Simulation of global solar radiation based on cloud observations

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Simulation of global solar radiation

based on cloud observations

Jimmy S.G. Ehnberg

*

_{, Math H.J. Bollen}

Department of Electric Power Engineering, Chalmers University of Technology, Go¨teborg SE-412 96, Sweden Received 6 August 2003; received in revised form 23 August 2004; accepted 23 August 2004

Available online 14 October 2004 Communicated by: Associate Editor Pierre Ineichen

Abstract

Astochastic model for simulating global solar radiation on a horizontal surface has been developed for use in power systems reliability calculations. The importance of an appropriate model for global solar radiation has increased with the increased use of photovoltaic power generation. The global solar radiation shows not only regular yearly and daily variations but also a random behaviour. The yearly and daily variations can be described in a deterministic way while the random behaviour has a high correlation with the state of the atmosphere. The astronomic effects can easily be described mathematical with only some minor simplifications but the atmospheric effects are more complicated to describe. The transmittivity of solar radiation in the atmosphere depends on various factors, e.g. humidity, air pressure and cloud type. By using cloud observations as input for the simulations, the local meteorological conditions can be accounted for. The model is usable for any geographical location if cloud observations are available at the location or at locations with similar climatological conditions. This is especially useful for development countries where long-term solar radiation measurement can be hard to obtain. Cloud observations can be performed without any expensive equipment and have been a standard parameter for many years throughout the world. Standard observations are done according to the Oktas-scale. It is the interval between observations that sets the resolution of the simulation: the obser-vations are normally only every hour or every third hour. The model can easily be combined with cloud coverage simu-lations, has been proposed, for a more general model. For some calculations higher resolution may be needed. This can be obtained by including a stochastic model for the short-term variations and simple model has been proposed. Errors and limitations of the model are estimated and discussed.

Keywords:Solar power; Markov model; Cloud coverage

1. Introduction

In rural areas of developing countries the interest in electrification has increased over the last few years result-ing in an improved standard of livresult-ing. This is confirmed by many ongoing research programs. The problem of electrification of remote rural areas may be solved by 0038-092X/$ - see front matter Ó2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.solener.2004.08.016 *

Corresponding author. Tel.: +46 31 772 16 30; fax: +46 31 772 16 33.

E-mail address: [email protected] (J.S.G. Ehnberg).

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the use of autonomous electric power systems, supplied exclusively by environmental friendly power sources, e.g. solar power. Solar power is a suitable source because of the high solar radiation, available in many developing countries and the low maintenance requirements. The interest for solar power has also increased in other parts of the world because of the need of more environmental friendly power generation to secure both the future power demand and the survival of our planet.

Several models have been proposed for generation of global radiation. The random nature of global solar radiation is included in all proposals, but the way of implementing this in a model varies signiﬁcantly. In (Amato et al., 1986; Albizzati et al., 1997; Balouktsis et al. 1989) they model daily global solar radiation (thus the yearly variations) but a higher resolution of the sim-ulation is needed for photovoltaic power generation in an autonomous electric power system. Such model would be applicable in a system with a storage capability higher than the daily load demand. The models of (Amato et al., 1986;Balouktsis et al., 1989) requires sev-eral years of solar radiation measurements, which are for most locations not available. The model proposed by (Albizzati et al., 1997) is adapted for clear sky condi-tions but the authors mentioned the importance of the cloud coverage.

In (Graham and Hollands, 1990) hourly radiation has been modelled but the model could be diﬃcult to ap-ply due to the data requirements. Monthly average val-ues of global radiation are needed which can only be obtained from long time measurements.

Another model is proposed in (Balouktsis and

Tsali-des, 1986) but the problem with the input of the model

remains. Alocation-dependent factor is used which de-pends on the probability distribution of the solar radia-tion. This model can again only be used when a large amount of solar radiation data is available.

Outside the atmosphere the solar radiation can

accu-rately be determined (Stull, 1995) and the atmosphere will

induce the randomness (Graham and Hollands, 1990).

The transmittivity of solar radiation in the atmosphere depends on various factors, e.g. humidity, air pressure and cloud type. Afactor that has a great impact on the

transmittivity is the cloud coverage (Nielsen et al., 1981;

Albizzati et al., 1997). By assuming a deterministic rela-tion between cloud coverage and hourly global solar radi-ation, the need for measurement of the latter disappears. Cloud observations can be used because of the simplicity of measuring, no expensive equipment is needed. The le-vel of cloudiness is expressed in Oktas, which describes how many eight parts of the sky that are covered with

clouds (Jones, 1992). By combining the solar radiation

model with a model of simulating cloud coverage the sim-ulation method could be even more suitable.

Models for simulating cloud coverage and the solar radiation have been investigated by several authors,

e.g. (Gu et al., 2001; Badescu, 2002). In (Gu et al.,

2001) the focus is on spectral simulation, which has a

great importance for the use of solar panels but requires detailed and extensive knowledge of the atmosphere and

its contents. (Badescu, 2002) has reviewed some simple

models and concluded that even simple or very simple (words used by the author) models can be useful. The author has proposed a new kind of sky model but has lost the simplicity of the model that is needed for use in power system studies.

Nomenclature

ds vector of the solar declination angles over a

time period determined byd(rad)

Ur the tilt of the earthÕs axis relative the orbital

plane of the earth around the sun,

Ur= 0.409 rad

C C= 2p(rad)

d vector of days of the year for the time period

of the simulation (days)

dr the day of the year at summer solstice, 22nd

of June for non-leap years (days)

dy total number of days in a year (days)

w vector of local elevation angle angles over a

time period determined byd(rad)

tUTC Co-ordinated Universal Time (h)

td hours in a day (h)

/ latitude of the location, positive north of the

equator (rad)

ke longitude of the location, positive west of

Greenwich (rad)

Gh vector of global solar radiation

(hourly-scale) over a time period determined by d

(W/m2)

Gmin global solar radiation (minute-scale) over a

time period determined byd(W/m2)

N number of Oktas

L, a, ai, i= 0, 1, 3 empirical determined constants

(W/m2)

e statistically varying term,eN(m,r2₎

kij estimated transition probability fromitoj

fij number of transitions from stateitoj

b

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The solar radiation distribution is expected to be similar

in areas with similar climatological conditions (Balouktsis

and Tsalides, 1986). That means that this method could be used when cloud observations are available for an area with similar climatological conditions. In reliability lations for power systems without storage capacity, simu-lation data with higher resolution than one hour is needed in some cases. This is the case when short-duration inter-ruptions (less than one half hour) are a concern.

In this paper a model for simulating six minutes val-ues of global solar radiation without any geographical restrictions is proposed and discussed. The method uses cloud coverage observations as input. Amethod of generating cloud coverage by using a discrete Markov model is also proposed.

2. Astronomical part of the model

Astronomical eﬀects are due to the earth rotation around the sun and the rotation of the earth around its axis. The seasonal and daily variations can be

de-scribed by Eqs.(1) and (2)(Stull, 1995).

The equation for the seasonal eﬀects(1)is an

approx-imation under the assumption of circular orbit of the earth around the sun. This assumption is allowed be-cause the excentricity is only 0.07 and the results are

only used in stochastic ways. Eq.(2)describes the daily

eﬀects and is dependent on the geographical location

through latitude and longitude. Eq. (2) contains time

dependence. Correlation is needed with local time to be used in power systems studies where comparisons with often time depended load are of importance. ds¼Urcos

CðddrÞ dy

ð1Þ

sinw¼sin/sindscos/cosdscos

CtUTC

td

ke

ð2Þ

3. Atmospherical part of the model

The randomness in this simulation model for global solar radiation is introduced in this section. The impor-tance of the randomness in the atmosphere is also

dis-cussed in (Graham and Hollands, 1990). An

empirically determined relationship between the global

solar radiation and the cloud coverage, Eq.(3)was

ob-tained by (Nielsen et al., 1981), after many years of

cloud observations, solar elevation measurements and global solar radiation measurements. The obtained rela-tionship reads as follows:

G¼ a0ðNÞ þa1ðNÞsinwþa3ðNÞsin 3 wLðNÞ aðNÞ ð3Þ

The values of the constantsL(N),a(N) andai,i= 0, 1, 3,

in Eq. (3)are given in Table 1and the elevation angle

from Eq.(2)can be used.

InFig. 1the global solar radiation is presented as a function of the solar elevation angle for the nine possible values of cloud coverage in the Oktas-scale. Note that even for a fully clouded sky, a non-negligible part of the solar radiation reaches the solar panel (about 25 %). In (Nielsen et al., 1981) it was found that the

stand-ard error of the estimation was less then 80 W/m2 and

has a square correlation coeﬃcient of approx. 0.9 when compared with other sites. The model does not include extreme values. However, this has low inﬂuence when using time average values.

If the global radiation is below zero in Eq.(3)it should

be set to zero according to Eq.(4). If the radiation is

neg-ative it is from the surface of the earth upwards. This radi-ation has another frequency spectrum and will not generate any power from solar panels. This situation will occur during nighttime and for low elevation angles.

if w_i<0 orGhðiÞ<0 thenGhðiÞ ¼0 8i ð4Þ

Table 1

The empirical determined coeﬃcients for(3)

N a0 a1 a3 a L 0 112.6 653.2 174.0 0.73 95.0 1 112.6 686.5 120.9 0.72 89.2 2 107.3 650.2 127.1 0.72 78.2 3 97.8 608.3 110.6 0.72 67.4 4 85.1 552.0 106.3 0.72 57.1 5 77.1 511.5 58.5 0.70 45.7 6 71.2 495.4 37.9 0.70 33.2 7 31.8 287.5 94.0 0.69 16.5 8 13.7 154.2 64.9 0.69 4.3 0 0.2 0.4 0.6 0.8 1 –200 0 200 400 600 800 1000 1200

Sine of the Solar Elevation Angle

Global Solar Radiation [w/m

2 ] 0 1 2 3 4 5 6 7 8

Fig. 1. The relationship between global solar radiation and the solar elevations angle for diﬀerent cloud coverage.

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By examining global solar radiation measurements it can be seen that the radiation varies within a one-hour period. Introducing a statistically varying term

accord-ing to Eq.(5)could simulate this phenomenon. This

sta-tistical term (e) was set to have the same distribution as

the short duration variations seen in the measurements.

Gmin¼Ghþe ð5Þ

The statistically varying term can be estimated through cross validation, the so-called ‘‘hold out method’’

pro-posed by (Hjort, 1995). The deviation from the hourly

mean values for daytime can be ﬁtted to a normal distri-bution and the mean value and the standard deviation estimated.

4. Cloud coverage simulation

To be able to perform a simulation over a longer per-iod then the available cloud coverage data some kind of generation of stochastic cloud coverage data is needed. If it can be assumed that the current level of cloud cover-age is only depended on the previous value, a discrete Markov model can be used. With this assumption a mod-el with nine diﬀerent states (0–8), corresponding to the nine levels in the Oktas scale is proposed. The transitions

probability matrix (Kb) can be estimated from measured

cloud coverage data. The transition probabilities can by estimated in many ways but the most intuitively is: ^ kij¼ fij P8 k¼0fik ð6Þ b K¼ ^ k00 ^k01 ^k08 ^ k10 ^k11 ^k18 .. . .. . . . . .. . ^ k80 ^k81 ^k88 2 6 6 6 6 4 3 7 7 7 7 5 ð7Þ

This is a maximum likelihood estimation if the chain has

stationarity, as shown by (Macdonald and Zucchini,

1997). Astart value can be arbitrary chosen since the

starting condition has no inﬂuence in the long run. The deviation between measured and simulated cloud

coverage data is shown in Fig. 2and maximum

devia-tion is 1.5 %, for 0 Oktas. Cloud coverage data for Go¨te-borg was used in the ﬁgure.

5. Case study,Go¨teborg

To show the applicability of the model a case study has been made. The case study was made for Go¨teborg (Lat. 57.72°N, Long. 11.97°E). Go¨teborg is normally minus one hour from Greenwich, which is included in the following calculations.

The transition matrix for the Markov model for the cloud coverage simulation was estimated from measure-ments of cloud coverage obtained during the period from 1973 to 1999. The transition probabilities were

esti-mated according to Eq.(6). They were estimated every

three hours because the available cloud coverage obser-vations was done with this interval. The estimated

tran-sition matrix (Kb) for Go¨teborg is presented below, with

values given as a percentage.

b K¼ 53:8 22:5 7:1 4:7 2:7 2:3 1:7 2:6 2:6 15:5 45:5 14:0 9:1 4:3 3:7 3:2 3:0 1:5 7:0 24:5 23:4 15:3 8:8 7:2 6:2 5:4 2:2 3:8 13:4 17:7 20:3 12:6 10:6 9:0 9:1 3:4 2:2 8:5 12:1 15:9 16:2 14:4 13:4 13:2 4:2 1:5 5:1 8:1 12:2 12:6 17:3 18:7 18:3 6:2 1:0 3:0 5:2 7:4 9:5 14:2 22:2 28:0 9:5 0:6 2:0 2:3 3:0 3:9 6:3 11:3 50:3 20:4 0:5 0:7 0:8 1:1 1:3 2:0 3:8 13:5 76:3 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ð8Þ

The mean value of the deviation between the simulated

and the observed cloud coverage values is shown inFig.

2. The mean deviation for every cloud coverage value

and the standard deviation are shown. The mean values and standard deviations are calculated from ten inde-pendently made simulations. From the ﬁgure the conclu-sion is drawn that the error is around 1 %, and that the model is an acceptable model of the measurement.

This cloud coverage simulation was used together with the calculated solar elevation angle to obtain the global solar radiation for Go¨teborg according to Eq.

(3). Since the meteorological data used for the

estima-tion of the transiestima-tion probability matrix was only every three hours, a linear interpolation was used to achieve

0 2 4 6 8 –2 –1.5 –1 –0.5 0 0.5 1 1.5

Cloud coverage [Oktas]

Deviation [%]

Fig. 2. The deviation between measurements and simulations for each cloud coverage value. The error band indicates the standard deviation.

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hourly values.Fig. 3 shows the result by showing the maximum value for each day during one year. The upper and the lower bound of the distribution correspond to clear sky (0 Oktas) and totally cloud covered sky (8

Ok-tas). Fig. 4 shows the global solar radiation for a few

days in February.

By adding a statistically varying term, as in Eq.(5),

simulated values with a higher time resolution were ob-tained. For the statistical term, a normal distribution was used with a mean value of zero and a standard

devi-ation of 40 W/m2. For the results, shown inFig. 5,

meas-ured six-minute mean values of global solar radiation for a single day in June in 1999 were used. Hourly mean values were calculated and then the statistically varying term was added and the result is shown in the same ﬁg-ure as the measﬁg-ured values. Asolid line represents the measured values while the calculated values are

pre-sented by a dotted line. The comparison was done for a single day in June.

6. Conclusions

Amodel for simulation of global solar radiation based on cloud observations is presented. The advantages of the model are that there are no geographical restrictions to its use and that there is no need for global solar radiation measurements. The model does however require cloud coverage observations over a longer period, but those data are in most cases easier to obtain than solar radiation data. Astochastic model to generate cloud observations for use in simulations was developed. This model only needs to be adapted to a climate similar to the one at the desired geographical location. The errors and limita-tions of the model eﬀect the usefulness. However, for time average estimations the inﬂuence is low and in this model only time average estimations are of interest.

Acknowledgments

This project is ﬁnancially supported by the Alliance for Global Substantiality. The authors also wish to thank SMHI (Swedish Meteorological and Hydrological Institute) for providing meteorological data.

References

Albizzati, E., Rossetti, G., Alfano, O., 1997. Measurements and predictions of solar radiation incident on horizontal sur-faces at Santa Fe, Argentina (31°390_{S, 60}_°₄₃0_{W). Renewable} Energy 4, 469–478. 0 50 100 150 200 250 300 350 0 100 200 300 400 500 600 700 800 900

Day of the year

2]

Fig. 3. The maximum daily value of the global solar radiation in Go¨teborg during one year.

48 49 50 51 52 0 50 100 150 200 250 300

Day of the year

2]

Fig. 4. The global solar radiation for a few days in February in Go¨teborg. 0 5 10 15 20 0 100 200 300 400 500 600 Time [hours]

2]

Fig. 5. Comparison between calculated and measured values for a day in June.

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Amato, U., Andretta, B., Bartoli, B., Coluzzi, B., Cuomo, V., 1986. Markov processes and Fourier analysis as a tool to describe and simulate daily solar radiation. Int. J. Solar Energy 37, 179–194.

Badescu, V., 2002. Anew kind of cloudy sky model to compute instantaneous values of diﬀuse and global solar irradiance. Theor. Appl. Climatol. 72, 127–136.

Balouktsis, A., Tsalides, P., 1986. Stochastic simulation model of hourly total solar radiation. Int. J. Solar Energy 37, 119–126. Balouktsis, A., Tsanakas, D., Vachtsevanos, G., 1989. Stoch-astic modelling of daily global solar radiation. Int. J. Solar Energy 7, 1–10.

Graham, V., Hollands, K., 1990. Amethod to generate synthetic hourly solar radiation globally. Int. J. Solar Energy 44, 333–341.

Gu, L., Fuentes, J.D., Garstang, M., Da Silva, J.T., Heitz, R., Sigler, J., Shugart, H.H., 2001. Cloud modulation of solar

irradiance at a pasture site in southern Brazil. Agric. Forest Meteorol. 106, 117–129.

Hjort, U., 1995. Computer Intensive Statistical Methods: Validation, Model Selection and Bootstrap. Chapman & Hall, London.

Jones, P., 1992. Cloud-cover distribution and correlations. J. Appl. Meterol. 31, 732–741.

Macdonald, I., Zucchini, W., 1997. Hidden Markov and other Models for Discrete valued Time Series. Chapman & Hall, London.

Nielsen, L., Prahm, L., Berkowicz, R., Conradsen, K., 1981. Net incoming radiation estimated from hourly global radiation and/or cloud observations. J. Climatol. 1, 255– 272.

Stull, R.B., 1995. Meteorology Today For Scientists and Engineers: ATechnical Companion Book. West Publishing Company, Minneapolis/St. Paul.