# Simulation of global solar radiation based on cloud observations

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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

6 162 J.S.G. Ehnberg, M.H.J. Bollen / Solar Energy 78 (25) Amato, U., Andretta, B., Bartoli, B., Coluzzi, B., Cuomo, V., Markov processes and Fourier analysis as a tool to describe and simulate daily solar radiation. Int. J. Solar Energy 37, Badescu, V., 22. Anew kind of cloudy sky model to compute instantaneous values of diffuse and global solar irradiance. Theor. Appl. Climatol. 72, Balouktsis, A., Tsalides, P., Stochastic simulation model of hourly total solar radiation. Int. J. Solar Energy 37, Balouktsis, A., Tsanakas, D., Vachtsevanos, G., Stochastic modelling of daily global solar radiation. Int. J. Solar Energy 7, 1 1. Graham, V., Hollands, K., 199. Amethod to generate synthetic hourly solar radiation globally. Int. J. Solar Energy 44, Gu, L., Fuentes, J.D., Garstang, M., Da Silva, J.T., Heitz, R., Sigler, J., Shugart, H.H., 21. Cloud modulation of solar irradiance at a pasture site in southern Brazil. Agric. Forest Meteorol. 16, Hjort, U., Computer Intensive Statistical Methods: Validation, Model Selection and Bootstrap. Chapman & Hall, London. Jones, P., Cloud-cover distribution and correlations. J. Appl. Meterol. 31, Macdonald, I., Zucchini, W., Hidden Markov and other Models for Discrete valued Time Series. Chapman & Hall, London. Nielsen, L., Prahm, L., Berkowicz, R., Conradsen, K., Net incoming radiation estimated from hourly global radiation and/or cloud observations. J. Climatol. 1, Stull, R.B., Meteorology Today For Scientists and Engineers: ATechnical Companion Book. West Publishing Company, Minneapolis/St. Paul.

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