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Innovative Methods for Time Series
Modelling of Climate Variables
John Boland*, Kathryn Ward and Malgorzata Korolkowiecz
School of Mathematics and Statistics,
University of South Australia,, Email*: john.boland@unisa.edu.au
Overview
We will discuss modelling the level and volatility of climate variables, specifically solar radiation and wind speed using non-standard time series techniques. They are non-standard only in that they are normally not used for modelling these variables. In fact, it is not wind speed per se that we model, but rather the output from single wind farms or a collection of wind farms.
Measurements of the components of solar radiation - global, diffuse and direct traditionally have been made at only a limited number of sites. In recent times, for various reasons including the increased use of satellite images, this coverage has decreased further. To use simulation models to predict output from systems under the influence of solar radiation, hourly data values are usually needed. There are various approaches to forecasting solar radiation, using alternatively, Markov models, state space models, neural nets or Box-Jenkins methods. We use the latter, coupled with models of the deterministic component identified using spectral analysis, which we also denote as classical time series modelling structures [1]. There is a specific reason for choosing this methodology. It is the approach that gives the most knowledge of the underlying physical nature of the phenomenon. We describe the behaviour of global solar radiation on the daily time scale. In so doing, we identify the various components inherent in the time series, seasonality, autoregressive structure, and the statistical properties of the white noise.
Wind power is becoming a prominent source of much sought after renewable energy. In order to integrate wind energy into electricity grids we need to be able to predict with accuracy how much energy a wind farm will produce. Due to the volatile nature of wind farm output we need a method that will be able to cope with high fluctuations as well as periods of relatively steady activity.
Methods
The traditional time series approach described above is of great value, but may miss some of the inter-dependence of the volatility terms. Financial time series analysis has focused on specific models for this volatility, using techniques described as auto-regressive conditional heteroscedastic (ARCH) and Generalised ARCH (or GARCH) models and their variants [2]. Essentially, these are used to describe series that have weak or no auto-regressive structure but still display dependence. These models are normally applied to series such as stock market indices, but we describe the use of them for solar radiation time series. If one wishes to construct a solar power electricity generator, then forecasting volatility as well as level is essential for participation in the electricity market. We describe how the methods adopted from financial time series can aid in forecasting the volatility for solar radiation series.
Wind farm output requires a different methodology since there are times where the output is zero, and not systematically as with solar radiation, but randomly. Hidden Markov Models (HMMs) [3] capture the nature of highly volatile systems without needing a lot of information. HMMs have been used in financial markets for a number of years to predict stock volatility [4]. We will use HMMs to capture the nature of the system in order to predict future wind farm output and volatility.
Preliminary Results
After modeling the seasonality of daily totals of solar radiation with Fourier series, we turn to using ARMA methods for the deseasoned data. The example site exhibits an autoregressive structure, as is expected, and is in the form
The shock EMBED Equation.3 demonstrates no correlation, but the shock squared series has a long range autoregressive structure, suitable for modeling as a low lag ARMA process, typical of a GARCH model. It is this form that demonstrates the volatility clustering inherent in series following the ARCH maxim.
We are in the first stages of determining the state space for the HMM formulation for modelling wind farm output, and constructing the model.
Conclusions
The results of preliminary investigations of daily solar radiation time series show that there is promise for modeling the volatility of these time series using the ARCH paradigm. One can then better estimate the error bounds on forecasts than under the assumption of homoscedsticity. This gives credence to the idea that on a sub-diurnal time scale, we may well be better able to estimate the error bounds as well as the level of solar radiation if we assume the series is non-systematically heteroscedastic. We intend to investigate this possibility for delivery at the conference. We will also present the state of progress with modelling wind farm output using HMMs.
References
Boland J. (2008) Time series and statistical modelling of solar radiation, Recent Advances in Solar Radiation Modelling, Viorel Badescu (Ed.), Springer-Verlag,(in press).
Bollerslev T. (1986) Generalised autoregressive conditional heteroskedasticity, Journal of Econometrics, 31, pp. 307-327.
R. J. Elliott, L. Aggoun, and J. B. Moore, Hidden Markov Models, Estimation and Control. New York: Springer-Verlag, 1995
L. R. Rabiner, (1989) A tutorial on hidden Markov models and selected applications in speech recognition, Proc. IEEE, vol. 77, pp. 257285.
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