Prediction Intervals for Short-Term Load Forecasting Neuro-Fuzzy Models

In the paper the problem of estimation of the prediction intervals (error bars) for the family neuro-fuzzy Short-Term Load Forecasting (STLF) models is discussed. We investigate two neuro-fuzzy networks: Fuzzy Basis Function (FBF) Networks, and linear neuro-fuzzy model with Tagagi-Sugeno reasoning. The paper contains comparison of selected most important methods for error bars calculation (analytical delta method, and bootstrap), and discusses the obtained results in context STLF. Streszczenie. W artykule zaprezentowane zostały metody wyznaczania przedziałów prognozy dla rodziny neuronowo rozmytych modeli krótkoterminowego prognozowania obciążenia sieci. Przebadane zostały dwa rodzaje sieci neuronowo-rozmytych: sieci Fuzzy Basis Function (FBF) i liniowe neuronowe modele rozmyte z wnioskowaniem typu Takagi-Sugeno. Artykuł obejmuje porównanie najistotniejszych metod szacowania przedziałów prognozy: analitycznej metody delta i bootstrapu), dyskutując wyniki w kontekście krótkoterminowych prognoz obciążenia sieci. (Metody wyznaczania przedziałów prognozy dla rodziny neuronowo rozmytych modeli krótkoterminowego prognozowania obciążenia sieci) Introduction Accurate prediction of the loads has a significant impact on economic and reliable operation of an electric power system. A lot of decisions and operating procedures in power companies require estimates of the energy demand in the future. In short time horizon, future loads are influenced mainly by their past values and weather factors (for instance temperature, pressure, humidity, etc.). The later relationship is commonly complex, implicit and nonlinear. Because of this short-term load forecasting (STLF) tools, based on nonlinear modelling methods, have ability to outperform classical short-term load forecasting models, especially during rapid changes in weather conditions. Good results have been achieved using neural networks [6-7], and neuro-fuzzy models [1][5][8]. In the paper the application of the family neuro-fuzzy models to short-term load forecasting problems is discussed. We investigate the problem of estimation of prediction intervals (error bars) for the energy demand forecasts. There were developed several method of error bars assignment for nonlinear regression models. They showed reasonable good accuracy in STLF tasks for neural networks models (multilayered perceptrons) [2-4], [6-7]. The goal of this paper is to verify their usefulness in case of neuro-fuzzy networks. The other approaches to prediction intervals estimation for neuro-fuzzy systems follow general theory of uncertainty assessment in nonlinear models. They are based mainly in empirical error estimation on test data set [10] or analytical assessment on training set [9] (co called delta method). In this paper we investigate both kinds of approaches but in more complex form, with more exact results. The empirical approach considered here, in opposition to [10] is based on bootstrap approach, with resampling procedure of the test set. The delta method in [9] uses approximation of the model weights covariance matrix with outer product of first order derivatives of the model on training set patterns (so called outer product approximation or Levenberg-Marquard approximation). We have developed for this purposes full error Hessian algorithms for neuro-fuzzy networks. In case of STLF it gives significant improvement of the results. The first considered model, Fuzzy Basis Function (FBF) Network is based on additive fuzzy logic system, with Gaussian fuzzy sets in premises of the rules, Larsen product aggregation rule, and utilizes crisp numbers in consequents of the rules. As a result, the FBF models are functionally equivalent Radial Basis Function neural networks, and show similar approximation capabilities. The equation of FBF network can be written as: