Solar Cells, 18 (1986) 213-222 213

SOLAR IRRADIANCE CONVERSION MODELS*

RICHARD PEREZ and RONALD STEWART

ASRC, State University of New York at Albany, NY (U.S.A.)

(Accepted July 3, 1985)

1. Introduction

1.1. Problem definition There are three basic levels of modeling which might be considered

when estimating, on an hourly basis, the irradiance received by a sloping surface in a given location, depending upon the data available to the user. These three types of models are as follows.

(1) Conversion or transposition models that use available direct and horizontal global or diffuse hourly irradiance records to compute hourly global irradiance values on tilted planes [1 - 7].

(2) Models which use hourly global radiation as input to generate the diffuse and direct components rleeded to obtain values for tilted planes [1, S ,9 ] .

(3) Models which generate global and/or direct irradiance from other quantities, such as percent sunshine, cloud cover, satellite imagery, etc. [10- 12].

This paper focuses on the first type of model. It is therefore assumed that global and direct hourly data are either available (which is the case for over 40 locations in the U.S.A. [13, 14]) or accurately modeled.

The need for adequate modeling in this area is crucial for both HVAC and PV system designers, as instantaneous computat ional errors may reach 200 W m -2 under certain circumstances. A recent study by Menicucci [15], showed that transposition models could be the largest source of error, depending on the model used, associated with the simulation of PV systems.

1.2. Conversion models: input parameters, model components All the hourly transposition models presented herein are equivalent in

the sense that they rely on the same set of input parameters. Besides global irradiance Gh, direct normal irradiance I, (or diffuse irradiance D h ), the other inputs needed are: the location's latitude ~, the plane slope considered s, and azimuth 7, the julian date n, the true solar time ts, and the surround-

*Paper presented at the SERI Photovoltaics and Insolation Measurements Workshop, Vail, CO, U.S.A., June 30 - July 3, 1985.

0379-6787/86/$3.50 © Elsevier Sequoia/Printed in The Netherlands

214

ing ground albedo A. The output of these models consists of the global irradiance received by the tilted plane.

Most present models also divide the energy received into three components , which are computed independently. These are the direct beam, the sky-diffuse and the ground-reflected components .

Differences between models are found mainly in the computat ion of the sky-diffuse component . Indeed, the distribution of radiance throughout the sky is difficult to represent adequately, and the assumptions used to describe it have proved to be the main cause of error associated with these models.

The modeling of direct radiation is identical for all models. It is a straightforward problem, and is a negligible source of error, for flat plate collectors, if accurate input data are used.

The ground-reflected component may be difficult to model with precision, and large computat ional errors may occur under certain circumstances. However, this component carries in most instances, less weight than the two previous ones.

2. Conversion algorithms

The hourly energy Gc impinging on a tilting plane is given by

Gc = I c + D c + R c (1)

where I¢ is the direct component , and Dc and Rc are the sky-diffuse and the ground-reflected components respectively.

2.1. Direct component Given direct irradiance I as input, I¢ is obtained from

I c = I (max (0, cos0)} (2)

where 0 is the solar incidence angle on the plane considered; cos0 is given by [16]

cos 0 = sin 6 sin ~ cos s -- sin 5 cos 0 sin s cos 7

+ cos

+ cos

6 cos ~b cos s cos co + cos 5 sin @ sin s cos-y cos co

6 sins sin 7 sin 6) (3)

¢ are given as input, and 6 is the declination obtained from [16]. The hour angle 6) is obtained from t s knowing that

where s, 7, and the julian date co = 180 ° at 12 a.m. and decreases at the rate of 15 ° h -1.

In the case of non-fixed surfaces, where s and 7 are not known a priori, it is still possible to evaluate cos 0 by using the tracking characteristics (axis, rotational speed) as input. Examples are given below for specific configurations.

215

For the two-axis tracker the slope and incidence angle are given by

s = arccos (sin 6 sin ~ + cos 6 cos ¢ cos c0) (4)

0 = 0

In the case of the one-axis tracker (vertical axis, azimuth tracking) the slope is known, and the incidence angle is given by

0 = arccos (sin 6 sin ¢ + cos 6 cos ¢ cos co) -- s (5)

and for the one-axis tracker (north-south, latitude-set axis, constant rota- tional speed) slope and incidence are given by

s = arccos (cos ~ cos ¢) (6)

0 = 5

whereas for the one-axis tracker (north-south axis set at slope angle a, constant rotational speed) slope and azimuth are given by

s = a rccos (cos ~ cos ~) (7)

and

sin w cos ¢0 sin a - - - arccos (8)

}sin ¢01 sin (arccos (cos u) cos ~))

The incidence angle 0 is then obtained through eqn. (3). For the one-axis tracker (east-west horizontal axis, altitude tracking)

the slope is given by

cos 6 sin ¢ cos c~ -- sin 6 cos ¢ s = arctan (9)

sin 6 sin ¢ + cos 6 cos ¢ cos ¢o

The incidence angle 0 is obtained from eqn. (3), knowing that ~ = 0.

2. 2. Reflected component The classical approach to the modeling of this component [1], assumes

that a constant radiance originates from every point of the ground. Based on this assumption, Rc is given by

Rc = GhA(1 -- coss)/2 (10)

Values of 0.2 and 0.7 for A are suggested for bare ground and snow covered ground respectively [1].

Non-isotropic approaches have been proposed (e.g. [2, 7]) based on experimental data recorded at particular sites. However, published studies have not shown improvement on the isotropic assumptions for independent tests [17]. In fact, the primary concern of the user for the modeling of this component should be the determination of the appropriate albedo and of existing important differences of albedo with azimuth.

216

2.3. D i f f u s e c o m p o n e n t The most widely used method for the calculation of this component

has been to assume an isotropic distribution of radiance throughout the sky dome [ 1 ]. The diffuse radiation received by the slope is given by

Dciso = Dh(1 + coss) /2 (11)

The simplicity of the method and its reasonable accuracy have been the main reasons for its widespread use. However, as energy system modeling became more refined, the need for more accurate modeling of the solar input became obvious.

The first a t tempt at incorporating the observed directionality of diffuse radiation consisted of assuming that this originated entirely from the solar disc according to

De dir = Dh COS 0/COS Z (12 )

where Z is the solar zenith angle. This algorithm performed generally more poorly than the isotropic.

The next step consisted of modeling the diffuse component as a combination of the two previous approaches as

D c = F Dc~ o + (1 -- F) Dc dix (13)

LeQuere [18], suggested a value of 0.8 for F. Hay and Davies [4] took the concept a step further by formulating the degree of anisotropy (1 - - F ) as the ratio of terrestrial direct radiation to extraterrestrial radiation. Noticeable improvement on the isotropic approach was noted. Similar, although more complex approaches by Willmott [20], and Puri e t al. [5] were not found to create additional improvement [ 19].

Klucher [3] took a different approach by adapting the Temps and Coulson [2] clear day model to all sky conditions. This model multiplies the isotropic value by two factors reflecting the anisotropic effect of both circumsolar and horizon brightening.

D c = D c ~o(1 + K sina(s/2) (1 + K cos 2 0 sin 3 Z) (14)

As in the Hay and Davies model [4], the factor K, expressing the degree of anistropy, is a function of one parameter describing the amount of direct radiation received at the earth's surface.

K = 1 - - ( D h / G h ) : (15)

Perez e t al. [6] also included the two anisotropic effects but used a systematic approach including: (1) a simple representation of the sky dome (Fig. l ) ; (2) circumsolar and horizon anisotropy factors (F 1 and F2) established from experimental data and varying independently with insolation condition; (3) a three-dimensional representation of the latter making full use of the input available to the user. The three dimensions are, (a) Z, (b) Dh, and (c) e = (I + Dh ) /Dh . The equation is given by

217

~TH

Fig. 1. Representation of the three radiative components seen by a tilted plane (direct, diffuse and reflected) and representation of the sky dome used in the Perez algorithm (sky radiance is respectively equal to L, FIL and F2L for the main the circumsolar and the horizon zone).

De = (Dc,o + Dh(F1 - - 1) a(O) + b(s) (F2 - - 1)) (1 + c ( Z ) (F1 -- 1) (16)

+ d(F2 -- I))-'

where a(O) and b(s) are factors representing the solid angle covered by the circumsolar region and the horizon band, respectively, multiplied by their average incidence on the considered plane, while c(Z) and d are the equivalent of a(O) and b(s) for the horizontal.

3. Model performance

Four of the algorithms described above are compared. These include the isotropic, Hay, Klucher and Perez models.

Comparison is based on three types of observation: (1) observation of roo t mean square errors and mean bias error, generated by all models when tested against experimental data sets; (2) observation of real-time perfor- mance for typical insolation conditions, and (3) ability of models to perform adequate array design optimization.

218

!+ ! SOTROPIC . . . . X P~REZ [ ALeANy', !

e - ~ . . . . 7- . . . . CHE . . . . . EZ{ . . . . . . ) I ~ . . . . . . . k 40 m HAY • PEREZ{D£PENDENT)

• ~ OEPENDENT / . , | ~0~". / . . . . . ,

/ '~ J ,; ~ / ~ I '°9 ....... ~ o o ~ - - /*'--~",~

AIb. r r o C¢,: San AIb Tro. COt S(In A lb r f o Ca r Sa~A Ib rra Cor Son IAIb Tra Cot Son AIb r ro Co r : ~ AIb Tea Co¢ Son AIb Try; Co[Son

(a (b}

Fig. 2. Root mean square (a) and mean bias errors (b) obtained with each of the selected models, after tests against hourly data on fixed tilted sensors. Data sets are as follows: Alb., Albany, NY, 3 months; Tra., Trappes, France, 21 months; Car., Carpentras, France, 24 months; San, San Antonio, TX, 3 months.

3.1. Long term performance The overall performance of each model is presented for four climatic

environments in Fig. 2. Hourly test data sets were obtained from Albany, NY (humid continental climate), San Antonio, TX (semi-arid), Carpentras, France (mediterranean) and Trappes, France (maritime). These sets including ground-shielded radiance measurements in vertical planes facing south, north, east and west and a south facing fixed sloping plane, are respectively 3 months, 3 months, 24 months and 21 months long. Root mean square and mean bias errors have been plotted against location for each slope and orientation. Two versions of the Perez model are included: (1) a version developed from Albany data and tested independently against the other sets, and (2) a dependent version based on each site's data.

Examples of monthly variations of the algorithms' performance are plot ted in Fig. 3 for two climatically distinct sites, Goodnoe Hill, WA (mari-

\ \;~ / \ / i 20 \ \ \ - - ~ - - - \ z ..... . Y /

I I _ I i I I L ~ I ~L _ 3 - 40 j F M A M J J A S O N D J F M A M J J

( a ) ( b )

. . . . . Pe rez . . . . K l uche r Hoy I $0 t r 0p i ¢

Fig. 3. Monthly variations of the mean bias errors of the models after tests against hourly two-axis tracking global data from (a) Phoenix, AZ and (b) Goodnoe H i l l WA (analysis performed by SERI).

400 200

219

200'

-20C

-400

-600

-800 -

' ' ' , 5 . . . . . . . . . ,b 6 8 12 14 16 18 9 ( c ) Time of dcy ( b )

1 Time of dsy

Fig. 4. Modeled minus measured irradiance for two selected cases: (a) clear day, 90 ° west facing vertical surface; (b) partly cloudy day, 53 ° south facing surface.

time) and Phoenix, AZ (arid). Test data are normal incidence global measure- ments (two-axis tracking) performed by SERI. The ground component was assumed to be isotropic and albedo was set equal to 0.2. The Albany version of the Perez model was used in both cases.

3.2. Real-time performance The difference between modeled and measured irradiance impinging

on test sensors in Albany has been plotted in Fig. 4 for two illustrative examples; (1) 6/12/80, clear day, west facing vertical sensor and (2) 3/2/80, partly cloudy day, 53 ° tilt south facing sensor.

The maximum isotropic error reaches 150 W m -2 in both cases. This is indicative of the type of error that would have been observed throughout the day for a tracking plane.

3.3. Array design suitability The choice of opt imum array configuration is strongly influenced by

climate as seen in Fig. 5 where the relative energy received by four designs (concentrator, one-axis, two-axis tracking and fixed tilt south facing) has been plotted for Albany, NY and Phoenix, AZ. Model performance in t h i s area is summarized in Table 1. Measured energy ratios between two con- figurations are compared to modeled ratios. The Albany version of the Perez model was used.

220

1.6

1.4

1.2

1.0 J

Co 2.0

RATIO

F M A M d J A S 0 N D

RATIO

1.81

1.6

1.2

1.0 ' F M A M J J A S 0 N O

(b

Fig. 5. Monthly variations in the relative energy received using different array mounting configurations for (a) Phoenix, AZ and (b) Albany, NY. Curves (1), two-axis flat plate• concentrator; curves (2), two-axis fiat plate/single-axis flat plate; curves (3), two-axis flat plate/fixed flat plate.

4. Conclusions

There exist today methods which have demonstrated a substantial performance improvement in the modeling of the diffuse irradiance component , when compared to the classical methods. It is important to note that these methods do no t require any additional input.

There is still need for improvement , however, in the following areas: (1) bet ter understanding of cl imate/environment effects on model performance; (2) simplification of algorithm end use (notably the Perez algorithm), and (3) adaptability of models to: (a) applications with restricted fields of view such as low concentrat ion arrays, and (b) simulation of more spectrally sensitive materials than crystalline silicon modules.

221

TABLE 1

Ability of the models to simulate overall energy ratios between different PV mounting configurations for Phoenix, AZ (12 months of hourly data) and Goodnoe Hills, WA (7 months of data)

Ratio a Measured Difference (%) ratio

Liu and Hay and Klucher [3] Perez Jordan [1] Davies [4] et al. [6]

A 1.27 -- 6.9 --2.8 --3.7 --2.4 B 1 . 3 8 - - 1 . 8 - - 0 . 3 + 0 . 4 + 1 . 0 C 1.41 --11.4 --9.7 --6.3 --3.0 D 1.37 -- 8.2 --7.5 --4.4 --2.0

aA, Phoenix AZ, track flat plate/concentrator; B, Phoenix, AZ, track flat plate/fixed tilt; C, Goodnoe Hills, WA, track flat plate/concentrator; D, Goodnoe Hills WA, track fiat plate/fixed tilt.

Acknowledgment

The authors thank the Solar Energy Research Institute for providing much of the data analysis presented in this paper.

References

1 B. Y. H. Liu and R. C. Jordan, The interrelationship and characteristic distribution of direct, diffuse and total solar radiation, Sol. Energy, 4 (1960) 1 - 19.

2 R. C. Temps and K. L. Coulson, Solar radiation incident upon slopes of different orientations, Sol. Energy, 19 (1977) 179 - 184.

3 T. M. Klucher, Evaluation of models to predict insolation on tilted surfaces, Sol. Energy, 23 (1978) 111 - 114.

4 J. E. Hay and J. A. Davies, Calculation of the solar radiation incident on an inclined surface, in J. E. Hay and T. K. Won (eds.), ]>roe. 1st Canadian Solar Radiation Data Workshop, Toronto, pp. 59 - 72.

5 V.M. Puri, R. Jimenez and M. Menzer, Total and non-isotropic diffuse insolation on tilted surfaces, Sol. Energy, 25 (1980) 85 - 90.

6 R. R. Perez, J. T. Scott and R. Stewart, An anisotropic model for diffuse radiation incident on slopes of different orientations, and possible applications to CPCs. Proc. American Solar Energy Society, Minneapolis, MN, 1983, pp. 883- 888.

7 P. Ineichen, O. Guisar and A. RazaFindraibe, Mesures d'Ensoleillement A Gen~ve, Pub. 13, GAP, University of Geneva, Geneva, 1985.

9 M. lqbal, Prediction of hourly diffuse solar radiation from measured hourly global radiation on a horizontal surface, Sol. Energy, 24 (1980) 491 - 503.

9 D.G. Erbs, S. A. Klein and J. A. Duffie, Estimation of the diffuse radiation fraction for hourly, daily and monthly average global radiation, Sol. Energy, 28 (1982) 293 - 302. O. M. Essenwanger, Synthetic solar radiation statistics on cloudy and sunny days, 10

222

Proc. American Society of Heating, Refrigerating and Air Conditioning Engineers, (1985) Honolulu, HI.

11 W. Moser, and E. Raschke, Mapping of global radiation and of cloudiness from meteosat image data, Meteorol. Rundsch., 36 (1983) 33 - 41.

12 R. K. Reed, An evaluation of cloud factors for estimating insolation over the ocean, NOAA TM ERL RMEL-8, 1976.

13 Solar Radiation Data, Monthly Summaries (1977-1980) , available from NCC, AsheviUe, NC.

14 R. L. Hulstrom, SEMRT Sites: 2nd Annu. Rep., SERI/SP-250-1978, SERI, Golden, CO, 1979.

15 D. F. Menicucci, The determination of opt imum flat plate PV module mounting configuration. Proc. American Society of Heating, Refrigerating and Air Condition- ing Engineers, (1985) Annual Meeting, Honolulu, HI.

16 J. A. Duffie and W. A. Beckman, Solar Energy and Thermal Processes, Wiley, New York, NY, 1979.

17 A. Lebru, Cahier du CSTB 1847, no. 239. Centre Scientifique et Technique du Batiment, Sophia-Antipolis, France, 1983.

18 J. LeQuere, Rapport sur la comparaison des methodes de calcul des besoins de chauffage des logements, CSTB TEA-S 87, CSTB, Paris, France, 1980.

19 B. Bourges and F. Lasnier, Frequency cumulative curves of global radiation, development and validation of computation methods for inclined surfaces, M6t~rologie et Energies Renouvelables, AFME, Valbome, France, 1984.

20 C . J . Wilmott, On the climatic optimization of the tilt and azimuth of flat-plate solar collectors, Sol. Energy, 28 (1982) 205 - 216.