Measurement and modeling of solar irradiance components on horizontal and tilted planes

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Solar Energy 84 (2010) 2068–2084 www.elsevier.com/locate/solener

Measurement and modeling of solar irradiance components on horizontal and tilted planes Andrea Padovan, Davide Del Col ⇑ Dipartimento di Fisica Tecnica, Universita` degli Studi di Padova, Via Venezia 1, 35131 Padova, Italy Received 23 March 2009; received in revised form 20 September 2010; accepted 22 September 2010 Available online 20 October 2010 Communicated by: Associate Editor Frank Vignola

Abstract In this work new measurements of global and diffuse solar irradiance on the horizontal plane and global irradiance on planes tilted at 20° and 30° oriented due South and at 45° and 65° oriented due East are used to discuss the modeling of solar radiation. Irradiance data are collected in Padova (45.4°N, 11.9°E, 12 m above sea level), Italy. Some diffuse fraction correlations have been selected to model the hourly diffuse radiation on the horizontal plane. The comparison with the present experimental data shows that their prediction accuracy strongly depends on the sky characteristics. The hourly irradiance measurements taken on the tilted planes are compared with the estimations given by one isotropic and three anisotropic transposition models. The use of an anisotropic model, based on a physical description of the diffuse radiation, provides a much better accuracy, especially when measurements of the diffuse irradiance on the horizontal plane are not available and thus transposition models have to be applied in combination with a diffuse fraction correlation. This is particularly significant for the planes oriented away from South. Ó 2010 Elsevier Ltd. All rights reserved. Keywords: Measurement of solar irradiance; Diffuse radiation; Tilted planes

1. Introduction The design, simulation and performance evaluation of solar energy systems require to assess the effective solar radiation resource on sloping surfaces. The estimation of solar irradiance on tilted planes can be performed by applying the transposition models, which convert the solar irradiance on the horizontal plane to that on the tilted plane. The classical approach of these models is to treat separately the solar radiation components; hence their use requires knowledge of the direct and the diffuse horizontal irradiances. The complete characterization of the solar radiation on the horizontal plane can be obtained by measuring ⇑ Corresponding author. Tel.: +39 049 827 6891; fax: +39 049 827 6896.

E-mail address: [email protected] (D. Del Col). 0038-092X/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.solener.2010.09.009

the global irradiance and either the beam or the diffuse component; the other term can then be derived from the measured quantities. A common and inexpensive system is to measure the diffuse irradiance with a shadow band that occults the sensor of the pyranometer from the direct radiation. This procedure requires the introduction of a correction factor for experimental readings, because the shadow band screens a part of diffuse radiation, too. When only the global irradiance on the horizontal plane is measured, a common procedure is the estimation of the diffuse component by means of diffuse fraction correlations. They are simple to use, but each correlation is developed from the measurements taken at some specific stations; therefore their accuracy must be carefully evaluated if they are applied for locations different from the ones included in the databases used for their development.

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2069

Nomenclature a AM AI b B Bn CD D e Err f f0 F F1 F2 Fij G I0,h I0,n k kt MAE max MBE ME Np r

coefficient in Perez et al. (1990a) model air mass anisotropy index in HDKR (Reindl et al., 1990b) model coefficient in Perez et al. (1990) model direct beam solar irradiance (W/m2) normal incident beam irradiance = Bh/cos(h) (W/m2) Drummond (1956) correction diffuse solar irradiance (W/m2) error (%) or (W/m2) difference between calculated and measured irradiance (W/m2) fraction of the hemispherical irradiance blocked by the shadow band, Drummond (1956) formula modulating factor in HDKR (Reindl et al., 1990b) model modulating factor in Klucher (1979) model circumsolar brightening coefficient in Perez et al. (1990a) model horizon brightening coefficient in Perez et al. (1990a) model coefficients in Perez et al. (1990a) model global solar irradiance (W/m2) extraterrestrial horizontal irradiance (W/m2) normal incident extraterrestrial irradiance (W/m2) diffuse fraction = ratio between diffuse and global horizontal irradiance clearness index = ratio between global and extraterrestrial horizontal irradiance mean absolute error P = (1/Np)[ (|Gb,cal  Gb,meas|/Gb,meas)]100 (%) maximum P mean bias error = (Gb,cal  Gb,meas)/Np (W/m2) mean error P = (1/Np){ [(Gb,cal  Gb,meas)/Gb,meas]}100 (%) number of experimental points radius of the shadow band (m)

Modeling of diffuse radiation on horizontal and tilted planes is a critical aspect, because of the anisotropic behaviour of diffuse radiation, originated from two main effects of anisotropy in the atmosphere: the circumsolar brightening, due to forward scattering of solar radiation by aerosols and concentrated in the part of the sky around the sun, and the horizon brightening concentrated near the horizon and more pronounced in clear atmospheres (Perez et al., 1986; Duffie and Beckman, 2006). Several experimental studies, as for instance Olmo et al. (1999), Vartiainen (2000), Notton et al. (2006a,b), Evseev

R transposition factor RMSE root Pmean square error = [ (Gb,cal  Gb,meas)2/Np]1/2 (W/m2) u standard uncertainty (%) or (W/m2) U expanded uncertainty with 95% level of confidence (%) or (W/m2) w width of the shadow band (m) Z zenith angle (rad) ([°] in Table 5) Greek symbols a solar elevation angle (rad) b tilt angle (rad) d sun declination angle (rad) D sky brightness parameter = (Dh AM)/I0,n e sky clearness parameter = (Dh+Bn)/Dh e0 sky clearness parameter as defined by Perez et al. (1990a), Eq. (6) / latitude (rad) h incidence angle of solar radiation on the horizontal plane (rad) or (°) hb incidence angle of solar radiation on the tilted plane (rad) or (°) q reflectance x hour angle (rad) Subscripts A type A uncertainty b direct beam B type B uncertainty cal calculated d diffuse h horizontal i ith term j jth term meas measured n normal direction r reflected s sunrise or sunset b tilted plane 0 extraterrestrial

and Kudish (2009), have investigated the prediction of solar irradiance on tilted planes. But, these contributions do not focus on the experimental accuracy of data used to evaluate solar radiation models. Myers (2005), Gueymard and Myers (2009) have reported as the measurement of solar irradiance can be affected by a large uncertainty under some sky conditions; both these works highlight the importance of discussing the experimental uncertainty of irradiance data. Following this approach, Gueymard (2009) has investigated the prediction accuracy when using different procedures to obtain the

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2. Solar irradiance measurement 2.1. Instrumentation and data The instrumentation for the measurement of solar irradiance has been installed on the roof of the Dipartimento di Fisica Tecnica at the University of Padova (Padovan and Del Col, 2008). The installation site is located inside the university campus, close to the centre of the town. The annual solar radiation energy received on the horizontal plane is about 1300 kWh/m2. Six datasets have been collected from September 5, 2007 to August 27, 2009: global and diffuse irradiance on the horizontal plane, global irradiance on 20° and 30° tilted planes oriented due South and global irradiance on Eastoriented surfaces, tilted at 45° and 65°. The instrumentation consists of two Kipp&Zonen CM11 pyranometers, classified as secondary standard, and two Delta Ohm LP PYRA 02 pyranometers, classified as first class. The measurements of the global and the diffuse irradiance on the horizontal plane have been performed with the LP PYRA 02 pyranometers. The diffuse irradiance is measured placing the sensor under a black coated shadow band, which needs a periodical manual adjustment as the declination of the sun changes. The two CM11 pyranometers, instead, have been used to measure the global irradiance on the South-oriented and Eastoriented tilted planes. The right position of the shadow band and the status of the glass dome of pyranometers is periodically checked to

prevent dust, dew, water droplets and other sources of inaccuracy. The irradiance measurements are acquired with a time step of 10 s: the maximum, minimum and mean values of irradiance and the standard deviation of the measurements are recorded every 5 min. The hourly radiation data are obtained from a numerical integration of the 5-min mean values. In the reduction of the experimental data a correction has been applied to refer the measurements to the true solar time. Moreover, a proper correction is applied to the hourly diffuse irradiance values to account for the reduction of sky diffuse radiation reaching the radiometer, because of the obstruction of the band. By measuring the global and the diffuse irradiances on the horizontal plane, the following relationship is used to obtain the direct irradiance on the horizontal plane: Gh ¼ Bh þ Dh

ð1Þ

where Gh, Bh and Dh are the global, beam and diffuse irradiances on the horizontal plane. In Fig. 1 the typical daily curves of solar radiation are plotted as a function of the true solar time for a clear sky day; 5-min data are reported in the graph. The standard deviation of the measurements is an index of the sky condition: clear sky and complete overcast sky days are characterized by a uniform and low value of standard deviation. On the contrary, a scattering in the values of standard deviation means the fast alternation of sun and clouds. Fig. 2 shows the standard deviation of the global irradiance measurements for a clear sky day (September 8, 2007)

900 Global horizontal Diffuse horizontal

800 700 600

2

horizontal radiation data to introduce as input in the transposition models. In this work new experimental data of hourly global and diffuse solar irradiance, taken in Padova (45.4°N, 11.9°E, 12 m above the sea level), north-east of Italy, are used to investigate the modeling of solar radiation on horizontal and inclined planes. The accuracy of measurements is discussed to get a more comprehensive understanding of the performance of models. This is a relevant aspect, especially considering the recent findings on the uncertainty of solar radiation data. The accuracy of the diffuse fraction correlations by Orgill and Hollands (1977), Erbs et al. (1982), Reindl et al. (1990a) and CLIMED 2 (De Miguel et al., 2001) is tested against the new diffuse irradiance measurements on the horizontal plane. The horizontal irradiance components are then used as input in the transposition models to predict the irradiance on South-oriented planes tilted at 20° and 30° and on East-oriented planes inclined of 45° and 65°. The isotropic Liu and Jordan (1963) model and the anisotropic models by Klucher (1979), Perez et al. (1990) and HDKR (Reindl et al., 1990b) are assessed against the new tilted planes measurements. The accuracy of predictions is discussed when the horizontal irradiance components are measured as well as estimations of the horizontal diffuse irradiance have to be used.

Irradiance [ W/m ]

2070

500 400 300 200 100 0 6.00

8.00

10.00

12.00

14.00

16.00

18.00

True solar time Fig. 1. Daily curves of global and diffuse horizontal irradiance for a clear sky day in Padova, Italy (September 8, 2007).

A. Padovan, D. Del Col / Solar Energy 84 (2010) 2068–2084

300 September 8, 2007 September 9, 2007

2

Standard Deviation [ W/m ]

250

200

150

100

50

0 6.00

8.00

10.00

12.00

14.00

16.00

18.00

True solar time Fig. 2. Standard deviation of the global irradiance measurements.

Table 1 Filtering criteria of experimental irradiance data. Radiometric parameter

Filter

Global radiation Beam radiation

G P 50 W/m2 h 6 80° hb 6 80° 0 < Dh/Gh < 1.1 0 < Dh < 0.8I0,h 0 < Gh < 1.2I0,h

Diffuse radiation Diffuse radiation Global radiation

2071

of data, characterized by an incidence angle greater than 90°, is included in the database, even with solar irradiance lower than 50 W/m2. When the incidence angle exceeds 90° the sun is behind the surface and the direct radiation does not reach the tilted plane. With regard to the uncertainty associated with large incidence angles, the manufacturer of the CM11 pyranometer provides the relative error of the instrument as a function of the zenith angle (Kipp&Zonen, 2000): the maximum error reported is around 6% at 80° zenith angle, but no information is given for higher values of zenith angle. Gueymard and Myers (2009) observed that a significant error occurs in the measurement of horizontal global irradiance for values of air mass greater than 6, corresponding to zenith angles higher than 80°. Reindl et al. (1990b) also omitted in their study hourly data with a zenith angle higher than 80° in order to eliminate the source of uncertainty associated with large incidence angles. The other checks reported in Table 1 are addressed to discover possible data showing a physical violation. It should be pointed out that a limited number of data having a diffuse fraction greater than one is accepted because of the experimental uncertainty of the measurements. Data points referred to measurements taken during rainy conditions are rejected or, if used, treated separately. As a result of the filters applied, the hourly data used in this work and their distribution in the various seasons are specified in Table 2. 2.2. Experimental uncertainty in the measurement with pyranometers

Data is not processed when (kt < 0.20 and k < 0.90) or (kt > 0.60 and k > 0.80) (Reindl et al. (1990a).

and a partially cloudy day (September 9, 2007) as a function of the true solar time: in the second case the data show high variability of irradiance. The standard deviation refers to the 30 readings acquired during each 5-min time step. Some filters have been applied to hourly data to check the experimental accuracy of the database (Table 1). The restrictions regarding the minimum global irradiance and the maximum incidence angle aim at removing data affected by a large experimental uncertainty. In the case of the East-oriented tilted planes, a limited number

In all black thermopile pyranometers the black painted disk increases its temperature when it absorbs solar radiation. A thermopile measures the temperature difference between the black surface near the centre of the disk and a reference surface that does not receive solar radiation. The pyranometer sensitivity (responsivity) is the ratio of the output signal voltage to the irradiance. A constant sensitivity value is provided by the calibration certificate of each pyranometer. But, during applications, environmental conditions can be quite different compared to the calibration test conditions and the pyranometer responsivity can vary; therefore, all possible sources of error involved in the measurement must be evaluated.

Table 2 Number of hourly measurements taken in Padova (45.4°N, 11.9°E). Winter

Spring

Summer

Autumn

Total

Horizontal

Global radiation Diffuse radiation

483 483

432 432

567 567

463 463

1945 1945

South-oriented

Global radiation, 20° tilt Global radiation, 30° tilt

– 483

– 407

53 239

– 463

53 1592

East-oriented

Global radiation, 45° tilt Global radiation, 65° tilt

224 –

– –

– 227

– –

224 227

2072

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Table 3 reports a list of the sources of error for the all black pyranometers installed here, as declared by the manufacturers: the values reported represent the largest error due to each component and they depend on the classification of the pyranometer. The directional error accounts for both solar elevation and azimuth effects, which are usually known as Lambert cosine response and azimuth response, respectively. It depends on the imperfections of the glass dome and on the angular reflection properties of the black paint. The error value, reported in Table 3, derives by assuming that the normal incidence responsivity is valid for all the directions when measuring a beam irradiance whose normal value is 1000 W/m2. The directional error is significant especially when the direct beam radiation, which is the largest contributor to global radiation under clear skies, is present (Gueymard and Myers, 2009). Thermopile response is sensitive to temperature: the maximum deviation of the pyranometer sensitivity for a change in the ambient temperature within an interval of 50 K is specified in Table 3. The non-linearity error accounts for the variation of the sensitivity with irradiance. The spectral response depends on the spectral absorptance of the black coating and on the spectral transmittance of the glass domes. The tilt error considers the deviation of the responsivity due to a change in the tilt angle from 0° to 90° at 1000 W/m2 irradiance. Zero offset occurs when the sensor does not absorb radiation with wavelengths in the spectral range of the instrument and there is still a signal. Two types of zero offset can occur: type A zero offset, which is related to the thermal energy flux exchanged among the absorbing sensor, the glass dome and the sky and type B zero offset, which depends on the body temperature of the pyranometer. In the CM11 instrument a second non-illuminated compensation element minimizes the type B zero offset. However, this possible error is taken into account in the present uncertainty analysis of global and diffuse irradiance data. The manufacturer, in agreement with the WMO (World Meteorological Organization, 2006), always specifies an achievable experimental uncertainty with 95% level of confidence that shall be expected. In the case of hourly measurements it is equal to 3% and 8% for a secondary standard and a first class pyranometer, respectively. But, Table 3 Accuracy of all black pyranometers installed.

Directional response Temperature response Non-linearity Spectral response Tilt response Zero offset A Zero offset B

CM11

LPPYRA02

<±10 W/m2 <±1% <±0.5% <±2% <±0.2% <7 W/m2 <±2 W/m2

<±18 W/m2 <±4% <±1% <±5% <±2% <15 W/m2 <±4 W/m2

to have a more comprehensive understanding of the accuracy of data, the experimental uncertainty should be evaluated for each irradiance measurement. The present analysis of the experimental uncertainty has been lead in agreement with the guide lines provided by ISO (1995). This approach was also adopted in the papers by Mathioulakis et al. (1999), Kratzenberg et al. (2006), Padovan and Del Col (2008). The experimental uncertainty of a measurement is based on the combination of type A and type B evaluations of uncertainty (ISO, 1995). Type A uncertainty comes from the statistical analysis of the repeated measurements under steady state conditions. Type B uncertainty, instead, is related to the instrument and data logger used and it is provided by the manufacturer or specific calibration procedures. With regard to type B uncertainty, when the manufacturer specifies the maximum error given by the instrument, e, one can assume that it is equally probable for the value of the measured quantity to lie anywhere within the interval ±e. Hence, according to ISO (1995), the distribution of possible values can be considered rectangular and the standard uncertainty, uB, is given as: e uB ¼ pffiffiffi 3

ð2Þ

The uncertainty is then calculated by means of the law of uncertainties combination: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ð3Þ u ¼ ðuA Þ þ ðuB Þ where uA and uB are the type A and type B uncertainties, respectively. In the case of pyranometers the manufacturer provides separately the contributions due to the various sources of error (Table 3) and the total type B uncertainty can be obtained using the law of uncertainties combination. A coverage factor equal to 2 is here assumed to expand the uncertainty to a confidence level of 95%. The contribution of the type A uncertainty can be evaluated only when no variation of solar irradiance occurs during the recording time interval. In this case, such term is negligible compared to the error of the instrument. For instance, during a clear sky day, the highest variation of irradiance recorded by the LP PYRA 02 pyranometer within each 5-min time step was equal to 19 W/m2, while the expanded type B uncertainty due to the instrument was ±37 W/m2. Therefore, in those conditions, the uncertainty contribution due to scattering during data recording can be considered negligible compared to the error of the radiometer and the mere type B uncertainty can describe the accuracy of the irradiance measurement. The same results have been obtained for the diffuse radiation, which has shown a lower variability of irradiance. An inter-calibration check of pyranometers is periodically performed by placing all four radiometers on the horizontal plane. Fig. 3 reports for a clear sky day the

A. Padovan, D. Del Col / Solar Energy 84 (2010) 2068–2084

tribution. The portion of the hemispherical radiation screened by the band is calculated by the following equation:

Uncertainty LP PYRA02 Uncertainty CM11 Difference between readings

30

20

f ¼ Percentage deviation [%]

2073

10

2w ðcos3 ðdÞÞðxs sinð/Þ sinðdÞ þ cosð/Þ cosðdÞ pr  sinðxs ÞÞ

ð4Þ

where r is the radius of the shadow band, w is the width of the band, xs is the sunrise (sunset) hour angle, / is the latitude and d is the declination. The isotropic correction by Drummond (1956), CD, is then obtained as:

0

-10

CD ¼

1 1f

ð5Þ

-20

-30 7.30

9.30

11.30

13.30

15.30

17.30

True solar time Fig. 3. Uncertainty band lines for a secondary standard (CM11) and a first class (LP PYRA 02) pyranometer and percentage deviation between the readings taken with the two instruments.

percentage difference between the readings of global irradiance taken with a CM11 secondary standard pyranometer and a LP PYRA 02 first class pyranometer as a function of the true solar time; the CM11 readings are taken as reference. The lines represent the expanded uncertainty (95% confidence level) and they are obtained from the combination of the sources of error listed in Table 3. The tilt response is not considered in this case because of the horizontal installation of the pyranometers. The higher values of percentage difference between the readings of the two sensors occur in the first morning hours and in the evening, when the solar irradiance is quite low and the zenith angle is high. For most of the day the expanded uncertainty is around 3.5% for the CM11 pyranometer and around 8.5% for the first class instrument. This is in agreement with the hourly expanded uncertainty declared by the manufacturers, but, in the first and last hours of the day, the uncertainty is considerably higher. As reported in the graph of Fig. 3, the inter-calibration check has shown that the deviation between the readings of different radiometers was always within the experimental uncertainty band lines, confirming the agreement between the pyranometers used to collect the present database. 2.3. Diffuse irradiance measurement: shadow band correction models Two methods have been taken into account to correct the experimental diffuse irradiance readings for the amount of sky hemisphere eclipsed by the band. The first is the Drummond (1956) correction, which can be applied everywhere. It is a geometrical calculation based on the assumption that the diffuse radiation has an isotropic dis-

Drummond (1964) noted that an additional correction was necessary, equal to 7% and 3% for clear and overcast sky conditions, respectively. This was due to the anisotropic distribution shown by the diffuse radiation. The second correction method, considered in the present study, is the model by LeBaron et al. (1990). It is a semiphysical method accounting for both the band geometry and the sky condition, that considers how the diffuse radiation is distributed. The effect of the shadow band reflection is considered minimal because of the black paint of the band. Following the scheme of Perez et al. (1987), four parameters are used to define the sky conditions: the isotropic geometrical correction (CD), the zenith angle (Z), and the parameters e and D, meaning the sky clearness and the sky brightness, respectively. The zenith angle is an index of the position of the sun in the sky. All the parameters can be calculated by the measurement of the global and the uncorrected diffuse irradiance. In this way a variety of states can be identified: 256 categories are created as a function of the geometric and sky characteristics. The model was developed using hourly data collected in Albany (New York) and Bluefield (West Virginia). However, the model is not dependent on the geographic location. LeBaron et al. (1990) found that their model provided a significantly better correction than the Drummond (1956) formula, particularly under partially cloudy sky, which represents the most anisotropic distribution. Fig. 4 shows a comparison between the corrections provided by LeBaron et al. (1990) and Drummond (1956) for the dates considered in Fig. 2: average hourly values of irradiance are reported. As discussed before, the considered days represent clear and partially cloudy conditions. The LeBaron et al. (1990) model provides a higher correction value under partially cloudy sky and the irradiance results increased by about 8% compared to the value corrected using the Drummond (1956) formula. Later, other experimental investigations, as the work by Lopez et al. (2004), have confirmed the higher overall accuracy of the correction provided by LeBaron et al. (1990) compared to the one by Drummond (1956). Therefore, the LeBaron et al. (1990) model will be used in the next sections to correct the hourly diffuse irradiance

2074

A. Padovan, D. Del Col / Solar Energy 84 (2010) 2068–2084

300 LeBaron et al. Drummond

2

Diffuse Irradiance [ W/m ]

250

200

150

100

50

Sept 9, 2007

Sept 8, 2007 0 0.00

12.00 True solar time

0.00

12.00 True solar time

0.00

Fig. 4. Comparison between the LeBaron et al. (1990) and Drummond (1956) correction methods.

measurements. The effect of the Drummond (1956) correction on the prediction of tilted irradiance will be evaluated, too. 3. Modeling of diffuse irradiance on the horizontal plane When diffuse irradiance is not directly measured, it can be estimated by means of diffuse fraction correlations. These correlations predict the diffuse fraction, k, that is the ratio of diffuse to global horizontal irradiance, as a function of the clearness index, kt, defined as the ratio of global to extraterrestrial horizontal irradiance. Orgill and Hollands (1977) noted the need for a specific correlation to predict the hourly diffuse radiation. Erbs et al. (1982) developed a correlation to estimate the hourly diffuse fraction using diffuse radiation data recorded at four US stations and obtained from the measurements taken with a pyrheliometer and a pyranometer. Data from an Australian site, measured with a shaded pyranometer, were used for the validation tests. Reindl et al. (1990a) studied the influence of several variables in addition to the clearness index. In fact, if for example the clearness index is equal to 0.5, the diffuse fraction can vary between 0.2 and nearby 1; therefore, further variables must be incorporated in the predicting methods. They considered data from four European and two US locations: in the first ones Kipp solarimeters (shade ring were used to monitor diffuse radiation) were used, while in the case of the US stations Eppley pyranometers and pyrheliometers were used. On the whole they developed three correlations. The first is based on the clearness index, the solar altitude, the ambient air temperature and the relative humidity. The second is based on the clearness index and the solar altitude. The third correlation is based on the clearness index

only. In the present study the second correlation, named in this work as “Reindl 1”, and the third correlation, named as “Reindl 2”, are considered. De Miguel et al. (2001) presented a hourly diffuse fraction correlation, named CLIMED 2, based on data of Mediterranean stations. The accuracy of the Orgill and Hollands (1977), Erbs et al. (1982), “Reindl 1” and “Reindl 2” (Reindl et al., 1990a) and CLIMED 2 (De Miguel et al., 2001) correlations have been evaluated using the present data of diffuse irradiance. The LeBaron et al. (1990) model has been applied to correct the experimental measurements. Table 4 reports the root mean square error, RMSE, the mean bias error, MBE, the mean absolute error, MAE, and the mean error, ME, obtained in the prediction of the hourly diffuse irradiance on the horizontal plane. The average value of the measured diffuse irradiance is also reported in the caption of the table; the percentage values of RMSE and MBE can then be deduced. Data characterized by rainy conditions are not considered in the analysis. All the correlations provide an overestimation of the measurements, as it can be observed from the values of MBE and ME; this is probably related to their similar structure. However, the Erbs et al. (1982) model gives the most accurate results. In the case of the Orgill and Hollands (1977), Erbs et al. (1982) and CLIMED 2 models, the values of RMSE are close to the ones obtained by Notton et al. (2004) for a European site located in the North-Mediterranean area; besides, they did not observe a significantly improved accuracy with models that also use the solar altitude as predicting variable. The percentage values of RMSE and MBE obtained here for CLIMED 2 are equal to 36% and +11.5%, respectively; De Miguel et al. (2001) reported values of RMSE and MBE equal to 58% and +11.8%, respectively, for the site of Athens, Greece, and 38% and +6.2%, respectively, for the site of Seville, Spain. Fig. 5 reports the model error, Err, defined as the difference between calculated and measured diffuse irradiance, against the measured irradiance. The Erbs et al. (1982) correlation has been taken as representative model to obtain the predicted values of irradiance used in the graph of Fig. 5. The band lines of the expanded uncertainty with

Table 4 Prediction of diffuse irradiance on the horizontal plane. The average measured diffuse irradiance is 139 W/m2.

Orgill and Hollands (1977) Erbs et al. (1982) “Reindl 1”, Reindl et al. (1990a) “Reindl 2”, Reindl et al. (1990a) CLIMED 2, De Miguel et al. (2001)

RMSE (W/m2)

MBE (W/m2)

MAE (%)

ME (%)

53 48 52

+23 +13 +20

46.1 38.5 41.3

+36.5 +28.5 +32.1

51

+16

42.7

+32.5

50

+16

41.6

+31.7

A. Padovan, D. Del Col / Solar Energy 84 (2010) 2068–2084 Err < - U

300

2075 Err< - U

1.1

Err > U

Err > U

1

- U < Err < U Uncertainty band

0.9

Rain Erbs et al.

0.8 100 Diffuse Fraction (k)

2

Dh,cal - Dh,meas [ W/m ]

200

- U < Err < U

0

-100

0.7 0.6 0.5 \

0.4 0.3

-200

0.2 0.1

-300 0

100

200

300

400

0

500

0

2

Dh,meas [ W/m ] Fig. 5. Diffuse irradiance: model error (Err) and experimental uncertainty band lines against measured irradiance. Calculated values are obtained from the Erbs et al. (1982) model.

95% level of confidence, ±U, are also drawn. This band accounts only for the uncertainty component due to the error of the instrument (type B); besides, the component due to the fluctuations of irradiance during data recording provides a negligible contribution under stationary sky conditions (Section 2.2). The following regions have been distinguished: data points with values of Err outside the experimental uncertainty band (Err > +U and Err < U) and data points with a model error within the experimental uncertainty band lines (U < Err < + U). It can be observed that the model provides an evident overestimation, particularly when the diffuse irradiance is low (very high density of square points below about 150 W/m2), while it underestimates more data points as the measured irradiance increases. In Fig. 6 the measured data, expressed as diffuse fraction, are plotted as a function of the experimental clearness index: the same symbols as used in Fig. 5 are kept to identify the data points. The Erbs et al. (1982) correlation is drawn for comparison. Measurements that refer to rainy conditions are also reported in order to show as they do not have a significant influence on the model evaluation. Therefore, they are not considered because of their high experimental uncertainty. At low values of clearness index the model performs well, but, as clearness index increases, the scattering of data points increases too. For a more comprehensive understanding, the influence of the parameters sky clearness, sky brightness and zenith angle has been investigated, as suggested by several studies (Perez et al., 1987, 1990a; LeBaron et al., 1990). The diagram of Fig. 7 shows the distribution of experimental data of hourly diffuse irradiance as a function of sky clearness and sky brightness; in the this graph sky clearness is used

0.1

0.2

0.3 0.4 0.5 0.6 0.7 Clearness Index (k t)

0.8

0.9

1

Fig. 6. Hourly diffuse fraction vs. clearness index: experimental data and Erbs et al. (1982) correlation. Data are subdivided by comparing the model error to the uncertainty band (see Fig. 5).

as defined by Perez et al. (1990a) to eliminate the dependence on the zenith angle and it is here named as e0 . e0 ¼

ðDh þ Bn Þ=Dh þ Z 3 1:041 1 þ Z 3 1:041

ð6Þ

where Dh is the horizontal diffuse irradiance, Bn is the normal incidence beam irradiance, and Z is the zenith angle. The sky brightness, D, is calculated as: D¼

Dh AM I 0;n

ð7Þ

where AM is the air mass, obtained as described in Duffie and Beckman (2006), and I0,n is the normal incidence extraterrestrial irradiance. The same symbols as already used in Figs. 5 and 6 are kept to distinguish the data points as a function of the accuracy exhibited by the Erbs et al. (1982) correlation. From the analysis of the graph in Fig. 7, one can see that data points display a systematic distribution. All the data characterized by values of e0 higher than 7 are overestimated by the Erbs et al. (1982) correlation: clear sky conditions fall within this category. This result was expected since clear sky data are characterized by high clearness index and low values of diffuse irradiance. As sky clearness decreases, the sky turbidity increases and the prediction capability of the correlation improves, although overestimation of experimental data is observed again, particularly in the case of the lower values of sky brightness. When e0 becomes lower than 5, Erbs et al. (1982) underestimates the data corresponding to the higher sky brightness. Finally, overcast sky data having values of e0 between 1 and 1.3 are predicted within the experimental

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A. Padovan, D. Del Col / Solar Energy 84 (2010) 2068–2084 Global 30° tilted - South

1000

Global horizontal Diffuse horizontal Global 45° tilted - East

2

Irradiance [ W/m ]

800

600

400

200

0 6.00

Fig. 7. Experimental sky brightness (D) vs. experimental sky clearness (e0 ). Different markers are used for data points depending on the model error compared to the experimental uncertainty (U). The Erbs et al. (1982) correlation is used for prediction.

uncertainty band, with the exception of measured data characterized by the higher values of D. Experimental data have been then grouped as a function of the zenith angle and the values of RMSE, MBE, MAE and ME, obtained for each group from the comparison with Erbs et al. (1982), are reported in Table 5. A dependence of the bias error on zenith angle has been found, since a larger overestimation occurs at high values of zenith angle; this may be attributed to the fact that the Erbs et al. (1982) correlation does not incorporate the zenith angle as input parameter. Similar findings were reported by Perez et al. (1990b) when they used the Erbs et al. (1982) correlation to predict the diffuse irradiance; they obtained the beam component as the difference between the measured global irradiance and the calculated diffuse irradiance and they compared the estimated beam irradiance with the measured value, finding a different behaviour in the bias error between high and low zenith angles. 4. Modeling of global irradiance on tilted planes

8.00

10.00 12.00 14.00 True solar time

16.00

18.00

Fig. 8. Global irradiance measured on the horizontal plane, on the South and East-oriented tilted planes and diffuse irradiance measured on the horizontal plane (February 18, 2009).

East-oriented tilted plane can be about two times the irradiance on the tilted surface oriented due South. This is interesting for the sun tracking applications. The solar radiation incident on the inclined plane is modeled as being composed of three parts: the direct beam radiation, the sky diffuse radiation and the radiation reflected from the ground. The irradiance on the tilted plane is calculated as a function of the irradiance components on the horizontal plane (Duffie and Beckman, 2006): Gb ¼ Bh Rb þ Dh Rd þ Gh qRr

ð8Þ

where Rb is the ratio between the beam irradiance on the tilted plane and the beam irradiance on the horizontal plane, Rd is the ratio of the diffuse irradiance on the tilted plane to the diffuse irradiance on the horizontal plane and the product qRr accounts for the ground reflected radiation incident on the tilted surface; the coefficient q is the reflectance. The term Rb is calculated as: Rb ¼

Bn cos hb Bn cos h

ð9Þ

4.1. Tilted plane models Solar radiation on a surface with optimal orientation and inclination can be considerably higher than the one on the horizontal plane. Fig. 8 shows, for a winter day, the solar irradiance curves measured on the horizontal plane, on the 30° tilted South-facing plane and on the plane oriented to East and inclined of 45°. During the whole day, the irradiance measured on the South-oriented tilted surface is higher than the one on the horizontal plane, while in the first part of the morning the irradiance on the

where Bn is the normal incident beam irradiance, while h and hb are the incidence angles of solar radiation on the horizontal and tilted planes, respectively. The term cos(hb) is obtained as (Duffie and Beckman, 2006): cosðhb Þ ¼ sinðdÞ sinð/Þ cosðbÞ  sinðdÞ cosð/Þ sinðbÞ  cosðcÞ þ cosðdÞ cosð/Þ cosðbÞ cosðxÞ þ cosðdÞ sinð/Þ sinðbÞ cosðcÞ cosðxÞ þ cosðdÞ sinðbÞ sinðcÞ sinðxÞ

ð10Þ

A. Padovan, D. Del Col / Solar Energy 84 (2010) 2068–2084 Table 5 Accuracy of the Erbs et al. (1982) correlation as a function of the zenith angle (Z).

20° < Z < 40° 40° < Z < 60° 60° < Z < 80°

Average Irradiance (W/m2)

RMSE (W/m2)

MBE (W/m2)

MAE (%)

ME (%)

211 176 104

67 52 39

+1 +9 +19

31.6 32.4 43.7

+14.4 +20.6 +36.3

where d is the declination of the sun, / is the latitude, b is the tilt angle, c is the surface azimuth angle and x is the hour angle. The expression of the term cos(h) can be obtained by Eq. (10) when the tilt angle, b, is set equal to zero. The incidence angle is calculated as mean value in the hourly time interval. In this study the ground reflected radiation is considered to be isotropic and so the term Rr is calculated as (Duffie and Beckman, 2006): Rr ¼

ð1  cos bÞ 2

ð11Þ

A value of q equal to 0.2 is here assumed because of the lack of a specific measurement. Kambezidis et al. (1994) showed that seasonally varying or anisotropic albedo models did not improve the prediction of the solar irradiation on South-oriented surfaces with a low tilt angle (equal to 50° in their measurements) compared to the use of a constant ground albedo (q) set equal to 0.2. In the case of the present measurements on the South-oriented 30° tilted plane, the ground reflected irradiance, calculated with Eq. (11) and a value of reflectance equal to 0.2, does not exceed 2% of the global irradiance measured on that inclined surface. The modeling of the sky diffuse radiation is the most critical part. Table 6 reports the equations to calculate the transposition factor of diffuse radiation, Rd, with the isotropic model by Liu and Jordan (1963) and the anisotropic models by Klucher (1979), Perez et al. (1990a) and HDKR (Reindl et al., 1990b). The Liu and Jordan (1963) model considers all the diffuse radiation uniformly distributed over the sky dome. The three anisotropic models, instead, describe the diffuse radiation as being composed of three parts: the isotropic portion received uniformly from the entire sky dome and the anisotropic contributions given by the circumsolar and the horizon radiations. Different approaches are followed to model the effect of the sky conditions on the distribution of the diffuse radiation in the three components. The Klucher (1979) model uses a modulating function, F, to describe the variation from clear to overcast skies. In the Perez et al. (1990a) model, the anisotropic circumsolar and horizon radiations are included in the terms F1 and F2, respectively; both the terms are correlated to the sky conditions, described by the parameters D, Z and e0 , by means of empirical correlations. The values of the

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Table 6 Transposition models: equations for the calculation of tilted plane diffuse irradiance. Model

Transposition factor: Rd ¼ Db =Dh

Liu and Jordan (1963) Klucher (1979)

bÞ Rd ¼ ð1þcos 2 ð1þcos bÞ Rd ¼ ½1 þ F sen3 ðb=2Þ 2 ½1 þ F cos2 ðhb Þsen3 ðp=2  aÞ  2 h F ¼1 D Gh

Perez et al. (1990a)

bÞ Rd ¼ ð1þcos ð1  F 1 Þ þ F 1 ab þ F 2 senðbÞ 2 a ¼ maxð0; cos hb Þ b ¼ maxð0:087; cos ZÞ F 1 ¼ F 11 þ F 12 D þ F 13 Z F 2 ¼ F 21 þ F 22 D þ F 23 Z   b ð1 þ f 0 sin3 ðb=2ÞÞ þ AI Rb Rd ¼ ð1  AI Þ 1þcos 2 qffiffiffiffi AI ¼ IB0nn ; f 0 ¼ GBhh

HDKR, Reindl et al. (1990b)

coefficients Fij depend on the value of sky clearness (e0 ) and they are reported in the original paper by Perez et al. (1990a). The HDKR (Reindl et al., 1990b) model comes from the Hay and Davies (1980) model with the introduction of a horizon brightening term similar to that of Klucher (1979); consequently this model is named as HDKR (Hay, Davies, Klucher, Reindl) in Duffie and Beckman (2006). The modulating factor, f0 , accounts for the effect of the cloudiness, while the coefficient AI is the anisotropy index and it defines the portion of horizontal diffuse radiation to be treated as circumsolar. 4.2. Assessment of predictive methods In this section the performances of the transposition models by Liu and Jordan (1963), Klucher (1979), Perez et al. (1990a) and HDKR (Reindl et al., 1990b) are evaluated against the data taken on the tilted planes oriented South and East. The effect of the uncertainty in the measurement and estimation of the horizontal diffuse irradiance is also discussed. Fig. 9 shows the comparison between the hourly irradiance measured on the South-oriented planes tilted at 20° and 30° and the irradiance calculated with the models by Liu and Jordan (1963) and Perez et al. (1990a). The calculated values have been obtained using both the global and the diffuse irradiances measured on the horizontal plane; the LeBaron et al. (1990) correction has been applied to the diffuse irradiance. The experimental uncertainty band of the measurements on the tilted plane is also plotted: it describes the uncertainty of the measured irradiance on the tilted plane and it does not account for the experimental uncertainty of the irradiance measurements taken on the horizontal plane and used as input in the models. A second band, corresponding to the ±20% deviation between model predictions and data, is also drawn with the purpose of highlighting such agreement or disagreement.

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A. Padovan, D. Del Col / Solar Energy 84 (2010) 2068–2084

1200

1200 SOUTH

SOUTH

Uncertainty band

Uncertainty band

1000

1000

+ 20%

800

2

Gβ ,cal [ W/m ]

2

Gβ ,cal [ W/m ]

+ 20%

800 - 20%

600

400

- 20%

600

400

200

200 Liu and Jordan

0 0

Perez et al.

0 200

400

600

800

1000

1200

0

200

Gβ,meas [ W/m 2 ]

400

600

800

1000

1200

Gβ,meas [ W/m 2 ]

Fig. 9. Calculated vs. measured hourly global irradiance on tilted planes due South (20° and 30° tilt angle). The experimental uncertainty band and the band lines for the ±20% deviation between model and data are also shown. Measurements of global and diffuse horizontal irradiance are used in the models; the diffuse horizontal irradiance is corrected with the LeBaron et al. (1990) method.

For high values of irradiance (clear sky days and thus high values of clearness index) most data are within the experimental uncertainty band. But, when the irradiance is lower than about 600 W/m2, the Liu and Jordan (1963) model exhibits some underestimation. Table 7 reports a summary on the performance of the models for the South-oriented tilted planes when the diffuse irradiance is measured and corrected with the LeBaron et al. (1990) and the Drummond (1956) methods, respectively; on the whole this database includes 53 measurements taken on the plane tilted of 20° and 1592 measurements collected on the 30° inclined plane. All the models provide a fairly accurate prediction of the data. From Table 7 one can calculate the percentage value of MBE for each model; when the LeBaron et al. (1990) correction is applied, the percentage MBE is equal to 2.4% for Liu and Jordan (1963), +2.2% for Klucher (1979), +3.1% for Perez et al. (1990a) and +1.4% for HDKR. Similar findings were reported by Kambezidis et al. (1994), when they applied the transposition models considering the ground reflected radiation as isotropic.

The adoption of the Drummond (1956) formula does not provide significant differences compared to the use of the LeBaron et al. (1990) model. The little improvement observed for the Liu and Jordan (1963) model may be attributed to the underestimation of the horizontal diffuse irradiance provided by the Drummond (1956) method; in fact, this yields an overestimation of the direct radiation, which can result in higher values of calculated irradiance on the tilted plane. In Fig. 10 the global irradiance calculated on the 45° and 65° tilted planes, East-facing, is plotted as a function of the measured irradiance; the calculated values are obtained with the Liu and Jordan (1963) and the Perez et al. (1990a) models. The data characterized by an incidence angle, hb, lower than 80° and higher than 90° are divided in two datasets. When the incidence angle exceeds 90° the sun is behind the surface. The Liu and Jordan (1963) model overestimates by more than 20% the experimental data corresponding to hb > 90°. Moreover, it underestimates the data characterized by hb < 80° when the irradiance is higher than around 600 W/m2. The Perez et al. (1990a)

Table 7 Prediction of global irradiance on South-oriented tilted (20° and 30°) planes, using experimental values of horizontal diffuse irradiance (the correction by LeBaron et al., 1990, or Drummond, 1956, is applied). The average measured global irradiance is 481 W/m2. RMSE (W/m2)

MBE (W/m2)

MAE (%)

ME (%)

LeBaron et al. (1990) correction Liu and Jordan (1963) Klucher (1979) Perez et al. (1990a) HDKR, Reindl et al. (1990b)

26 22 24 22

11 +10 +15 +7

5.3 4.3 4.8 4.6

2.2 +1.9 +3.4 +1.3

Drummond (1956) correction Liu and Jordan (1963) Klucher (1979) Perez et al. (1990a) HDKR, Reindl et al. (1990b)

22 23 25 21

9 +14 +18 +10

4.5 4.3 5.2 4.2

1.5 +3 +4.3 +2.2

A. Padovan, D. Del Col / Solar Energy 84 (2010) 2068–2084 1000

1000 900 800

EAST, EAST, θβ < 80°

900

EAST, EAST, θβ > 90° Uncertainty UncertaintyBand band

800

2

600

Gβ ,cal [ W/m ]

2

EAST, θβ < 80° EAST, EAST, θβ > 90° EAST, Uncertainty Band Uncertainty band

700

700

Gβ ,cal [ W/m ]

2079

+ 20%

500 400

- 20%

300

600 + 20%

500 400

- 20%

300

200

200 Liu and Jordan

100 0

0 100 200 300 400 500 600 700 800 900 1000

Perez et al.

100 0

0 100 200 300 400 500 600 700 800 900 1000

2

Gβ,meas [ W/m 2 ]

Gβ,meas [ W/m ]

Fig. 10. Calculated vs. measured hourly global irradiance on tilted planes due East (45° and 65° tilt angle). The experimental uncertainty band and the band lines for the ±20% deviation between model and data are also shown. Measurements of global and diffuse horizontal irradiance are used in the models; the diffuse irradiance is corrected with the LeBaron et al. (1990) method. Direct beam does not reach the surface when hb > 90°.

model shows a much better agreement with measured data and most data points fall within the experimental uncertainty band at low as well as at high values of irradiance. Table 8 reports the values of RMSE, MBE, MAE and ME obtained from all the data measured on the East-oriented surfaces (45° and 65° tilted). In comparison with the surface oriented due South, the prediction capability of the Klucher (1979) model becomes significantly worse. The Liu and Jordan (1963) model gives a negative value of MBE, equal to 11 W/m2, but a ME value equal to +5%; this is due to the definition of the parameter ME, which evaluates the percentage error obtained for each measurement. The data characterized by incidence angles lower than 80° and greater than 90° are also treated separately. When hb > 90° the tilted planes do not receive the direct radiation and the distribution of the diffuse radiation

can be quite different compared to the case of hb < 80°. The model by Perez et al. (1990a) exhibits the higher accuracy when the incidence angle is greater than 90°; on the contrary, the Klucher (1979) model gives a strong overestimation and this can be attributed to the structure of the model, which tends to overestimate for slopes that do not face the sun, as discussed by Perez et al. (1986). In Fig. 11 the percentage difference between the calculated and the measured value of irradiance on the tilted planes is plotted as a function of the clearness index for all the transposition models considered here. In the case of the East-oriented planes, the data characterized by incidence angles greater than 90° have been omitted in this figure. For values of kt lower than about 0.25 complete overcast conditions occur, as it can be seen from Fig. 6; the Klucher (1979) and the HDKR models approximate

Table 8 Prediction of global irradiance on East-oriented tilted (45° and 65°) planes, using experimental values of horizontal diffuse irradiance (the correction by LeBaron et al., 1990, is applied). The deviations are calculated for all data and for the two datasets depending on the incidence angle. RMSE (W/m2)

MBE (W/m2)

MAE (%)

ME (%)

All data: Average irradiance = 354 W/m Liu and Jordan (1963) Klucher (1979) Perez et al. (1990a) HDKR, Reindl et al. (1990b)

39 30 19 23

11 +11 +9 +1

13.3 15.5 6.8 10.2

+5 +13 +4.6 +2.1

hb < 80°: Average irradiance = 439 W/m2 Liu and Jordan (1963) Klucher (1979) Perez et al. (1990a) HDKR, Reindl et al. (1990b)

40 19 20 22

22 +3 +9 1

7.1 4.8 4.8 5.4

3 +1.9 +3.3 +1

hb > 90°: Average irradiance = 75 W/m2 Liu and Jordan (1963) Klucher (1979) Perez et al. (1990a) HDKR, Reindl et al. (1990b)

36 51 13 26

+27 +41 +7 +6

34 51 13.5 26.3

+32 +50 +8.7 +5.6

2

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A. Padovan, D. Del Col / Solar Energy 84 (2010) 2068–2084

60

60 SOUTH

SOUTH

50

EAST

EAST

40

40

Liu and Jordan

30 20 10 0 -10 -20 -30 -40

(Gβ ,cal - Gβ ,meas) / Gβ ,meas [ % ]

(Gβ ,cal - Gβ,meas) / Gβ ,meas [ % ]

50

-50

Perez et al.

30 20 10 0 -10 -20 -30 -40 -50

-60

-60 0

0.2

0.4

0.6

0.8

1

0

0.2

Clearness Index (kt) 60

0.8

1

60 SOUTH

50

SOUTH

50

EAST

EAST

40

Klucher

30 20 10 0 -10 -20 -30 -40 -50

(Gβ ,cal - Gβ,meas) / Gβ ,meas [ % ]

40

(Gβ ,cal - Gβ ,meas) / G β ,meas [ % ]

0.4 0.6 Clearness Index (k t)

HDKR

30 20 10 0 -10 -20 -30 -40 -50

-60

-60 0

0.2

0.4 0.6 Clearness Index (kt)

0.8

1

0

0.2

0.4 0.6 Clearness Index (k t)

0.8

1

Fig. 11. Percentage of relative difference between calculated and measured global irradiance on South and East-oriented tilted planes as a function of the clearness index. Measurements of global and diffuse horizontal irradiance are used in the models. For the East-oriented surfaces the data characterized by incidence angles higher than 90° are omitted.

to the response of the Liu and Jordan (1963) model, as it was expected. When the clearness index is between 0.25 and 0.6, the Liu and Jordan (1963) model underestimates more than 10% the measured irradiance; the anisotropic models by Klucher (1979) and HDKR do not show any trend with the clearness index, while the Perez et al. (1990a) model exhibits some overestimation. It should be pointed out that this range of clearness index represents mainly partially cloudy skies. Higher values of clearness index are associated with clear sky conditions, instead. During clear sky, the measurement of diffuse irradiance can be affected by a large uncertainty, particularly in winter days, because of the high offset error due to the infrared radiation losses (Gueymard and Myers, 2009). This can be observed in Fig. 5, when considering the values of irradiance below 100 W/m2. By differentiating the transposition equation, Eq. (8), and neglecting the error in the measurement of horizontal global irradiance, the effect of

the uncertainty in the measurement of diffuse horizontal irradiance can be evaluated. For example, in the case of a tracking surface on a winter day, if the term Rb is 2.5 and the term Rd is around 1, an uncertainty on the horizontal diffuse irradiance of 40% can lead to an uncertainty of 3% for an estimated tilted irradiance of 1000 W/m2. The diagram in Fig. 12 compares the hourly irradiance measured on the South-oriented tilted surfaces to the irradiance calculated with the Liu and Jordan (1963) and the Perez et al. (1990a) models. In this case the diffuse irradiance is not obtained from measurements; the Erbs et al. (1982) correlation has been used for the estimation of the diffuse irradiance on the horizontal plane, instead. This correlation resulted the most accurate among the models tested for the diffuse irradiance. Fig. 13 reports the hourly irradiance calculated on the East-oriented planes, tilted at 45° and 65°, against the irradiance measured on those planes. Data characterized by

A. Padovan, D. Del Col / Solar Energy 84 (2010) 2068–2084

1200

1200

SOUTH

SOUTH

Uncertainty band

Uncertainty band

1000

1000 + 20%

+ 20%

Gβ ,cal [ W/m 2 ]

800

2

G β ,cal [ W/m ]

2081

- 20%

600

400

800 - 20%

600

400

200

200 Liu and Jordan - Erbs et al.

Perez et al. - Erbs et al.

0

0 0

200

400

600

800

1000

0

1200

Gβ,meas [ W/m2 ]

200

400

600

800

1000

1200

2 Gβ,meas [ W/m ]

Fig. 12. Calculated vs. measured hourly irradiance on tilted planes due South (20° and 30° tilt angle). The experimental uncertainty band and the band lines for the ±20% deviation between model and data are also shown. Measured global horizontal irradiance is only used in the models, while the corresponding diffuse irradiance is estimated by the Erbs et al. (1982) correlation.

1000 900 800

1000 EAST, θβ < 80° EAST,

EAST, θβ < 80° EAST, EAST, EAST, θβ > 90°

900

UncertaintyBand band Uncertainty

800 + 20% - 20%

500 400

2

600

700

Gβ ,cal [ W/m ]

2

Gβ ,cal [ W/m ]

700

500

200

200

0

- 20%

400 300

Liu and Jordan - Erbs et al.

+ 20%

600

300

100

EAST, θβ > 90° EAST, Uncertainty Band band Uncertainty

100

Perez et al. - Erbs et al.

0

0 100 200 300 400 500 600 700 800 900 1000 Gβ,meas [ W/m 2 ]

0 100 200 300 400 500 600 700 800 900 1000 Gβ,meas [ W/m 2 ]

Fig. 13. Calculated vs. measured hourly irradiance on tilted planes due East (45° and 65° tilt angle). The experimental uncertainty band and the band lines for the ±20% deviation between model and data are also shown. Measured global horizontal irradiance is only used in the models, while the corresponding diffuse irradiance is estimated by the Erbs et al. (1982) correlation. Direct beam does not reach the surface when hb > 90°.

incidence angles lower than 80° and higher than 90° are divided in two datasets, again. From Figs. 12 and 13 one can observe that both the Liu and Jordan (1963) and the Perez et al. (1990a) models perform worse compared to the case when the diffuse irradiance on the horizontal plane is measured and its value used for prediction (Figs. 9 and 10); this is even more evident for the isotropic model, which can underestimate experimental data by more than 20%. These data points can be identified in Fig. 14, where the percentage difference between calculated and measured irradiance on the tilted planes is plotted as a function of the clearness index. In this

figure data corresponding to hb > 90° are omitted. When the clearness index is between 0.2 and 0.6 the Liu and Jordan (1963) model provides the worst prediction, severely underestimating the data. However, the performance of all the models degrades, particularly under partially cloudy skies, since the diffuse fraction can be quite high and a large error in the estimation of the diffuse irradiance can result in a large error in the prediction of the global irradiance on the tilted plane. Tables 9 and 10 report a summary on the performance of all the transposition models implemented along with the Erbs et al. (1982), “Reindl 1” and “Reindl 2” (Reindl

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A. Padovan, D. Del Col / Solar Energy 84 (2010) 2068–2084

60

60

Liu and Jordan - Erbs et al.

50 40

40 30 20 10 0 -10 -20 -30 SOUTH

-40

(Gβ ,cal - Gβ,meas) / Gβ ,meas [ % ]

(Gβ ,cal - Gβ ,meas) / Gβ,meas [ % ]

50

EAST

-50

30 20 10 0 -10 -20 SOUTH

-30

EAST

-40

Perez et al.- Erbs et al.

-50

-60

-60 0

0.2

0.4

0.6

0.8

1

0

0.2

60

60

50

50

40

40

30 20 10 0 -10 -20 SOUTH

-30

EAST

-40

Klucher - Erbs et al.

-50

(Gβ ,cal - Gβ ,meas) / Gβ,meas [ % ]

(Gβ ,cal - Gβ,meas) / Gβ ,meas [ % ]

Clearness Index (kt)

0.4 0.6 Clearness Index (kt)

0.8

1

30 20 10 0 -10 -20 SOUTH

-30

EAST

-40

HDKR - Erbs et al.

-50 -60

-60 0

0.2

0.4 0.6 Clearness Index (kt)

0.8

1

0

0.2

0.4 0.6 Clearness Index (kt)

0.8

1

Fig. 14. Percentage of relative difference between calculated and measured global irradiance on South and East-oriented tilted planes as a function of the clearness index. Measured global horizontal irradiance is only used in the models, while the corresponding diffuse irradiance is estimated by the Erbs et al. (1982) correlation. For the East-oriented surfaces the data characterized by incidence angles higher than 90° are omitted. Table 9 Prediction of global irradiance on South-oriented tilted (20° and 30°) planes. The diffuse irradiance on the horizontal plane is calculated by means of three correlations. The average measured global irradiance on the tilted plane is 481 W/m2. RMSE (W/m2)

MBE (W/m2)

MAE (%)

ME (%)

Erbs et al. (1982) Liu and Jordan (1963) Klucher (1979) Perez et al. (1990a) HDKR, Reindl et al. (1990b)

41 34 33 31

20 +6 +10 +2

7.5 6.8 6.6 6.1

3.9 +1.3 +2.2 +0.3

“Reindl 1”, Reindl et al. (1990a) Liu and Jordan (1963) Klucher (1979) Perez et al. (1990a) HDKR, Reindl et al. (1990a)

38 32 32 29

19 +9 +12 +4

7.1 6.9 6.7 6

2.8 +2.8 +3.4 +1.7

“Reindl 2”, Reindl et al. (1990a) Liu and Jordan (1963) Klucher (1979) Perez et al. (1990a) HDKR, Reindl et al. (1990b)

42 34 33 30

21 +7 +10 +2

7.5 7.2 6.8 6.2

3.4 +2.2 +2.8 +1.1

A. Padovan, D. Del Col / Solar Energy 84 (2010) 2068–2084

2083

Table 10 Prediction of global irradiance on East-oriented tilted (45° and 65°) planes. The diffuse irradiance on the horizontal plane is calculated by means of three correlations. The average measured global irradiance on the tilted plane is 354 W/m2. RMSE (W/m2)

MBE (W/m2)

MAE (%)

ME (%)

Erbs et al. (1982) Liu and Jordan (1963) Klucher (1979) Perez et al. (1990a) HDKR, Reindl et al. (1990b)

71 59 40 49

18 +8 +2 5

26.7 30.5 15.6 18.7

+12.6 +22.2 +9.2 +8.8

“Reindl 1”, Reindl et al. (1990a) Liu and Jordan (1963) Klucher (1979) Perez et al. (1990a) Reindl et al. (1990b)

68 56 36 42

14 +14 +6 1

25.8 30.3 15 17.2

+13.6 +24 +10.3 +9.8

“Reindl 2”, Reindl et al. (1990a) Liu and Jordan (1963) Klucher (1979) Perez et al. (1990a) HDKR, Reindl et al. (1990b)

70 57 37 44

17 +10 +3 4

26.9 31.3 15.8 18

+13.5 +23.9 +10.2 +9.6

et al., 1990a) diffuse fraction correlations for the South-oriented and the East-oriented tilted surfaces, respectively. The first database, used in the comparison, includes data taken on planes tilted at 20° and 30° with South orientation. The second database covers data measured on surfaces tilted at 45° and 65° with East orientation. In comparison with the case when diffuse irradiance is measured (Tables 7 and 8) the accuracy of the models here decreases, particularly for the East-facing planes; in this case the Perez et al. (1990a) model performs significantly better than the others. 5. Conclusions (1) Modeling of diffuse irradiance on the horizontal plane All the diffuse fraction correlations, considered in this paper, overestimate on average the diffuse irradiance measurements. The Erbs et al. (1982) correlation has been taken as representative method to discuss the disagreement between experimental and predicted data. The results show that the accuracy of Erbs et al. (1982) depends on the sky conditions, overestimating the measured irradiance under clear skies, while exhibiting an opposite trend in the case of higher sky brightness. The overestimation of experimental data increases when increasing zenith angle. (2) Modeling of global irradiance on the tilted plane When the global and the diffuse irradiances on the horizontal plane are measured, the Liu and Jordan (1963) and the anisotropic models predict the data on the tilted planes oriented due South with similar accuracy. In the case of the East-oriented planes, a higher disagreement between estimated and measured data is found and the anisotropic models by Perez et al.

(1990a) and HDKR (Reindl et al., 1990b) provide significantly more accurate predictions. The use of the Drummond (1956) formula in the shadow band correction procedure for diffuse irradiance measurements does not affect the prediction accuracy of transposition models, although it could underestimate the horizontal diffuse irradiance by about 8% compared to the LeBaron et al. (1990) anisotropic correction. If the transposition models are applied using measured global irradiance and estimated diffuse irradiance on the horizontal plane, the error exhibited by all the models increases. For example, when using the “Reindl 1” (Reindl et al., 1990a) correlation, which predicts the diffuse irradiance on the horizontal plane with a mean absolute error around 40%, the MAE value, obtained in the prediction of the global irradiance on the tilted planes with the HDKR model, increases from 4.6% to 6% for the South orientation and from 10.2% to 17.2% for the East orientation. (3) Final remarks The use of the anisotropic transposition models by Perez et al. (1990a) or HDKR (Reindl et al., 1990b) is highly recommended when only global solar radiation data on the horizontal plane are available for the site or when the orientation of the tilted plane is away from due South. The present results can be a useful tool for the selection of the most adequate models to be used in the design of conventional solar energy systems and new solar devices for electricity generation and thermal applications. Future research must be addressed to improve the estimation of diffuse irradiance on the horizontal plane, since the existing diffuse fraction correlations have not provided encouraging results. This is a key point also for getting better prediction of global irradiance on tilted planes. Besides, models assessment using short term data is needed for

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