Hourly solar radiation forecasting using optimal coefficient 2-D linear filters and feed-forward neural networks

Abstract

In this work, the hourly solar radiation data collected during the period August 1, 2005–July 30, 2006 from the solar observation station in Iki Eylul campus area of Eskisehir region are studied. A two-dimensional (2-D) representation model of the hourly solar radiation data is proposed. The model provides a unique and compact visualization of the data for inspection, and enables accurate forecasting using image processing methods. Using the hourly solar radiation data mentioned above, the image model is formed in raster scan form with rows and columns corresponding to days and hours, respectively. Logically, the between-day correlations along the same hour segment provide the vertical correlations of the image, which is not available in the regular 1-D representation. To test the forecasting efficiency of the model, nine different linear filters with various filter-tap configurations are optimized and tested. The results provide the necessary correlation model and prediction directions for obtaining the optimum prediction template for forecasting. Next, the 2-D forecasting performance is tested through feed-forward neural networks (NN) using the same data. The optimal linear filters and NN models are compared in the sense of root mean square error (RMSE). It is observed that the 2-D model has pronounced advantages over the 1-D representation for both linear and NN prediction methods. Due to the capability of depicting the nonlinear behavior of the input data, the NN models are found to achieve better forecasting results than linear prediction filters in both 1-D and 2-D.

Introduction

Hourly solar radiation data forecasting has significant consequences in most solar applications such as energy system sizing and meteorological estimation. Accurate forecasting improves the efficiency of the outputs of these applications. Classically, the solar radiation data can be regarded as a random time series produced by a stochastic process, and its prediction depends on accurate mathematical modeling of the underlying stochastic process. Using an accurate model, the prediction is mathematically the conditional expectation of the data given the model and the past data samples. On the other hand, the computation of such conditional expectation, which is in general non-linear, requires the knowledge of the distribution of the samples including higher order statistics. Since the available or recorded data is finite, such distributions can be estimated or fit into pre-set stochastic models such as Auto-Regressive (AR) (Maafi and Adane, 1989), Auto-Regressive Moving-Average (ARMA) (Mellit et al., 2005), Markov (Amato et al., 1986, Aguiar et al., 1988), or learning adaptive systems such as Neural Networks (NNs) (Cao and Cao, 2006). NNs can be trained to predict results from available examples, and they are also able to deal with nonlinear problems.

The determination and optimization of mathematical model parameters is normally done by training algorithms. Once the training is complete, the predictor can be settled to a fixed function for further prediction or forecasting. In recent years, a number of researchers have used NNs for prediction of hourly global solar radiation data (Sfetsos and Coonıck, 2000, Cao and Cao, 2006.). There are several studies about modeling the solar radiation data in the literature (Chena et al., 2007, Cucumo et al., 2007, Kaplanis, 2006, Aguiar and Collares-Pereira, 1992, Kaplanis and Kaplani, 2007). In these works, the data is treated in its raw form as a 1-D time series therefore the inter-day dependencies are not exploited.

This paper presents a novel and low-complexity approach for hourly solar radiation forecasting. The approach is based on a new representation which renders the data in a matrix to form a 2-D image-like model, as explained in Section 2. Although 2-D rendering of a time-series data was already used for some other types of 1-D signals, this kind of an approach is novel in the area of solar radiation signal processing. In fact, the 2-D representation of solar radiation data was first introduced in (Hocaoğlu et al., 2007) using a limited number of prediction methods. In this work, a comprehensive and comparative study is carried out to test the 2-D representation efficiency with several linear and neural prediction filters. The single direction (1-D) and multiple direction (2-D) prediction templates are tested, compared, and optimized to find the optimal template for the representation.

The proposed 2-D representation clearly enables the visualization and comprehension of seasonal inter-day dependencies along the same hour segments of the days. This is due to the fact that the column elements of consecutive days corresponding to the same hour of the day are located as vertically neighbor in 2-D data. Signal compression theory indicates that locating relatively correlated samples in geometrically near positions in a representation greatly improves predictability of the data. This intuitive observation is also justified in the case of solar radiation data by the conducted experiments, which yield the fact that full 2-D prediction templates provide better prediction compared to 1-D prediction. Mathematically, the stochastic part of the solar radiation data is due to transient atmospheric phenomena. These non-predictable variations are often secondary to the oscillatory variations determined by solar geometry which are entirely predictable as a function of latitude, date, and time. The latitude of a geometrical place stays constant, leaving the varying parameters as date and time. The idea of 2-D rendering is motivated by the fact that there is a pronounced periodicity of the solar radiation data with periods of exactly 1 day. If the data is presented in 1-D form along an axis of hours, the daily oscillations and date-wise (seasonal) oscillations become occluded and the data get difficult to interpret. It is illustrated here that splitting the date and time variables into two separate axes better exploits the predictable sinusoidal behavior along both time and date parameters.

After describing the 2-D rendering method in Section 2, the first experiments conducted to test the 2-D model efficiency utilized optimal linear image prediction filters (Gonzalez and Woods, 2002), as explained in Section 3. In order to take into account the adaptive nature of complex (due to non-predictable fluctuations) and non-stationary (oscillatory) time series, neural networks are also applied to the forecasting problem in Section 4. The training algorithms for NNs are also discussed in this Section. In Section 5, performance evaluation methods for predictors are described and the numerical forecasting results that are obtained from both optimal linear filters and neural network models are presented in Section 6.

Throughout this work, our own solar recordings were used for the implementation of the proposed methods. This was possible due to the wind and solar observation station established at Iki Eylul campus of Anadolu University, Turkey, in order to determine the wind and solar energy potentials of the region. The data collected in this observation station between the dates of July 1, 2005 and September 30, 2006 were evaluated via the CALLaLOG 98 software and by algorithms implemented in MATLAB. The data is saved in a data logger at 1 h time intervals (Kurban and Hocaoglu, 2006). The continuity of data acquisition was constantly monitored in our labs. Therefore, the quality of the data was assured. The accuracy of the data was also cross-checked by the data obtained from the National Meteorological Station values for Eskisehir region.