RECENT TRENDS IN TIME SERIES MODELING AND PREDICTION OF WIND DATA: STATISTICAL AND FUZZY REASONING APPROACH

Abstract

We developed stochastic time series models such as ARMA( p,q), non- seasonal ARIMA, seasonal ARIMA (SARIMA) and MTM models to simulate and forecast hourly averaged wind speed sequences on twenty year data ,.i.e, 1985-2004 of Quetta, Pakistan. Stochastic Time Series Models take into account several basic features of wind speed including autocorrelation, non-Gaussian distribution and non-stationarity. The positive correlation between consecutive wind speed observations is taken into account by fitting ARMA process to wind speed data. The data are normalized to make their distributions approximately Gaussian and standardized to remove scattering of transformed data (stationary,.i.e., without chaos).Diurnal variations has been taken into account to observe forecasts and its dependence on lead times. We find the ARMA (p,q) model suitable for prediction interval and probability forecasts. But the MTM model is relatively better as a simulator compared to ARMA modeling. The suitability of ARMA (p,q) model for both long range (1-6 hours) and short range (1-2 hours) indicates that forecast values are the deciding components for an appropriate wind energy conversion systems, WECS. ARMA processes work with non-stationary (chaotic) data. Non-seasonal ARIMA models and the prediction equations for each month and indeed for each season of a twenty year wind data are presented. The seasonal ARIMA (SARIMA) and its prediction equations for each month of a twenty year data are also studied. With non- stationarity or chaos in data, stochastic simulator in the ARIMA processes does not effectively work although its prediction equations are good enough to forecast relatively short range reliable values. Various statistical techniques are used on twenty five years, .i.e., 1980-2004 data of average humidity, rainfall, maximum and minimum temperatures, respectively. The relationships to regression analysis time series (RATS) are developed for determining the overall trend of these climate parameters on the basis of which forecast models can be corrected and modified. We followed the coefficient of determination,.i.e., a measure of goodness of fit, to our polynomial regression analysis time series (PRATS). The correlation to multiple linear regression (MLR) and multiple linear regression analysis time series (MLRATS) are also developed from deciphering the interdependence of weather parameters. We used Spearman’s rank correlation and Goldfeld-Quandt tests to check the uniformity or non uniformity of variances in our fit to polynomial regression (PR). The Breusch-Pagan test was applied to MLR and MLRATS, respectively which yielded homoscedasticity (uniformity of variances in the distribution of data). We also employed Bartlett’s test for homogeneity of variances on a twenty five years data of rainfall and humidity, respectively which showed that the variances in rainfall data are not homogenous while in case of humidity, are homogenous. Our results on regression and regression analysis time series show the best fit to prediction modeling on climatic data of Quetta, Pakistan. We performed design free fuzzy logic (FL) time series prediction modeling on a twenty year wind data, .i.e., 1985-2004 for Quetta, Pakistan. We followed design free fuzzy logic and obtained prediction of hourly wind data for spring (February, March and April). Non-stationarity or random walk in wind data exists but it does not influence prediction. Mackey Glass (MG) simulation of wind data indicated chaos or non periodicity. Moreover, stable attractors are observed in MG-time series, the origin of which is yet unknown. The attractors seen in MG simulation do not influence FL time series prediction. We studied singleton and non-singleton type-1 back propagation (BP) designed sixteen rule fuzzy logic system (FLS) on hourly averaged wind data of twenty years ,.i.e., 1985-2004. We found that the BP designed 16 rule non-singleton-type-1 FLS is relatively a better forecaster than singleton-type-1.We find hidden or unraveled uncertainties such as non-stationarity and stable attractors. These uncertainties make the data chaotic. The criterion of selecting root mean square error (RMSE) for establishing comparison is not suitable for chaotic data. Non-stationarity in the data can be properly handled with non- singleton type-1 FLS, therefore, there appears no reason to use a type-2 FLS. The stable attractors and non-stationarity in our data do not affect the predicted values as confirmed by Mackey Glass simulation. The chaos can be effectively resolved through parallel structure fuzzy system (PSFS) which exploits time-delays.. A variety of Artificial Neural Network models for prediction of hourly wind speed (which a few hours in advance is required to ensure efficient utilization of wind energy systems) at Quetta, Pakistan is studied and the results are compared. Satisfactory results are obtained with Feed Forward Back Propagation Neural Networks (FFBPNN). An empirical relationship is developed which shows the Gaussian profile for the number of neurons which varies with lag inputs, .i.e., nn = k exp(-il2) where nn shows the number of neurons, il the lag inputs, and k the sloping ratio. Feed Forward Neural Networks (FFNNs) can be corrected with optimization of empirical relationship for simulators followed by back propagation technique. The disadvantages of FFNNs comprise of heavy computational requirements, and non-existence of Artificial Neural Network(ANN) design methodologies for deciding the value of the learning rate and momentum. Neural Network (NN) modeling is not suitable for chaotic data characterized by randomness and non-stationarity.