by

r ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pkThðt ¼ 12Þ

p ð31Þ

where rt (ts = 12) is the monthly mean hourly ratio in the monthlyaverage per day of the global irradiation at mid-day true solar time.To simplify the determination of r values, Jain [7] established a cor-relation between r and theoretical day length SSo, and proposed thefollowing relation:

r ¼ 0:2SSo þ 0:378 ð32Þ

SSo can be calculated by the following expression:

SSo ¼2

15cos�1ð� tan gðuÞ tan gðdÞÞ ð33Þ

u and d are, respectively, the latitude of the considered site and thesolar declination.

2.2.1.2. Model ofCollares-Pereira and Rabl [16]. Using monthly aver-age data of solar radiation measurement from five United Statesstations and based on Whillier’s [18] and Hottel’s and Whiller’s[19] works, Liu and Jordan [4] have established a generalized chartof rt as function of day length and the hour angle, and presentedthe chart curves in the following form:

rt ¼gh

Gh¼ f ðx;xsÞ ð34Þ

However, Collares-Pereira and Rabl [16] used the least squaresfit and presented the proposed chart curves in a semi empiricalform and gave the following expression:

rt ¼p

24aþ b cosðxÞ cosðxÞ � cosðxsÞ

sinðxsÞ � pxs180 cosðxsÞ

� �ð35Þ

yields the coefficients given by

a ¼ 0:409þ 0:5016 sinðxs � 60Þ ð36:aÞb ¼ 0:6609� 0:4767 sinðxs � 60Þ ð36:bÞ

where x is hour angle in degrees calculated in the mid-point of theconsidered hour and xs is the sunset hour angle in degrees calcu-lated by

xs ¼ cos�1ð�tgðuÞtgðdÞÞ ð37Þ

where u is the latitude of the considered site and d is the solar dec-lination calculated for the representative day of the month.

2.2.2. Diffuse irradiation model2.2.2.1. Model of Jain [7]. In the same paper, Jain proposed the fol-lowing relation for calculating the monthly mean hourly ratio indaily diffuse irradiation rd as function of solar time and themonthly mean theoretical day length:

rd ¼1ffiffiffiffiffiffi

2pp

rexp �ðt � 12Þ2

2r2

" #ð38Þ

So, to simplify the calculation of the r values, especially for thesites where the hourly diffuse irradiation data are not available, heproposed the following correlation:

r ¼ 0:222SSo þ 0:27 ð39Þ

2.2.2.2. Liu’s and Jordan’s [4] equation. As for the monthly meanhourly global irradiation, in their work, Liu and Jordan [4] showeda related set of curves for rd to solar hour angle and the sunset hourangle

rd ¼dh

Dh¼ f ðx;xsÞ ð40Þ

and the proposed chart curves have been converted to the followingexpression:

rd ¼p

24cosðxÞ � cosðxsÞ

sinðxsÞ � pxs180 cosðxsÞ

� �ð41Þ

3. Used data

The retained formulas were tested for different Algerianlocations where Bouzareah was considered as a coastal site, andGhardaïa and Adrar were considered as dry sites and arid sites.Thus, the geographical co-ordinates of these sites are listed outin Table 2.

For the three considered sites, monthly mean daily sunshineduration hours, relative humidity, minimum and maximum tem-perature data were obtained from National Office of Meteorology.The used monthly mean daily and hourly global and diffuse irradi-ation data were obtained from three recording data stationsinstalled, respectively, in the Renewable Energy ApplicationsCenter at Ghardaia site, in Saharan Renewable Energy ResearchCenter at Adrar site and in the Center of Renewable Energies Devel-opment at Bouzareah site. It should be noted that 5 min and 1 hare, respectively, used as recording step time for radiometricparameters at Ghardaïa site and Bouzaréah and Adrar sites.However, all measured diffuse and total solar radiation data areaveraged to monthly mean hourly and daily values.

M. Koussa et al. / Energy Conversion and Management 50 (2009) 1227–1235 1231

4. Method of statistical comparison

There are numerous works in literature journals which dealwith the assessment and comparison of solar radiation estimationmodels [20–23]. The most popular statistical parameters indicatorsare the mean bias error, MBE, and the root mean square error,RMSE. In this study, to evaluate the accuracy of the calculated data,from the models described above, we used three statistical tests,the mean percentage error, MPE, the mean bias error, MBE, andthe root mean square error, RMSE.

4.1. The mean percentage error

MPE ¼ 1n

Xn

1

Gi;calc � Gi;meas

Gi;meas

� �� 100 ð42Þ

where Gi,meas, Gi,cal and n are, respectively, the ith measured valuesand ith calculated values of global or diffuse solar radiation andthe number of values.

4.2. The mean bias error

MBE ¼ 1n

Xn

1

ðGi;calc � Gi;measÞ ð43Þ

4.3. The root mean square error

RMSE ¼ 1n

Xn

i¼1

ðGi;calc � Gi;measÞ2" #1

2

ð44Þ

So, to evaluate the performance of the hourly models we usedthe MPE and the normalized mean bias error, NMBE, and rootmean square error, NRMSE.

Table 3MPE (%), MBE (kWh/m2) and RMSE (kWh/m2) values between measured and estimated to

Adrar Ghardaia

MPE (%) MBE (kWh/m2) RMSE (kWh/m2) MPT (%)

Rietveld’s Eq. (3) 0.57 0.005 0.212 �3.22McCulloch’s Eq. (4) 0.058 0.035 0.215 �5.27Hay’s Eq. (5) 0.41 0.020 0.237 2.94Sayigh’s Eq. (11) �8.35 �0.473 0.837 �10.3Reddy’s Eq. (7) 27.01 2.892 3.800 15.6Swartman’s Eq. (13) 21.26 1.529 1.607 13.6Swartman’s Eq. (14) 19.25 1.330 1.477 14.7Garg’s Eq. (15) 3.89 0.212 0.242 0.4Hussain’s Eq. (17) �1.3 �0.111 0.215 �4.8Sambo’s Eq. (18) �17.8 �0.817 .864 �22.5

Table 4MPE (%), MBE (kWh/m2) and RMSE (kWh/m2) values between measured and estimated d

Bouzareah Ghardaia

MPE (%) MBE (kWh/m2) RMSE (kWh/m2) MPE (%)

Iqbal’s Eq. (19) 10.88 �0.195 0.249 14.18Iqbal’s Eq. (29) �3.12 �0.008 0.151 �23.05Hay’s Eq. (23) �4.30 �0.005 0.137 �14.94Hussain’s Eq. (2) 12.4 0.252 0.294 15.4Iqbal’s Eq. (20) 17.1 0.328 0.355 11.7Page’s Eq. (26) 8.4 0.139 0.193 �11.8Collares P.’s Eq. (27) �26.1 �0.277 0.353 22.0Liu’s & Jordan’s Eq. (25) �4.4 �0.075 0.147 �10.8Erbs et al.’s Eqs. (28.a & b) 7.6 0.141 0.185 0.9

4.4. The normalized mean bias error

NMBE½%� ¼ MBE1n

Pni¼1Gi;meas

� 100 ð45Þ

This test provides information on long-term performance. A lowNMBE value is desired. A negative value gives the average amountof underestimation in the calculated value. So, one drawback ofthese two mentioned tests is that overestimation of an individualobservation will cancel underestimation in a separate observation.

4.5. The normalized root mean square error

NRMSE½%� ¼ RMSE1n

Pni¼1Gi;meas

� 100 ð46Þ

5. Results and discussion

5.1. Monthly mean daily models

The performances of the 10 global and of the nine diffuse solarradiation models were evaluated using the three statistical indica-tors, i.e MPE, MBE and the RMSE, discussed early. The obtained re-sults are presented in Tables 3 and 4. Furthermore, as presented inthe corresponding literature, relations (7), (11), (13), and (14) areexpressed in British units system, and in our work the results areconverted into Wh/m2 (international unit system).

5.1.1. Total solar radiationFrom results presented in Table 3, we note that generally Rietveld

[1], Garg and Garg [3], Glover and McCulloch [2] have used more orless comparable levels of different statistical indicators in this work.So, their correlations yield, respectively, the best results for Adrar,Ghardaia and Bouzareah. Therefore, relations (3), (15) and (4) are

tal solar irradiations.

Bouzareah

MBE (kWh/m2) RMSE (%) MPE (%) MBE (kWh/m2) RMSE (kWh/m2)

�0.162 0.346 4.15 0.177 0.2700.277 0.394 �2.47 �0.085 0.2250.160 0.337 �5.47 �0.225 0.303�0.556 .974 7.4 0.219 1.072

0.969 1.443 �12.4 �0.619 1.2920.814 1.196 14.1 0.637 1.4200.896 1.200 10.7 0.444 1.3520.020 0.218 8.4 0.358 0.411�0.266 0.349 5.3 0.210 0.294�0.952 1.012 �9.7 �0.462 0.562

iffuse solar irradiations.

Adrar

MBE (kWh/m2) RMSE (kWh/m2) MPE (%) MBE (kWh/m2) RMSE (kWh/m2)

�0.240 0.293 12.21 �0.232 0.285�0.246 0.347 �32.94 �0.315 0.336�0.259 0.289 15.36 0.281 0.313

0.269 0.303 13.6 0.246 0.2760.198 0.221 16.7 0.294 0.311�0.133 0.147 �3.8 �0.048 0.125

0.441 0.509 22.3 0.476 0.562�0.125 0.130 �4.9 �0.057 0.099

0.05 0.112 6.8 0.123 0.172

1232 M. Koussa et al. / Energy Conversion and Management 50 (2009) 1227–1235

recommended respectively to be use for calculating the monthlymean horizontal global solar radiation, for Adrar, Ghardaia andBouzareah and for all sites included in the same climatic zones.

Hay’s [8] relation is preferred best. Furthermore, monthly val-ues of ground and atmospheric albedo are needed. These valuesare based on Canadian atmospheric conditions. So, they appearto perform well at the considered Algerian stations.

Hussain’s [12] models result in larger errors than Hay’s [8] rela-tion for Adrar and Bouzareah sites, but are better for the Ghardaiasite.

The above five models results can be explained by the fact thatthese models are in an empirical form and are based on the mainweather parameters such as relative humidity, dry temperatureand monthly average daily of bright sunshine and on some astro-nomical parameters such as solar constant, earth sun distance var-iation, the monthly mean daily extraterrestrial solar energy andtheoretical bright sunshine duration. Thus, the latitude and sea-sonal effects are simplified by the use of monthly average of thevarious astronomical and geographical parameters, and on theother hand, the climatic zones effects are represented by the mea-sured and used weather parameters in these two models.

The model proposed by Sayigh [10] is the next best one with thesmallest range of the MPE, MBE and RMSE values for Adrar andGhardaia considered as dry and arid locations. The negative valuesof MPE and MBE show that there is a constant underestimation. So,the magnitude of the three statistic indicators is generally largerfor Bouzareah, which is considered as a coastal site.

Reddy’s [9] equation and Swartman’s and Ogunlade’s [11] twomodels result in larger errors than the previous correlations, andthe MPEs are positive, showing that there is constant overestima-tion. As mentioned previously, it is clearly observed that thesemodels overestimate the total solar radiation throughout the year,and it is more apparent for result of Reddy’s [9] model, especiallyfor Ghardaia and Adrar locations.

Sayigh’s [10], Swartman’s and Ogunlade’s [11] and Reddy’s [9]relation results can be explained by the fact that the derivationof these models is based on data from Indian locations character-ized principally by the humidity factor wij and the number of rainydays which depend strongly on the considered sites climate andcan be different from those of the three considered Algerian sites,especially those of coastal site.

5.1.2. Diffuse irradiation modelsThe performance of the nine diffuse solar radiation correlations

was evaluated following the same procedure. The results ofthe comparison of the different retained models are presented inTable 4.

From the results presented in Table 4, we note that Liu’s andJordan’s [4] correlation yields the best results in term of MBEsand RMSEs.

Erbs et al.’s [5] model is the next best one. The low MBEs areparticularly remarkable for the three considered sites. Even thoughthe Liu and Jordan [4] and Erbs et al. [5] correlations were estab-lished by using data from the United State sites, they appear to

Table 5Monthly and overall performance of hourly global irradiation models (Ghardaia site).

January February March April May Ju

MPE (%) Eq. (35) �0.4 �5.9 �10.1 �9.0 �7.0 1Eq. (30) 3.2 �4.1 1.4 �0.7 �2.2 �

NMBE (%) Eq. (35) �.77 �1.88 �6.02 �11.89 �7.00Eq. (30) .04 �4.30 0.91 �1.19 �3.03 �

NMRSE Eq. (35) 6.88 8.63 7.39 18.58 13.55Eq. (30) 8.21 5.95 4.70 8.20 6.17

perform well at the considered sites, and with their universalapplicability, their use is recommended for estimating the monthlyaverage per day of diffuse irradiation, respectively, for Bouzareahand Ghardaia.

Page’s [6] relations derivation was based, respectively, on datafrom locations within latitude u = ±40�. So it is the next best model,especially for Adrar, but the correlation results in larger errors thanLiu’s and Jordan’s [4] and Erbs et al.’s [5] relations for Bouzareahand Ghardaia where the MPEs and MBEs are positive and negative,respectively, for coastal sites and arid sites. Its simplicity and uni-versal applicability should be emphasized upon. However, thismodel is recommended to reconstitute the monthly mean dailydiffuse irradiation for Adrar location and is applied for all sites inthe same climatic zone.

Iqbal’s [17] relation (29) and Hay’s [15] equation produce re-sults very close to each other. They have more or less comparablelevels of MBEs and RMSEs. They present the best results for Bouz-areah location, but produce the largest RMSEs, MBEs and MPEs forthe two Saharian sites. Furthermore, these correlations were estab-lished by using data from Canadian site, and to apply Hay’s [15]equation the monthly values of ground and atmospheric albedoare needed. However, they appear to perform very well at thecoastal site, but not at the arid sites. This is due to the fact thatthe ground and the atmospheric albedo and the climatic parame-ters are totally different and the diffuse radiation is directly relatedto these parameters. So this was not observed in the relation (5) re-sults where it was mentioned that Hay’s [8] relation performs wellat all the considered sites. Thus, the atmospheric diffusion and themultiple reflections have an inverse effect on each of the two dif-fuse and direct solar radiation components. Furthermore, thesetwo phenomena increase and decreases, respectively, the diffuseand the direct components, and their effect cannot appear clearlyon the global radiation component.

5.2. Monthly mean hourly models

5.2.1. Global irradiation modelsTo make a comparison between the models, the calculated and

measured data were compared for each month, and a statisticalsummary of the monthly and overall performance of the combina-tion of the different test indicators discussed previously as MPE,NMBE and the NRMSE is, respectively, presented in Tables 5–7and 8–10 for the hourly global and diffuse irradiations.

For the hourly global irradiation, the results presented in Tables5–7 show that Jain’s [7] model generally leads to the best results.For the three considered sites, the NRMSE values obtained by usingthis model do not exceed 12% except for October in Adrar locationwhere it is equal to 16.42% (see Table 6). Even though its derivationwas based on data from Canada locations, this equation appears toperform well at the considered Algerian stations.

Collares-Pereira’s and Rabl’s [16] model resulted in largestNRMSE with the highest value close, respectively, to 18.58% and18.18% for April at Ghardaia location (see Table 5) and for Octoberat Adrar location (see Table 6). Furthermore, the monthly values of

ne July August September October November December

2.1 �9.6 4.3 �8.8 6.5 16.0 16.61.8 �0.5 �0.5 �1.6 �0.5 �2.8 �1.02

8.82 �7.67 4.69 �5.70 7.55 10.51 11.322.03 �1.00 �1.31 �2.44 �0.09 �2.03 �0.65

6.95 8.98 10.04 6.50 11.82 12.31 12.526.40 8.13 8.27 4.50 3.75 4.84 7.73

Table 7Monthly and overall performance of hourly global irradiation models (Bouzareah site).

January February March April May June July August September October November December

MPE (%) Eq. (35) 0.3 2.7 2.6 �9.2 �0.6 �0.9 �6.2 6.5 �0.2 �21.0 4.82 �10.1Eq. (30) �10.1 �9.8 �9.7 �0..9 �0.8 4.5 1.9 13.7 1.6 �0.5 �4.1 3.6

NMBE (%) Eq. (35) 0.99 3.63 3.62 �.52 �8.67 �0.52 �2.60 2.50 �0.21 �11.21 3.76 �4.12Eq. (30) �7.86 �6.63 �6.62 �1.10 �0.65 2.16 1.28 1.81 0.30 �1.40 �3.39 2.28

NMRSE (%) Eq. (35) 6.32 6.50 6.44 4.87 10.36 11.80 5.84 7.11 7.75 12.71 12.53 7.84Eq. (30) 9.03 11.78 11.88 2.29 4.36 6.02 4.09 5.62 7.03 5.08 9.03 5.14

Table 8Monthly and overall performance of hourly diffuse radiation models (Ghardaia site).

January February March April May June July August September October November December

MPE (%) Eq. (41) �9.4 �10.9 0.6 �9.3 �7.6 3.5 4.3 �15.3 �3.1 �8.6 �6.5 �5.8Eq. (38) �17.0 �11.6 �5.9 �7.5 �10.8 �2.8 �0.6 �17.8 �10.1 �11.6 �14.4 �13.3

NMBE (%) Eq. (41) �11.10 �11.66 0.73 �8.97 �10.41 2.15 2.05 �12.64 �7.40 �9.26 �9.24 �5.30Eq. (38) �16.50 �15.43 �6.01 �12.44 �14.63 �2.99 �2.97 �16.38 �12.48 �13.56 �14.77 �11.01

NMRE (%) Eq. (41) 15.52 14.97 7.86 20.30 12.63 11.16 7.73 14.83 11.05 12.45 13.38 8.64Eq. (38) 19.15 19.43 12.2 24.45 18.33 13.36 13.88 19.26 16.76 17.35 18.78 13.48

Table 9Monthly and overall performance of hourly diffuse radiation models (Adrar site).

January February March April May June July August September October November December

MPE (%) Eq. (41) 7.0 5.3 1.5 �0.4 �0.5 �1.0 6.7 �1.1 �0.7 22.8 9.1 0.8Eq. (38) �1.5 2.3 �5.8 �7.4 �3.5 �5.0 3.3 �2.1 �6.7 26.2 0.3 �7.6

NMBE (%) Eq. (41) 6.46 5.84 �3.14 �3.02 �1.51 �8.22 3.73 �6.49 �1.48 18.23 9.39 0.24Eq. (38) �0.01 0.99 �8.31 �8.42 �5.90 �12.64 1.14 �10.27 �6.87 12.43 2.78 �5.86

NMRSE (%) Eq. (41) 12.04 12.60 10.27 7.58 12.33 17.74 7.29 10.69 7.15 22.55 11.76 4.06Eq. (38) 15.50 15.46 12.15 12.87 13.87 22.59 11.11 15.81 12.23 23.13 12.24 9.09

Table 6Monthly and overall performance of hourly global irradiation models (Adrar site).

January February March April May June July August September October November December

MPE (%) Eq. (35) 2.9 �5.3 �7.3 �4.1 1.3 �4.4 �13.2 �10.9 �8.3 �14.2 5.4 �10.9Eq. (30) 10.5 6.5 16.0 7.2 �4.6 1.9 1.1 �1.8 4.2 �1.4 0.3 4.5

NMBE (%) Eq. (35) 2.27 �3.50 �5.83 �0.87 1.70 �0.55 �9.17 �9.15 �3.78 �8.57 4.35 �8.07Eq. (30) 5.97 5.41 2.76 2.73 �4.24 0.66 3.04 �1.89 2.79 �3.49 0.73 0.32

NMRSE (%) Eq. (35) 7.05 8.28 7.34 6.47 6.34 12.13 14.24 13.36 5.50 18.18 9.73 10.23Eq. (30) 9.47 8.82 7.15 6.71 7.44 4.28 9.93 8.76 3.72 16.42 6.87 7.69

Table 10Monthly and overall performance of hourly diffuse radiation models (Bouzareah site).

January February March April May June July August September October November December

MPE (%) Eq. (41) 0.8 3.3 �0.3 5.5 2.0 �1.3 2.4 �2.6 �1.9 2.8 0.1 �0.7Eq. (38) �6.7 �5.2 �6.8 6.4 �3.2 �7.6 �3.8 �5.1 �9.1 �1.1 �7.7 �6.2

NMBE (%) Eq. (41) 2.64 2.45 �0.23 6.48 1.95 �1.06 2.65 �4.19 �1.87 2.79 0.27 �0.13Eq. (38) �3.49 �3.73 �5.07 2.20 �3.06 �6.20 �2.58 �8.52 �7.27 �1.86 �6.31 �5.92

NMRSE (%) Eq. (41) 6.19 5.66 7.46 11.92 5.92 6.21 5.89 6.77 5.98 7.42 8.41 4.92Eq. (38) 11.11 10.83 10.34 14.90 11.34 11.83 9.95 13.31 11.96 10.00 13.95 10.65

M. Koussa et al. / Energy Conversion and Management 50 (2009) 1227–1235 1233

the a (Eq. (36.a)) and b (Eq. (36.b)) are based on Hump Mountainlocation in United States of America, and probably cannot performvery well for the Algerian stations.

In addition, the NMBEs show the long-term performance of thetwo models. Jain’s [7] relation generally yields the best results. The

low NMBEs are particularly remarkable. So, the negative NMBEsvalues presented in Table 5 show that there is an underestimationfor Bouzareah location (see Table 7) during the period January–May and October–November and during May, August and Octoberfor Adrar location (see Table 6).

Fig. 1. Matlab-Simulink program description.

1234 M. Koussa et al. / Energy Conversion and Management 50 (2009) 1227–1235

Collares-Pereira’s and Rabl’s [16] equation presents NMBEs val-ues higher than those obtained by Jain’s [7] model. Even thoughwithout exceeding 10%, these are sometimes positive and some-times negative over the whole year and for the three consideredsites.

Also, since the MPE indicator provides information on the sizeof the discrepancy in the measurements and quantifies the system-atic component of the normalized difference for an individualobservation, the results show that generally the magnitude ofMPE presented by Jain [7] is the smallest for the three consideredsites, especially for Ghardaia location.

5.2.2. Diffuse irradiation modelsThe two monthly mean hourly diffuse relations performances

were evaluated following the same procedure. The resulting MPEs,NMBEs and NRMSEs are presented in Table 8–10.

The NMBEs results show the long-term performance of the twomodels. Jain’s [7] model presents an underestimation for almost allthe considered sites and months, except for February, July andOctober–November for Adrar (see Table 9), and for April for Bouz-areah (see Table 10). On the other hand, for the three locations (seeTables 8–10), Liu’s and Jordan’s [4] relation results are the best, andthe low NMBEs are particulary remarkable for Bouzareah, which donot exceed 7% over the whole year.

The NRMSE provide information on the short-term performanceof a model by allowing a term by term comparison of actual differ-ence between the estimated and measured values and generallythe lower NMRSE value correspond to the better model0s perfor-mance. However, from the results presented in Tables 8–10 it willbe noted that Liu’s and Jordan’s [4] equation yields the best results.So, a larger magnitude of the NMRSE was obtained over almost thewhole year for Ghardaia and Adrar considered as arid sites thanthose obtained for Bouzareah considered as a coastal location. Thiscan be explained by the fact that Ghardaïa and Adrar sites are sit-uated, respectively, in the north and in the center of Sahara, wherethe spring and autumn seasons are characterized by the wind sandphenomena which considerably increase the diffusion of the solar

radiation. Furthermore, in terms of MPE indicators, generally Liu’sand Jordan’s [4] model leads to better performances.

6. Simulink program

Using the obtained results, a Simulink program was developedfollowing the methodology described above:

1. using the theoretical data, as:– the considered site latitude;– the representative day number for each month;– all required theoretical parameters are calculated;

2. using sunshine hours duration as input in relation3, 4, and 15the monthly mean daily global irradiation is calculated forAdrar, Bouzareah or Ghardaia;

3. the calculated global irradiation are used, respectively, inrelation (26), (25) and (28) to obtain the monthly mean dailydiffuse irradiation for Adrar, Bouzareah and Ghardaia;

4. the monthly mean daily diffuse and global irradiation areused as input, respectively, in relations (30) and (41) to calcu-late the monthly mean hourly global and diffuse irradiation.

The obtained Simulink program is presented in Fig. 1.

7. Conclusions

First, we can affirm that for any given site, the direct use of amodel suggested in the literature can lead to erroneous values,and consequently can influence the dimensioning of the solar en-ergy conversion systems considerably. However, the choice of themodels strongly depends on the climatic characteristics of the con-sidered site compared to those on which its application is beingconsidered. This was observed from results obtained by Eqs. (7),(11), (13), (14), and (18) for the monthly mean global solar radia-tion, from those obtained by Eqs. (20), (24), and (27) for themonthly mean daily diffuse irradiation and by Eqs. (34) and (38)for the monthly average data per hour.

M. Koussa et al. / Energy Conversion and Management 50 (2009) 1227–1235 1235

However, the 10 monthly mean daily global irradiation correla-tions were compared statistically in terms of mean percentage, themean bias and root mean square errors. Rietveld’s [1], Glover’s andMcCulloch’s [2] and Garg’s and Garg’s [3] correlations are the mostaccurate in general. Only the sunshine hours, dry air temperatureand relative humidity data available in different meteorologicaloffices are, respectively, needed as input in the first two modelsand in the third model. Their use is recommended for estimatingthe horizontal monthly average per day of the global radiation,respectively, for Adrar, Bouzareah and Ghardaia and for all othersites in the same climatic zone.

The nine monthly mean daily diffuse correlations were comparedfollowing the procedure as described above. Liu’s and Jordan’s [4],Erbs et al.’s [5] and Page’s [6] correlations are the most accurate ingeneral and are simple to use, especially the Page [6] correlation. Onlyglobal solar radiation is needed as input data. These three correlationsare recommended for estimating the horizontal monthly average perday diffuse irradiation, respectively, Bouzareh, Ghardaia, Adrar.

On the other hand, the two hourly global radiation correlationsand the two diffuse equations were compared statistically in termsof mean percentage, the normalized mean bias and the normalizedroot mean square errors. However, relation (30) and relation (41)are, respectively, the most accurate in general to estimate themonthly mean daily of global and diffuse irradiations. Further-more, if only the monthly mean of the main meteorological param-eters is available, we can calculate the monthly daily and hourlyglobal and diffuse irradiations on a horizontal surface with a goodaccuracy. In this way, a Simulink program was developed to calcu-late the monthly mean daily and hourly global and diffuse datafrom only monthly mean daily sunshine hours duration, dry tem-perature and relative humidity.

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