Solar Energy 74 (2003) 235–244 Using probabilistic finite automata to simulate hourly series of global radiation a, b * ´ L. Mora-Lopez , M. Sidrach-de-Cardona a ´ ´ ´ ´ Dpto. Lenguajes y C. Computacion, E.T.S.I. Informatica, Universidad de Malaga, Campus de Teatinos, 29071 Malaga, Spain b ´ ´ ´ ´ Dpto. Fısica Aplicada II, E.T.S.I. Informatica, Universidad de Malaga, Campus de Teatinos, 29071 Malaga, Spain Abstract A model to generate synthetic series of hourly exposure of global radiation is proposed. This model has been constructed using a machine learning approach. It is based on the use of a subclass of probabilistic finite automata which can be used for variable-order Markov processes. This model allows us to represent the different relationships and the representative information observed in the hourly series of global radiation; the variable-order Markov process can be used as a natural way to represent different types of days, and to take into account the ‘‘variable memory’’ of cloudiness. A method to generate new series of hourly global radiation, which incorporates the randomness observed in recorded series, is also proposed. As input data this method only uses the mean monthly value of the daily solar global radiation. We examine if the recorded and simulated series are similar. It can be concluded that both series have the same statistical properties.  2003 Elsevier Ltd. All rights reserved. 1. Introduction next values of the series from their predecessors. The approach is as follows: first, the model must be identified; Different approaches have been followed to characterize to do this, the recorded series are statistically analyzed in the hourly series of solar global radiation. Taking into order to select the best model for the series. Then the account the nature of these series, we propose the use of a parameters of the model must be estimated. After this, a new model for their characterization and simulation. This new series of values can be generated using the estimated new model is easy to use once it has been constructed and model. For example, this approach has been followed in it allows us to represent the relationships observed in the Brinkworth (1977), Bendt et al. (1981), Aguiar et al. hourly series of global radiation. Moreover, it can be (1988), Aguiar and Collares-Pereira (1992), and Mora- ´ embedded in engineering software by including the esti- Lopez and Sidrach-de-Cardona (1997). mated probabilistic finite automata and the algorithm One of the problems with most of these methods is that presented in Section 5. Before explaining the model, we the probability distribution functions of the generated briefly review the existing models, paying special attention series are normal when stochastic models are used. This to their simplicity, requirements and limitations. problem can be solved for daily series using first-order Several studies have been carried out to obtain models Markov models (see Aguiar et al., 1988). For hourly series, which allow us to simulate the hourly series of solar global to circumvent this problem, a differenced series and ´ radiation. Traditionally, the analysis of time series has ARMA models can be used (e.g. Mora-Lopez and Sidrach- been carried out using stochastic process theory. One of de-Cardona, 1997); however, in this case the simulation of the most detailed analyses of statistical methods for time a new series uses a complex iterative process: the use of series research was performed by Box and Jenkins (1970). the differences operator makes it difficult to generate new The goal of data analysis by time series is to find models series of global radiation because it is necessary to which are able to reproduce the statistical characteristics of eliminate the negative values which appear in the series. the series. Moreover, these models allow us to predict the The aim of this work was to study the use of a mathematical model called probabilistic finite automata (PFA) as a means of representing the relationships ob- *Corresponding author. Tel.: 134-95-2132-802; fax: 134-95- served in hourly global solar radiation series. PFAs are 2131-397. ´ E-mail address: llanos@lcc.uma.es (L. Mora-Lopez). mathematical models developed within the fields of Artifi- 0038-092X / 03 / $ – see front matter  2003 Elsevier Ltd. All rights reserved. doi:10.1016 / S0038-092X(03)00149-X