Forecasting solar radiation on short time scales using a coupled autoregressive and dynamical system (CARDS) model

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1 Forecasting solar radiation on short time scales using a coupled autoregressive and dynamical system (CARDS) model John Boland, Ma lgorzata Korolkiewicz, Manju Agrawal & Jing Huang School of Mathematics and Statistics and Barbara Hardy Institute, University of South Australia Mawson Lakes Boulevard, Mawson Lakes, SA, 5095, Australia Abstract The electricity market in Australia operates on time scales out to several days at granularities of 5 and 30 minutes. The planning for electricity provision involves longer time scales. In order to cater for the intermittency issues with solar electricity, robust forecasts are necessary. For short time scales, we have developed a Combined AutoRegressive, Dynamical System (CARDS) forecasting tool. We will demonstrate its usefulness for one-step-ahead at the hourly and sub hourly time scales, and show that its performance is significantly better than any forecast model in the published literature. Keywords: Solar radiation forecasting, time series method, Lucheroni model, Box Jenkins, Dynamical Systems, 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 a standard autoregressive approach with a resonating model borrowed from work on dynamical systems, and also an additional component that greatly enhances forecasting ability. We apply it to short term solar energy forecasting. The Australian electricity market operates on short time scales, typically five minute and half hour for spot prices. The exigencies of the operation of the grid also require middle (up to two hourly) up to long term forecasting. The methods that are described here prove to be extremely useful at the shorter time scales. 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 (ANN) (Mihalakakou et al., 2000; Tymvios et al., 2005), satellite-derived cloud motion forecast models (Perez et al., 2010), numerical weather prediction models (Mathiesen and Kleissl 2011), the diagonal recurrent wavelet neural network model (Cao and Lin 2008), and stochastic prediction models (Kaplanis and Kaplani, 2010, Gueymard (2000)) that use the volatility of 1

2 the data over the whole time period. 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. 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 an attempt to address some of these problems, in this paper we introduce a different way to forecast global solar radiation on one-period ahead time scale. That is, given the observed radiation F t at time t, we forecast F t+1. In a recent paper (Huang et al 2013), a new forecasting method, based on a combination of an autoregressive (AR) model and a dynamical system model, was developed. The dynamical system model considered was 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 was used to obtain one-hour ahead solar radiation forecasts for Mildura, a small town in western Victoria, Australia and its performance was assessed using a range of forecasting accuracy measures. Comparisons were also made with a number of other models available in the literature. In this paper, a short description will be given of the development of that model. It will then be also applied to shorter time scales, specifically half hourly and five minute solar energy forecasting, using data collected at Whyalla, South Australia. 2 Forecasting on an hourly time scale 2.1 Data 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 8760 (24 365) hourly global radiation values in total. The data set used for showing the forecasting skill on shorter time scales is from a weather station operated at the Whyalla, South Australia, Latitude S, Longitude E, campus of the University of South Australia. 2.2 Determining the seasonality of a solar radiation series 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. Following Boland (1995; 2008), and using the results of Power Spectrum analysis, the Fourier series representation of a function S t can be written as: S t = α 0 + α 1 cos 2πt β 1 sin 2πt α 2 cos 4πt β 2 sin 4πt ( ) 2π(365n + m)t 2π(365n + m)t α nm cos + β nm sin (1) n=1 m= 1 2

3 Here, t is in hours, α 0 is the mean of the data, α 1, β 1 are coefficients of the yearly cycle, α 2, β 2 of twice yearly and α i, β 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. 1/year 2/year 1/day 2/day 2π 4π 2π 364 2π 365 2π 366 2π 729 2π 730 Freq Var % π Table 1: Fourier series model for 2000 Mildura data Table 1 shows the frequency of yearly, twice yearly, daily and twice daily cycles, with percentage of variance 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) Solar Radiation Fourier series Solar radiation ( ) Hour Figure 1: Three days (consecutive hourly observations) of Mildura 2000 data and the Fourier series fit Figure 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 = F t S t (2) 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. 2.3 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 = φ 0 + φ 1 R t 1 + φ 2 R t φ p R t p + e t (3) 3

4 where p is a non-negative integer and φ i are coefficients, while e t is assumed to be white noise with mean zero and variance σ 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 = R t R t 2 + e t (4) Deseasoned solar radiation ( ) Deseasoned AR(2) Hour Figure 2: Three days (consecutive hourly observations) of 2000 Mildura deseasoned data and the AR(2) fit Figure 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. 2.4 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 to which we refer as the Lucheroni model is given in the discretised version of the model for the deseasoned solar radiation time series R t : R t+1 = R t + z t t + ω t z t+1 = z t + [κ(z t + R t ) λ(3r 2 t z t + R 3 t ) ɛz t γr t b] t ɛ + a t. Here ω t and a t are noise terms, t is the time step. Equations (5) aim to exploit the fact that the current value of z t is useful to predict the future value R t+1. The parameters κ, λ, ɛ, γ and b can be estimated using the method of ordinary least squares (OLS). For our deseasoned data, estimated values for the parameters are: κ = 2.1, λ = , ɛ = 0.09, γ = 0.5 and b = 2. λ is virtually zero which indicates that the deseasoned residuals R t behave linearly. Further to this, a negative value of κ assures the stability of the inherent damped oscillator in equations (5). 2.5 Combining the Lucheroni and AR models and developing the CARDS model It is clear from Figure 3 that the Lucheroni model can effectively follow the pattern of the deseasoned data; in particular, it is able to capture the magnitude (5) 4

5 200 Deseasoned Lucheroni Deseasoned solar radiation ( ) Hour Figure 3: Three days (consecutive hourly observations) of 2000 Mildura deseasoned data and the Lucheroni model fit 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. The Lucheroni model is used for rising residual solar radiation values and AR(2) for falling values. Figure 4 shows that this Combination model gives a small improvement compared with the Lucheroni model shown in Figure 3. Deseasoned solar radiation ( ) Deseasoned Combination model Hour Figure 4: Three days (consecutive hourly observations) of 2000 Mildura deseasoned data and the Combination model fit While the combination of the Lucheroni and autoregressive models give better forecasting skill, more can be done. In order to define the CARDS model, we introduce the notion of a fixed component f t. Let 1 t = R t R t 1, then f t, the fixed component, is defined as f t = M t + 1 t 1, (6) 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 under the rules given in the flow chart in Figure 5 Following the rules in the flowchart can improve forecasts from the Combination model. Figure 6 illustrates what happens when the fixed component 5

6 f t 1 f t 2 < 0 & f t f t 1 < 0? No f t 1 f t 2 > 0 & f t f t 1 > 0? Yes No Yes 1 t 1 > A? M t+1 M t > 0? No Yes M t+1 M t < 0? Yes No No Yes Prediction = f t Prediction = M t Prediction = f t Figure 5: Decision rules for use of 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. Deseasoned solar radiation ( ) Deseasoned Combination CARDS Hour of the day Figure 6: One day 2000 Mildura deseasoned data, Combination model and the CARDS model fit More precisely, MeAPE, MBE, KSI, NRMSE analyses have also been used to see how much improvement was made for the Mildura 2000 series (See the Appendix for descriptions of the error measures). These are presented in Table 2. Table 2: Combination model+seasonal CARDS model+seasonal MeAPE 11.31% 7.53% MBE KSI 27.50% 17.92% NRMSE 18.92% 16.50% Comparison between the Combination plus seasonal and the CARDS model plus seasonal for error analyses in 2000 Mildura data 6

7 Table 2 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%. Before we discuss performance as compared with other models in the literature, it should be compared with the most trivial of models. This is the so-called persistence model, Ŝt+1 = S t, sometimes used as a benchmark. As an example, the nrmse for the hourly persistence model is 32.95%, whereas for Lucheroni plus seasonal (S) is 19.09%, AR+S is 18.59% and CARDS+S is 16.5%. Kostylev and Pavlovski (2011) summarised results of journal articles forecasting solar energy on various time scales from one hour ahead upwards to a number of days. From that paper, the best performing model for any type of modelling approach at the one hour time step had an NRMSE of 16-17% for clear days and 32% for cloudy, whereas for our model it is 16.5% for all days. Similarly, we conducted comparisons with the diagonal recurrent wavelet neural network model (Cao and Lin 2008), the hybrid time delay neural network and ARMA model of Wu and Chan (2011), and Kaplanis and Kaplani s (2010) stochastic model in the paper where we developed the forecasting tool in full (Huang et al 2013). The comparison provides evidence of the superior performance of the model discussed in the present paper. It can be stated then that the model presented here outperforms the best models available to date for the hourly time scale. In the next section, results will be presented for the use of the model at shorter time scales. 3 Forecasting at sub hourly time scales The first scale that is examined is the thirty minute time scale. Because of the smaller time scale, the Lucheroni model performs better than when it is used for the longer time scale data. The method for the combination model switching between the AR and Lucheroni models is different. Instead of using Lucheroni when the residual time series is increasing and AR for first step decreasing, the Lucheroni model is used for all situations, except when the change from time step to time step with Lucheroni is larger than with AR. In this case, the autoregressive model is used. For the next step, the fixed component is only used when the model is AR. The rules for using fixed component are not changed. Results are presented in Table 3 for the both the winter and summer periods separately. Results for winter are somewhat higher but still excellent. CARDS Winter CARDS Summer MeAPE 8.23% 3.52% MBE KSI 31.36% 6.95% NRMSE 29.7% 12.92% Table 3: The CARDS model plus seasonal for half hour data for Whyalla Figure 7 gives a visual comparison of the model fit for the CARDS model plus seasonal on a thirty minute time scale. When applied to forecasting on a five minute time scale, the results are even better than at any longer time scale. This is in agreement with the results of Kostylev and Pavlovski (2011) in better forecasting skill at shorter time scales. However, the results at this time scale are extremely good and may well be in keeping with the present error limits for forecasting conventional electricity generating plant. The results are given in Table 4 and displayed visually in Figure 8. 7

8 Figure 7: Data and model fit for six days at Whyalla for the thirty minute time scale, for daytime only CARDS Winter CARDS Summer MeAPE 5.37% 2.46% MBE KSI 45.01% 22.95% NRMSE 18.42% 12.04% Table 4: The CARDS model plus seasonal for five minute data Figure 8: Data and model for five minute time scale for Whyalla, for daytime only 4 Conclusions and Future Work This paper has described a new method for one-hour ahead global solar radiation forecasting, and applied it to both hourly and sub-hourly time scales. 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 and Whyalla was used in this evaluation. The model was constructed from the 2000 hourly Mildura data, tested on 2001 hourly Mildura data, and on the Whyalla data on shorter time scales. The performance at all time scales is impressive and presents a significant advance in this area. When one examines the progression to shorter and shorter time scales, there is a sustained decrease in the median perecentage error, indicating an improvement. There appears also to be a small but appreciable improvement in the NRMSE, 8

9 in keeping with the results of Kostylev and Pavlovski (2011), but leveling off. The Mean Bias Error stays somewhat constant, which is encouraging in that one does not wish to either over estimate or under estimate the forecast. The KSI appears to be increasing but may still be within reasonable bounds. The authors are at present developing a hypothesis test for this statistic. In future, the project goals include providing error bounds on the forecasts, as well as merging this statistical forecast technique with physically based forecasting methods to improve the forecast further. We also intend to demonstrate the benefits of spatial diversity of locations in lowering the intermittency and improving the forecasting ability. We want to add wind farms as well since they have a distinctively different temporal pattern. This will show that attention needs to be given to how to add both temporal and spatial diversity to maximise the penetration of renewable energy sources into the electricity supply system. Acknowledgement: 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. A Appendix 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: ( ) ŷ i y i MeAP E = MEDIAN 100 y i (7) MBE = 1 n NRMSE = n (ŷ i y i ) (8) i=1 n i=1 (ŷ i y i ) 2 where ŷ i are predicted values, y i are measured values and ȳ is the average of measured values. KSI(%) = 100 n ȳ xmax x min D n dx (9) α critical (10) where x max and x min are the extreme values of the independent variable. This interval is split into n sub-intervals. The D n are the differences between 9

10 the cumulative distribution functions (CDF) for each sub-interval. α critical is calculated as α critical = V c (x max x min ). The critical value V c depends on population size N and is calculated for a 99% level of confidence as V c = 1.63/ N, N 35. The higher the KSI value, the worse the fit of model to data. References [1] Boland J.W. 1995, Time-Series Analysis of Climatic Variables, Solar Energy, vol. 55, PP [2] Boland J.W. 2008, Time series and statistical modelling of solar radiation, Recent Advances in Solar Radiation Modelling, Viorel Badescu (Ed.), Springer-Verlag, pp [3] Cao J. C. and Lin X. C. (2008) Application of the diagonal recurrent wavelet neural network to solar irradiation forecast assisted with fuzzy technique, Artifical Intelligence, vol.21, pp [4] Espinar, B., Ramirez, L., Drews, A., Georg Beyer, H., Zarzalejo, LF. Polo, J., & Martin, L. 2009, Analysis of different comparison parameters applied to solar radiation data from satellite and German radiometric stations, Solar Energy, vol. 83, pp [5] Gueymard, C. 2000, Prediction and performance assessment of mean hourly global radiation, Solar Energy, vol. 68, No. 3, pp [6] Jing Huang, Malgorzata Korolkiewicz, Manju Agrawal and John Boland, 2013 Forecasting solar radiation on an hourly time scale using a coupled autoregressive and dynamical system (CARDS) model, Solar Energy, (in [7] press). Kaplanis, S. & Kaplani, E. 2010, Stochastic prediction of hourly global solar radiation for Patra, Greece, Applied Energy, vol. 87, pp [8] Kostylev V. and Pavlovski A. (2011) Solar power forecasting performance, 1st International Workshop on the Integration of Solar Power into Power Systems, Aarhus, Denmark, 24 October 2011 [9] Lucheroni, C. 2007, Resonating models for the electric power market, Physical Review E, vol.76, no.5, p [10] Massey Jr, F.J. 1951, The Kolmogorov-Smirnov test for goodness of fit, Journal of the American Statistical Association, vol. 46, pp [11] Patrick Mathiesen, Kleissl, J. (2011) Evaluation of numerical weather prediction for intra-day solar forecasting in the continental United States, Solar Energy 85, 5, pp [12] Mihalakakou, G., Santamouris, M. & Asimakopoulos, D.N. 2000, The total solar radiation time series simulation in Athens, using neural networks, Theoretical and Applied Climatology, vol. 66, pp [13] Perez, R., Kivalov, S., Schlemmer, J., Hemker, K.Jr., Renn, D. & Hoff, T.E. 2010, Validation of short and medium term operational solar radiation forecasts in the US, Solar Energy, vol. 84, pp [14] Tsay, R.S. 2005, Analysis of Financial Time Series, 2nd edn, A John Wiley & Sons, INC., Hoboken, New Jersey. [15] Tymvios, F.S., Jacovides, C.P., Michaelides, S.C. & Scouteli, C. 2005, Comparative study of Angström s and artificial neural networks methodologies in estimating global solar radiation, Solar Energy, vol. 78, pp [16] Wu, J. & Chan, C.K. 2011, Prediction of hourly solar radiation using a novel hybrid model of ARMA and TDNN, Solar Energy, vol. 85, pp