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Renewable Energy 33 (2008) 14061412

Technical Note

Evaluation of 12 models to estimate hourly diffuse irradiation on

inclined surfaces

Ali Mohammad Nooriana, Isaac Moradib,, Gholam Ali Kamalia

aAtmospheric Science and Meteorological Research Center (ASMERC), IRIMO, Tehran, IranbDepartment of Physical Geography, Faculty of Geography, University of Tehran, Tehran, Iran

Received 26 October 2006; accepted 30 June 2007

Available online 7 September 2007

Abstract

This study evaluates the performance of 12 models to estimate hourly diffuse solar irradiation on inclined surfaces from those

measured on horizontal surfaces. Total solar irradiation incident on a tilted surface consists of three components including: beam, diffuse

and reflected from the ground. On a semi-hourly basis, the beam component can be calculated by the ratio of the incidence angle to the

solar zenith angle. The reflected component has a small effect on calculations and may be calculated with an isotropic model. In contrast,

models for estimating the diffuse component show major differences, which justify the validation study that this paper discusses. Twelve

models were tested against recorded south- and west-facing slope irradiances at Karaj (351550N; 501560E), Iran. The following models

were included: Badescu [Ba], Tian et al. [Ti], Perez et al. [P9], Reindl et al. [Re], Koronakis [Kr], Perez et al. [P8], Skartveit and Olseth

[SO], Steven and Unsworth [SU], Hay [Ha], Klucher [Kl], Temps and Coulson [TC], and Liu and Jordan [LJ].

The relative root mean square error (RMSE), for the south-facing surface ranges from 10.16% to 54.89% for the SO and TC models,

respectively. For the west-facing surface, RMSE ranges from 30.71% for the P9 model to 63.53% for the TC model. Statistical indices

show that all models produce large errors for the west-facing surface. Statistical indices for the south-facing surface show reasonably

good agreement with measured data.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Diffuse irradiation; Isotropic models; Anisotropic models; Radiation modeling; Iran

1. Introduction

Solar energy is a sustainable, safe and abundant energy

resource. Estimating solar irradiation incident on inclined

surfaces of various orientations is necessary in order to

calculate the building heat gain from the building envelope

as well as the electric power generated by photovoltaics

(PV) [1], design solar systems and to evaluate their long-term average performance. Despite the fact that many

meteorological/radiometric stations measure global and

diffuse irradiation received on horizontal surfaces the data

on inclined surfaces are not available and are also

estimated with different models from those measured on

horizontal surfaces[2].

Total radiation incident on a tilted plane consists of

three components: beam radiation, diffuse radiation and

reflected radiation from the ground. On an hourly basis the

direct and reflected components can be computed with

good accuracy by using simple algorithms but the nature of

the diffuse component is more complicated and the desired

algorithms need assessment and evaluation.

Perez et al. [3,4] developed two new and more accuratebut considerably simpler versions of the original Perez

diffuse irradiance model[5]. The original version has been

used world wide to estimate short time step (hourly or less)

irradiance on tilted surfaces based on global and direct (or

diffuse) irradiance measured on horizontal surfaces. Li and

Lam [6] evaluated the anisotropic models of Klucher [7],

Hay [8] and Perez et al. [3] applied to two-years of

measured data in Hong Kong (19961997). All three

models produced large errors for north-facing surfaces.

Predictions for south-facing surfaces showed reasonably

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www.elsevier.com/locate/renene

0960-1481/$- see front matter r 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.renene.2007.06.027

Corresponding author. Tel.: +98 21 44580661; fax: +98 21 44580670.

E-mail addresses: [email protected] (A.M. Noorian),isaac_moradi@

yahoo.com (I. Moradi), [email protected] (G.A. Kamali).
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good agreement with measured data. Kamali et al. [9]

evaluated eight diffuse models to estimate solar irradiation

on tilted surfaces using daily measured solar irradiation

data in Iran and recommended the Reindl et al.[10]model

for estimating solar irradiation on tilted surfaces. Notton

et al. [11] evaluated the combination of some well-known

models to estimate the hourly global solar irradiation on a

tilted surface from those on a horizontal surface. They

recommend the Klucher [7] model separately or in

combination with other models to estimate diffuse solar

irradiation on surfaces tilted towards the equator.The objective of this study is to compare the results of 12

widely used models, for estimating total solar radiation on

tilted surfaces, with measurements at Karaj, Iran for west-

and south-facing surfaces inclined at 40 and 45,

respectively. The models used the same method for

calculating beam and ground reflected radiation on the

tilted surface. The only difference was in the treatment of

diffuse radiation.

2. Data

The measuring station was located on the roof-top of the

Faculty of Agriculture, University of Tehran, Karaj

35550N; 50560E), Iran. The measurements have been

carried out using two CM5 pyranometers with solarimeter

integrator CC2 manufactured by Kipp and Zonen, Hol-

land. The solar radiation data measured by the authors,

during the period from June to October 2002 were used in

this study. This consisted of semi-hourly global solar

radiation on a horizontal and two tilted surfaces (45

south-facing and 40 west-facing). Also for testing the

diffuse model, semi-hourly global and diffuse radiation on

a horizontal surface was measured by the previously

mentioned radiometers from February to June 2006. This

represents the only measured data available for tilted

surfaces in Iran[12].

3. Theory and models

The hourly total irradiation incident on a tilted surface

GTh is composed of direct BTd, ground reflected RTh

and sky-diffuse DTh components:

GTh BThDThRTh, (1)

The amount of direct radiation on a surface tilted S

degrees from the horizontal and rotated aT degrees from

the northsouth axis can be calculated by multiplying the

direct horizontal irradiation by the ratio of cosy= cosZ,where y is solar incidence angle on a tilted plane and Z is

solar zenith angle. Also, the measuring station was located

on a roof-top with very low reflectance, and the reflected

component was very much lower than the direct and the

diffuse components so an isotropic model can be used to

compute the reflected component on the tilted surface. So,

Eq. (1) can be written again as follows:

GTh Bh: cosy= cosZ rdDh Gh:r:1cosS=2,

(2)where Bh, Dh and Gh are hourly direct, diffuse and total

solar radiation on a horizontal surface, either measured

directly or estimated from each other,rd is the ratio of the

hourly diffuse irradiation incident on a tilted surface to

that on a horizontal surface, r is ground reflectivity, and Z

and y are calculated by well-known formulae:

cosZ sinf sind cosf cosd coso,

cosy sinS sinZ cosaS aT cosS cosZ, (3)

where f is the latitude of the location, d is the solar

declination, ando is the solar hour angle[13]. Calculation

of the diffuse ratio is more complicated and many

ARTICLE IN PRESS

Nomenclature

Bh hourly direct solar radiation on a horizontal

surface MJ m2

BTh hourly direct solar radiation on a tilted surface

MJ m

2

Dh hourly diffuse solar radiation on a horizontal

surface MJ m2

DTh hourly diffuse solar radiation on a tilted surface

MJ m2

Gh hourly total solar radiation on a horizontal

surface MJ m2

Goh hourly extraterrestrial solar radiation on a

horizontal surface MJ m2

GTh hourly total solar radiation on a tilted surface

MJ m2

kT atmospheric transmissivity and equals the ratio

ofGh=Goh (dimensionless)

rb beam ratio factor, the ratio of cosy= cosZ(dimensionless)

rd ratio of the hourly diffuse solar radiation

incident on a tilted surface to that on a

horizontal surface (dimensionless)

RTh hourly reflected solar radiation on a tiltedsurface MJ m2

S tilted plane slope angle (deg.)

Z solar zenith angle (deg.)

aS solar azimuth angle (deg.)

aT tilted plane azimuth angle (deg.)

d solar de clination (deg.)

f latitude of the site and positive for the northern

hemisphere (deg.)

r ground reflectivity (dimensionless)

y solar incidence angle on tilted surface (deg.)

o solar hour angle and clockwise from south is

positive (deg.)

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the west-facing surface the absolute MBE ranges from

2.43% (Kl model) to 38.52% (TC model). For all

the models studied covering south-facing surfaces, except

for SU (47.18%) and TC (54.89%), the RMSE values

do not differ a great deal. The SO, Ha, Re and P9 models

give the most accurate predictions for the south-facing

surface, and P9 model performs better for the west-facing

surface. The MBE shows that all models, with the

exception of SU model, underestimate values for west-

facing surfaces.

Graphical residual analysis of each model (residuals are

estimates of experimental error obtained by subtracting the

observed irradiation from the estimated one) for the south-

facing-surface is reported in Fig. 1 which shows that the

values of residuals for the SO, Ha, Re, P9 and P8 models,

exhibit generally small differences and are very close to

zero. This suggests that these models estimate of irradia-

tion compare well with measured data. Residual analysis

for the west-facing surface (Fig. 2) shows that only the

values of residuals for P9 model show relatively small

values with a distribution near to normal distribution. The

residual analysis for both surfaces confirms the result of the

statistical indices, RMSE and MBE.

For the west-facing surface, although the correlation

coefficients between the measured and estimated irradia-

tion are significant at 0.05 level (Table 3) the frequency of

the residuals (Fig. 2) shows that the high correlation is due

to the scattering of residuals around zero and high positiveand negative values of the residuals neutralize the effect of

each other. Statistical indices show that all models produce

large errors for the west-facing surface. The statistical and

graphical results for the south-facing surface show reason-

ably good agreement with measured data.

5. Conclusions

An evaluation of the predicted tilted solar irradiance

based on 12 inclined surface models and measured

horizontal solar data in Karaj, Iran has been carried out.

It has been observed that relatively high MBE and RMSEvalues are found for the west-facing surfaces, which receive

much less direct radiation than south-facing surfaces; and

the tilted components predicted for south-facing surface

are more accurate. The Skartveit and Olseth[20], Hay[8],

Reindl et al.[10]and Perez et al. [3]models give the most

accurate predictions for the south-facing surface and the

Perez et al. [3] model performs best for the west-facing

surface. The RMSE values for the south-facing surface

range from 10.16% to 54.89% and for west-facing surface

range from 30.71% to 63.53%. In general, the Perez et al.

[3] model shows the best agreement with the measured

tilted data.

Acknowledgments

The authors would like to give their sincere thanks to

Dr. Helfried Scheifinger (Vienna) for his helpful advices

and recommendations, Dr. Peter Mayes (NJDEP) for

improving the English text, and anonymous referees for

carefully reading the manuscript and giving a number of

useful suggestions for improvement.

Appendix A

A.1. Computing the semi-hourly total extraterrestrial solar

irradiation, Goh

For a given time, Goh in unit of MJ m2 is given by[23]:

Goh 37:595

d2 cosf cosdsino2sino1

p

180o2o1 sinf sind

n o,

(A.1)

wheredis the earthsun distance in astronomical units,f is

latitude,d is solar declination, o1 is the hour angle for the

beginning time, o2 is the hour angle for the ending time

and all angles are given in degrees. The solar constant is

taken as 1367 W m2 [24,25].

ARTICLE IN PRESS

Table 2

Comparison of regression (model abmeasured) and correlation

(R) coefficients, root mean square (RMSE) and mean bias errors (MBE) of

different hourly diffuse models for south-facing surface

Model a b R RMSE MBE %RMSE %MBE

LJ 0.07 0.93 0.99 0.07 0.03 13.4 6.16

Kr 0.08 0.95 0.99 0.08 0.05 14.89 9.54

Ba 0.06 0.88 0.99 0.08 0.01 13.95 1.00

Ti 0.06 0.88 0.99 0.08 0.01 13.95 1.00

Ha 0.06 0.94 1.00 0.06 0.02 10.37 4.47

SO 0.06 0.94 1.00 0.06 0.02 10.16 4.27

Re 0.06 0.95 1.00 0.06 0.03 10.87 6.07

SU 0.09 1.23 0.99 0.26 0.22 47.18 40.62

P8 0.02 0.94 0.99 0.07 0.01 12.44 2.07

P9 0.04 0.94 0.99 0.06 0.01 11.17 1.44

Kl 0.08 0.96 0.99 0.08 0.06 15.43 10.40

TC 0.1 0.5 0.94 0.3 0.18 54.89 31.86

RMSE and MBE are in units of MJ m2.

Table 3

Same asTable 2but for west-facing surface

Model a b R RMSE MBE %RMSE %MBE

LJ 0.12 0.72 0.89 0.24 0.04 42.49 7.70

Kr 0.12 0.74 0.89 0.24 0.03 42.10 5.20

Ba 0.1 0.69 0.89 0.25 0.08 44.17 13.45

Ti 0.1 0.68 0.89 0.26 0.08 44.57 14.46

Ha 0.08 0.76 0.92 0.21 0.06 37.18 10.04

SO 0.08 0.76 0.92 0.21 0.06 37.25 10.23

Re 0.09 0.76 0.92 0.21 0.05 36.85 8.95

SU 0.13 1.03 0.95 0.24 0.15 40.90 26.26

P8 0.02 0.76 0.95 0.24 0.16 41.80 28.23

P9 0.05 0.81 0.95 0.18 0.07 30.71 11.50

Kl 0.11 0.78 0.91 0.22 0.01 37.97 2.43

TC 0.07 0.5 0.89 0.37 0.22 63.53 38.52



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d180

p 0:0069180:399912 cosG0:070257 sinG

0:006758 cos 2G0:000907 sin 2G0:002697 cos 3G0:00148 sin 3G, A:2

1

d2 1:000110:034221 cosG0:00128 sinG

0:000719 cosG0:00077 sin 2G, A:3

G360

365dn 1, (A.4)

o 18012TST=12, (A.5)

wherednis day of the year, e.g. 1 Jan 1, 20 Feb 51, etc.

and TST is true solar time.

A.2. Computing hourly diffuse and direct components from

hourly global irradiance

A simple, physically based method proposed by Miguel

et al. [15] was used for estimating hourly diffuse and direct

components from hourly global irradiance. For three different

ranges of atmospheric transmissivity kT Gh=Goh, theresulting correlations are given by the following expression:

Dh

Gh

0:9950:081kT if kTo0:21;

0:7242:738kT8:32k2T4:967k

3T if 0:21pkTp0:76;

0:180 if kT40:76:

8>:

(A.6)

ThenBh can be calculated as follows:

Bh GhDh. (A.7)

ARTICLE IN PRESS

Fig. 1. Percent frequency of the deviations of the daily global radiation on south-facing tilted surface, calculated with different models to the experimental

data.

A.M. Noorian et al. / Renewable Energy 33 (2008) 140614121410


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A.3. Statistical methods of comparison

The statistical error tests are as follows [26]:

RMSE ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiSC

i M

i2

ns

, (A.8)

MBE

PCi Mi

n , (A.9)

where Ci and Mi are the ith calculated and measured

values on a tilted surface, respectively, and n is the total

number of observations for a specific period of time.

Relative RMSE (%RMSE) and MBE (%MBE), which

are a dimensionless measure of RMSE and MBE, can be

defined as follows.

%RMSERMSE

M100, (A.10)

%MBE MBE

M100, (A.11)

where M is the mean of measured values on the tilted

surface.

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Fig. 2. Same asFig. 1but for west-facing surface.



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