Simulation of global solar radiation based on cloud observations

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1 Solar Energy 78 (25) 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, Göteborg SE , Sweden Received 6 August 23; received in revised form 23 August 24; accepted 23 August 24 Available online 14 October 24 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 longterm 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 observations are normally only every hour or every third hour. The model can easily be combined with cloud coverage simulations, 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. Ó 24 Elsevier Ltd. All rights reserved. Keywords: Solar power; Markov model; Cloud coverage 1. Introduction * Corresponding author. Tel.: ; fax: address: jimmy.ehnberg@elteknik.chalmers.se (J.S.G. Ehnberg). In rural areas of developing countries the interest in electrification has increased over the last few years resulting in an improved standard of living. This is confirmed by many ongoing research programs. The problem of electrification of remote rural areas may be solved by 38-92X/$ - see front matter Ó 24 Elsevier Ltd. All rights reserved. doi:1.116/j.solener

2 158 J.S.G. Ehnberg, M.H.J. Bollen / Solar Energy 78 (25) Nomenclature d s vector of the solar declination angles over a time period determined by d (rad) U r the tilt of the earthõs axis relative the orbital plane of the earth around the sun, U r =.49 rad C C=2p (rad) d vector of days of the year for the time period of the simulation (days) d r the day of the year at summer solstice, 22nd of June for non-leap years (days) d y total number of days in a year (days) w vector of local elevation angle angles over a time period determined by d (rad) t UTC Co-ordinated Universal Time (h) t d hours in a day (h) / latitude of the location, positive north of the equator (rad) k e longitude of the location, positive west of Greenwich (rad) G h vector of global solar radiation (hourlyscale) over a time period determined by d (W/m 2 ) G min global solar radiation (minute-scale) over a time period determined by d (W/m 2 ) N number of Oktas L, a, a i, i =, 1, 3 empirical determined constants (W/m 2 ) e statistically varying term, e N (m,r 2 ) k ij estimated transition probability from i to j f ij number of transitions from state i to j bk estimated transition probability matrix 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 simulation 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 several 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 conditions but the authors mentioned the importance of the cloud coverage. In (Graham and Hollands, 199) hourly radiation has been modelled but the model could be difficult to apply due to the data requirements. Monthly average values of global radiation are needed which can only be obtained from long time measurements. Another model is proposed in (Balouktsis and Tsalides, 1986) but the problem with the input of the model remains. Alocation-dependent factor is used which depends on the probability distribution of the solar radiation. This model can again only be used when a large amount of solar radiation data is available. Outside the atmosphere the solar radiation can accurately be determined (Stull, 1995) and the atmosphere will induce the randomness (Graham and Hollands, 199). 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 relation between cloud coverage and hourly global solar radiation, 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 level 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 simulation 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., 21; Badescu, 22). In (Gu et al., 21) 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, 22) 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.

3 J.S.G. Ehnberg, M.H.J. Bollen / Solar Energy 78 (25) 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 simulations for power systems without storage capacity, simulation data with higher resolution than one hour is needed in some cases. This is the case when short-duration interruptions (less than one half hour) are a concern. In this paper a model for simulating six minutes values 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 described by Eqs. (1) and (2) (Stull, 1995). The equation for the seasonal effects (1) is an approximation under the assumption of circular orbit of the earth around the sun. This assumption is allowed because the excentricity is only.7 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. d s ¼ U r cos Cðd d rþ ð1þ d y sin w ¼ sin / sin d s cos / cos d s cos C t UTC t d 3. Atmospherical part of the model k e ð2þ The randomness in this simulation model for global solar radiation is introduced in this section. The importance of the randomness in the atmosphere is also discussed in (Graham and Hollands, 199). An empirically determined relationship between the global solar radiation and the cloud coverage, Eq. (3) was obtained by (Nielsen et al., 1981), after many years of cloud observations, solar elevation measurements and global solar radiation measurements. The obtained relationship reads as follows: G ¼ a ðnþþa 1 ðnþ sin w þ a 3 ðnþsin 3 w LðNÞ ð3þ aðnþ The values of the constants L(N), a(n) and a i, i =,1,3, in Eq. (3) are given in Table 1 and the elevation angle from Eq. (2) can be used. In Fig. 1 the 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 standard error of the estimation was less then 8 W/m 2 and has a square correlation coefficient of approx..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 negative it is from the surface of the earth upwards. This radiation 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 < org h ðiþ < then G h ðiþ ¼ 8i ð4þ Table 1 The empirical determined coefficients for (3) N a a 1 a 3 a L Sine of the Solar Elevation Angle Fig. 1. The relationship between global solar radiation and the solar elevations angle for different cloud coverage

4 16 J.S.G. Ehnberg, M.H.J. Bollen / Solar Energy 78 (25) By examining global solar radiation measurements it can be seen that the radiation varies within a one-hour period. Introducing a statistically varying term according to Eq. (5) could simulate this phenomenon. This statistical term (e) was set to have the same distribution as the short duration variations seen in the measurements. G min ¼ G h þ e ð5þ The statistically varying term can be estimated through cross validation, the so-called hold out method proposed by (Hjort, 1995). The deviation from the hourly mean values for daytime can be fitted to a normal distribution and the mean value and the standard deviation estimated. Deviation [%] Cloud coverage [Oktas] 4. Cloud coverage simulation To be able to perform a simulation over a longer period 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 coverage is only depended on the previous value, a discrete Markov model can be used. With this assumption a model with nine different states ( 8), corresponding to the nine levels in the Oktas scale is proposed. The transitions probability matrix ( bk) can be estimated from measured cloud coverage data. The transition probabilities can by estimated in many ways but the most intuitively is: ^k ij ¼ f ij P 8 k¼ f ik 2 3 ^k ^k1 ^k8 ^k 1 ^k11 ^k18 bk ¼ ^k 8 ^k81 ^k88 ð6þ 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. 2 and maximum deviation is 1.5 %, for Oktas. Cloud coverage data for Göteborg was used in the figure. 5. Case study,göteborg To show the applicability of the model a case study has been made. The case study was made for Göteborg (Lat N, Long E). Göteborg is normally minus one hour from Greenwich, which is included in the following calculations. Fig. 2. The deviation between measurements and simulations for each cloud coverage value. The error band indicates the standard deviation. The transition matrix for the Markov model for the cloud coverage simulation was estimated from measurements of cloud coverage obtained during the period from 1973 to The transition probabilities were estimated according to Eq. (6). They were estimated every three hours because the available cloud coverage observations was done with this interval. The estimated transition matrix ( bk) for Göteborg is presented below, with values given as a percentage :8 22:5 7:1 4:7 2:7 2:3 1:7 2:6 2:6 15:5 45:5 14: 9:1 4:3 3:7 3:2 3: 1:5 7: 24:5 23:4 15:3 8:8 7:2 6:2 5:4 2:2 3:8 13:4 17:7 2:3 12:6 1:6 9: 9:1 3:4 bk ¼ 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: 3: 5:2 7:4 9:5 14:2 22:2 28: 9: :6 2: 2:3 3: 3:9 6:3 11:3 5:3 2:45 :5 :7 :8 1:1 1:3 2: 3:8 13:5 76:3 ð8þ The mean value of the deviation between the simulated and the observed cloud coverage values is shown in Fig. 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 independently made simulations. From the figure the conclusion 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 Göteborg according to Eq. (3). Since the meteorological data used for the estimation of the transition probability matrix was only every three hours, a linear interpolation was used to achieve

5 J.S.G. Ehnberg, M.H.J. Bollen / Solar Energy 78 (25) Time [hours] Fig. 5. Comparison between calculated and measured values for a day in June Day of the year Fig. 3. The maximum daily value of the global solar radiation in Göteborg during one year Day of the year Fig. 4. The global solar radiation for a few days in February in Göteborg. 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 ( Oktas) and totally cloud covered sky (8 Oktas). 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 obtained. For the statistical term, a normal distribution was used with a mean value of zero and a standard deviation of 4 W/m 2. For the results, shown in Fig. 5, measured 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 figure as the measured values. Asolid line represents the measured values while the calculated values are presented 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 limitations 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., Measurements and predictions of solar radiation incident on horizontal surfaces at Santa Fe, Argentina (31 39 S, 6 43 W). Renewable Energy 4,

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