Hourly solar radiation forecasting using optimal coefficient 2-D linear filters and feed-forward neural networks
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.
Section snippets
A novel representation for the solar radiation data
In this section, the novel 2-D representation (Hocaoğlu et al., 2007) is explained. In order to visualize the efficiency of our new representation, one year data (Agust 1, 2005- July 30, 2006) data are first considered as one-dimensional time series, and then as a 2-D image signal. In order to observe one year behavior of radiations, the data samples are reordered to span from January to December, and then plotted in Fig. 1. Next, the same data are rendered as a two-dimensional matrix (Eq. (1))
Optimal 2-D linear prediction filter design
The proposed 2-D representation has significant advantages which makes prediction and forecasting more robust and accurate. In Hocaoğlu et al. (2007), a preliminary set of linear prediction filters were tested with the 2-D representation. In this work, the experiments are extended to evaluate and compare linear prediction filters with different template sizes and structures. Throughout this section, we first construct the basis of 2-D linear prediction. Due to predictive image coding
Artificial neural networks
An alternative method to use in the proposed 2-D representation is the adaptive or nonlinear method that converges to a predictor for the solar radiation data. The classical and well known method for this purpose is the application of NNs which have been widely used in many areas, such as control, data compression, forecasting, optimization, pattern recognition, classification, speech, vision, etc. NNs have been utilized to overcome the limitations of the conventional and linear approaches to
Prediction performance assessment
The prediction success of a mathematical model normally depends on the correlation among samples of the data. A strong correlation is an indicator of successful prediction with small prediction error. It is also critical to observe or determine which data samples are correlated with each other. In this work, it is argued that hourly variations of recorded solar radiation are correlated as well as the recordings corresponding to the same hour along consecutive days. Therefore, the proposed 2-D
Experimental results
The correlation coefficients between solar radiations at any hour of any day i (XHi,Di) and day j (XHj,Dj) are calculated and tabulated for up to 3 h (horizontal) and two days (vertical) difference. This corresponds to a vertical distance of up to two pixels and horizontal distance of up to three pixels in the 2-D representation. The results are presented in Table 1. It can be seen that the correlation decreases as the pixel difference increases in both directions. It is worthy to note, however,
Conclusions
In this work, a novel 2-D approach, which was recently proposed (Hocaoğlu et al., 2007) for hourly solar radiation forecasting, is comprehensively evaluated through linear and neural network prediction models. The hourly solar radiation data are rendered as a 2-D image and its correlation properties are examined. It is initially observed that 2-D representations give more insight to the solar pattern than the regular 1-D interpretation. Furthermore, the 1-D and 2-D interpretations are found to
Acknowledgements
This work is supported in parts by TUBITAK (Turkish NSF) Grant no: 107M212, and Anadolu Univ. Research Fund Contract no: 040258.
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