Forecasting solar radiation on an hourly time scale using a Coupled AutoRegressive and Dynamical System (CARDS) model

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1 Available online at Solar Energy 87 (2013) Forecasting solar radiation on an hourly time scale using a Coupled AutoRegressive and Dynamical System (CARDS) model Jing Huang, Małgorzata Korolkiewicz, Manju Agrawal, John Boland School of Mathematics and Statistics and Barbara Hardy Institute, University of South Australia, Mawson Lakes Boulevard, Mawson Lakes, SA 5095, Australia Received 5 November 2011; received in revised form 18 October 2012; accepted 23 October 2012 Available online 21 November 2012 Communicated by: Associate Editor Christian Gueymard Abstract The trends of solar radiation are not easy to capture and become especially hard to predict when weather conditions change dramatically, such as with clouds blocking the sun. At present, the better performing methods to forecast solar radiation are time series methods, artificial neural networks and stochastic models. This paper will describe a new and efficient method capable of forecasting 1-h ahead solar radiation during cloudy days. The method combines an autoregressive (AR) model with a dynamical system model. In addition, the difference of solar radiation values at present and lag one time step is used as a correction to a predicted value, improving the forecasting accuracy by 30% compared to models without this correction. Ó 2012 Elsevier Ltd. All rights reserved. Keywords: Solar radiation forecasting; Time series method; Lucheroni model; Combination model; Fixed component; CARDS model 1. Introduction When methods for forecasting solar radiation time series were first developed, the principal applications were for estimating performance of rooftop photovoltaic or hot water systems. If there were significant errors in the forecast, the consequences were not severe. In recent times there has been increasing development of larger solar installations, both large scale photovoltaic and also concentrated solar thermal. In order to first influence financial backers to participate in their development, and also to potentially compete in the electricity markets, better forecasting models are required than simple Box Jenkins models, such as those outlined in Boland (2008). In this paper we explore a range of possible improvements in forecasting skill, and suggest a Combination model linking Corresponding author. Tel.: ; fax: address: huajy017@mymail.unisa.edu.au (J. Huang). a standard autoregressive approach with a resonating model borrowed from work on dynamical systems, and also an additional component that greatly enhances forecasting ability. There are many distinctive models for forecasting global solar radiation, such as those based on time series methods (Boland, 2008; Wu and Chan, 2011), Artificial Neural Networks (ANNs) (Mihalakakou et al., 2000; Tymvios et al., 2005; Cao and Lin, 2008; Mellit et al., 2010), satellite-derived cloud motion forecast models (Perez et al., 2010) and stochastic prediction models (Kaplanis and Kaplani, 2010) that use the volatility of the data over the whole time period. The inclusion of long-term volatility estimates into the design of stochastic prediction models leads to a method that works quite well overall (Kaplanis and Kaplani, 2010). However, predictions at individual points can actually be worse because of the inclusion of the stochastic component. In Gueymard (2000), instead of predictions for individual hourly periods for a specific day and X/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved.

2 J. Huang et al. / Solar Energy 87 (2013) year, long-term distributions of daily irradiation are used to obtain the mean hourly distribution of global radiation over the average day of each average month. Some of these models work very effectively in normal weather conditions, but not as well in extreme weather situations. For example, the Autoregressive and Moving Average (ARMA) model, which is considered to be a stable model can produce a large error (Wu and Chan, 2011) in extreme conditions. Some ANN models have been found to produce less accurate solar radiation forecasts when the weather conditions changed dramatically (Mihalakakou et al., 2000). Even a hybrid model, such as an ARMA plus a Time Delay Neural Network (TDNN) (Wu and Chan, 2011), does not always improve the accuracy of solar radiation prediction in extreme weather. Note that Kostylev and Pavlovski (2011) have defined the benchmark to which one may compare the performance of a forecasting tool. They surveyed the literature on forecasting at various time scales, from a number of days, down to hourly forecasts. They found that the best performing model on an hourly time scale had an NRMSE of 16 17% for mostly clear days and 32% for mostly cloudy days. In this paper, a new method for one-period ahead forecasting of global solar radiation is developed. That is, given the observed radiation F t at time t, a forecast is made for F t+1. The proposed method is based on a combination of autoregressive (AR) model and a dynamical system model. The dynamical system model considered here is the resonating model introduced in Lucheroni (2007), here termed the Lucheroni model, which is particularly well suited to high frequency data analysis, such as solar radiation. The resulting forecasting model is used to obtain 1-h ahead solar radiation forecasts for Mildura, a small town in western Victoria, Australia and its performance is assessed using a range of forecasting accuracy measures. Comparisons are also made with a number of other models using similar time scales available in the literature. In future work, other time scales will be considered as well. These will include half hour and 5-min time scales, which are the critical scales in the operation of the Australian electricity market. The paper is organized as follows. Section 2 describes the data and Fourier series model of that data while gives the seasonal component, after the seasonal component is subtracted from the data, the residual time series is analyzed in Section 3. The Lucheroni model is described in Section 4 and the combination of AR and Lucheroni models in Section 5. Error analysis and comparison of these three models are given in Section 6. A new method to improve forecasting performance, based on a fixed component, is described and applied to the Combination model in Section 7. The result is the Coupled AutoRegressive Dynamical System (CARDS) model. Comparisons of the CARDS model with other models are made in Section 8. The final section is devoted to conclusions and a brief discussion of future work. 2. Determining the seasonality of a solar radiation series The global solar radiation data used in the development of the prediction procedure is from Mildura, during the year Mildura is a small city in north western Victoria, Australia, Latitude S, Longitude E and the time zone is AEST (UTC+10). The data set consists of (24 365) hourly global radiation values in total. For the time period considered, 54% of days were sunny; the remaining days included cloudy periods. The average hourly global radiation was W/m 2 and the maximum value was 1146 W/m 2. When analysing a time series data set, the first step is to consider whether it contains a trend, or seasonality, or both. In this case, seasonality is the only significant feature of global solar radiation of the Mildura 2000 data, thus the first step is to deseason the data. From Tolstov (1976), the Fourier series of a function S t is given by S t ¼ 1 2 a 0 þ X1 a n cosðntþþ X1 b n sinðntþ where a 0 ¼ 1 N a n ¼ 1 N b n ¼ 1 N X 2N S t k¼1 X 2N k¼1 X 2N k¼1 n¼1 n¼1 ð1þ ð2þ pnk S t cos ; n ¼ 0; 1; 2 Nt ð3þ S t sin pnk ; n ¼ 1; 2; 3 Nt ð4þ However, for the purposes of calculations, this model can be transformed into another form. Following Boland (1995, 2008), and using the results of Power Spectrum analysis, S t can be written as: S t ¼ a 0 þ a 1 cos 2pt þ b 1 sin 2pt þ a 2 cos 4pt þ b 2 sin 4pt þ X11 X 3 X 1 2pð356n þ mþt 2pð365n þ mþt a i cos þ b i sin i¼3 n¼1 m¼ 1 Here, a 0 is the mean of the data, a 1, b 1 are coefficients of the yearly cycle, a 2, b 2 of twice yearly and a i, b i are coefficients of the daily cycle and its harmonics and associated beat frequencies. An inspection of the Power Spectrum would show that we need to include the harmonics of the daily cycle (n = 2, 3 as well as n = 1) and also the beat frequencies (m = 1, 1). The latter modulate the amplitude to fit the time of year, in other words describe the beating of the yearly and daily cycles. Table 1 shows the frequency of yearly, twice yearly, daily and twice daily cycles, with percentage of variance ð5þ

3 138 J. Huang et al. / Solar Energy 87 (2013) Table 1 Fourier series model in 2000 Mildura data, all significant contributors to the variance are indicated in bold. 1/year 2/year 1/day 2/day Freq 2p 4p 2p364 2p365 2p366 2p729 Var % p730 2p731 Fig. 1. Three days (consecutive hourly observations) of Mildura 2000 data and the Fourier series fit. Fig. 2. Three days (consecutive hourly observations) of 2000 Mildura deseasoned data and the AR(2) fit. of the original time series explained by the contribution of the Fourier series component at that frequency. All significant values are indicated in bold. As shown in Table 1, the yearly cycle explains 7.31% of the variance of the series, while the daily and twice daily cycles explain over 70% of the variance of the series. Therefore, global solar radiation of Mildura 2000 has a very strong daily cycle and a less prominent yearly cycle. The same conclusions are reached for other locations when using the Power Spectrum method shown in Boland (2008). Fig. 1 shows that Fourier series alone is not enough to model global solar radiation, because for some days it underestimates and for other days it overestimates the data. To deseason the data, residuals are calculated according to the equation R t ¼ F t S t where F t is the solar radiation measurement at time t, and R t is the residual which will be modelled in three different ways, so we can forecast. 3. Autoregressive AR (p) model The usual procedure to follow next is to determine if the residual series follows some autoregressive moving average process ARMA (p, q) (Boland, 2008). We find in this case that an autoregressive model at some order represents the series best. From Tsay (2005), the equation of the AR (p) model is R t ¼ / 0 þ / 1 R t 1 þ / 2 R t 2 þþ/ p R t p þ e t ð6þ ð7þ where p is a non-negative integer and / i are coefficients, while e t is assumed to be white noise with mean zero and variance r 2 e. Using Minitab statistical software to analyse the data, for Mildura 2000 solar radiation data the following AR(2) model is obtained: R t ¼ 0:8896R t 1 0:097R t 2 þ e t Fig. 2 shows that this AR(2) model works particularly well when solar radiation residuals are decreasing. However, the AR(2) model has underestimated all the high peaks of the data. So, it is important to find another model to help capture these peaks. 4. Lucheroni model Lucheroni (2007) presented a resonating model for the power market which exploits the simultaneous presence of a Hopf critical point, periodic forcing and noise in a two-dimensional first order non-autonomous stochastic differential equation system for the logprice and the derivative of logprice. The model which originates from biophysics (known in the literature as the FitzHugh Nagumo system, (FitzHugh, 1961; Linder et al., 2004) performs very well for power market modelling, see Lucheroni (2007, 2009). We adapt the same version as presented in Lucheroni (2007, 2009), to the deseasoned solar radiation data R. The model to which we refer as the Lucheroni model is described below. The continuous version of the model can be written as ð8þ

4 J. Huang et al. / Solar Energy 87 (2013) Fig. 3. Three days (consecutive hourly observations) of 2000 Mildura deseasoned data and the Lucheroni model fit. Fig. 4. Three days (consecutive hourly observations) of 2000 Midura deseasoned data and the Combination model fit. _R ¼ z ð9þ _z ¼ jðz þ RÞ kð3r 2 z þ R 3 Þ z cr bþf; where j, k,, c and b are adjustable parameters, and f is the noise term. In the Eq. (9), _R denotes the derivative of R with respect to time, and hence _z stands for the second derivative of R with respect to time. We shall use the following discretised version of the model for our deseasoned solar radiation time series R t : R tþ1 ¼ R t þ z t Dt þ x t z tþ1 ¼ z t þ½jðz t þ R t Þ kð3r 2 t z t þ R 3 t Þ z t cr t bš Dt þ a t: ð10þ Here x t and a t are noise terms, Dt is the time step. Eqs. (10) aim to exploit the fact that the current value of z t is useful to predict the future value R t+1. The parameters j, k,, c and b can be estimated using the method of ordinary least squares (OLSs). For our deseasoned data, estimated values for the parameters are: j = 2.1, k = , = 0.09, c = 0.5 and b = 2. k is virtually zero which indicates that the deseasoned residuals R t behave linearly. Further to this, a negative value of j assures the stability of the inherent damped oscillator in Eqs. (10). It is clear from Fig. 3 that the Lucheroni model can effectively follow the pattern of the deseasoned data; in particular, it is able to capture the magnitude of peaks and troughs almost perfectly. However, it has a shift problem. When residuals are decreasing, the Lucheroni model does not perform very well when compared with the AR(2) model, and forecasting errors tend to be larger. This motivates us to combine the AR(2) and Lucheroni models together to get an improved forecasting profile. 5. Enhancement I: The combination of the Lucheroni and AR(2) models As observed in Section 4, the Lucheroni model captures peaks very well, while the AR(2) model produces less error when the solar radiation is decreasing. We want to use Lucheroni for rising residual solar radiation values and AR(2) for falling values, but how to decide when to switch? The first step difference of deseasoned data R t R t 1 ¼r 1 t is used to decide which model should be chosen at time t +1.IfR t R t 1 is positive, the Lucheroni model will be used for modelling the residual data at time t +1.IfR t R t 1 is negative, the AR(2) model will be used to obtain the estimate of R t+1. But when the previous difference R t 1 R t 2 also is negative, we switch back to the Lucheroni model. Thus, we take into account a proxy for the curvature by considering what is happening over three time steps. Fig. 4 shows that the Combination model gives a small improvement compared with the Lucheroni model shown in Fig. 3. More examples and error analysis will be provided in the next section. 6. Comparison between the AR(2), Lucheroni and Combination models For the purposes of formal error analysis of the proposed models, the following measures are considered: median absolute percentage error (MeAPE), mean bias error (MBE), Kolmogorov Smirnov test integral (KSI), normalized root mean square error (NRMSE). MeAPE captures the size of the errors, while MBE is used to determine whether any particular model is more biased than another. NRMSE measures overall model quality related to regression fit. KSI is a new model validation measure based on the Kolmogorov Smirnov test (Massey, 1951) which has the advantage of being non-parametric. The KSI measure was proposed in Espinar et al. (2009) to assess the similarity of the cumulative distribution functions (CDFs) of actual and modelled data over the whole range of observed values. Definitions of all the measures are as follows:

5 140 J. Huang et al. / Solar Energy 87 (2013) Table 2 Error analyses for these three different models in 2000 Mildura data. Lucheroni + seasonal AR + seasonal Combination + seasonal MeAPE (%) MBE KSI (%) NRMSE (%) MeAPE ¼ MEDIAN ^y i y i 100 MBE ¼ 1 n NRMSE ¼ X n i¼1 ð^y i y i Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P n i¼1 ð^yi y iþ 2 n y y i ð11þ ð12þ ð13þ (a) (b) (c) Fig. 5. Plot of the CDFs for two compared data sets (left) and the differences D n between those (right) at Mildura. The dotted line marks the critical value V c. (a) AR(2) model plus seasonal components, (b) Lucheroni model plus seasonal components and (c) Combination model plus seasonal components.

6 J. Huang et al. / Solar Energy 87 (2013) Fig. 6. Comparison between measured and forecasts (AR(2) model plus seasonal) for the whole year of hourly global solar radiation. The red line (larger line) represents the unit line y = x and the black line is the regression line. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Fig. 8. Comparison between measured and forecasts (Combination model plus seasonal) for the whole year of hourly global solar radiation. The red line (larger line) represents the unit line y = x and the black line is the regression line. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Fig. 7. Comparison between measured and forecasts (Lucheroni model plus seasonal) for the whole year of hourly global solar radiation. The red line (larger line) represents the unit line y = x and the black line is the regression line. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) where ^y i are predicted values, y i are measured values and y is the average of measured values. KSIð%Þ ¼100 R xmax D n dx x min a critical ð14þ where x max and x min are the extreme values of the independent variable, and a critical is calculated as a critical = V c (x max x min ). The critical value V c depends on population size N andp is calculated for a 99% level of confidence as V c ¼ 1:63= ffiffiffiffi N ; N P 35. The Dn is the differences between the cumulative distribution functions (CDFs) for each interval. The higher the KSI value, the worse the fit of model to data. It is worth noting that in assessing a model against the actual data, NRMSE measures how close the points are clustered around the regression line for the relationship Fig. 9. Lucheroni model fit (72 consecutive hours) compared with the Combination model and deseasoned data. between the observed and predicted values, while KSI and MBE assess the distribution of points around the unit line, ^y ¼ y. Thus by considering a set of diverse measures the aim is to allow for a more complete comparison of the proposed models. For example, additional information on the CDFs carried by KSI and MBE can be used to distinguish between models with similar MeAPE or NRMSE values. For error analyses we only consider the data for solar altitude greater than 10, because during these periods of solar radiation, there is potentially significant solar radiation to generate electricity. Results of error analyses are shown in Table 2. The two most commonly used measures, MeAPE and NRMSE, indicate similar forecasting ability of the three models, the Lucheroni model plus the seasonal component, the AR(2) model plus the seasonal component, and the Combination model plus the seasonal component. However, these measures assess forecasting performance from the point of view of global optimality. For practical

7 142 J. Huang et al. / Solar Energy 87 (2013) purposes, it is not only important whether a proposed model fits solar radiation data overall, but also how well it performs locally or in particular regions of the data. As noted earlier, MBE and KSI provide additional information about the empirical distribution compared to the proposed model distribution. In the case of solar radiation at Mildura, KSI and MBE measures point to the AR(2) plus seasonal model having significantly worse fit to the data compared with the other two models. This is further illustrated in Fig. 5 which shows observed and modelled CDFs, as well as differences between them over the whole range of the data. Clearly, Lucheroni plus seasonal and Combination plus seasonal models obtain estimates closer to the measured values. Figs. 6 8 show residuals with fitted regression line as well as the unit line for the three models. For AR(2) (Fig. 6), the regression line is much further away from the unit line than for the other two models (Figs. 7 and 8). Combining the reasons just outlined, the AR(2) model itself is excluded from further analysis. Since the Lucheroni and Combination models give similar results in Table 2, as well as showing comparable quality of overall fit in Figs. 5, 7 and 8, it is not possible to distinguish between these two models. However, there are advantages to the Combination model and these can be observed in specific regions of the time series. For example, Fig. 9 shows that the Combination model performs better than the Lucheroni model in the regions of the deseasoned data marked by circles. While the improvement from the Combination model may not be large in a global sense, this model does appear to perform better than the Lucheroni model alone when the series is decreasing. Therefore, based on its global and local performance, we conclude that the Combination model is the best choice. In the next section, we propose a further improvement to the Combination model that gives rise to the CARDS model, and illustrate how it contributes to the accuracy of forecasting global solar radiation in the Combination model. The fixed component must be continuously increasing which means that the first difference of the fixed component is increasing for at least two consecutive time steps, that is, f t 1 f t 2 > 0 and f t f t 1 >0. The first continuous increasing point at time t is decided by the sign of r 1 t 1. If it is positive, the fixed component is deemed to be valuable and M t is replaced by f t. Otherwise M t will not be changed. The last continuing f t is decided by the calculated value of M t+1.ifm t+1 M t < 0, then we revert back to M t (that is, M t is not replaced by f t ). Between the first and last continuous increasing points, all M t should be replaced by f t. Case 2: When the fixed component is decreasing, that is, f t f t 1 <0. The fixed component must be continuously decreasing which means that the first difference of the fixed component is decreasing for at least two consecutive time steps, that is, f t 1 f t 2 < 0 and f t f t 1 <0. The first continuous decreasing point at time t is decided by the sign of r 1 t 1. If it is negative, the fixed component is valuable and M t is replaced by f t. Otherwise M t will not be changed. The last continuous decreasing point f t is decided by the calculated value of M t+1.ifm t+1 M t > 0, then we revert back to M t (that is, M t is not replaced by f t ). Between the first and last continuous decreasing points, the absolute value of the previous difference must be greater than a threshold value A. Here, A is decided by the 10th percentile of all middle points of previous differences for a year (for example, for the year 2000 data, A is and A for the year 2001 data is However, using as threshold instead of does not affect the results). 7. Enhancement II: The CARDS model for forecasting solar radiation In order to define the CARDS model, we introduce the notion of a fixed component f t. Let r 1 t ¼ R t R t 1 ; then f t, the fixed component, is defined as f t ¼ M t þr 1 t 1 ; ð15þ where M t is the prediction obtained from the Combination model at time t. The fixed component f t is intended to replace M t. However, not all the predictions from the Combination model are replaced by the fixed component values. The Combination model predictions are replaced by the fixed component only if the following situations occur. Case 1: When the fixed component is increasing, that is, f t f t 1 >0. Fig. 10. Decision rules for use of the fixed component.

8 J. Huang et al. / Solar Energy 87 (2013) Fig. 11. One day 2000 Mildura deseasoned data, Combination model and the CARDS model. Table 3 Comparison between the combination plus seasonal and the CARDS model plus seasonal for error analyses in 2000 Mildura data. Combination model + seasonal MeAPE (%) MBE KSI (%) NRMSE (%) CARDS model + seasonal The rules described in the above dot points are summarised in the following flow chart (Fig. 10): Following the above rules can improve forecasts from the Combination model. Fig. 11 illustrates what happens when the fixed component replaces some of the predictions in the Combination model. The CARDS model, which is the Combination model with the fixed component, better follows the variation in the observed data series. More precisely, MeAPE, MBE, KSI, NRMSE analyses have also been used to see how much improvement was made for the Mildura 2000 series. These are presented in Table 3. Fig. 13. Comparison between measured and forecasts (CARDS model plus seasonal) for the whole year of hourly global solar radiation. The red line (larger line) represents the unit line y = x and the black line is the regression line. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Table 3 shows that when the Combination model uses the fixed component to replace some of the predictions, results of the estimation can be significantly improved. For example, MeAPE is improved from 11.31% to 7.53%, an improvement of 33.4%. Fig. 12 shows that compared with Fig. 5, when the fixed component is added D n and estimated values are even closer to the measured values and further away from the critical value. Similarly, Fig. 13 shows that compared with Fig. 8, when the fixed component is added, the regression line moves even closer to the unit line. In Figs there are several examples to show, in different weather situations and seasons, how the CARDS model plus seasonal component performs compared to the actual measurement of the hourly global solar radiation. For example, in Fig. 16 on the 25th of January, (a3), the Combination model plus seasonal component is missing all the changes on that day and the CARDS model plus seasonal component shows much improvement in fol- Fig. 12. Plot of the CDF for two compared data sets (left) and the differences D n between those (right) at Mildura. The dotted line marks the critical value V c.

9 144 J. Huang et al. / Solar Energy 87 (2013) (a1) (b1) (c1) (d1) Fig. 14. Very clear day, (a1) 4th of January, (b1) 2nd of April, (c1) 11th of July and (d1) 2nd of October. lowing the changing of measured values. These examples provide further evidence that the CARDS model improves the predictive ability in difficult weather condition. So, from the above analysis and examples, it can be concluded that the CARDS model is a more efficient way to forecast 1-h ahead solar radiation. 8. Comparison with other models As mentioned in the Introduction, there are several different approaches for modelling hourly time scale solar radiation in the literature. These are artificial neural networks (Mihalakakou et al., 2000; Tymvios et al., 2005; Wu and Chan, 2011; Cao and Lin, 2008; Mellit et al., 2010), time series methods (Boland, 2008), the hybrid model which is a combination of ARMA and Time Delay Neural Network models (Wu and Chan, 2011), and a stochastic prediction model (Kaplanis and Kaplani, 2010). As this section will demonstrate, for our data the CARDS model, introduced in Section 7, produces better results. We will provide evidence to highlight the advantages of our approach Neural networks Several different neural network models and adaptations have been used for forecasting hourly global solar radiation. One such model is the diagonal recurrent wavelet neural network (DRWNN) (Cao and Lin, 2008). This combines the dynamic nature of the diagonal recurrent neural network (DRNN) with the enhanced ability of the wavelet neural network (WNN) to model nonlinear functions. They used data from January 1, 2001 to November 30, 2002 for training the model and the data from December 2002 for testing. The error measures used to evaluate the performance were the mean relative error (MRE) and coefficient of determination R 2. The best configuration of their approach had an MRE of 9.23% and R 2 = In our case, the results were MRE = 2.20% and R 2 = for Midura 2001 data. It should be noted that Cao and Lin (2008) seem to be reporting the performance based on only two clear days (in Figs. 8, 9 and Table 3 in their paper, they compare the performance of their model with others for December 30 and 31 of 2002), whereas the CARDS model performance was measured over a whole year.

10 J. Huang et al. / Solar Energy 87 (2013) (a2) (b2) (c2) (d2) Fig. 15. Overcast day but with no obvious clouds blocking the sun, (a2) 24th of January, (b2) 4th of April, (c2) 12th of July and (d2) 13th of October. Another paper that discussed the use of neural networks for forecasting was that of Mellit et al. (2010). What they actually did was to propose an adaptive model, the a- model, and compared its performance to a neural network model. In fact, the results showed that the neural network model they used for comparison, performed better. The model was trained using 8000 data points and tested on 765 data values. The neural network model had correlation coefficient r of 98.53%, while the CARDS has correlation coefficient r value of 96.18%. However, Mellit et al. (2010) used a short period for comparison of about 32 days for their error analysis, 70% of which were sunny days. The analysis for the CARDS model was done over a full year, with more varied conditions The hybrid model (TDNN and ARMA) The hybrid model, which is a combination of ARMA and Time Delay Neural Network models, has been shown to perform better overall than either an ARMA or a TDNN model (Wu and Chan, 2011). The basic idea of the hybrid model is using an ARMA model first to fit the linear component, which means the residual series only contains a nonlinear component. Then they use the TDNN model to fit the residual. They used normalized root mean square error (NRMSE) to evaluate their hybrid model with ARMA and TDNN (to facilitate visualization, they take the interval of 2 days). For the CARDS model, we used the same out-of-sample testing procedure as in Wu and Chan (2011), the sample for testing being Mildura 2001 data set whereas the model was constructed on Mildura 2000 data. Figs. 17 and 18 show that the CARDS model obtains a smaller NRMSE than the hybrid model. The average of NRMSE from the CARDS model plus seasonal component is 0.167, but the hybrid model gives an average of approximately 0.3. So, the CARDS model provides a better prediction for hourly global solar radiation Kaplanis and Kaplani s model The model developed in Kaplanis and Kaplani (2010) is a type of stochastic model for predicting the hourly profile of the intensity of global solar radiation. It takes into account hourly average and hourly standard deviation over the preceding 6 years data. To synthetically generate a value of solar radiation for time t + 1 from value at time t, they consider the previous 3 h observations as well as the hourly standard deviation profiles for the previous

11 146 J. Huang et al. / Solar Energy 87 (2013) (a3) (b3) (c3) (d3) Fig. 16. Overcast day with clouds blocking the sun intermittently, (a3) 25th of January, (b3) 13th of April, (c3) 29th of July and (d3) 7th of October. 3 h. The model is based on the assumption that the difference between the measured value at a n hour h 1 and the averaged value of solar radiation for the same hour h 1,as recorded over the years, follows a Gaussian probability density function. In essence, Kaplanis and Kaplani s model consists of the averaged value for the hour plus terms similar to a third order Taylor-series expression with multipliers 1, 1/4, and 1/9 respectively. The first order term represents the present trend of the hourly measurements, the second and third order terms respectively represent the contribution to the prediction by its rate of change during the two previous hours and the contribution to the prediction by its rate of change during the 2 h prior to the hour of prediction. The zero order term, of course, provides expected hourly fluctuations based on the past history of the stored data for the hour of a day. To involve stochasticity in the model, the first, second and third order terms are respectively multiplied by three Gaussian random numbers having zero mean and standard deviation equal to one, and satisfying certain bias selection rules. For the details see Kaplanis and Kaplani (2010). We used 6 years of hourly solar radiation data for Mildura ( ) to build Kaplanis and Kaplani s model, and tested the model on the Mildura 2000 data. For the purposes of comparison and because of the stochastic nature of Kaplanis and Kaplani s model, we executed the model ten times. Then, using hourly data we compared Kaplanis and Kaplani s model with the CARDS model plus seasonal component based on the daily profile. Figs show 4 days chosen at random from summer and winter, respectively, to compare hourly predictions based on the worst and best realizations out of the ten of the Kaplanis and Kaplani s model with those from the CARDS model. The definition of the best realization is that with the value closest to the measured daily solar radiation value and opposite for the worst day. The CARDS plus seasonal component does not always give the best estimated daily value. For example, in Fig. 22, the smallest difference between measured and estimated daily value is from Kaplanis and Kaplani s model 5th realization which is 5.5, but the worst also comes from Kaplanis and Kaplani s model which is For CARDS model the difference is However, the best daily value of Kaplanis and Kaplani s model cannot pick up the hour to hour variation in the real data. This is an obvious point since the model is designed to generate possible realizations rather than forecasts. In Kaplanis and Kaplani (2010), daily total solar energy is computed for all realizations and then the average of these daily values is compared with the measured totals. We decided that even though we have

12 J. Huang et al. / Solar Energy 87 (2013) Fig. 19. Estimated solar radiation for a randomly chosen sunny summer day. CARDS model plus seasonal component and the best and worst out of ten realizations of Kaplanis and Kaplani s model. Fig. 17. Taken from Wu and Chan (2011), Fig. 16, p The NRMSE of the prediction of different models on first 5 month s solar radiation and the NRMSE of the prediction of the CARDS models plus seasonal on first 5 month s solar radiation, Mildura Fig. 20. Estimated solar radiation for a randomly chosen cloudy summer day. CARDS model plus seasonal component and the best and worst out of ten realizations of Kaplanis and Kaplani s model. Fig. 18. Taken from Wu and Chan (2011), Fig. 18, p The NRMSE of the prediction of different models on second 5 month s solar radiation and the NRMSE of the prediction of the CARDS models plus seasonal on second 5 month s solar radiation, Mildura Fig. 21. Estimated solar radiation for a randomly chosen sunny winter day. CARDS model plus seasonal component and the best and worst out of ten realizations of Kaplanis and Kaplani s model.

13 148 J. Huang et al. / Solar Energy 87 (2013) Fig. 22. Estimated solar radiation for a randomly chosen cloudy winter day. CARDS model plus seasonal component and the best and worst out of ten realizations of Kaplanis and Kaplani s model. Kaplanis and Kaplani do, would be useful. For the analysis, we use both graphical and numerical summaries. In Figs. 23 and 24, we present a comparison of the CARDS model aggregated over the day and the measured data, over the whole year in the first and in the most variable time of the year, the spring, in the second. We have superimposed the results from averaging the daily totals estimated from the ten realisations of the Kaplanis and Kaplani s model. Note that we are not directly comparing the two approaches, since one is based on stochastic realization and ours is a forecasting model. But, it is instructive to note that the averaging of the stochastic realizations yields the correct general pattern over the year but not the day to day variation. The CARDS model, though, both correctly forecasts on an hourly basis, and when aggregated correctly replicates the daily radiation. To further corroborate this, the MBE and MeAPE for the CARDS daily aggregations are 4.58% and 1.39%. 9. Conclusions and future work Fig. 23. Comparison between daily basis of CARDS model plus seasonal component and daily average of ten realizations of Kaplanis and Kaplani s model for whole year of Mildura 2000 data. This paper has introduced a new method for 1-h ahead global solar radiation forecasting. We began by evaluating the prediction ability of the AR(2) model, the Lucheroni model and a combination of the two. The Combination model was selected as the best performer of the three options. The model was enhanced further by addition of the fixed component, thus constructing the CARDS model. Global solar radiation data for Mildura was used in this evaluation. The results of error analyses show that the CARDS model has decreased the Combination model forecasting error by 33.4% for MeAPE. We also presented a number of examples of comparisons with other models available in the literature to illustrate advantages of our model. These certify that the CARDS model is a good method to forecast hourly global solar radiation. Note that the results found for the CARDS model compare very favourably with what Kostylev and Pavlovski (2011) found from their survey of the literature. From that paper, the best performing model at the 1 h time step had an NRMSE of 16 17% for mostly clear days and 32% for mostly cloudy, whereas for the CARDS model it is 16.5% for all days. In the future, the prediction accuracy of the proposed model can be further tested using global solar radiation data on different time scales and for different locations. It will be also applied to other types of data, such as wind energy data. Acknowledgements Fig. 24. Comparison between daily basis of CARDS model plus seasonal component and daily average of ten realizations of Kaplanis and Kaplani s model for October December of Mildura 2000 data. developed an hourly forecasting tool, a comparison between predicted and measured on a daily total basis as This research was supported by ARC Discovery Grant DP , Strategic integration of renewable energy systems into the electricity grid, Linkage Grant LP , Unlocking the grid: the future of the electricity distribution network, and Australian Solar Institute Grant, Forecasting and characterizing grid connected solar energy and developing synergies with wind.

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