G. Lopez1*, F. J. Batlles2 and J. L. Bosch2

1 Departamento de Ingenieria Electrica y Termica, Escuela Politecnica Superior, Universidad de Huelva,

21819, Huelva, Spain

2 Departamento de Fisica Aplicada, Universidad de Almeria, Almeria, Spain
Corresponding Author, gabriel. lopez@die. uhu. es

Abstract

In this work we analyse a daily global irradiance time series into the framework of chaotic dynamic systems in order to examine a possible underlying nonlinear behaviour. Employed methods are based on a phase space reconstruction from the measured data and are devoted to the calculation of the properties of an underlying attractor, such as the Lyapunov exponents. Researches on these dynamical system invariants will point out the presence of chaos. We also use local lineal models as a test for nonlinearity.

The global solar radiation data were measured at the radiometric station of the University of Almeria (Spain) during eight years. Results have shown the non-existence of any attractor in the phase space for the global irradiance time series. Negative Lyapunov exponents exclude a chaotic behaviour that might allow a better short term prediction than autoregressive models, and the idea of the existence of a nonlinear differential equation system. These results match with those obtained from applying local linear models for prediction, of which estimations suggest that the data are best described by a linear stochastic process.

Keywords: solar radiation, forecasting, chaos, nonlinear time series.

1. Introduction

Information on the availability of solar radiation is needed in many applications dealing with the exploitation of solar energy. Particularly, global solar radiation is one of the most important input parameters for any solar energy system and different techniques have been developed to model and forecast it. Once the solar energy system (like concentrated solar power plants) is running, prediction of power load, normally done on an hourly basis with a prediction horizon between 1 and 24 hours, is instrumental for planning and operation of the total power system, e. g. for buying or selling power or for solving the unit commitment and dispatch problems.

Studies about solar radiation time series and other meteorological variables with autoregressive or stochastic models [1-3] have experienced an important growth in the last years, since synthetic sequences statistically indistinguishable from the original ones are needed to design the solar devices properly. However, using this type of model, prediction from past values is limited due to the random character exhibited by these time series. The main factor of randomness affecting global solar radiation data is due to variations in the sky cloud cover, which make the radiation on cloudy days difficult to predict.

Appearing of several nonlinear dynamical models developed by Lorenz [4], with a fully irregular and complex behaviour (generically named chaos), allows to study the nature of fluctuations in solar radiation data from a new methodology. Time series from many system evolutions, apparently evolving in a random way, can now be predicted with higher precision than with traditional ARMA models, at least for short term predictions. Detection of chaos in a time series would thus allow a better modelling of time series against statistically based models.

In this paper, we analyse a daily global solar irradiance time series into the framework of chaotic dynamic systems in order to examine a possible underlying nonlinear behaviour. Existence of chaos should help to improve the short term predictions performed by means of the traditional
statistical techniques. Employed methods are based on a phase space reconstruction from the measured data and are devoted to the calculation of the properties of an underlying attractor, such as the Lyapunov exponents. We also use local lineal models as a test for nonlinearity.

2. Experimental Data

The horizontal global solar radiation time series (symbolized by {xt}, being t the time step in days) consisted of 2945 daily values. They were obtained by integration from experimental values averaged every ten minutes during the years 1990-1992 and every five minutes during the years 1993-1998. The measurements were recorded at the radiometric station of the University of Almeria (36.8° N, 2.44° W) in south-eastern Spain by means of a Kipp and Zonen CM-11 pyranometer. Measurements from other two pyranometers (LI-COR, model LI-200) were also available. They were used to detect and replace any anomalous value measured by the CM-11 pyranometer. The calibration constants of the pyranometers were checked periodically by our research group. Measurements by the LI-200 pyranometers were also corrected following [5].

3. Methodology and results